3.6.24 \(\int \frac {1}{(d+e x)^2 (a+c x^2)^4} \, dx\) [524]

Optimal. Leaf size=430 \[ \frac {e \left (5 c^3 d^6+23 a c^2 d^4 e^2+47 a^2 c d^2 e^4-35 a^3 e^6\right )}{16 a^3 \left (c d^2+a e^2\right )^4 (d+e x)}+\frac {a e+c d x}{6 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )^3}-\frac {a e \left (c d^2-7 a e^2\right )-c d \left (5 c d^2+13 a e^2\right ) x}{24 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \left (a+c x^2\right )^2}-\frac {a e \left (5 c d^2-7 a e^2\right ) \left (c d^2+5 a e^2\right )-3 c d \left (5 c^2 d^4+18 a c d^2 e^2+29 a^2 e^4\right ) x}{48 a^3 \left (c d^2+a e^2\right )^3 (d+e x) \left (a+c x^2\right )}+\frac {\sqrt {c} \left (5 c^4 d^8+28 a c^3 d^6 e^2+70 a^2 c^2 d^4 e^4+140 a^3 c d^2 e^6-35 a^4 e^8\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{16 a^{7/2} \left (c d^2+a e^2\right )^5}+\frac {8 c d e^7 \log (d+e x)}{\left (c d^2+a e^2\right )^5}-\frac {4 c d e^7 \log \left (a+c x^2\right )}{\left (c d^2+a e^2\right )^5} \]

[Out]

1/16*e*(-35*a^3*e^6+47*a^2*c*d^2*e^4+23*a*c^2*d^4*e^2+5*c^3*d^6)/a^3/(a*e^2+c*d^2)^4/(e*x+d)+1/6*(c*d*x+a*e)/a
/(a*e^2+c*d^2)/(e*x+d)/(c*x^2+a)^3+1/24*(-a*e*(-7*a*e^2+c*d^2)+c*d*(13*a*e^2+5*c*d^2)*x)/a^2/(a*e^2+c*d^2)^2/(
e*x+d)/(c*x^2+a)^2+1/48*(-a*e*(-7*a*e^2+5*c*d^2)*(5*a*e^2+c*d^2)+3*c*d*(29*a^2*e^4+18*a*c*d^2*e^2+5*c^2*d^4)*x
)/a^3/(a*e^2+c*d^2)^3/(e*x+d)/(c*x^2+a)+8*c*d*e^7*ln(e*x+d)/(a*e^2+c*d^2)^5-4*c*d*e^7*ln(c*x^2+a)/(a*e^2+c*d^2
)^5+1/16*(-35*a^4*e^8+140*a^3*c*d^2*e^6+70*a^2*c^2*d^4*e^4+28*a*c^3*d^6*e^2+5*c^4*d^8)*arctan(x*c^(1/2)/a^(1/2
))*c^(1/2)/a^(7/2)/(a*e^2+c*d^2)^5

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Rubi [A]
time = 0.36, antiderivative size = 430, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {755, 837, 815, 649, 211, 266} \begin {gather*} -\frac {a e \left (c d^2-7 a e^2\right )-c d x \left (13 a e^2+5 c d^2\right )}{24 a^2 \left (a+c x^2\right )^2 (d+e x) \left (a e^2+c d^2\right )^2}-\frac {a e \left (5 c d^2-7 a e^2\right ) \left (5 a e^2+c d^2\right )-3 c d x \left (29 a^2 e^4+18 a c d^2 e^2+5 c^2 d^4\right )}{48 a^3 \left (a+c x^2\right ) (d+e x) \left (a e^2+c d^2\right )^3}+\frac {e \left (-35 a^3 e^6+47 a^2 c d^2 e^4+23 a c^2 d^4 e^2+5 c^3 d^6\right )}{16 a^3 (d+e x) \left (a e^2+c d^2\right )^4}+\frac {\sqrt {c} \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (-35 a^4 e^8+140 a^3 c d^2 e^6+70 a^2 c^2 d^4 e^4+28 a c^3 d^6 e^2+5 c^4 d^8\right )}{16 a^{7/2} \left (a e^2+c d^2\right )^5}+\frac {a e+c d x}{6 a \left (a+c x^2\right )^3 (d+e x) \left (a e^2+c d^2\right )}-\frac {4 c d e^7 \log \left (a+c x^2\right )}{\left (a e^2+c d^2\right )^5}+\frac {8 c d e^7 \log (d+e x)}{\left (a e^2+c d^2\right )^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^2*(a + c*x^2)^4),x]

[Out]

(e*(5*c^3*d^6 + 23*a*c^2*d^4*e^2 + 47*a^2*c*d^2*e^4 - 35*a^3*e^6))/(16*a^3*(c*d^2 + a*e^2)^4*(d + e*x)) + (a*e
+ c*d*x)/(6*a*(c*d^2 + a*e^2)*(d + e*x)*(a + c*x^2)^3) - (a*e*(c*d^2 - 7*a*e^2) - c*d*(5*c*d^2 + 13*a*e^2)*x)
/(24*a^2*(c*d^2 + a*e^2)^2*(d + e*x)*(a + c*x^2)^2) - (a*e*(5*c*d^2 - 7*a*e^2)*(c*d^2 + 5*a*e^2) - 3*c*d*(5*c^
2*d^4 + 18*a*c*d^2*e^2 + 29*a^2*e^4)*x)/(48*a^3*(c*d^2 + a*e^2)^3*(d + e*x)*(a + c*x^2)) + (Sqrt[c]*(5*c^4*d^8
+ 28*a*c^3*d^6*e^2 + 70*a^2*c^2*d^4*e^4 + 140*a^3*c*d^2*e^6 - 35*a^4*e^8)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(16*a^
(7/2)*(c*d^2 + a*e^2)^5) + (8*c*d*e^7*Log[d + e*x])/(c*d^2 + a*e^2)^5 - (4*c*d*e^7*Log[a + c*x^2])/(c*d^2 + a*
e^2)^5

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 649

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[(-a)*c]

Rule 755

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(a*e + c*d*x)*
((a + c*x^2)^(p + 1)/(2*a*(p + 1)*(c*d^2 + a*e^2))), x] + Dist[1/(2*a*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^
m*Simp[c*d^2*(2*p + 3) + a*e^2*(m + 2*p + 3) + c*e*d*(m + 2*p + 4)*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[
{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]

Rule 815

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x)^m*((f + g*x)/(a + c*x^2)), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Integer
Q[m]

Rule 837

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(
m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] +
Dist[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^
2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g},
x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rubi steps

