\(\int \frac {1}{(d+e x) (a+c x^2)^4} \, dx\) [523]

Optimal result

Integrand size = 17, antiderivative size = 295 \[ \int \frac {1}{(d+e x) \left (a+c x^2\right )^4} \, dx=\frac {a e+c d x}{6 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^3}+\frac {6 a^2 e^3+c d \left (5 c d^2+11 a e^2\right ) x}{24 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )^2}+\frac {8 a^3 e^5+c d \left (5 c^2 d^4+16 a c d^2 e^2+19 a^2 e^4\right ) x}{16 a^3 \left (c d^2+a e^2\right )^3 \left (a+c x^2\right )}+\frac {\sqrt {c} d \left (5 c^3 d^6+21 a c^2 d^4 e^2+35 a^2 c d^2 e^4+35 a^3 e^6\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{16 a^{7/2} \left (c d^2+a e^2\right )^4}+\frac {e^7 \log (d+e x)}{\left (c d^2+a e^2\right )^4}-\frac {e^7 \log \left (a+c x^2\right )}{2 \left (c d^2+a e^2\right )^4} \]

[Out]

1/6*(c*d*x+a*e)/a/(a*e^2+c*d^2)/(c*x^2+a)^3+1/24*(6*a^2*e^3+c*d*(11*a*e^2+5*c*d^2)*x)/a^2/(a*e^2+c*d^2)^2/(c*x
^2+a)^2+1/16*(8*a^3*e^5+c*d*(19*a^2*e^4+16*a*c*d^2*e^2+5*c^2*d^4)*x)/a^3/(a*e^2+c*d^2)^3/(c*x^2+a)+e^7*ln(e*x+
d)/(a*e^2+c*d^2)^4-1/2*e^7*ln(c*x^2+a)/(a*e^2+c*d^2)^4+1/16*d*(35*a^3*e^6+35*a^2*c*d^2*e^4+21*a*c^2*d^4*e^2+5*
c^3*d^6)*arctan(x*c^(1/2)/a^(1/2))*c^(1/2)/a^(7/2)/(a*e^2+c*d^2)^4

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {755, 837, 815, 649, 211, 266} \[ \int \frac {1}{(d+e x) \left (a+c x^2\right )^4} \, dx=\frac {6 a^2 e^3+c d x \left (11 a e^2+5 c d^2\right )}{24 a^2 \left (a+c x^2\right )^2 \left (a e^2+c d^2\right )^2}+\frac {8 a^3 e^5+c d x \left (19 a^2 e^4+16 a c d^2 e^2+5 c^2 d^4\right )}{16 a^3 \left (a+c x^2\right ) \left (a e^2+c d^2\right )^3}+\frac {\sqrt {c} d \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (35 a^3 e^6+35 a^2 c d^2 e^4+21 a c^2 d^4 e^2+5 c^3 d^6\right )}{16 a^{7/2} \left (a e^2+c d^2\right )^4}+\frac {a e+c d x}{6 a \left (a+c x^2\right )^3 \left (a e^2+c d^2\right )}-\frac {e^7 \log \left (a+c x^2\right )}{2 \left (a e^2+c d^2\right )^4}+\frac {e^7 \log (d+e x)}{\left (a e^2+c d^2\right )^4} \]

[In]

Int[1/((d + e*x)*(a + c*x^2)^4),x]

[Out]

(a*e + c*d*x)/(6*a*(c*d^2 + a*e^2)*(a + c*x^2)^3) + (6*a^2*e^3 + c*d*(5*c*d^2 + 11*a*e^2)*x)/(24*a^2*(c*d^2 +
a*e^2)^2*(a + c*x^2)^2) + (8*a^3*e^5 + c*d*(5*c^2*d^4 + 16*a*c*d^2*e^2 + 19*a^2*e^4)*x)/(16*a^3*(c*d^2 + a*e^2
)^3*(a + c*x^2)) + (Sqrt[c]*d*(5*c^3*d^6 + 21*a*c^2*d^4*e^2 + 35*a^2*c*d^2*e^4 + 35*a^3*e^6)*ArcTan[(Sqrt[c]*x
)/Sqrt[a]])/(16*a^(7/2)*(c*d^2 + a*e^2)^4) + (e^7*Log[d + e*x])/(c*d^2 + a*e^2)^4 - (e^7*Log[a + c*x^2])/(2*(c
*d^2 + a*e^2)^4)

