3.584 \(\int \frac {1}{(d+e x)^3 (a+c x^2)^{5/2}} \, dx\)
Optimal. Leaf size=327 \[ \frac {c d e \sqrt {a+c x^2} \left (-81 a^2 e^4+28 a c d^2 e^2+4 c^2 d^4\right )}{6 a^2 (d+e x) \left (a e^2+c d^2\right )^4}+\frac {e \sqrt {a+c x^2} \left (-15 a^2 e^4+24 a c d^2 e^2+4 c^2 d^4\right )}{6 a^2 (d+e x)^2 \left (a e^2+c d^2\right )^3}-\frac {a e \left (2 c d^2-5 a e^2\right )-c d x \left (9 a e^2+2 c d^2\right )}{3 a^2 \sqrt {a+c x^2} (d+e x)^2 \left (a e^2+c d^2\right )^2}+\frac {a e+c d x}{3 a \left (a+c x^2\right )^{3/2} (d+e x)^2 \left (a e^2+c d^2\right )}-\frac {5 c e^4 \left (6 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{2 \left (a e^2+c d^2\right )^{9/2}} \]
[Out]
1/3*(c*d*x+a*e)/a/(a*e^2+c*d^2)/(e*x+d)^2/(c*x^2+a)^(3/2)-5/2*c*e^4*(-a*e^2+6*c*d^2)*arctanh((-c*d*x+a*e)/(a*e
^2+c*d^2)^(1/2)/(c*x^2+a)^(1/2))/(a*e^2+c*d^2)^(9/2)+1/3*(-a*e*(-5*a*e^2+2*c*d^2)+c*d*(9*a*e^2+2*c*d^2)*x)/a^2
/(a*e^2+c*d^2)^2/(e*x+d)^2/(c*x^2+a)^(1/2)+1/6*e*(-15*a^2*e^4+24*a*c*d^2*e^2+4*c^2*d^4)*(c*x^2+a)^(1/2)/a^2/(a
*e^2+c*d^2)^3/(e*x+d)^2+1/6*c*d*e*(-81*a^2*e^4+28*a*c*d^2*e^2+4*c^2*d^4)*(c*x^2+a)^(1/2)/a^2/(a*e^2+c*d^2)^4/(
e*x+d)
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Rubi [A] time = 0.37, antiderivative size = 327, normalized size of antiderivative = 1.00,
number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used =
{741, 823, 835, 807, 725, 206} \[ \frac {c d e \sqrt {a+c x^2} \left (-81 a^2 e^4+28 a c d^2 e^2+4 c^2 d^4\right )}{6 a^2 (d+e x) \left (a e^2+c d^2\right )^4}+\frac {e \sqrt {a+c x^2} \left (-15 a^2 e^4+24 a c d^2 e^2+4 c^2 d^4\right )}{6 a^2 (d+e x)^2 \left (a e^2+c d^2\right )^3}-\frac {a e \left (2 c d^2-5 a e^2\right )-c d x \left (9 a e^2+2 c d^2\right )}{3 a^2 \sqrt {a+c x^2} (d+e x)^2 \left (a e^2+c d^2\right )^2}+\frac {a e+c d x}{3 a \left (a+c x^2\right )^{3/2} (d+e x)^2 \left (a e^2+c d^2\right )}-\frac {5 c e^4 \left (6 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{2 \left (a e^2+c d^2\right )^{9/2}} \]
Antiderivative was successfully verified.