\begin {align*} \int \frac {1}{(d+e x)^2 \left (a+c x^2\right )^4} \, dx &=\frac {a e+c d x}{6 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )^3}-\frac {\int \frac {-5 c d^2-7 a e^2-6 c d e x}{(d+e x)^2 \left (a+c x^2\right )^3} \, dx}{6 a \left (c d^2+a e^2\right )}\\ &=\frac {a e+c d x}{6 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )^3}-\frac {a e \left (c d^2-7 a e^2\right )-c d \left (5 c d^2+13 a e^2\right ) x}{24 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \left (a+c x^2\right )^2}+\frac {\int \frac {c \left (15 c^2 d^4+34 a c d^2 e^2+35 a^2 e^4\right )+4 c^2 d e \left (5 c d^2+13 a e^2\right ) x}{(d+e x)^2 \left (a+c x^2\right )^2} \, dx}{24 a^2 c \left (c d^2+a e^2\right )^2}\\ &=\frac {a e+c d x}{6 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )^3}-\frac {a e \left (c d^2-7 a e^2\right )-c d \left (5 c d^2+13 a e^2\right ) x}{24 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \left (a+c x^2\right )^2}-\frac {a e \left (5 c d^2-7 a e^2\right ) \left (c d^2+5 a e^2\right )-3 c d \left (5 c^2 d^4+18 a c d^2 e^2+29 a^2 e^4\right ) x}{48 a^3 \left (c d^2+a e^2\right )^3 (d+e x) \left (a+c x^2\right )}-\frac {\int \frac {-3 c^2 \left (5 c^3 d^6+13 a c^2 d^4 e^2+11 a^2 c d^2 e^4+35 a^3 e^6\right )-6 c^3 d e \left (5 c^2 d^4+18 a c d^2 e^2+29 a^2 e^4\right ) x}{(d+e x)^2 \left (a+c x^2\right )} \, dx}{48 a^3 c^2 \left (c d^2+a e^2\right )^3}\\ &=\frac {a e+c d x}{6 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )^3}-\frac {a e \left (c d^2-7 a e^2\right )-c d \left (5 c d^2+13 a e^2\right ) x}{24 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \left (a+c x^2\right )^2}-\frac {a e \left (5 c d^2-7 a e^2\right ) \left (c d^2+5 a e^2\right )-3 c d \left (5 c^2 d^4+18 a c d^2 e^2+29 a^2 e^4\right ) x}{48 a^3 \left (c d^2+a e^2\right )^3 (d+e x) \left (a+c x^2\right )}-\frac {\int \left (\frac {3 c^2 e^2 \left (5 c^3 d^6+23 a c^2 d^4 e^2+47 a^2 c d^2 e^4-35 a^3 e^6\right )}{\left (c d^2+a e^2\right ) (d+e x)^2}-\frac {384 a^3 c^3 d e^8}{\left (c d^2+a e^2\right )^2 (d+e x)}-\frac {3 c^3 \left (5 c^4 d^8+28 a c^3 d^6 e^2+70 a^2 c^2 d^4 e^4+140 a^3 c d^2 e^6-35 a^4 e^8-128 a^3 c d e^7 x\right )}{\left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}\right ) \, dx}{48 a^3 c^2 \left (c d^2+a e^2\right )^3}\\ &=\frac {e \left (5 c^3 d^6+23 a c^2 d^4 e^2+47 a^2 c d^2 e^4-35 a^3 e^6\right )}{16 a^3 \left (c d^2+a e^2\right )^4 (d+e x)}+\frac {a e+c d x}{6 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )^3}-\frac {a e \left (c d^2-7 a e^2\right )-c d \left (5 c d^2+13 a e^2\right ) x}{24 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \left (a+c x^2\right )^2}-\frac {a e \left (5 c d^2-7 a e^2\right ) \left (c d^2+5 a e^2\right )-3 c d \left (5 c^2 d^4+18 a c d^2 e^2+29 a^2 e^4\right ) x}{48 a^3 \left (c d^2+a e^2\right )^3 (d+e x) \left (a+c x^2\right )}+\frac {8 c d e^7 \log (d+e x)}{\left (c d^2+a e^2\right )^5}+\frac {c \int \frac {5 c^4 d^8+28 a c^3 d^6 e^2+70 a^2 c^2 d^4 e^4+140 a^3 c d^2 e^6-35 a^4 e^8-128 a^3 c d e^7 x}{a+c x^2} \, dx}{16 a^3 \left (c d^2+a e^2\right )^5}\\ &=\frac {e \left (5 c^3 d^6+23 a c^2 d^4 e^2+47 a^2 c d^2 e^4-35 a^3 e^6\right )}{16 a^3 \left (c d^2+a e^2\right )^4 (d+e x)}+\frac {a e+c d x}{6 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )^3}-\frac {a e \left (c d^2-7 a e^2\right )-c d \left (5 c d^2+13 a e^2\right ) x}{24 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \left (a+c x^2\right )^2}-\frac {a e \left (5 c d^2-7 a e^2\right ) \left (c d^2+5 a e^2\right )-3 c d \left (5 c^2 d^4+18 a c d^2 e^2+29 a^2 e^4\right ) x}{48 a^3 \left (c d^2+a e^2\right )^3 (d+e x) \left (a+c x^2\right )}+\frac {8 c d e^7 \log (d+e x)}{\left (c d^2+a e^2\right )^5}-\frac {\left (8 c^2 d e^7\right ) \int \frac {x}{a+c x^2} \, dx}{\left (c d^2+a e^2\right )^5}+\frac {\left (c \left (5 c^4 d^8+28 a c^3 d^6 e^2+70 a^2 c^2 d^4 e^4+140 a^3 c d^2 e^6-35 a^4 e^8\right )\right ) \int \frac {1}{a+c x^2} \, dx}{16 a^3 \left (c d^2+a e^2\right )^5}\\ &=\frac {e \left (5 c^3 d^6+23 a c^2 d^4 e^2+47 a^2 c d^2 e^4-35 a^3 e^6\right )}{16 a^3 \left (c d^2+a e^2\right )^4 (d+e x)}+\frac {a e+c d x}{6 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )^3}-\frac {a e \left (c d^2-7 a e^2\right )-c d \left (5 c d^2+13 a e^2\right ) x}{24 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \left (a+c x^2\right )^2}-\frac {a e \left (5 c d^2-7 a e^2\right ) \left (c d^2+5 a e^2\right )-3 c d \left (5 c^2 d^4+18 a c d^2 e^2+29 a^2 e^4\right ) x}{48 a^3 \left (c d^2+a e^2\right )^3 (d+e x) \left (a+c x^2\right )}+\frac {\sqrt {c} \left (5 c^4 d^8+28 a c^3 d^6 e^2+70 a^2 c^2 d^4 e^4+140 a^3 c d^2 e^6-35 a^4 e^8\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{16 a^{7/2} \left (c d^2+a e^2\right )^5}+\frac {8 c d e^7 \log (d+e x)}{\left (c d^2+a e^2\right )^5}-\frac {4 c d e^7 \log \left (a+c x^2\right )}{\left (c d^2+a e^2\right )^5}\\ \end {align*}

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Mathematica [A]
time = 0.27, size = 336, normalized size = 0.78 \begin {gather*} \frac {-\frac {48 e^7 \left (c d^2+a e^2\right )}{d+e x}+\frac {3 c \left (c d^2+a e^2\right ) \left (5 c^3 d^6 x+23 a c^2 d^4 e^2 x+47 a^2 c d^2 e^4 x+a^3 e^5 (48 d-19 e x)\right )}{a^3 \left (a+c x^2\right )}+\frac {2 c \left (c d^2+a e^2\right )^2 \left (5 c^2 d^4 x+18 a c d^2 e^2 x+a^2 e^3 (24 d-11 e x)\right )}{a^2 \left (a+c x^2\right )^2}+\frac {8 c \left (c d^2+a e^2\right )^3 \left (c d^2 x+a e (2 d-e x)\right )}{a \left (a+c x^2\right )^3}+\frac {3 \sqrt {c} \left (5 c^4 d^8+28 a c^3 d^6 e^2+70 a^2 c^2 d^4 e^4+140 a^3 c d^2 e^6-35 a^4 e^8\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{a^{7/2}}+384 c d e^7 \log (d+e x)-192 c d e^7 \log \left (a+c x^2\right )}{48 \left (c d^2+a e^2\right )^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^2*(a + c*x^2)^4),x]

[Out]

((-48*e^7*(c*d^2 + a*e^2))/(d + e*x) + (3*c*(c*d^2 + a*e^2)*(5*c^3*d^6*x + 23*a*c^2*d^4*e^2*x + 47*a^2*c*d^2*e
^4*x + a^3*e^5*(48*d - 19*e*x)))/(a^3*(a + c*x^2)) + (2*c*(c*d^2 + a*e^2)^2*(5*c^2*d^4*x + 18*a*c*d^2*e^2*x +
a^2*e^3*(24*d - 11*e*x)))/(a^2*(a + c*x^2)^2) + (8*c*(c*d^2 + a*e^2)^3*(c*d^2*x + a*e*(2*d - e*x)))/(a*(a + c*
x^2)^3) + (3*Sqrt[c]*(5*c^4*d^8 + 28*a*c^3*d^6*e^2 + 70*a^2*c^2*d^4*e^4 + 140*a^3*c*d^2*e^6 - 35*a^4*e^8)*ArcT
an[(Sqrt[c]*x)/Sqrt[a]])/a^(7/2) + 384*c*d*e^7*Log[d + e*x] - 192*c*d*e^7*Log[a + c*x^2])/(48*(c*d^2 + a*e^2)^
5)