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*} \text {integral}& = \frac {a e+c d x}{6 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^3}-\frac {\int \frac {-5 c d^2-6 a e^2-5 c d e x}{(d+e x) \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 ) \left (a+c x^2\right )^3}+\frac {6 a^2 e^3+c d \left (5 c d^2+11 a e^2\right ) x}{24 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )^2}+\frac {\int \frac {3 c \left (5 c^2 d^4+11 a c d^2 e^2+8 a^2 e^4\right )+3 c^2 d e \left (5 c d^2+11 a e^2\right ) x}{(d+e x) \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 ) \left (a+c x^2\right )^3}+\frac {6 a^2 e^3+c d \left (5 c d^2+11 a e^2\right ) x}{24 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )^2}+\frac {8 a^3 e^5+c d \left (5 c^2 d^4+16 a c d^2 e^2+19 a^2 e^4\right ) x}{16 a^3 \left (c d^2+a e^2\right )^3 \left (a+c x^2\right )}-\frac {\int \frac {-3 c^2 \left (5 c^3 d^6+16 a c^2 d^4 e^2+19 a^2 c d^2 e^4+16 a^3 e^6\right )-3 c^3 d e \left (5 c^2 d^4+16 a c d^2 e^2+19 a^2 e^4\right ) x}{(d+e x) \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 ) \left (a+c x^2\right )^3}+\frac {6 a^2 e^3+c d \left (5 c d^2+11 a e^2\right ) x}{24 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )^2}+\frac {8 a^3 e^5+c d \left (5 c^2 d^4+16 a c d^2 e^2+19 a^2 e^4\right ) x}{16 a^3 \left (c d^2+a e^2\right )^3 \left (a+c x^2\right )}-\frac {\int \left (-\frac {48 a^3 c^2 e^8}{\left (c d^2+a e^2\right ) (d+e x)}-\frac {3 c^3 \left (5 c^3 d^7+21 a c^2 d^5 e^2+35 a^2 c d^3 e^4+35 a^3 d e^6-16 a^3 e^7 x\right )}{\left (c d^2+a e^2\right ) \left (a+c x^2\right )}\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 ) \left (a+c x^2\right )^3}+\frac {6 a^2 e^3+c d \left (5 c d^2+11 a e^2\right ) x}{24 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )^2}+\frac {8 a^3 e^5+c d \left (5 c^2 d^4+16 a c d^2 e^2+19 a^2 e^4\right ) x}{16 a^3 \left (c d^2+a e^2\right )^3 \left (a+c x^2\right )}+\frac {e^7 \log (d+e x)}{\left (c d^2+a e^2\right )^4}+\frac {c \int \frac {5 c^3 d^7+21 a c^2 d^5 e^2+35 a^2 c d^3 e^4+35 a^3 d e^6-16 a^3 e^7 x}{a+c x^2} \, dx}{16 a^3 \left (c d^2+a e^2\right )^4} \\ & = \frac {a e+c d x}{6 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^3}+\frac {6 a^2 e^3+c d \left (5 c d^2+11 a e^2\right ) x}{24 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )^2}+\frac {8 a^3 e^5+c d \left (5 c^2 d^4+16 a c d^2 e^2+19 a^2 e^4\right ) x}{16 a^3 \left (c d^2+a e^2\right )^3 \left (a+c x^2\right )}+\frac {e^7 \log (d+e x)}{\left (c d^2+a e^2\right )^4}-\frac {\left (c e^7\right ) \int \frac {x}{a+c x^2} \, dx}{\left (c d^2+a e^2\right )^4}+\frac {\left (c d \left (5 c^3 d^6+21 a c^2 d^4 e^2+35 a^2 c d^2 e^4+35 a^3 e^6\right )\right ) \int \frac {1}{a+c x^2} \, dx}{16 a^3 \left (c d^2+a e^2\right )^4} \\ & = \frac {a e+c d x}{6 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^3}+\frac {6 a^2 e^3+c d \left (5 c d^2+11 a e^2\right ) x}{24 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )^2}+\frac {8 a^3 e^5+c d \left (5 c^2 d^4+16 a c d^2 e^2+19 a^2 e^4\right ) x}{16 a^3 \left (c d^2+a e^2\right )^3 \left (a+c x^2\right )}+\frac {\sqrt {c} d \left (5 c^3 d^6+21 a c^2 d^4 e^2+35 a^2 c d^2 e^4+35 a^3 e^6\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{16 a^{7/2} \left (c d^2+a e^2\right )^4}+\frac {e^7 \log (d+e x)}{\left (c d^2+a e^2\right )^4}-\frac {e^7 \log \left (a+c x^2\right )}{2 \left (c d^2+a e^2\right )^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 265, normalized size of antiderivative = 0.90 \[ \int \frac {1}{(d+e x) \left (a+c x^2\right )^4} \, dx=\frac {\frac {8 \left (c d^2+a e^2\right )^3 (a e+c d x)}{a \left (a+c x^2\right )^3}+\frac {2 \left (c d^2+a e^2\right )^2 \left (6 a^2 e^3+5 c^2 d^3 x+11 a c d e^2 x\right )}{a^2 \left (a+c x^2\right )^2}+\frac {3 \left (c d^2+a e^2\right ) \left (8 a^3 e^5+5 c^3 d^5 x+16 a c^2 d^3 e^2 x+19 a^2 c d e^4 x\right )}{a^3 \left (a+c x^2\right )}+\frac {3 \sqrt {c} d \left (5 c^3 d^6+21 a c^2 d^4 e^2+35 a^2 c d^2 e^4+35 a^3 e^6\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{a^{7/2}}+48 e^7 \log (d+e x)-24 e^7 \log \left (a+c x^2\right )}{48 \left (c d^2+a e^2\right )^4} \]