[In]
Int[1/((d + e*x)^3*(a + c*x^2)^(5/2)),x]
[Out]
(a*e + c*d*x)/(3*a*(c*d^2 + a*e^2)*(d + e*x)^2*(a + c*x^2)^(3/2)) - (a*e*(2*c*d^2 - 5*a*e^2) - c*d*(2*c*d^2 +
9*a*e^2)*x)/(3*a^2*(c*d^2 + a*e^2)^2*(d + e*x)^2*Sqrt[a + c*x^2]) + (e*(4*c^2*d^4 + 24*a*c*d^2*e^2 - 15*a^2*e^
4)*Sqrt[a + c*x^2])/(6*a^2*(c*d^2 + a*e^2)^3*(d + e*x)^2) + (c*d*e*(4*c^2*d^4 + 28*a*c*d^2*e^2 - 81*a^2*e^4)*S
qrt[a + c*x^2])/(6*a^2*(c*d^2 + a*e^2)^4*(d + e*x)) - (5*c*e^4*(6*c*d^2 - a*e^2)*ArcTanh[(a*e - c*d*x)/(Sqrt[c
*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(2*(c*d^2 + a*e^2)^(9/2))
Rule 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 725
Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]
Rule 741
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 807
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((e*f - d*g
)*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e^2
), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0]
&& EqQ[Simplify[m + 2*p + 3], 0]
Rule 823
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] + Di
st[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])
Rule 835
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((e*f - d*g)
*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[
(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; Fr
eeQ[{a, c, d, e, f, g, p}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || Integer
sQ[2*m, 2*p])
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^3 \left (a+c x^2\right )^{5/2}} \, dx &=\frac {a e+c d x}{3 a \left (c d^2+a e^2\right ) (d+e x)^2 \left (a+c x^2\right )^{3/2}}-\frac {\int \frac {-2 c d^2-5 a e^2-4 c d e x}{(d+e x)^3 \left (a+c x^2\right )^{3/2}} \, dx}{3 a \left (c d^2+a e^2\right )}\\ &=\frac {a e+c d x}{3 a \left (c d^2+a e^2\right ) (d+e x)^2 \left (a+c x^2\right )^{3/2}}-\frac {a e \left (2 c d^2-5 a e^2\right )-c d \left (2 c d^2+9 a e^2\right ) x}{3 a^2 \left (c d^2+a e^2\right )^2 (d+e x)^2 \sqrt {a+c x^2}}+\frac {\int \frac {-3 a c e^2 \left (2 c d^2-5 a e^2\right )+2 c^2 d e \left (2 c d^2+9 a e^2\right ) x}{(d+e x)^3 \sqrt {a+c x^2}} \, dx}{3 a^2 c \left (c d^2+a e^2\right )^2}\\ &=\frac {a e+c d x}{3 a \left (c d^2+a e^2\right ) (d+e x)^2 \left (a+c x^2\right )^{3/2}}-\frac {a e \left (2 c d^2-5 a e^2\right )-c d \left (2 c d^2+9 a e^2\right ) x}{3 a^2 \left (c d^2+a e^2\right )^2 (d+e x)^2 \sqrt {a+c x^2}}+\frac {e \left (4 c^2 d^4+24 a c d^2 e^2-15 a^2 e^4\right ) \sqrt {a+c x^2}}{6 a^2 \left (c d^2+a e^2\right )^3 (d+e x)^2}-\frac {\int \frac {2 a c^2 d e^2 \left (2 c d^2-33 a e^2\right )-c^2 e \left (4 c^2 d^4+24 a c d^2 e^2-15 a^2 e^4\right ) x}{(d+e x)^2 \sqrt {a+c x^2}} \, dx}{6 a^2 c \left (c d^2+a e^2\right )^3}\\ &=\frac {a e+c d x}{3 a \left (c d^2+a e^2\right ) (d+e x)^2 \left (a+c x^2\right )^{3/2}}-\frac {a e \left (2 c d^2-5 a e^2\right )-c d \left (2 c d^2+9 a e^2\right ) x}{3 a^2 \left (c d^2+a e^2\right )^2 (d+e x)^2 \sqrt {a+c x^2}}+\frac {e \left (4 c^2 d^4+24 a c d^2 e^2-15 a^2 e^4\right ) \sqrt {a+c x^2}}{6 a^2 \left (c d^2+a e^2\right )^3 (d+e x)^2}+\frac {c d e \left (4 c^2 d^4+28 a c d^2 e^2-81 a^2 e^4\right ) \sqrt {a+c x^2}}{6 a^2 \left (c d^2+a e^2\right )^4 (d+e x)}+\frac {\left (5 c e^4 \left (6 c d^2-a e^2\right )\right ) \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{2 \left (c d^2+a e^2\right )^4}\\ &=\frac {a e+c d x}{3 a \left (c d^2+a e^2\right ) (d+e x)^2 \left (a+c x^2\right )^{3/2}}-\frac {a e \left (2 c d^2-5 a e^2\right )-c d \left (2 c d^2+9 a e^2\right ) x}{3 a^2 \left (c d^2+a e^2\right )^2 (d+e x)^2 \sqrt {a+c x^2}}+\frac {e \left (4 c^2 d^4+24 a c d^2 e^2-15 a^2 e^4\right ) \sqrt {a+c x^2}}{6 a^2 \left (c d^2+a e^2\right )^3 (d+e x)^2}+\frac {c d e \left (4 c^2 d^4+28 a c d^2 e^2-81 a^2 e^4\right ) \sqrt {a+c x^2}}{6 a^2 \left (c d^2+a e^2\right )^4 (d+e x)}-\frac {\left (5 c e^4 \left (6 c d^2-a e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{c d^2+a e^2-x^2} \, dx,x,\frac {a e-c d x}{\sqrt {a+c x^2}}\right )}{2 \left (c d^2+a e^2\right )^4}\\ &=\frac {a e+c d x}{3 a \left (c d^2+a e^2\right ) (d+e x)^2 \left (a+c x^2\right )^{3/2}}-\frac {a e \left (2 c d^2-5 a e^2\right )-c d \left (2 c d^2+9 a e^2\right ) x}{3 a^2 \left (c d^2+a e^2\right )^2 (d+e x)^2 \sqrt {a+c x^2}}+\frac {e \left (4 c^2 d^4+24 a c d^2 e^2-15 a^2 e^4\right ) \sqrt {a+c x^2}}{6 a^2 \left (c d^2+a e^2\right )^3 (d+e x)^2}+\frac {c d e \left (4 c^2 d^4+28 a c d^2 e^2-81 a^2 e^4\right ) \sqrt {a+c x^2}}{6 a^2 \left (c d^2+a e^2\right )^4 (d+e x)}-\frac {5 c e^4 \left (6 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{2 \left (c d^2+a e^2\right )^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.90, size = 296, normalized size = 0.91 \[ \frac {1}{6} \left (\frac {\sqrt {a+c x^2} \left (\frac {2 c \left (a e^2+c d^2\right ) \left (-a^2 e^3+3 a c d e (d-e x)+c^2 d^3 x\right )}{a \left (a+c x^2\right )^2}+\frac {4 c \left (-3 a^3 e^5+3 a^2 c d e^3 (5 d-4 e x)+7 a c^2 d^3 e^2 x+c^3 d^5 x\right )}{a^2 \left (a+c x^2\right )}-\frac {3 e^5 \left (a e^2+c d^2\right )}{(d+e x)^2}-\frac {33 c d e^5}{d+e x}\right )}{\left (a e^2+c d^2\right )^4}+\frac {15 c e^4 \left (a e^2-6 c d^2\right ) \log \left (\sqrt {a+c x^2} \sqrt {a e^2+c d^2}+a e-c d x\right )}{\left (a e^2+c d^2\right )^{9/2}}+\frac {15 c e^4 \left (6 c d^2-a e^2\right ) \log (d+e x)}{\left (a e^2+c d^2\right )^{9/2}}\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[1/((d + e*x)^3*(a + c*x^2)^(5/2)),x]