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Maple [A]
time = 0.79, size = 473, normalized size = 1.10

method	result	size
default	\(-\frac {e^{7}}{\left (e^{2} a +c \,d^{2}\right )^{4} \left (e x +d \right )}+\frac {8 c d \,e^{7} \ln \left (e x +d \right )}{\left (e^{2} a +c \,d^{2}\right )^{5}}-\frac {c \left (\frac {\frac {c^{2} \left (19 a^{4} e^{8}-28 a^{3} c \,d^{2} e^{6}-70 a^{2} c^{2} d^{4} e^{4}-28 a \,c^{3} d^{6} e^{2}-5 c^{4} d^{8}\right ) x^{5}}{16 a^{3}}+\left (-3 d \,e^{7} a \,c^{2}-3 e^{5} d^{3} c^{3}\right ) x^{4}+\frac {c \left (17 a^{4} e^{8}-20 a^{3} c \,d^{2} e^{6}-60 a^{2} c^{2} d^{4} e^{4}-28 a \,c^{3} d^{6} e^{2}-5 c^{4} d^{8}\right ) x^{3}}{6 a^{2}}+\left (-7 d \,e^{7} a^{2} c -8 e^{5} d^{3} a \,c^{2}-c^{3} d^{5} e^{3}\right ) x^{2}+\frac {\left (29 a^{4} e^{8}-20 a^{3} c \,d^{2} e^{6}-90 a^{2} c^{2} d^{4} e^{4}-52 a \,c^{3} d^{6} e^{2}-11 c^{4} d^{8}\right ) x}{16 a}-\frac {d e \left (13 e^{6} a^{3}+18 e^{4} d^{2} a^{2} c +6 d^{4} e^{2} c^{2} a +d^{6} c^{3}\right )}{3}}{\left (c \,x^{2}+a \right )^{3}}+\frac {64 a^{3} d \,e^{7} \ln \left (c \,x^{2}+a \right )+\frac {\left (35 a^{4} e^{8}-140 a^{3} c \,d^{2} e^{6}-70 a^{2} c^{2} d^{4} e^{4}-28 a \,c^{3} d^{6} e^{2}-5 c^{4} d^{8}\right ) \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}}}{16 a^{3}}\right )}{\left (e^{2} a +c \,d^{2}\right )^{5}}\)	\(473\)
risch	\(\text {Expression too large to display}\)	\(5462\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^2/(c*x^2+a)^4,x,method=_RETURNVERBOSE)

[Out]

-e^7/(a*e^2+c*d^2)^4/(e*x+d)+8*c*d*e^7*ln(e*x+d)/(a*e^2+c*d^2)^5-c/(a*e^2+c*d^2)^5*((1/16*c^2*(19*a^4*e^8-28*a
^3*c*d^2*e^6-70*a^2*c^2*d^4*e^4-28*a*c^3*d^6*e^2-5*c^4*d^8)/a^3*x^5+(-3*a*c^2*d*e^7-3*c^3*d^3*e^5)*x^4+1/6*c*(
17*a^4*e^8-20*a^3*c*d^2*e^6-60*a^2*c^2*d^4*e^4-28*a*c^3*d^6*e^2-5*c^4*d^8)/a^2*x^3+(-7*a^2*c*d*e^7-8*a*c^2*d^3
*e^5-c^3*d^5*e^3)*x^2+1/16*(29*a^4*e^8-20*a^3*c*d^2*e^6-90*a^2*c^2*d^4*e^4-52*a*c^3*d^6*e^2-11*c^4*d^8)/a*x-1/
3*d*e*(13*a^3*e^6+18*a^2*c*d^2*e^4+6*a*c^2*d^4*e^2+c^3*d^6))/(c*x^2+a)^3+1/16/a^3*(64*a^3*d*e^7*ln(c*x^2+a)+(3
5*a^4*e^8-140*a^3*c*d^2*e^6-70*a^2*c^2*d^4*e^4-28*a*c^3*d^6*e^2-5*c^4*d^8)/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))
))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1150 vs. \(2 (400) = 800\).
time = 0.57, size = 1150, normalized size = 2.67 \begin {gather*} -\frac {4 \, c d e^{7} \log \left (c x^{2} + a\right )}{c^{5} d^{10} + 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} + 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} + a^{5} e^{10}} + \frac {8 \, c d e^{7} \log \left (x e + d\right )}{c^{5} d^{10} + 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} + 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} + a^{5} e^{10}} + \frac {{\left (5 \, c^{5} d^{8} + 28 \, a c^{4} d^{6} e^{2} + 70 \, a^{2} c^{3} d^{4} e^{4} + 140 \, a^{3} c^{2} d^{2} e^{6} - 35 \, a^{4} c e^{8}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{16 \, {\left (a^{3} c^{5} d^{10} + 5 \, a^{4} c^{4} d^{8} e^{2} + 10 \, a^{5} c^{3} d^{6} e^{4} + 10 \, a^{6} c^{2} d^{4} e^{6} + 5 \, a^{7} c d^{2} e^{8} + a^{8} e^{10}\right )} \sqrt {a c}} + \frac {16 \, a^{3} c^{3} d^{6} e + 80 \, a^{4} c^{2} d^{4} e^{3} + 208 \, a^{5} c d^{2} e^{5} + 3 \, {\left (5 \, c^{6} d^{6} e + 23 \, a c^{5} d^{4} e^{3} + 47 \, a^{2} c^{4} d^{2} e^{5} - 35 \, a^{3} c^{3} e^{7}\right )} x^{6} - 48 \, a^{6} e^{7} + 3 \, {\left (5 \, c^{6} d^{7} + 23 \, a c^{5} d^{5} e^{2} + 47 \, a^{2} c^{4} d^{3} e^{4} + 29 \, a^{3} c^{3} d e^{6}\right )} x^{5} + 8 \, {\left (5 \, a c^{5} d^{6} e + 23 \, a^{2} c^{4} d^{4} e^{3} + 55 \, a^{3} c^{3} d^{2} e^{5} - 35 \, a^{4} c^{2} e^{7}\right )} x^{4} + 8 \, {\left (5 \, a c^{5} d^{7} + 23 \, a^{2} c^{4} d^{5} e^{2} + 43 \, a^{3} c^{3} d^{3} e^{4} + 25 \, a^{4} c^{2} d e^{6}\right )} x^{3} + 3 \, {\left (11 \, a^{2} c^{4} d^{6} e + 57 \, a^{3} c^{3} d^{4} e^{3} + 161 \, a^{4} c^{2} d^{2} e^{5} - 77 \, a^{5} c e^{7}\right )} x^{2} + {\left (33 \, a^{2} c^{4} d^{7} + 139 \, a^{3} c^{3} d^{5} e^{2} + 227 \, a^{4} c^{2} d^{3} e^{4} + 121 \, a^{5} c d e^{6}\right )} x}{48 \, {\left (a^{6} c^{4} d^{9} + 4 \, a^{7} c^{3} d^{7} e^{2} + 6 \, a^{8} c^{2} d^{5} e^{4} + 4 \, a^{9} c d^{3} e^{6} + a^{10} d e^{8} + {\left (a^{3} c^{7} d^{8} e + 4 \, a^{4} c^{6} d^{6} e^{3} + 6 \, a^{5} c^{5} d^{4} e^{5} + 4 \, a^{6} c^{4} d^{2} e^{7} + a^{7} c^{3} e^{9}\right )} x^{7} + {\left (a^{3} c^{7} d^{9} + 4 \, a^{4} c^{6} d^{7} e^{2} + 6 \, a^{5} c^{5} d^{5} e^{4} + 4 \, a^{6} c^{4} d^{3} e^{6} + a^{7} c^{3} d e^{8}\right )} x^{6} + 3 \, {\left (a^{4} c^{6} d^{8} e + 4 \, a^{5} c^{5} d^{6} e^{3} + 6 \, a^{6} c^{4} d^{4} e^{5} + 4 \, a^{7} c^{3} d^{2} e^{7} + a^{8} c^{2} e^{9}\right )} x^{5} + 3 \, {\left (a^{4} c^{6} d^{9} + 4 \, a^{5} c^{5} d^{7} e^{2} + 6 \, a^{6} c^{4} d^{5} e^{4} + 4 \, a^{7} c^{3} d^{3} e^{6} + a^{8} c^{2} d e^{8}\right )} x^{4} + 3 \, {\left (a^{5} c^{5} d^{8} e + 4 \, a^{6} c^{4} d^{6} e^{3} + 6 \, a^{7} c^{3} d^{4} e^{5} + 4 \, a^{8} c^{2} d^{2} e^{7} + a^{9} c e^{9}\right )} x^{3} + 3 \, {\left (a^{5} c^{5} d^{9} + 4 \, a^{6} c^{4} d^{7} e^{2} + 6 \, a^{7} c^{3} d^{5} e^{4} + 4 \, a^{8} c^{2} d^{3} e^{6} + a^{9} c d e^{8}\right )} x^{2} + {\left (a^{6} c^{4} d^{8} e + 4 \, a^{7} c^{3} d^{6} e^{3} + 6 \, a^{8} c^{2} d^{4} e^{5} + 4 \, a^{9} c d^{2} e^{7} + a^{10} e^{9}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^2/(c*x^2+a)^4,x, algorithm="maxima")