[In]

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

[Out]

((8*(c*d^2 + a*e^2)^3*(a*e + c*d*x))/(a*(a + c*x^2)^3) + (2*(c*d^2 + a*e^2)^2*(6*a^2*e^3 + 5*c^2*d^3*x + 11*a*
c*d*e^2*x))/(a^2*(a + c*x^2)^2) + (3*(c*d^2 + a*e^2)*(8*a^3*e^5 + 5*c^3*d^5*x + 16*a*c^2*d^3*e^2*x + 19*a^2*c*
d*e^4*x))/(a^3*(a + c*x^2)) + (3*Sqrt[c]*d*(5*c^3*d^6 + 21*a*c^2*d^4*e^2 + 35*a^2*c*d^2*e^4 + 35*a^3*e^6)*ArcT
an[(Sqrt[c]*x)/Sqrt[a]])/a^(7/2) + 48*e^7*Log[d + e*x] - 24*e^7*Log[a + c*x^2])/(48*(c*d^2 + a*e^2)^4)

Maple [A] (verified)

Time = 2.28 (sec) , antiderivative size = 390, normalized size of antiderivative = 1.32

method	result	size
default	\(\frac {c \left (\frac {\frac {c^{2} d \left (19 e^{6} a^{3}+35 d^{2} e^{4} a^{2} c +21 d^{4} e^{2} c^{2} a +5 c^{3} d^{6}\right ) x^{5}}{16 a^{3}}+\left (\frac {1}{2} a \,e^{7} c +\frac {1}{2} d^{2} e^{5} c^{2}\right ) x^{4}+\frac {c d \left (17 e^{6} a^{3}+33 d^{2} e^{4} a^{2} c +21 d^{4} e^{2} c^{2} a +5 c^{3} d^{6}\right ) x^{3}}{6 a^{2}}+\left (\frac {5}{4} a^{2} e^{7}+\frac {3}{2} a \,d^{2} e^{5} c +\frac {1}{4} c^{2} d^{4} e^{3}\right ) x^{2}+\frac {d \left (29 e^{6} a^{3}+61 d^{2} e^{4} a^{2} c +43 d^{4} e^{2} c^{2} a +11 c^{3} d^{6}\right ) x}{16 a}+\frac {e \left (11 e^{6} a^{3}+18 d^{2} e^{4} a^{2} c +9 d^{4} e^{2} c^{2} a +2 c^{3} d^{6}\right )}{12 c}}{\left (c \,x^{2}+a \right )^{3}}+\frac {-\frac {8 a^{3} e^{7} \ln \left (c \,x^{2}+a \right )}{c}+\frac {\left (35 a^{3} d \,e^{6}+35 a^{2} c \,d^{3} e^{4}+21 a \,c^{2} d^{5} e^{2}+5 c^{3} d^{7}\right ) \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}}}{16 a^{3}}\right )}{\left (e^{2} a +c \,d^{2}\right )^{4}}+\frac {e^{7} \ln \left (e x +d \right )}{\left (e^{2} a +c \,d^{2}\right )^{4}}\)	\(390\)
risch	\(\text {Expression too large to display}\)	\(1139\)

[In]