[Out]
((Sqrt[a + c*x^2]*((-3*e^5*(c*d^2 + a*e^2))/(d + e*x)^2 - (33*c*d*e^5)/(d + e*x) + (4*c*(-3*a^3*e^5 + c^3*d^5*
x + 7*a*c^2*d^3*e^2*x + 3*a^2*c*d*e^3*(5*d - 4*e*x)))/(a^2*(a + c*x^2)) + (2*c*(c*d^2 + a*e^2)*(-(a^2*e^3) + c
^2*d^3*x + 3*a*c*d*e*(d - e*x)))/(a*(a + c*x^2)^2)))/(c*d^2 + a*e^2)^4 + (15*c*e^4*(6*c*d^2 - a*e^2)*Log[d + e
*x])/(c*d^2 + a*e^2)^(9/2) + (15*c*e^4*(-6*c*d^2 + a*e^2)*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2
]])/(c*d^2 + a*e^2)^(9/2))/6
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fricas [B] time = 4.05, size = 2628, normalized size = 8.04 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x+d)^3/(c*x^2+a)^(5/2),x, algorithm="fricas")
[Out]
[1/12*(15*(6*a^4*c^2*d^4*e^4 - a^5*c*d^2*e^6 + (6*a^2*c^4*d^2*e^6 - a^3*c^3*e^8)*x^6 + 2*(6*a^2*c^4*d^3*e^5 -
a^3*c^3*d*e^7)*x^5 + (6*a^2*c^4*d^4*e^4 + 11*a^3*c^3*d^2*e^6 - 2*a^4*c^2*e^8)*x^4 + 4*(6*a^3*c^3*d^3*e^5 - a^4
*c^2*d*e^7)*x^3 + (12*a^3*c^3*d^4*e^4 + 4*a^4*c^2*d^2*e^6 - a^5*c*e^8)*x^2 + 2*(6*a^4*c^2*d^3*e^5 - a^5*c*d*e^
7)*x)*sqrt(c*d^2 + a*e^2)*log((2*a*c*d*e*x - a*c*d^2 - 2*a^2*e^2 - (2*c^2*d^2 + a*c*e^2)*x^2 - 2*sqrt(c*d^2 +
a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a))/(e^2*x^2 + 2*d*e*x + d^2)) + 2*(6*a^2*c^4*d^8*e + 70*a^3*c^3*d^6*e^3 + 1
4*a^4*c^2*d^4*e^5 - 53*a^5*c*d^2*e^7 - 3*a^6*e^9 + (4*c^6*d^7*e^2 + 32*a*c^5*d^5*e^4 - 53*a^2*c^4*d^3*e^6 - 81
*a^3*c^3*d*e^8)*x^5 + (8*c^6*d^8*e + 64*a*c^5*d^6*e^3 - 16*a^2*c^4*d^4*e^5 - 87*a^3*c^3*d^2*e^7 - 15*a^4*c^2*e
^9)*x^4 + 2*(2*c^6*d^9 + 19*a*c^5*d^7*e^2 + 65*a^2*c^4*d^5*e^4 - 24*a^3*c^3*d^3*e^6 - 72*a^4*c^2*d*e^8)*x^3 +
2*(6*a*c^5*d^8*e + 63*a^2*c^4*d^6*e^3 - 7*a^3*c^3*d^4*e^5 - 74*a^4*c^2*d^2*e^7 - 10*a^5*c*e^9)*x^2 + (6*a*c^5*
d^9 + 42*a^2*c^4*d^7*e^2 + 110*a^3*c^3*d^5*e^4 + 13*a^4*c^2*d^3*e^6 - 61*a^5*c*d*e^8)*x)*sqrt(c*x^2 + a))/(a^4
*c^5*d^12 + 5*a^5*c^4*d^10*e^2 + 10*a^6*c^3*d^8*e^4 + 10*a^7*c^2*d^6*e^6 + 5*a^8*c*d^4*e^8 + a^9*d^2*e^10 + (a
^2*c^7*d^10*e^2 + 5*a^3*c^6*d^8*e^4 + 10*a^4*c^5*d^6*e^6 + 10*a^5*c^4*d^4*e^8 + 5*a^6*c^3*d^2*e^10 + a^7*c^2*e
^12)*x^6 + 2*(a^2*c^7*d^11*e + 5*a^3*c^6*d^9*e^3 + 10*a^4*c^5*d^7*e^5 + 10*a^5*c^4*d^5*e^7 + 5*a^6*c^3*d^3*e^9
+ a^7*c^2*d*e^11)*x^5 + (a^2*c^7*d^12 + 7*a^3*c^6*d^10*e^2 + 20*a^4*c^5*d^8*e^4 + 30*a^5*c^4*d^6*e^6 + 25*a^6
*c^3*d^4*e^8 + 11*a^7*c^2*d^2*e^10 + 2*a^8*c*e^12)*x^4 + 4*(a^3*c^6*d^11*e + 5*a^4*c^5*d^9*e^3 + 10*a^5*c^4*d^
7*e^5 + 10*a^6*c^3*d^5*e^7 + 5*a^7*c^2*d^3*e^9 + a^8*c*d*e^11)*x^3 + (2*a^3*c^6*d^12 + 11*a^4*c^5*d^10*e^2 + 2