[Out]

-4*c*d*e^7*log(c*x^2 + a)/(c^5*d^10 + 5*a*c^4*d^8*e^2 + 10*a^2*c^3*d^6*e^4 + 10*a^3*c^2*d^4*e^6 + 5*a^4*c*d^2*
e^8 + a^5*e^10) + 8*c*d*e^7*log(x*e + d)/(c^5*d^10 + 5*a*c^4*d^8*e^2 + 10*a^2*c^3*d^6*e^4 + 10*a^3*c^2*d^4*e^6
+ 5*a^4*c*d^2*e^8 + a^5*e^10) + 1/16*(5*c^5*d^8 + 28*a*c^4*d^6*e^2 + 70*a^2*c^3*d^4*e^4 + 140*a^3*c^2*d^2*e^6
- 35*a^4*c*e^8)*arctan(c*x/sqrt(a*c))/((a^3*c^5*d^10 + 5*a^4*c^4*d^8*e^2 + 10*a^5*c^3*d^6*e^4 + 10*a^6*c^2*d^
4*e^6 + 5*a^7*c*d^2*e^8 + a^8*e^10)*sqrt(a*c)) + 1/48*(16*a^3*c^3*d^6*e + 80*a^4*c^2*d^4*e^3 + 208*a^5*c*d^2*e
^5 + 3*(5*c^6*d^6*e + 23*a*c^5*d^4*e^3 + 47*a^2*c^4*d^2*e^5 - 35*a^3*c^3*e^7)*x^6 - 48*a^6*e^7 + 3*(5*c^6*d^7
+ 23*a*c^5*d^5*e^2 + 47*a^2*c^4*d^3*e^4 + 29*a^3*c^3*d*e^6)*x^5 + 8*(5*a*c^5*d^6*e + 23*a^2*c^4*d^4*e^3 + 55*a
^3*c^3*d^2*e^5 - 35*a^4*c^2*e^7)*x^4 + 8*(5*a*c^5*d^7 + 23*a^2*c^4*d^5*e^2 + 43*a^3*c^3*d^3*e^4 + 25*a^4*c^2*d
*e^6)*x^3 + 3*(11*a^2*c^4*d^6*e + 57*a^3*c^3*d^4*e^3 + 161*a^4*c^2*d^2*e^5 - 77*a^5*c*e^7)*x^2 + (33*a^2*c^4*d
^7 + 139*a^3*c^3*d^5*e^2 + 227*a^4*c^2*d^3*e^4 + 121*a^5*c*d*e^6)*x)/(a^6*c^4*d^9 + 4*a^7*c^3*d^7*e^2 + 6*a^8*
c^2*d^5*e^4 + 4*a^9*c*d^3*e^6 + a^10*d*e^8 + (a^3*c^7*d^8*e + 4*a^4*c^6*d^6*e^3 + 6*a^5*c^5*d^4*e^5 + 4*a^6*c^
4*d^2*e^7 + a^7*c^3*e^9)*x^7 + (a^3*c^7*d^9 + 4*a^4*c^6*d^7*e^2 + 6*a^5*c^5*d^5*e^4 + 4*a^6*c^4*d^3*e^6 + a^7*
c^3*d*e^8)*x^6 + 3*(a^4*c^6*d^8*e + 4*a^5*c^5*d^6*e^3 + 6*a^6*c^4*d^4*e^5 + 4*a^7*c^3*d^2*e^7 + a^8*c^2*e^9)*x
^5 + 3*(a^4*c^6*d^9 + 4*a^5*c^5*d^7*e^2 + 6*a^6*c^4*d^5*e^4 + 4*a^7*c^3*d^3*e^6 + a^8*c^2*d*e^8)*x^4 + 3*(a^5*
c^5*d^8*e + 4*a^6*c^4*d^6*e^3 + 6*a^7*c^3*d^4*e^5 + 4*a^8*c^2*d^2*e^7 + a^9*c*e^9)*x^3 + 3*(a^5*c^5*d^9 + 4*a^
6*c^4*d^7*e^2 + 6*a^7*c^3*d^5*e^4 + 4*a^8*c^2*d^3*e^6 + a^9*c*d*e^8)*x^2 + (a^6*c^4*d^8*e + 4*a^7*c^3*d^6*e^3
+ 6*a^8*c^2*d^4*e^5 + 4*a^9*c*d^2*e^7 + a^10*e^9)*x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1862 vs. \(2 (400) = 800\).
time = 33.96, size = 3750, normalized size = 8.72 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^2/(c*x^2+a)^4,x, algorithm="fricas")

[Out]