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

[Out]

c/(a*e^2+c*d^2)^4*((1/16*c^2*d*(19*a^3*e^6+35*a^2*c*d^2*e^4+21*a*c^2*d^4*e^2+5*c^3*d^6)/a^3*x^5+(1/2*a*e^7*c+1
/2*d^2*e^5*c^2)*x^4+1/6*c*d*(17*a^3*e^6+33*a^2*c*d^2*e^4+21*a*c^2*d^4*e^2+5*c^3*d^6)/a^2*x^3+(5/4*a^2*e^7+3/2*
a*d^2*e^5*c+1/4*c^2*d^4*e^3)*x^2+1/16*d*(29*a^3*e^6+61*a^2*c*d^2*e^4+43*a*c^2*d^4*e^2+11*c^3*d^6)/a*x+1/12*e*(
11*a^3*e^6+18*a^2*c*d^2*e^4+9*a*c^2*d^4*e^2+2*c^3*d^6)/c)/(c*x^2+a)^3+1/16/a^3*(-8*a^3*e^7/c*ln(c*x^2+a)+(35*a
^3*d*e^6+35*a^2*c*d^3*e^4+21*a*c^2*d^5*e^2+5*c^3*d^7)/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))))+e^7*ln(e*x+d)/(a*e
^2+c*d^2)^4

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 880 vs. \(2 (277) = 554\).