5*a^5*c^4*d^8*e^4 + 30*a^6*c^3*d^6*e^6 + 20*a^7*c^2*d^4*e^8 + 7*a^8*c*d^2*e^10 + a^9*e^12)*x^2 + 2*(a^4*c^5*d^
11*e + 5*a^5*c^4*d^9*e^3 + 10*a^6*c^3*d^7*e^5 + 10*a^7*c^2*d^5*e^7 + 5*a^8*c*d^3*e^9 + a^9*d*e^11)*x), -1/6*(1
5*(6*a^4*c^2*d^4*e^4 - a^5*c*d^2*e^6 + (6*a^2*c^4*d^2*e^6 - a^3*c^3*e^8)*x^6 + 2*(6*a^2*c^4*d^3*e^5 - a^3*c^3*
d*e^7)*x^5 + (6*a^2*c^4*d^4*e^4 + 11*a^3*c^3*d^2*e^6 - 2*a^4*c^2*e^8)*x^4 + 4*(6*a^3*c^3*d^3*e^5 - a^4*c^2*d*e
^7)*x^3 + (12*a^3*c^3*d^4*e^4 + 4*a^4*c^2*d^2*e^6 - a^5*c*e^8)*x^2 + 2*(6*a^4*c^2*d^3*e^5 - a^5*c*d*e^7)*x)*sq
rt(-c*d^2 - a*e^2)*arctan(sqrt(-c*d^2 - a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a)/(a*c*d^2 + a^2*e^2 + (c^2*d^2 + a
*c*e^2)*x^2)) - (6*a^2*c^4*d^8*e + 70*a^3*c^3*d^6*e^3 + 14*a^4*c^2*d^4*e^5 - 53*a^5*c*d^2*e^7 - 3*a^6*e^9 + (4
*c^6*d^7*e^2 + 32*a*c^5*d^5*e^4 - 53*a^2*c^4*d^3*e^6 - 81*a^3*c^3*d*e^8)*x^5 + (8*c^6*d^8*e + 64*a*c^5*d^6*e^3
- 16*a^2*c^4*d^4*e^5 - 87*a^3*c^3*d^2*e^7 - 15*a^4*c^2*e^9)*x^4 + 2*(2*c^6*d^9 + 19*a*c^5*d^7*e^2 + 65*a^2*c^
4*d^5*e^4 - 24*a^3*c^3*d^3*e^6 - 72*a^4*c^2*d*e^8)*x^3 + 2*(6*a*c^5*d^8*e + 63*a^2*c^4*d^6*e^3 - 7*a^3*c^3*d^4
*e^5 - 74*a^4*c^2*d^2*e^7 - 10*a^5*c*e^9)*x^2 + (6*a*c^5*d^9 + 42*a^2*c^4*d^7*e^2 + 110*a^3*c^3*d^5*e^4 + 13*a
^4*c^2*d^3*e^6 - 61*a^5*c*d*e^8)*x)*sqrt(c*x^2 + a))/(a^4*c^5*d^12 + 5*a^5*c^4*d^10*e^2 + 10*a^6*c^3*d^8*e^4 +
10*a^7*c^2*d^6*e^6 + 5*a^8*c*d^4*e^8 + a^9*d^2*e^10 + (a^2*c^7*d^10*e^2 + 5*a^3*c^6*d^8*e^4 + 10*a^4*c^5*d^6*
e^6 + 10*a^5*c^4*d^4*e^8 + 5*a^6*c^3*d^2*e^10 + a^7*c^2*e^12)*x^6 + 2*(a^2*c^7*d^11*e + 5*a^3*c^6*d^9*e^3 + 10
*a^4*c^5*d^7*e^5 + 10*a^5*c^4*d^5*e^7 + 5*a^6*c^3*d^3*e^9 + a^7*c^2*d*e^11)*x^5 + (a^2*c^7*d^12 + 7*a^3*c^6*d^
10*e^2 + 20*a^4*c^5*d^8*e^4 + 30*a^5*c^4*d^6*e^6 + 25*a^6*c^3*d^4*e^8 + 11*a^7*c^2*d^2*e^10 + 2*a^8*c*e^12)*x^
4 + 4*(a^3*c^6*d^11*e + 5*a^4*c^5*d^9*e^3 + 10*a^5*c^4*d^7*e^5 + 10*a^6*c^3*d^5*e^7 + 5*a^7*c^2*d^3*e^9 + a^8*
c*d*e^11)*x^3 + (2*a^3*c^6*d^12 + 11*a^4*c^5*d^10*e^2 + 25*a^5*c^4*d^8*e^4 + 30*a^6*c^3*d^6*e^6 + 20*a^7*c^2*d
^4*e^8 + 7*a^8*c*d^2*e^10 + a^9*e^12)*x^2 + 2*(a^4*c^5*d^11*e + 5*a^5*c^4*d^9*e^3 + 10*a^6*c^3*d^7*e^5 + 10*a^
7*c^2*d^5*e^7 + 5*a^8*c*d^3*e^9 + a^9*d*e^11)*x)]
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giac [B] time = 0.64, size = 2044, normalized size = 6.25 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x+d)^3/(c*x^2+a)^(5/2),x, algorithm="giac")
[Out]
5*(6*c^2*d^2*e^4 - a*c*e^6)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + a))*e + sqrt(c)*d)/sqrt(-c*d^2 - a*e^2))/((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)*sqrt(-c*d^2 - a*e^2)) + 1/3*((2*((c^18*
d^29 + 19*a*c^17*d^27*e^2 + 138*a^2*c^16*d^25*e^4 + 538*a^3*c^15*d^23*e^6 + 1243*a^4*c^14*d^21*e^8 + 1617*a^5*