[1/96*(30*c^7*d^9*x^5 + 80*a*c^6*d^9*x^3 + 66*a^2*c^5*d^9*x - 3*(5*c^7*d^9*x^6 + 15*a*c^6*d^9*x^4 + 15*a^2*c^5
*d^9*x^2 + 5*a^3*c^4*d^9 - 35*(a^4*c^3*x^7 + 3*a^5*c^2*x^5 + 3*a^6*c*x^3 + a^7*x)*e^9 - 35*(a^4*c^3*d*x^6 + 3*
a^5*c^2*d*x^4 + 3*a^6*c*d*x^2 + a^7*d)*e^8 + 140*(a^3*c^4*d^2*x^7 + 3*a^4*c^3*d^2*x^5 + 3*a^5*c^2*d^2*x^3 + a^
6*c*d^2*x)*e^7 + 140*(a^3*c^4*d^3*x^6 + 3*a^4*c^3*d^3*x^4 + 3*a^5*c^2*d^3*x^2 + a^6*c*d^3)*e^6 + 70*(a^2*c^5*d
^4*x^7 + 3*a^3*c^4*d^4*x^5 + 3*a^4*c^3*d^4*x^3 + a^5*c^2*d^4*x)*e^5 + 70*(a^2*c^5*d^5*x^6 + 3*a^3*c^4*d^5*x^4
+ 3*a^4*c^3*d^5*x^2 + a^5*c^2*d^5)*e^4 + 28*(a*c^6*d^6*x^7 + 3*a^2*c^5*d^6*x^5 + 3*a^3*c^4*d^6*x^3 + a^4*c^3*d
^6*x)*e^3 + 28*(a*c^6*d^7*x^6 + 3*a^2*c^5*d^7*x^4 + 3*a^3*c^4*d^7*x^2 + a^4*c^3*d^7)*e^2 + 5*(c^7*d^8*x^7 + 3*
a*c^6*d^8*x^5 + 3*a^2*c^5*d^8*x^3 + a^3*c^4*d^8*x)*e)*sqrt(-c/a)*log((c*x^2 - 2*a*x*sqrt(-c/a) - a)/(c*x^2 + a
)) - 2*(105*a^4*c^3*x^6 + 280*a^5*c^2*x^4 + 231*a^6*c*x^2 + 48*a^7)*e^9 + 2*(87*a^4*c^3*d*x^5 + 200*a^5*c^2*d*
x^3 + 121*a^6*c*d*x)*e^8 + 8*(9*a^3*c^4*d^2*x^6 + 40*a^4*c^3*d^2*x^4 + 63*a^5*c^2*d^2*x^2 + 40*a^6*c*d^2)*e^7
+ 8*(57*a^3*c^4*d^3*x^5 + 136*a^4*c^3*d^3*x^3 + 87*a^5*c^2*d^3*x)*e^6 + 12*(35*a^2*c^5*d^4*x^6 + 104*a^3*c^4*d
^4*x^4 + 109*a^4*c^3*d^4*x^2 + 48*a^5*c^2*d^4)*e^5 + 12*(35*a^2*c^5*d^5*x^5 + 88*a^3*c^4*d^5*x^3 + 61*a^4*c^3*
d^5*x)*e^4 + 8*(21*a*c^6*d^6*x^6 + 56*a^2*c^5*d^6*x^4 + 51*a^3*c^4*d^6*x^2 + 24*a^4*c^3*d^6)*e^3 + 8*(21*a*c^6
*d^7*x^5 + 56*a^2*c^5*d^7*x^3 + 43*a^3*c^4*d^7*x)*e^2 + 2*(15*c^7*d^8*x^6 + 40*a*c^6*d^8*x^4 + 33*a^2*c^5*d^8*
x^2 + 16*a^3*c^4*d^8)*e - 384*((a^3*c^4*d*x^7 + 3*a^4*c^3*d*x^5 + 3*a^5*c^2*d*x^3 + a^6*c*d*x)*e^8 + (a^3*c^4*
d^2*x^6 + 3*a^4*c^3*d^2*x^4 + 3*a^5*c^2*d^2*x^2 + a^6*c*d^2)*e^7)*log(c*x^2 + a) + 768*((a^3*c^4*d*x^7 + 3*a^4
*c^3*d*x^5 + 3*a^5*c^2*d*x^3 + a^6*c*d*x)*e^8 + (a^3*c^4*d^2*x^6 + 3*a^4*c^3*d^2*x^4 + 3*a^5*c^2*d^2*x^2 + a^6
*c*d^2)*e^7)*log(x*e + d))/(a^3*c^8*d^11*x^6 + 3*a^4*c^7*d^11*x^4 + 3*a^5*c^6*d^11*x^2 + a^6*c^5*d^11 + (a^8*c
^3*x^7 + 3*a^9*c^2*x^5 + 3*a^10*c*x^3 + a^11*x)*e^11 + (a^8*c^3*d*x^6 + 3*a^9*c^2*d*x^4 + 3*a^10*c*d*x^2 + a^1
1*d)*e^10 + 5*(a^7*c^4*d^2*x^7 + 3*a^8*c^3*d^2*x^5 + 3*a^9*c^2*d^2*x^3 + a^10*c*d^2*x)*e^9 + 5*(a^7*c^4*d^3*x^
6 + 3*a^8*c^3*d^3*x^4 + 3*a^9*c^2*d^3*x^2 + a^10*c*d^3)*e^8 + 10*(a^6*c^5*d^4*x^7 + 3*a^7*c^4*d^4*x^5 + 3*a^8*
c^3*d^4*x^3 + a^9*c^2*d^4*x)*e^7 + 10*(a^6*c^5*d^5*x^6 + 3*a^7*c^4*d^5*x^4 + 3*a^8*c^3*d^5*x^2 + a^9*c^2*d^5)*
e^6 + 10*(a^5*c^6*d^6*x^7 + 3*a^6*c^5*d^6*x^5 + 3*a^7*c^4*d^6*x^3 + a^8*c^3*d^6*x)*e^5 + 10*(a^5*c^6*d^7*x^6 +
3*a^6*c^5*d^7*x^4 + 3*a^7*c^4*d^7*x^2 + a^8*c^3*d^7)*e^4 + 5*(a^4*c^7*d^8*x^7 + 3*a^5*c^6*d^8*x^5 + 3*a^6*c^5
*d^8*x^3 + a^7*c^4*d^8*x)*e^3 + 5*(a^4*c^7*d^9*x^6 + 3*a^5*c^6*d^9*x^4 + 3*a^6*c^5*d^9*x^2 + a^7*c^4*d^9)*e^2
+ (a^3*c^8*d^10*x^7 + 3*a^4*c^7*d^10*x^5 + 3*a^5*c^6*d^10*x^3 + a^6*c^5*d^10*x)*e), 1/48*(15*c^7*d^9*x^5 + 40*
a*c^6*d^9*x^3 + 33*a^2*c^5*d^9*x + 3*(5*c^7*d^9*x^6 + 15*a*c^6*d^9*x^4 + 15*a^2*c^5*d^9*x^2 + 5*a^3*c^4*d^9 -
35*(a^4*c^3*x^7 + 3*a^5*c^2*x^5 + 3*a^6*c*x^3 + a^7*x)*e^9 - 35*(a^4*c^3*d*x^6 + 3*a^5*c^2*d*x^4 + 3*a^6*c*d*x
^2 + a^7*d)*e^8 + 140*(a^3*c^4*d^2*x^7 + 3*a^4*c^3*d^2*x^5 + 3*a^5*c^2*d^2*x^3 + a^6*c*d^2*x)*e^7 + 140*(a^3*c
^4*d^3*x^6 + 3*a^4*c^3*d^3*x^4 + 3*a^5*c^2*d^3*x^2 + a^6*c*d^3)*e^6 + 70*(a^2*c^5*d^4*x^7 + 3*a^3*c^4*d^4*x^5
+ 3*a^4*c^3*d^4*x^3 + a^5*c^2*d^4*x)*e^5 + 70*(a^2*c^5*d^5*x^6 + 3*a^3*c^4*d^5*x^4 + 3*a^4*c^3*d^5*x^2 + a^5*c
^2*d^5)*e^4 + 28*(a*c^6*d^6*x^7 + 3*a^2*c^5*d^6*x^5 + 3*a^3*c^4*d^6*x^3 + a^4*c^3*d^6*x)*e^3 + 28*(a*c^6*d^7*x
^6 + 3*a^2*c^5*d^7*x^4 + 3*a^3*c^4*d^7*x^2 + a^4*c^3*d^7)*e^2 + 5*(c^7*d^8*x^7 + 3*a*c^6*d^8*x^5 + 3*a^2*c^5*d
^8*x^3 + a^3*c^4*d^8*x)*e)*sqrt(c/a)*arctan(x*sqrt(c/a)) - (105*a^4*c^3*x^6 + 280*a^5*c^2*x^4 + 231*a^6*c*x^2
+ 48*a^7)*e^9 + (87*a^4*c^3*d*x^5 + 200*a^5*c^2*d*x^3 + 121*a^6*c*d*x)*e^8 + 4*(9*a^3*c^4*d^2*x^6 + 40*a^4*c^3
*d^2*x^4 + 63*a^5*c^2*d^2*x^2 + 40*a^6*c*d^2)*e^7 + 4*(57*a^3*c^4*d^3*x^5 + 136*a^4*c^3*d^3*x^3 + 87*a^5*c^2*d
^3*x)*e^6 + 6*(35*a^2*c^5*d^4*x^6 + 104*a^3*c^4*d^4*x^4 + 109*a^4*c^3*d^4*x^2 + 48*a^5*c^2*d^4)*e^5 + 6*(35*a^
2*c^5*d^5*x^5 + 88*a^3*c^4*d^5*x^3 + 61*a^4*c^3*d^5*x)*e^4 + 4*(21*a*c^6*d^6*x^6 + 56*a^2*c^5*d^6*x^4 + 51*a^3
*c^4*d^6*x^2 + 24*a^4*c^3*d^6)*e^3 + 4*(21*a*c^6*d^7*x^5 + 56*a^2*c^5*d^7*x^3 + 43*a^3*c^4*d^7*x)*e^2 + (15*c^
7*d^8*x^6 + 40*a*c^6*d^8*x^4 + 33*a^2*c^5*d^8*x^2 + 16*a^3*c^4*d^8)*e - 192*((a^3*c^4*d*x^7 + 3*a^4*c^3*d*x^5
+ 3*a^5*c^2*d*x^3 + a^6*c*d*x)*e^8 + (a^3*c^4*d^2*x^6 + 3*a^4*c^3*d^2*x^4 + 3*a^5*c^2*d^2*x^2 + a^6*c*d^2)*e^7
)*log(c*x^2 + a) + 384*((a^3*c^4*d*x^7 + 3*a^4*c^3*d*x^5 + 3*a^5*c^2*d*x^3 + a^6*c*d*x)*e^8 + (a^3*c^4*d^2*x^6
+ 3*a^4*c^3*d^2*x^4 + 3*a^5*c^2*d^2*x^2 + a^6*c*d^2)*e^7)*log(x*e + d))/(a^3*c^8*d^11*x^6 + 3*a^4*c^7*d^11*x^
4 + 3*a^5*c^6*d^11*x^2 + a^6*c^5*d^11 + (a^8*c^3*x^7 + 3*a^9*c^2*x^5 + 3*a^10*c*x^3 + a^11*x)*e^11 + (a^8*c^3*
d*x^6 + 3*a^9*c^2*d*x^4 + 3*a^10*c*d*x^2 + a^11*d)*e^10 + 5*(a^7*c^4*d^2*x^7 + 3*a^8*c^3*d^2*x^5 + 3*a^9*c^2*d
^2*x^3 + a^10*c*d^2*x)*e^9 + 5*(a^7*c^4*d^3*x^6...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**2/(c*x**2+a)**4,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 856 vs. \(2 (400) = 800\).
time = 1.43, size = 856, normalized size = 1.99 \begin {gather*} -\frac {4 \, c d e^{7} \log \left (c - \frac {2 \, c d}{x e + d} + \frac {c d^{2}}{{\left (x e + d\right )}^{2}} + \frac {a e^{2}}{{\left (x e + d\right )}^{2}}\right )}{c^{5} d^{10} + 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} + 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} + a^{5} e^{10}} + \frac {{\left (5 \, c^{5} d^{8} e^{2} + 28 \, a c^{4} d^{6} e^{4} + 70 \, a^{2} c^{3} d^{4} e^{6} + 140 \, a^{3} c^{2} d^{2} e^{8} - 35 \, a^{4} c e^{10}\right )} \arctan \left (\frac {{\left (c d - \frac {c d^{2}}{x e + d} - \frac {a e^{2}}{x e + d}\right )} e^{\left (-1\right )}}{\sqrt {a c}}\right ) e^{\left (-2\right )}}{16 \, {\left (a^{3} c^{5} d^{10} + 5 \, a^{4} c^{4} d^{8} e^{2} + 10 \, a^{5} c^{3} d^{6} e^{4} + 10 \, a^{6} c^{2} d^{4} e^{6} + 5 \, a^{7} c d^{2} e^{8} + a^{8} e^{10}\right )} \sqrt {a c}} - \frac {e^{15}}{{\left (c^{4} d^{8} e^{8} + 4 \, a c^{3} d^{6} e^{10} + 6 \, a^{2} c^{2} d^{4} e^{12} + 4 \, a^{3} c d^{2} e^{14} + a^{4} e^{16}\right )} {\left (x e + d\right )}} + \frac {15 \, c^{7} d^{7} e + 79 \, a c^{6} d^{5} e^{3} + 185 \, a^{2} c^{5} d^{3} e^{5} - 295 \, a^{3} c^{4} d e^{7} - \frac {3 \, {\left (25 \, c^{7} d^{8} e^{2} + 130 \, a c^{6} d^{6} e^{4} + 300 \, a^{2} c^{5} d^{4} e^{6} - 618 \, a^{3} c^{4} d^{2} e^{8} + 19 \, a^{4} c^{3} e^{10}\right )} e^{\left (-1\right )}}{x e + d} + \frac {6 \, {\left (25 \, c^{7} d^{9} e^{3} + 135 \, a c^{6} d^{7} e^{5} + 327 \, a^{2} c^{5} d^{5} e^{7} - 691 \, a^{3} c^{4} d^{3} e^{9} - 76 \, a^{4} c^{3} d e^{11}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac {2 \, {\left (75 \, c^{7} d^{10} e^{4} + 440 \, a c^{6} d^{8} e^{6} + 1162 \, a^{2} c^{5} d^{6} e^{8} - 2212 \, a^{3} c^{4} d^{4} e^{10} - 1277 \, a^{4} c^{3} d^{2} e^{12} + 68 \, a^{5} c^{2} e^{14}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}} + \frac {3 \, {\left (25 \, c^{7} d^{11} e^{5} + 165 \, a c^{6} d^{9} e^{7} + 490 \, a^{2} c^{5} d^{7} e^{9} - 742 \, a^{3} c^{4} d^{5} e^{11} - 1139 \, a^{4} c^{3} d^{3} e^{13} - 47 \, a^{5} c^{2} d e^{15}\right )} e^{\left (-4\right )}}{{\left (x e + d\right )}^{4}} - \frac {3 \, {\left (5 \, c^{7} d^{12} e^{6} + 38 \, a c^{6} d^{10} e^{8} + 131 \, a^{2} c^{5} d^{8} e^{10} - 140 \, a^{3} c^{4} d^{6} e^{12} - 517 \, a^{4} c^{3} d^{4} e^{14} - 250 \, a^{5} c^{2} d^{2} e^{16} + 29 \, a^{6} c e^{18}\right )} e^{\left (-5\right )}}{{\left (x e + d\right )}^{5}}}{48 \, {\left (c d^{2} + a e^{2}\right )}^{5} a^{3} {\left (c - \frac {2 \, c d}{x e + d} + \frac {c d^{2}}{{\left (x e + d\right )}^{2}} + \frac {a e^{2}}{{\left (x e + d\right )}^{2}}\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^2/(c*x^2+a)^4,x, algorithm="giac")