Time = 12.85 (sec) , antiderivative size = 1784, normalized size of antiderivative = 6.05 \[ \int \frac {1}{(d+e x) \left (a+c x^2\right )^4} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[1/96*(16*a^3*c^3*d^6*e + 72*a^4*c^2*d^4*e^3 + 144*a^5*c*d^2*e^5 + 88*a^6*e^7 + 6*(5*c^6*d^7 + 21*a*c^5*d^5*e^
2 + 35*a^2*c^4*d^3*e^4 + 19*a^3*c^3*d*e^6)*x^5 + 48*(a^3*c^3*d^2*e^5 + a^4*c^2*e^7)*x^4 + 16*(5*a*c^5*d^7 + 21
*a^2*c^4*d^5*e^2 + 33*a^3*c^3*d^3*e^4 + 17*a^4*c^2*d*e^6)*x^3 + 24*(a^3*c^3*d^4*e^3 + 6*a^4*c^2*d^2*e^5 + 5*a^
5*c*e^7)*x^2 + 3*(5*a^3*c^3*d^7 + 21*a^4*c^2*d^5*e^2 + 35*a^5*c*d^3*e^4 + 35*a^6*d*e^6 + (5*c^6*d^7 + 21*a*c^5
*d^5*e^2 + 35*a^2*c^4*d^3*e^4 + 35*a^3*c^3*d*e^6)*x^6 + 3*(5*a*c^5*d^7 + 21*a^2*c^4*d^5*e^2 + 35*a^3*c^3*d^3*e
^4 + 35*a^4*c^2*d*e^6)*x^4 + 3*(5*a^2*c^4*d^7 + 21*a^3*c^3*d^5*e^2 + 35*a^4*c^2*d^3*e^4 + 35*a^5*c*d*e^6)*x^2)
*sqrt(-c/a)*log((c*x^2 + 2*a*x*sqrt(-c/a) - a)/(c*x^2 + a)) + 6*(11*a^2*c^4*d^7 + 43*a^3*c^3*d^5*e^2 + 61*a^4*
c^2*d^3*e^4 + 29*a^5*c*d*e^6)*x - 48*(a^3*c^3*e^7*x^6 + 3*a^4*c^2*e^7*x^4 + 3*a^5*c*e^7*x^2 + a^6*e^7)*log(c*x
^2 + a) + 96*(a^3*c^3*e^7*x^6 + 3*a^4*c^2*e^7*x^4 + 3*a^5*c*e^7*x^2 + a^6*e^7)*log(e*x + d))/(a^6*c^4*d^8 + 4*
a^7*c^3*d^6*e^2 + 6*a^8*c^2*d^4*e^4 + 4*a^9*c*d^2*e^6 + a^10*e^8 + (a^3*c^7*d^8 + 4*a^4*c^6*d^6*e^2 + 6*a^5*c^
5*d^4*e^4 + 4*a^6*c^4*d^2*e^6 + a^7*c^3*e^8)*x^6 + 3*(a^4*c^6*d^8 + 4*a^5*c^5*d^6*e^2 + 6*a^6*c^4*d^4*e^4 + 4*
a^7*c^3*d^2*e^6 + a^8*c^2*e^8)*x^4 + 3*(a^5*c^5*d^8 + 4*a^6*c^4*d^6*e^2 + 6*a^7*c^3*d^4*e^4 + 4*a^8*c^2*d^2*e^
6 + a^9*c*e^8)*x^2), 1/48*(8*a^3*c^3*d^6*e + 36*a^4*c^2*d^4*e^3 + 72*a^5*c*d^2*e^5 + 44*a^6*e^7 + 3*(5*c^6*d^7
+ 21*a*c^5*d^5*e^2 + 35*a^2*c^4*d^3*e^4 + 19*a^3*c^3*d*e^6)*x^5 + 24*(a^3*c^3*d^2*e^5 + a^4*c^2*e^7)*x^4 + 8*
(5*a*c^5*d^7 + 21*a^2*c^4*d^5*e^2 + 33*a^3*c^3*d^3*e^4 + 17*a^4*c^2*d*e^6)*x^3 + 12*(a^3*c^3*d^4*e^3 + 6*a^4*c
^2*d^2*e^5 + 5*a^5*c*e^7)*x^2 + 3*(5*a^3*c^3*d^7 + 21*a^4*c^2*d^5*e^2 + 35*a^5*c*d^3*e^4 + 35*a^6*d*e^6 + (5*c
^6*d^7 + 21*a*c^5*d^5*e^2 + 35*a^2*c^4*d^3*e^4 + 35*a^3*c^3*d*e^6)*x^6 + 3*(5*a*c^5*d^7 + 21*a^2*c^4*d^5*e^2 +
35*a^3*c^3*d^3*e^4 + 35*a^4*c^2*d*e^6)*x^4 + 3*(5*a^2*c^4*d^7 + 21*a^3*c^3*d^5*e^2 + 35*a^4*c^2*d^3*e^4 + 35*
a^5*c*d*e^6)*x^2)*sqrt(c/a)*arctan(x*sqrt(c/a)) + 3*(11*a^2*c^4*d^7 + 43*a^3*c^3*d^5*e^2 + 61*a^4*c^2*d^3*e^4
+ 29*a^5*c*d*e^6)*x - 24*(a^3*c^3*e^7*x^6 + 3*a^4*c^2*e^7*x^4 + 3*a^5*c*e^7*x^2 + a^6*e^7)*log(c*x^2 + a) + 48
*(a^3*c^3*e^7*x^6 + 3*a^4*c^2*e^7*x^4 + 3*a^5*c*e^7*x^2 + a^6*e^7)*log(e*x + d))/(a^6*c^4*d^8 + 4*a^7*c^3*d^6*
e^2 + 6*a^8*c^2*d^4*e^4 + 4*a^9*c*d^2*e^6 + a^10*e^8 + (a^3*c^7*d^8 + 4*a^4*c^6*d^6*e^2 + 6*a^5*c^5*d^4*e^4 +
4*a^6*c^4*d^2*e^6 + a^7*c^3*e^8)*x^6 + 3*(a^4*c^6*d^8 + 4*a^5*c^5*d^6*e^2 + 6*a^6*c^4*d^4*e^4 + 4*a^7*c^3*d^2*
e^6 + a^8*c^2*e^8)*x^4 + 3*(a^5*c^5*d^8 + 4*a^6*c^4*d^6*e^2 + 6*a^7*c^3*d^4*e^4 + 4*a^8*c^2*d^2*e^6 + a^9*c*e^
8)*x^2)]

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x) \left (a+c x^2\right )^4} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 655 vs. \(2 (277) = 554\).