c^13*d^19*e^10 + 528*a^6*c^12*d^17*e^12 - 2244*a^7*c^11*d^15*e^14 - 5049*a^8*c^10*d^13*e^16 - 5819*a^9*c^9*d^1
1*e^18 - 4334*a^10*c^8*d^9*e^20 - 2166*a^11*c^7*d^7*e^22 - 707*a^12*c^6*d^5*e^24 - 137*a^13*c^5*d^3*e^26 - 12*
a^14*c^4*d*e^28)*x/(a^2*c^17*d^32 + 16*a^3*c^16*d^30*e^2 + 120*a^4*c^15*d^28*e^4 + 560*a^5*c^14*d^26*e^6 + 182
0*a^6*c^13*d^24*e^8 + 4368*a^7*c^12*d^22*e^10 + 8008*a^8*c^11*d^20*e^12 + 11440*a^9*c^10*d^18*e^14 + 12870*a^1
0*c^9*d^16*e^16 + 11440*a^11*c^8*d^14*e^18 + 8008*a^12*c^7*d^12*e^20 + 4368*a^13*c^6*d^10*e^22 + 1820*a^14*c^5
*d^8*e^24 + 560*a^15*c^4*d^6*e^26 + 120*a^16*c^3*d^4*e^28 + 16*a^17*c^2*d^2*e^30 + a^18*c*e^32) + 3*(5*a^2*c^1
6*d^26*e^3 + 59*a^3*c^15*d^24*e^5 + 318*a^4*c^14*d^22*e^7 + 1034*a^5*c^13*d^20*e^9 + 2255*a^6*c^12*d^18*e^11 +
3465*a^7*c^11*d^16*e^13 + 3828*a^8*c^10*d^14*e^15 + 3036*a^9*c^9*d^12*e^17 + 1683*a^10*c^8*d^10*e^19 + 605*a^
11*c^7*d^8*e^21 + 110*a^12*c^6*d^6*e^23 - 6*a^13*c^5*d^4*e^25 - 7*a^14*c^4*d^2*e^27 - a^15*c^3*e^29)/(a^2*c^17
*d^32 + 16*a^3*c^16*d^30*e^2 + 120*a^4*c^15*d^28*e^4 + 560*a^5*c^14*d^26*e^6 + 1820*a^6*c^13*d^24*e^8 + 4368*a
^7*c^12*d^22*e^10 + 8008*a^8*c^11*d^20*e^12 + 11440*a^9*c^10*d^18*e^14 + 12870*a^10*c^9*d^16*e^16 + 11440*a^11
*c^8*d^14*e^18 + 8008*a^12*c^7*d^12*e^20 + 4368*a^13*c^6*d^10*e^22 + 1820*a^14*c^5*d^8*e^24 + 560*a^15*c^4*d^6
*e^26 + 120*a^16*c^3*d^4*e^28 + 16*a^17*c^2*d^2*e^30 + a^18*c*e^32))*x + 3*(a*c^17*d^29 + 16*a^2*c^16*d^27*e^2
+ 105*a^3*c^15*d^25*e^4 + 376*a^4*c^14*d^23*e^6 + 781*a^5*c^13*d^21*e^8 + 792*a^6*c^12*d^19*e^10 - 363*a^7*c^
11*d^17*e^12 - 2640*a^8*c^10*d^15*e^14 - 4653*a^9*c^9*d^13*e^16 - 4928*a^10*c^8*d^11*e^18 - 3509*a^11*c^7*d^9*
e^20 - 1704*a^12*c^6*d^7*e^22 - 545*a^13*c^5*d^5*e^24 - 104*a^14*c^4*d^3*e^26 - 9*a^15*c^3*d*e^28)/(a^2*c^17*d
^32 + 16*a^3*c^16*d^30*e^2 + 120*a^4*c^15*d^28*e^4 + 560*a^5*c^14*d^26*e^6 + 1820*a^6*c^13*d^24*e^8 + 4368*a^7
*c^12*d^22*e^10 + 8008*a^8*c^11*d^20*e^12 + 11440*a^9*c^10*d^18*e^14 + 12870*a^10*c^9*d^16*e^16 + 11440*a^11*c
^8*d^14*e^18 + 8008*a^12*c^7*d^12*e^20 + 4368*a^13*c^6*d^10*e^22 + 1820*a^14*c^5*d^8*e^24 + 560*a^15*c^4*d^6*e
^26 + 120*a^16*c^3*d^4*e^28 + 16*a^17*c^2*d^2*e^30 + a^18*c*e^32))*x + (3*a^2*c^16*d^28*e + 68*a^3*c^15*d^26*e
^3 + 575*a^4*c^14*d^24*e^5 + 2688*a^5*c^13*d^22*e^7 + 8063*a^6*c^12*d^20*e^9 + 16676*a^7*c^11*d^18*e^11 + 2465
1*a^8*c^10*d^16*e^13 + 26400*a^9*c^9*d^14*e^15 + 20361*a^10*c^8*d^12*e^17 + 10956*a^11*c^7*d^10*e^19 + 3773*a^
12*c^6*d^8*e^21 + 608*a^13*c^5*d^6*e^23 - 75*a^14*c^4*d^4*e^25 - 52*a^15*c^3*d^2*e^27 - 7*a^16*c^2*e^29)/(a^2*
c^17*d^32 + 16*a^3*c^16*d^30*e^2 + 120*a^4*c^15*d^28*e^4 + 560*a^5*c^14*d^26*e^6 + 1820*a^6*c^13*d^24*e^8 + 43