[Out]

-4*c*d*e^7*log(c - 2*c*d/(x*e + d) + c*d^2/(x*e + d)^2 + a*e^2/(x*e + d)^2)/(c^5*d^10 + 5*a*c^4*d^8*e^2 + 10*a
^2*c^3*d^6*e^4 + 10*a^3*c^2*d^4*e^6 + 5*a^4*c*d^2*e^8 + a^5*e^10) + 1/16*(5*c^5*d^8*e^2 + 28*a*c^4*d^6*e^4 + 7
0*a^2*c^3*d^4*e^6 + 140*a^3*c^2*d^2*e^8 - 35*a^4*c*e^10)*arctan((c*d - c*d^2/(x*e + d) - a*e^2/(x*e + d))*e^(-
1)/sqrt(a*c))*e^(-2)/((a^3*c^5*d^10 + 5*a^4*c^4*d^8*e^2 + 10*a^5*c^3*d^6*e^4 + 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^
2*e^8 + a^8*e^10)*sqrt(a*c)) - e^15/((c^4*d^8*e^8 + 4*a*c^3*d^6*e^10 + 6*a^2*c^2*d^4*e^12 + 4*a^3*c*d^2*e^14 +
a^4*e^16)*(x*e + d)) + 1/48*(15*c^7*d^7*e + 79*a*c^6*d^5*e^3 + 185*a^2*c^5*d^3*e^5 - 295*a^3*c^4*d*e^7 - 3*(2
5*c^7*d^8*e^2 + 130*a*c^6*d^6*e^4 + 300*a^2*c^5*d^4*e^6 - 618*a^3*c^4*d^2*e^8 + 19*a^4*c^3*e^10)*e^(-1)/(x*e +
d) + 6*(25*c^7*d^9*e^3 + 135*a*c^6*d^7*e^5 + 327*a^2*c^5*d^5*e^7 - 691*a^3*c^4*d^3*e^9 - 76*a^4*c^3*d*e^11)*e
^(-2)/(x*e + d)^2 - 2*(75*c^7*d^10*e^4 + 440*a*c^6*d^8*e^6 + 1162*a^2*c^5*d^6*e^8 - 2212*a^3*c^4*d^4*e^10 - 12
77*a^4*c^3*d^2*e^12 + 68*a^5*c^2*e^14)*e^(-3)/(x*e + d)^3 + 3*(25*c^7*d^11*e^5 + 165*a*c^6*d^9*e^7 + 490*a^2*c
^5*d^7*e^9 - 742*a^3*c^4*d^5*e^11 - 1139*a^4*c^3*d^3*e^13 - 47*a^5*c^2*d*e^15)*e^(-4)/(x*e + d)^4 - 3*(5*c^7*d
^12*e^6 + 38*a*c^6*d^10*e^8 + 131*a^2*c^5*d^8*e^10 - 140*a^3*c^4*d^6*e^12 - 517*a^4*c^3*d^4*e^14 - 250*a^5*c^2
*d^2*e^16 + 29*a^6*c*e^18)*e^(-5)/(x*e + d)^5)/((c*d^2 + a*e^2)^5*a^3*(c - 2*c*d/(x*e + d) + c*d^2/(x*e + d)^2
+ a*e^2/(x*e + d)^2)^3)