Time = 0.30 (sec) , antiderivative size = 655, normalized size of antiderivative = 2.22 \[ \int \frac {1}{(d+e x) \left (a+c x^2\right )^4} \, dx=-\frac {e^{7} \log \left (c x^{2} + a\right )}{2 \, {\left (c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )}} + \frac {e^{7} \log \left (e x + d\right )}{c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}} + \frac {{\left (5 \, c^{4} d^{7} + 21 \, a c^{3} d^{5} e^{2} + 35 \, a^{2} c^{2} d^{3} e^{4} + 35 \, a^{3} c d e^{6}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{16 \, {\left (a^{3} c^{4} d^{8} + 4 \, a^{4} c^{3} d^{6} e^{2} + 6 \, a^{5} c^{2} d^{4} e^{4} + 4 \, a^{6} c d^{2} e^{6} + a^{7} e^{8}\right )} \sqrt {a c}} + \frac {24 \, a^{3} c^{2} e^{5} x^{4} + 8 \, a^{3} c^{2} d^{4} e + 28 \, a^{4} c d^{2} e^{3} + 44 \, a^{5} e^{5} + 3 \, {\left (5 \, c^{5} d^{5} + 16 \, a c^{4} d^{3} e^{2} + 19 \, a^{2} c^{3} d e^{4}\right )} x^{5} + 8 \, {\left (5 \, a c^{4} d^{5} + 16 \, a^{2} c^{3} d^{3} e^{2} + 17 \, a^{3} c^{2} d e^{4}\right )} x^{3} + 12 \, {\left (a^{3} c^{2} d^{2} e^{3} + 5 \, a^{4} c e^{5}\right )} x^{2} + 3 \, {\left (11 \, a^{2} c^{3} d^{5} + 32 \, a^{3} c^{2} d^{3} e^{2} + 29 \, a^{4} c d e^{4}\right )} x}{48 \, {\left (a^{6} c^{3} d^{6} + 3 \, a^{7} c^{2} d^{4} e^{2} + 3 \, a^{8} c d^{2} e^{4} + a^{9} e^{6} + {\left (a^{3} c^{6} d^{6} + 3 \, a^{4} c^{5} d^{4} e^{2} + 3 \, a^{5} c^{4} d^{2} e^{4} + a^{6} c^{3} e^{6}\right )} x^{6} + 3 \, {\left (a^{4} c^{5} d^{6} + 3 \, a^{5} c^{4} d^{4} e^{2} + 3 \, a^{6} c^{3} d^{2} e^{4} + a^{7} c^{2} e^{6}\right )} x^{4} + 3 \, {\left (a^{5} c^{4} d^{6} + 3 \, a^{6} c^{3} d^{4} e^{2} + 3 \, a^{7} c^{2} d^{2} e^{4} + a^{8} c e^{6}\right )} x^{2}\right )}} \]

[In]

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

[Out]

-1/2*e^7*log(c*x^2 + a)/(c^4*d^8 + 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 + 4*a^3*c*d^2*e^6 + a^4*e^8) + e^7*log(
e*x + d)/(c^4*d^8 + 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 + 4*a^3*c*d^2*e^6 + a^4*e^8) + 1/16*(5*c^4*d^7 + 21*a*
c^3*d^5*e^2 + 35*a^2*c^2*d^3*e^4 + 35*a^3*c*d*e^6)*arctan(c*x/sqrt(a*c))/((a^3*c^4*d^8 + 4*a^4*c^3*d^6*e^2 + 6
*a^5*c^2*d^4*e^4 + 4*a^6*c*d^2*e^6 + a^7*e^8)*sqrt(a*c)) + 1/48*(24*a^3*c^2*e^5*x^4 + 8*a^3*c^2*d^4*e + 28*a^4
*c*d^2*e^3 + 44*a^5*e^5 + 3*(5*c^5*d^5 + 16*a*c^4*d^3*e^2 + 19*a^2*c^3*d*e^4)*x^5 + 8*(5*a*c^4*d^5 + 16*a^2*c^
3*d^3*e^2 + 17*a^3*c^2*d*e^4)*x^3 + 12*(a^3*c^2*d^2*e^3 + 5*a^4*c*e^5)*x^2 + 3*(11*a^2*c^3*d^5 + 32*a^3*c^2*d^
3*e^2 + 29*a^4*c*d*e^4)*x)/(a^6*c^3*d^6 + 3*a^7*c^2*d^4*e^2 + 3*a^8*c*d^2*e^4 + a^9*e^6 + (a^3*c^6*d^6 + 3*a^4
*c^5*d^4*e^2 + 3*a^5*c^4*d^2*e^4 + a^6*c^3*e^6)*x^6 + 3*(a^4*c^5*d^6 + 3*a^5*c^4*d^4*e^2 + 3*a^6*c^3*d^2*e^4 +
a^7*c^2*e^6)*x^4 + 3*(a^5*c^4*d^6 + 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^2*e^4 + a^8*c*e^6)*x^2)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 562 vs. \(2 (277) = 554\).