68*a^7*c^12*d^22*e^10 + 8008*a^8*c^11*d^20*e^12 + 11440*a^9*c^10*d^18*e^14 + 12870*a^10*c^9*d^16*e^16 + 11440*
a^11*c^8*d^14*e^18 + 8008*a^12*c^7*d^12*e^20 + 4368*a^13*c^6*d^10*e^22 + 1820*a^14*c^5*d^8*e^24 + 560*a^15*c^4
*d^6*e^26 + 120*a^16*c^3*d^4*e^28 + 16*a^17*c^2*d^2*e^30 + a^18*c*e^32))/(c*x^2 + a)^(3/2) - (22*(sqrt(c)*x -
sqrt(c*x^2 + a))^2*c^(5/2)*d^3*e^4 + 10*(sqrt(c)*x - sqrt(c*x^2 + a))^3*c^2*d^2*e^5 - 34*(sqrt(c)*x - sqrt(c*x
^2 + a))*a*c^2*d^2*e^5 - 11*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a*c^(3/2)*d*e^6 - (sqrt(c)*x - sqrt(c*x^2 + a))^3*
a*c*e^7 + 11*a^2*c^(3/2)*d*e^6 - (sqrt(c)*x - sqrt(c*x^2 + a))*a^2*c*e^7)/((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)*((sqrt(c)*x - sqrt(c*x^2 + a))^2*e + 2*(sqrt(c)*x - sqrt(c*x^2 + a))*
sqrt(c)*d - a*e)^2)
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maple [B] time = 0.06, size = 1088, normalized size = 3.33 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(e*x+d)^3/(c*x^2+a)^(5/2),x)
[Out]
-1/2/e/(a*e^2+c*d^2)/(x+d/e)^2/(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(3/2)-7/2*c*d/(a*e^2+c*d^2)^2/
(x+d/e)/(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(3/2)+35/6*e*c^2*d^2/(a*e^2+c*d^2)^3/(-2*(x+d/e)*c*d/
e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(3/2)+35/6*c^3*d^3/(a*e^2+c*d^2)^3/a/(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d
^2)/e^2)^(3/2)*x+35/3*c^3*d^3/(a*e^2+c*d^2)^3/a^2/(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2)*x+35/
2*e^3*c^2*d^2/(a*e^2+c*d^2)^4/(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2)+35/2*e^2*c^3*d^3/(a*e^2+c
*d^2)^4/a/(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2)*x-35/2*e^3*c^2*d^2/(a*e^2+c*d^2)^4/((a*e^2+c*
d^2)/e^2)^(1/2)*ln((-2*(x+d/e)*c*d/e+2*(a*e^2+c*d^2)/e^2+2*((a*e^2+c*d^2)/e^2)^(1/2)*(-2*(x+d/e)*c*d/e+(x+d/e)
^2*c+(a*e^2+c*d^2)/e^2)^(1/2))/(x+d/e))-11/2*c^2*d/(a*e^2+c*d^2)^2/a/(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^
2)/e^2)^(3/2)*x-11*c^2*d/(a*e^2+c*d^2)^2/a^2/(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2)*x-5/6*e/(a
*e^2+c*d^2)^2*c/(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(3/2)-5/2*e^3/(a*e^2+c*d^2)^3*c/(-2*(x+d/e)*c
*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2)-5/2*e^2/(a*e^2+c*d^2)^3*c^2*d/a/(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2
+c*d^2)/e^2)^(1/2)*x+5/2*e^3/(a*e^2+c*d^2)^3*c/((a*e^2+c*d^2)/e^2)^(1/2)*ln((-2*(x+d/e)*c*d/e+2*(a*e^2+c*d^2)/
e^2+2*((a*e^2+c*d^2)/e^2)^(1/2)*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2))/(x+d/e))