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Mupad [B]
time = 2.49, size = 1876, normalized size = 4.36 \begin {gather*} \frac {\frac {-3\,a^3\,e^7+13\,a^2\,c\,d^2\,e^5+5\,a\,c^2\,d^4\,e^3+c^3\,d^6\,e}{3\,\left (c\,d^2+a\,e^2\right )\,\left (a^3\,e^6+3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2+c^3\,d^6\right )}+\frac {x\,\left (121\,a^2\,c\,d\,e^4+106\,a\,c^2\,d^3\,e^2+33\,c^3\,d^5\right )}{48\,a\,\left (a^3\,e^6+3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2+c^3\,d^6\right )}+\frac {x^6\,\left (-35\,a^3\,c^3\,e^7+47\,a^2\,c^4\,d^2\,e^5+23\,a\,c^5\,d^4\,e^3+5\,c^6\,d^6\,e\right )}{16\,a^3\,\left (a^4\,e^8+4\,a^3\,c\,d^2\,e^6+6\,a^2\,c^2\,d^4\,e^4+4\,a\,c^3\,d^6\,e^2+c^4\,d^8\right )}+\frac {x^3\,\left (25\,a^2\,c^2\,d\,e^4+18\,a\,c^3\,d^3\,e^2+5\,c^4\,d^5\right )}{6\,a^2\,\left (a^3\,e^6+3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2+c^3\,d^6\right )}+\frac {x^5\,\left (29\,a^2\,c^3\,d\,e^4+18\,a\,c^4\,d^3\,e^2+5\,c^5\,d^5\right )}{16\,a^3\,\left (a^3\,e^6+3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2+c^3\,d^6\right )}+\frac {x^4\,\left (-35\,a^3\,c^2\,e^7+55\,a^2\,c^3\,d^2\,e^5+23\,a\,c^4\,d^4\,e^3+5\,c^5\,d^6\,e\right )}{6\,a^2\,\left (c\,d^2+a\,e^2\right )\,\left (a^3\,e^6+3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2+c^3\,d^6\right )}+\frac {x^2\,\left (-77\,a^3\,c\,e^7+161\,a^2\,c^2\,d^2\,e^5+57\,a\,c^3\,d^4\,e^3+11\,c^4\,d^6\,e\right )}{16\,a\,\left (c\,d^2+a\,e^2\right )\,\left (a^3\,e^6+3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2+c^3\,d^6\right )}}{e\,a^3\,x+d\,a^3+3\,e\,a^2\,c\,x^3+3\,d\,a^2\,c\,x^2+3\,e\,a\,c^2\,x^5+3\,d\,a\,c^2\,x^4+e\,c^3\,x^7+d\,c^3\,x^6}-\frac {\ln \left (25\,c^9\,d^{20}\,{\left (-a^7\,c\right )}^{3/2}-1225\,a^{17}\,e^{20}\,\sqrt {-a^7\,c}+25\,a^{10}\,c^{11}\,d^{20}\,x-291237\,a\,d^4\,e^{16}\,{\left (-a^7\,c\right )}^{5/2}-184696\,c\,d^6\,e^{14}\,{\left (-a^7\,c\right )}^{5/2}+140106\,a^9\,d^2\,e^{18}\,{\left (-a^7\,c\right )}^{3/2}+1225\,a^{20}\,c\,e^{20}\,x+2069\,a^2\,c^7\,d^{16}\,e^4\,{\left (-a^7\,c\right )}^{3/2}+8568\,a^3\,c^6\,d^{14}\,e^6\,{\left (-a^7\,c\right )}^{3/2}+24514\,a^4\,c^5\,d^{12}\,e^8\,{\left (-a^7\,c\right )}^{3/2}+47740\,a^5\,c^4\,d^{10}\,e^{10}\,{\left (-a^7\,c\right )}^{3/2}+62370\,a^6\,c^3\,d^8\,e^{12}\,{\left (-a^7\,c\right )}^{3/2}+330\,a^{11}\,c^{10}\,d^{18}\,e^2\,x+2069\,a^{12}\,c^9\,d^{16}\,e^4\,x+8568\,a^{13}\,c^8\,d^{14}\,e^6\,x+24514\,a^{14}\,c^7\,d^{12}\,e^8\,x+47740\,a^{15}\,c^6\,d^{10}\,e^{10}\,x+62370\,a^{16}\,c^5\,d^8\,e^{12}\,x+184696\,a^{17}\,c^4\,d^6\,e^{14}\,x+291237\,a^{18}\,c^3\,d^4\,e^{16}\,x+140106\,a^{19}\,c^2\,d^2\,e^{18}\,x+330\,a\,c^8\,d^{18}\,e^2\,{\left (-a^7\,c\right )}^{3/2}\right )\,\left (c\,\left (4\,a^7\,d\,e^7+\frac {35\,a^3\,d^2\,e^6\,\sqrt {-a^7\,c}}{8}\right )-\frac {35\,a^4\,e^8\,\sqrt {-a^7\,c}}{32}+\frac {5\,c^4\,d^8\,\sqrt {-a^7\,c}}{32}+\frac {35\,a^2\,c^2\,d^4\,e^4\,\sqrt {-a^7\,c}}{16}+\frac {7\,a\,c^3\,d^6\,e^2\,\sqrt {-a^7\,c}}{8}\right )}{a^{12}\,e^{10}+5\,a^{11}\,c\,d^2\,e^8+10\,a^{10}\,c^2\,d^4\,e^6+10\,a^9\,c^3\,d^6\,e^4+5\,a^8\,c^4\,d^8\,e^2+a^7\,c^5\,d^{10}}+\frac {\ln \left (1225\,a^{17}\,e^{20}\,\sqrt {-a^7\,c}-25\,c^9\,d^{20}\,{\left (-a^7\,c\right )}^{3/2}+25\,a^{10}\,c^{11}\,d^{20}\,x+291237\,a\,d^4\,e^{16}\,{\left (-a^7\,c\right )}^{5/2}+184696\,c\,d^6\,e^{14}\,{\left (-a^7\,c\right )}^{5/2}-140106\,a^9\,d^2\,e^{18}\,{\left (-a^7\,c\right )}^{3/2}+1225\,a^{20}\,c\,e^{20}\,x-2069\,a^2\,c^7\,d^{16}\,e^4\,{\left (-a^7\,c\right )}^{3/2}-8568\,a^3\,c^6\,d^{14}\,e^6\,{\left (-a^7\,c\right )}^{3/2}-24514\,a^4\,c^5\,d^{12}\,e^8\,{\left (-a^7\,c\right )}^{3/2}-47740\,a^5\,c^4\,d^{10}\,e^{10}\,{\left (-a^7\,c\right )}^{3/2}-62370\,a^6\,c^3\,d^8\,e^{12}\,{\left (-a^7\,c\right )}^{3/2}+330\,a^{11}\,c^{10}\,d^{18}\,e^2\,x+2069\,a^{12}\,c^9\,d^{16}\,e^4\,x+8568\,a^{13}\,c^8\,d^{14}\,e^6\,x+24514\,a^{14}\,c^7\,d^{12}\,e^8\,x+47740\,a^{15}\,c^6\,d^{10}\,e^{10}\,x+62370\,a^{16}\,c^5\,d^8\,e^{12}\,x+184696\,a^{17}\,c^4\,d^6\,e^{14}\,x+291237\,a^{18}\,c^3\,d^4\,e^{16}\,x+140106\,a^{19}\,c^2\,d^2\,e^{18}\,x-330\,a\,c^8\,d^{18}\,e^2\,{\left (-a^7\,c\right )}^{3/2}\right )\,\left (\frac {5\,c^4\,d^8\,\sqrt {-a^7\,c}}{32}-\frac {35\,a^4\,e^8\,\sqrt {-a^7\,c}}{32}-c\,\left (4\,a^7\,d\,e^7-\frac {35\,a^3\,d^2\,e^6\,\sqrt {-a^7\,c}}{8}\right )+\frac {35\,a^2\,c^2\,d^4\,e^4\,\sqrt {-a^7\,c}}{16}+\frac {7\,a\,c^3\,d^6\,e^2\,\sqrt {-a^7\,c}}{8}\right )}{a^{12}\,e^{10}+5\,a^{11}\,c\,d^2\,e^8+10\,a^{10}\,c^2\,d^4\,e^6+10\,a^9\,c^3\,d^6\,e^4+5\,a^8\,c^4\,d^8\,e^2+a^7\,c^5\,d^{10}}+\frac {8\,c\,d\,e^7\,\ln \left (d+e\,x\right )}{{\left (c\,d^2+a\,e^2\right )}^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + c*x^2)^4*(d + e*x)^2),x)