Time = 0.28 (sec) , antiderivative size = 562, normalized size of antiderivative = 1.91 \[ \int \frac {1}{(d+e x) \left (a+c x^2\right )^4} \, dx=\frac {e^{8} \log \left ({\left | e x + d \right |}\right )}{c^{4} d^{8} e + 4 \, a c^{3} d^{6} e^{3} + 6 \, a^{2} c^{2} d^{4} e^{5} + 4 \, a^{3} c d^{2} e^{7} + a^{4} e^{9}} - \frac {e^{7} \log \left (c x^{2} + a\right )}{2 \, {\left (c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )}} + \frac {{\left (5 \, c^{4} d^{7} + 21 \, a c^{3} d^{5} e^{2} + 35 \, a^{2} c^{2} d^{3} e^{4} + 35 \, a^{3} c d e^{6}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{16 \, {\left (a^{3} c^{4} d^{8} + 4 \, a^{4} c^{3} d^{6} e^{2} + 6 \, a^{5} c^{2} d^{4} e^{4} + 4 \, a^{6} c d^{2} e^{6} + a^{7} e^{8}\right )} \sqrt {a c}} + \frac {8 \, a^{3} c^{3} d^{6} e + 36 \, a^{4} c^{2} d^{4} e^{3} + 72 \, a^{5} c d^{2} e^{5} + 44 \, a^{6} e^{7} + 3 \, {\left (5 \, c^{6} d^{7} + 21 \, a c^{5} d^{5} e^{2} + 35 \, a^{2} c^{4} d^{3} e^{4} + 19 \, a^{3} c^{3} d e^{6}\right )} x^{5} + 24 \, {\left (a^{3} c^{3} d^{2} e^{5} + a^{4} c^{2} e^{7}\right )} x^{4} + 8 \, {\left (5 \, a c^{5} d^{7} + 21 \, a^{2} c^{4} d^{5} e^{2} + 33 \, a^{3} c^{3} d^{3} e^{4} + 17 \, a^{4} c^{2} d e^{6}\right )} x^{3} + 12 \, {\left (a^{3} c^{3} d^{4} e^{3} + 6 \, a^{4} c^{2} d^{2} e^{5} + 5 \, a^{5} c e^{7}\right )} x^{2} + 3 \, {\left (11 \, a^{2} c^{4} d^{7} + 43 \, a^{3} c^{3} d^{5} e^{2} + 61 \, a^{4} c^{2} d^{3} e^{4} + 29 \, a^{5} c d e^{6}\right )} x}{48 \, {\left (c d^{2} + a e^{2}\right )}^{4} {\left (c x^{2} + a\right )}^{3} a^{3}} \]

[In]

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

[Out]

e^8*log(abs(e*x + d))/(c^4*d^8*e + 4*a*c^3*d^6*e^3 + 6*a^2*c^2*d^4*e^5 + 4*a^3*c*d^2*e^7 + a^4*e^9) - 1/2*e^7*
log(c*x^2 + a)/(c^4*d^8 + 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 + 4*a^3*c*d^2*e^6 + a^4*e^8) + 1/16*(5*c^4*d^7 +
21*a*c^3*d^5*e^2 + 35*a^2*c^2*d^3*e^4 + 35*a^3*c*d*e^6)*arctan(c*x/sqrt(a*c))/((a^3*c^4*d^8 + 4*a^4*c^3*d^6*e
^2 + 6*a^5*c^2*d^4*e^4 + 4*a^6*c*d^2*e^6 + a^7*e^8)*sqrt(a*c)) + 1/48*(8*a^3*c^3*d^6*e + 36*a^4*c^2*d^4*e^3 +
72*a^5*c*d^2*e^5 + 44*a^6*e^7 + 3*(5*c^6*d^7 + 21*a*c^5*d^5*e^2 + 35*a^2*c^4*d^3*e^4 + 19*a^3*c^3*d*e^6)*x^5 +
24*(a^3*c^3*d^2*e^5 + a^4*c^2*e^7)*x^4 + 8*(5*a*c^5*d^7 + 21*a^2*c^4*d^5*e^2 + 33*a^3*c^3*d^3*e^4 + 17*a^4*c^
2*d*e^6)*x^3 + 12*(a^3*c^3*d^4*e^3 + 6*a^4*c^2*d^2*e^5 + 5*a^5*c*e^7)*x^2 + 3*(11*a^2*c^4*d^7 + 43*a^3*c^3*d^5
*e^2 + 61*a^4*c^2*d^3*e^4 + 29*a^5*c*d*e^6)*x)/((c*d^2 + a*e^2)^4*(c*x^2 + a)^3*a^3)

Mupad [B] (verification not implemented)

Time = 10.85 (sec) , antiderivative size = 1470, normalized size of antiderivative = 4.98 \[ \int \frac {1}{(d+e x) \left (a+c x^2\right )^4} \, dx=\text {Too large to display} \]

[In]

int(1/((a + c*x^2)^4*(d + e*x)),x)

[Out]