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maxima [B] time = 2.66, size = 1176, normalized size = 3.60 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x+d)^3/(c*x^2+a)^(5/2),x, algorithm="maxima")
[Out]
35/6*c^3*d^3*x/((c*x^2 + a)^(3/2)*a*c^3*d^6 + 3*(c*x^2 + a)^(3/2)*a^2*c^2*d^4*e^2 + 3*(c*x^2 + a)^(3/2)*a^3*c*
d^2*e^4 + (c*x^2 + a)^(3/2)*a^4*e^6) + 35/2*c^3*d^3*x/(4*sqrt(c*x^2 + a)*a^2*c^3*d^6 + sqrt(c*x^2 + a)*a*c^4*d
^8/e^2 + 6*sqrt(c*x^2 + a)*a^3*c^2*d^4*e^2 + 4*sqrt(c*x^2 + a)*a^4*c*d^2*e^4 + sqrt(c*x^2 + a)*a^5*e^6) + 35/3
*c^3*d^3*x/(sqrt(c*x^2 + a)*a^2*c^3*d^6 + 3*sqrt(c*x^2 + a)*a^3*c^2*d^4*e^2 + 3*sqrt(c*x^2 + a)*a^4*c*d^2*e^4
+ sqrt(c*x^2 + a)*a^5*e^6) + 35/2*c^2*d^2/(sqrt(c*x^2 + a)*c^4*d^8/e^3 + 4*sqrt(c*x^2 + a)*a*c^3*d^6/e + 6*sqr
t(c*x^2 + a)*a^2*c^2*d^4*e + 4*sqrt(c*x^2 + a)*a^3*c*d^2*e^3 + sqrt(c*x^2 + a)*a^4*e^5) + 35/6*c^2*d^2/((c*x^2
+ a)^(3/2)*c^3*d^6/e + 3*(c*x^2 + a)^(3/2)*a*c^2*d^4*e + 3*(c*x^2 + a)^(3/2)*a^2*c*d^2*e^3 + (c*x^2 + a)^(3/2
)*a^3*e^5) - 11/2*c^2*d*x/((c*x^2 + a)^(3/2)*a*c^2*d^4 + 2*(c*x^2 + a)^(3/2)*a^2*c*d^2*e^2 + (c*x^2 + a)^(3/2)
*a^3*e^4) - 5/2*c^2*d*x/(3*sqrt(c*x^2 + a)*a^2*c^2*d^4 + sqrt(c*x^2 + a)*a*c^3*d^6/e^2 + 3*sqrt(c*x^2 + a)*a^3
*c*d^2*e^2 + sqrt(c*x^2 + a)*a^4*e^4) - 11*c^2*d*x/(sqrt(c*x^2 + a)*a^2*c^2*d^4 + 2*sqrt(c*x^2 + a)*a^3*c*d^2*
e^2 + sqrt(c*x^2 + a)*a^4*e^4) - 7/2*c*d/((c*x^2 + a)^(3/2)*c^2*d^4*x + 2*(c*x^2 + a)^(3/2)*a*c*d^2*e^2*x + (c
*x^2 + a)^(3/2)*a^2*e^4*x + (c*x^2 + a)^(3/2)*c^2*d^5/e + 2*(c*x^2 + a)^(3/2)*a*c*d^3*e + (c*x^2 + a)^(3/2)*a^
2*d*e^3) - 5/2*c/(sqrt(c*x^2 + a)*c^3*d^6/e^3 + 3*sqrt(c*x^2 + a)*a*c^2*d^4/e + 3*sqrt(c*x^2 + a)*a^2*c*d^2*e
+ sqrt(c*x^2 + a)*a^3*e^3) - 5/6*c/((c*x^2 + a)^(3/2)*c^2*d^4/e + 2*(c*x^2 + a)^(3/2)*a*c*d^2*e + (c*x^2 + a)^
(3/2)*a^2*e^3) - 1/2/((c*x^2 + a)^(3/2)*c*d^2*e*x^2 + (c*x^2 + a)^(3/2)*a*e^3*x^2 + 2*(c*x^2 + a)^(3/2)*c*d^3*
x + 2*(c*x^2 + a)^(3/2)*a*d*e^2*x + (c*x^2 + a)^(3/2)*c*d^4/e + (c*x^2 + a)^(3/2)*a*d^2*e) + 35/2*c^2*d^2*arcs
inh(c*d*x/(sqrt(a*c)*abs(e*x + d)) - a*e/(sqrt(a*c)*abs(e*x + d)))/((a + c*d^2/e^2)^(9/2)*e^5) - 5/2*c*arcsinh
(c*d*x/(sqrt(a*c)*abs(e*x + d)) - a*e/(sqrt(a*c)*abs(e*x + d)))/((a + c*d^2/e^2)^(7/2)*e^3)
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (c\,x^2+a\right )}^{5/2}\,{\left (d+e\,x\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/((a + c*x^2)^(5/2)*(d + e*x)^3),x)
[Out]
int(1/((a + c*x^2)^(5/2)*(d + e*x)^3), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a + c x^{2}\right )^{\frac {5}{2}} \left (d + e x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x+d)**3/(c*x**2+a)**(5/2),x)
[Out]
Integral(1/((a + c*x**2)**(5/2)*(d + e*x)**3), x)
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