[Out]

((c^3*d^6*e - 3*a^3*e^7 + 5*a*c^2*d^4*e^3 + 13*a^2*c*d^2*e^5)/(3*(a*e^2 + c*d^2)*(a^3*e^6 + c^3*d^6 + 3*a*c^2*
d^4*e^2 + 3*a^2*c*d^2*e^4)) + (x*(33*c^3*d^5 + 106*a*c^2*d^3*e^2 + 121*a^2*c*d*e^4))/(48*a*(a^3*e^6 + c^3*d^6
+ 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4)) + (x^6*(5*c^6*d^6*e - 35*a^3*c^3*e^7 + 23*a*c^5*d^4*e^3 + 47*a^2*c^4*d^2
*e^5))/(16*a^3*(a^4*e^8 + c^4*d^8 + 4*a*c^3*d^6*e^2 + 4*a^3*c*d^2*e^6 + 6*a^2*c^2*d^4*e^4)) + (x^3*(5*c^4*d^5
+ 18*a*c^3*d^3*e^2 + 25*a^2*c^2*d*e^4))/(6*a^2*(a^3*e^6 + c^3*d^6 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4)) + (x^5
*(5*c^5*d^5 + 18*a*c^4*d^3*e^2 + 29*a^2*c^3*d*e^4))/(16*a^3*(a^3*e^6 + c^3*d^6 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2
*e^4)) + (x^4*(5*c^5*d^6*e - 35*a^3*c^2*e^7 + 23*a*c^4*d^4*e^3 + 55*a^2*c^3*d^2*e^5))/(6*a^2*(a*e^2 + c*d^2)*(
a^3*e^6 + c^3*d^6 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4)) + (x^2*(11*c^4*d^6*e - 77*a^3*c*e^7 + 57*a*c^3*d^4*e^3
+ 161*a^2*c^2*d^2*e^5))/(16*a*(a*e^2 + c*d^2)*(a^3*e^6 + c^3*d^6 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4)))/(a^3*
d + c^3*d*x^6 + c^3*e*x^7 + a^3*e*x + 3*a^2*c*d*x^2 + 3*a*c^2*d*x^4 + 3*a^2*c*e*x^3 + 3*a*c^2*e*x^5) - (log(25
*c^9*d^20*(-a^7*c)^(3/2) - 1225*a^17*e^20*(-a^7*c)^(1/2) + 25*a^10*c^11*d^20*x - 291237*a*d^4*e^16*(-a^7*c)^(5
/2) - 184696*c*d^6*e^14*(-a^7*c)^(5/2) + 140106*a^9*d^2*e^18*(-a^7*c)^(3/2) + 1225*a^20*c*e^20*x + 2069*a^2*c^
7*d^16*e^4*(-a^7*c)^(3/2) + 8568*a^3*c^6*d^14*e^6*(-a^7*c)^(3/2) + 24514*a^4*c^5*d^12*e^8*(-a^7*c)^(3/2) + 477
40*a^5*c^4*d^10*e^10*(-a^7*c)^(3/2) + 62370*a^6*c^3*d^8*e^12*(-a^7*c)^(3/2) + 330*a^11*c^10*d^18*e^2*x + 2069*
a^12*c^9*d^16*e^4*x + 8568*a^13*c^8*d^14*e^6*x + 24514*a^14*c^7*d^12*e^8*x + 47740*a^15*c^6*d^10*e^10*x + 6237
0*a^16*c^5*d^8*e^12*x + 184696*a^17*c^4*d^6*e^14*x + 291237*a^18*c^3*d^4*e^16*x + 140106*a^19*c^2*d^2*e^18*x +
330*a*c^8*d^18*e^2*(-a^7*c)^(3/2))*(c*(4*a^7*d*e^7 + (35*a^3*d^2*e^6*(-a^7*c)^(1/2))/8) - (35*a^4*e^8*(-a^7*c
)^(1/2))/32 + (5*c^4*d^8*(-a^7*c)^(1/2))/32 + (35*a^2*c^2*d^4*e^4*(-a^7*c)^(1/2))/16 + (7*a*c^3*d^6*e^2*(-a^7*
c)^(1/2))/8))/(a^12*e^10 + a^7*c^5*d^10 + 5*a^11*c*d^2*e^8 + 5*a^8*c^4*d^8*e^2 + 10*a^9*c^3*d^6*e^4 + 10*a^10*
c^2*d^4*e^6) + (log(1225*a^17*e^20*(-a^7*c)^(1/2) - 25*c^9*d^20*(-a^7*c)^(3/2) + 25*a^10*c^11*d^20*x + 291237*
a*d^4*e^16*(-a^7*c)^(5/2) + 184696*c*d^6*e^14*(-a^7*c)^(5/2) - 140106*a^9*d^2*e^18*(-a^7*c)^(3/2) + 1225*a^20*
c*e^20*x - 2069*a^2*c^7*d^16*e^4*(-a^7*c)^(3/2) - 8568*a^3*c^6*d^14*e^6*(-a^7*c)^(3/2) - 24514*a^4*c^5*d^12*e^
8*(-a^7*c)^(3/2) - 47740*a^5*c^4*d^10*e^10*(-a^7*c)^(3/2) - 62370*a^6*c^3*d^8*e^12*(-a^7*c)^(3/2) + 330*a^11*c
^10*d^18*e^2*x + 2069*a^12*c^9*d^16*e^4*x + 8568*a^13*c^8*d^14*e^6*x + 24514*a^14*c^7*d^12*e^8*x + 47740*a^15*
c^6*d^10*e^10*x + 62370*a^16*c^5*d^8*e^12*x + 184696*a^17*c^4*d^6*e^14*x + 291237*a^18*c^3*d^4*e^16*x + 140106
*a^19*c^2*d^2*e^18*x - 330*a*c^8*d^18*e^2*(-a^7*c)^(3/2))*((5*c^4*d^8*(-a^7*c)^(1/2))/32 - (35*a^4*e^8*(-a^7*c
)^(1/2))/32 - c*(4*a^7*d*e^7 - (35*a^3*d^2*e^6*(-a^7*c)^(1/2))/8) + (35*a^2*c^2*d^4*e^4*(-a^7*c)^(1/2))/16 + (
7*a*c^3*d^6*e^2*(-a^7*c)^(1/2))/8))/(a^12*e^10 + a^7*c^5*d^10 + 5*a^11*c*d^2*e^8 + 5*a^8*c^4*d^8*e^2 + 10*a^9*
c^3*d^6*e^4 + 10*a^10*c^2*d^4*e^6) + (8*c*d*e^7*log(d + e*x))/(a*e^2 + c*d^2)^5

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