((11*a^2*e^5 + 2*c^2*d^4*e + 7*a*c*d^2*e^3)/(12*(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*(c^2*d^2*e^3 + 5*a*c*e^5))/(4*(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*(11*c^3*d^5 + 32
*a*c^2*d^3*e^2 + 29*a^2*c*d*e^4))/(16*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)) + (c^2*e^5*x^
4)/(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^3*(5*c^4*d^5 + 16*a*c^3*d^3*e^2 + 17*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 + 16*a*c^4*d^3*e^
2 + 19*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)))/(a^3 + c^3*x^6 + 3*a^
2*c*x^2 + 3*a*c^2*x^4) + (e^7*log(d + e*x))/(a*e^2 + c*d^2)^4 - (log(25*a^7*c^10*d^18*x - 2304*a^13*e^18*(-a^7
*c)^(1/2) - 25*a^4*c^9*d^18*(-a^7*c)^(1/2) + 5833*a^5*d^2*e^16*(-a^7*c)^(3/2) + 3612*c^5*d^12*e^6*(-a^7*c)^(3/
2) + 2304*a^16*c*e^18*x + 9660*a^2*c^3*d^8*e^10*(-a^7*c)^(3/2) + 8820*a^3*c^2*d^6*e^12*(-a^7*c)^(3/2) - 260*a^
5*c^8*d^16*e^2*(-a^7*c)^(1/2) - 1236*a^6*c^7*d^14*e^4*(-a^7*c)^(1/2) + 260*a^8*c^9*d^16*e^2*x + 1236*a^9*c^8*d
^14*e^4*x + 3612*a^10*c^7*d^12*e^6*x + 7126*a^11*c^6*d^10*e^8*x + 9660*a^12*c^5*d^8*e^10*x + 8820*a^13*c^4*d^6
*e^12*x + 7204*a^14*c^3*d^4*e^14*x + 5833*a^15*c^2*d^2*e^16*x + 7126*a*c^4*d^10*e^8*(-a^7*c)^(3/2) + 7204*a^4*
c*d^4*e^14*(-a^7*c)^(3/2))*(16*a^7*e^7 + 5*c^3*d^7*(-a^7*c)^(1/2) + 35*a^3*d*e^6*(-a^7*c)^(1/2) + 21*a*c^2*d^5
*e^2*(-a^7*c)^(1/2) + 35*a^2*c*d^3*e^4*(-a^7*c)^(1/2)))/(32*(a^11*e^8 + a^7*c^4*d^8 + 4*a^10*c*d^2*e^6 + 4*a^8
*c^3*d^6*e^2 + 6*a^9*c^2*d^4*e^4)) + (log(2304*a^13*e^18*(-a^7*c)^(1/2) + 25*a^7*c^10*d^18*x + 25*a^4*c^9*d^18
*(-a^7*c)^(1/2) - 5833*a^5*d^2*e^16*(-a^7*c)^(3/2) - 3612*c^5*d^12*e^6*(-a^7*c)^(3/2) + 2304*a^16*c*e^18*x - 9
660*a^2*c^3*d^8*e^10*(-a^7*c)^(3/2) - 8820*a^3*c^2*d^6*e^12*(-a^7*c)^(3/2) + 260*a^5*c^8*d^16*e^2*(-a^7*c)^(1/
2) + 1236*a^6*c^7*d^14*e^4*(-a^7*c)^(1/2) + 260*a^8*c^9*d^16*e^2*x + 1236*a^9*c^8*d^14*e^4*x + 3612*a^10*c^7*d
^12*e^6*x + 7126*a^11*c^6*d^10*e^8*x + 9660*a^12*c^5*d^8*e^10*x + 8820*a^13*c^4*d^6*e^12*x + 7204*a^14*c^3*d^4
*e^14*x + 5833*a^15*c^2*d^2*e^16*x - 7126*a*c^4*d^10*e^8*(-a^7*c)^(3/2) - 7204*a^4*c*d^4*e^14*(-a^7*c)^(3/2))*
(5*c^3*d^7*(-a^7*c)^(1/2) - 16*a^7*e^7 + 35*a^3*d*e^6*(-a^7*c)^(1/2) + 21*a*c^2*d^5*e^2*(-a^7*c)^(1/2) + 35*a^
2*c*d^3*e^4*(-a^7*c)^(1/2)))/(32*(a^11*e^8 + a^7*c^4*d^8 + 4*a^10*c*d^2*e^6 + 4*a^8*c^3*d^6*e^2 + 6*a^9*c^2*d^
4*e^4))