3.612 \(\int \frac{1}{(d+e x)^{5/2} \left (a+c x^2\right )} \, dx\)

Optimal. Leaf size=736 \[ -\frac{c^{3/4} e \left (2 \sqrt{c} d \sqrt{a e^2+c d^2}-a e^2+3 c d^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{2 \sqrt{2} \left (a e^2+c d^2\right )^{5/2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{c^{3/4} e \left (2 \sqrt{c} d \sqrt{a e^2+c d^2}-a e^2+3 c d^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{2 \sqrt{2} \left (a e^2+c d^2\right )^{5/2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{c^{3/4} e \left (-2 \sqrt{c} d \sqrt{a e^2+c d^2}-a e^2+3 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{\sqrt{2} \left (a e^2+c d^2\right )^{5/2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}-\frac{c^{3/4} e \left (-2 \sqrt{c} d \sqrt{a e^2+c d^2}-a e^2+3 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{\sqrt{2} \left (a e^2+c d^2\right )^{5/2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}-\frac{4 c d e}{\sqrt{d+e x} \left (a e^2+c d^2\right )^2}-\frac{2 e}{3 (d+e x)^{3/2} \left (a e^2+c d^2\right )} \]

[Out]

(-2*e)/(3*(c*d^2 + a*e^2)*(d + e*x)^(3/2)) - (4*c*d*e)/((c*d^2 + a*e^2)^2*Sqrt[d
+ e*x]) + (c^(3/4)*e*(3*c*d^2 - a*e^2 - 2*Sqrt[c]*d*Sqrt[c*d^2 + a*e^2])*ArcTan
h[(Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]] - Sqrt[2]*c^(1/4)*Sqrt[d + e*x])/Sqrt[S
qrt[c]*d - Sqrt[c*d^2 + a*e^2]]])/(Sqrt[2]*(c*d^2 + a*e^2)^(5/2)*Sqrt[Sqrt[c]*d
- Sqrt[c*d^2 + a*e^2]]) - (c^(3/4)*e*(3*c*d^2 - a*e^2 - 2*Sqrt[c]*d*Sqrt[c*d^2 +
a*e^2])*ArcTanh[(Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]] + Sqrt[2]*c^(1/4)*Sqrt[d
+ e*x])/Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]])/(Sqrt[2]*(c*d^2 + a*e^2)^(5/2)*
Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (c^(3/4)*e*(3*c*d^2 - a*e^2 + 2*Sqrt[c]
*d*Sqrt[c*d^2 + a*e^2])*Log[Sqrt[c*d^2 + a*e^2] - Sqrt[2]*c^(1/4)*Sqrt[Sqrt[c]*d
+ Sqrt[c*d^2 + a*e^2]]*Sqrt[d + e*x] + Sqrt[c]*(d + e*x)])/(2*Sqrt[2]*(c*d^2 +
a*e^2)^(5/2)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]) + (c^(3/4)*e*(3*c*d^2 - a*e^
2 + 2*Sqrt[c]*d*Sqrt[c*d^2 + a*e^2])*Log[Sqrt[c*d^2 + a*e^2] + Sqrt[2]*c^(1/4)*S
qrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]*Sqrt[d + e*x] + Sqrt[c]*(d + e*x)])/(2*Sqrt
[2]*(c*d^2 + a*e^2)^(5/2)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]])

_______________________________________________________________________________________

Rubi [A]  time = 4.28704, antiderivative size = 736, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421 \[ -\frac{c^{3/4} e \left (2 \sqrt{c} d \sqrt{a e^2+c d^2}-a e^2+3 c d^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{2 \sqrt{2} \left (a e^2+c d^2\right )^{5/2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{c^{3/4} e \left (2 \sqrt{c} d \sqrt{a e^2+c d^2}-a e^2+3 c d^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{2 \sqrt{2} \left (a e^2+c d^2\right )^{5/2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{c^{3/4} e \left (-2 \sqrt{c} d \sqrt{a e^2+c d^2}-a e^2+3 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{\sqrt{2} \left (a e^2+c d^2\right )^{5/2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}-\frac{c^{3/4} e \left (-2 \sqrt{c} d \sqrt{a e^2+c d^2}-a e^2+3 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{\sqrt{2} \left (a e^2+c d^2\right )^{5/2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}-\frac{4 c d e}{\sqrt{d+e x} \left (a e^2+c d^2\right )^2}-\frac{2 e}{3 (d+e x)^{3/2} \left (a e^2+c d^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^(5/2)*(a + c*x^2)),x]

[Out]

(-2*e)/(3*(c*d^2 + a*e^2)*(d + e*x)^(3/2)) - (4*c*d*e)/((c*d^2 + a*e^2)^2*Sqrt[d
+ e*x]) + (c^(3/4)*e*(3*c*d^2 - a*e^2 - 2*Sqrt[c]*d*Sqrt[c*d^2 + a*e^2])*ArcTan
h[(Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]] - Sqrt[2]*c^(1/4)*Sqrt[d + e*x])/Sqrt[S
qrt[c]*d - Sqrt[c*d^2 + a*e^2]]])/(Sqrt[2]*(c*d^2 + a*e^2)^(5/2)*Sqrt[Sqrt[c]*d
- Sqrt[c*d^2 + a*e^2]]) - (c^(3/4)*e*(3*c*d^2 - a*e^2 - 2*Sqrt[c]*d*Sqrt[c*d^2 +
a*e^2])*ArcTanh[(Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]] + Sqrt[2]*c^(1/4)*Sqrt[d
+ e*x])/Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]])/(Sqrt[2]*(c*d^2 + a*e^2)^(5/2)*
Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (c^(3/4)*e*(3*c*d^2 - a*e^2 + 2*Sqrt[c]
*d*Sqrt[c*d^2 + a*e^2])*Log[Sqrt[c*d^2 + a*e^2] - Sqrt[2]*c^(1/4)*Sqrt[Sqrt[c]*d
+ Sqrt[c*d^2 + a*e^2]]*Sqrt[d + e*x] + Sqrt[c]*(d + e*x)])/(2*Sqrt[2]*(c*d^2 +
a*e^2)^(5/2)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]) + (c^(3/4)*e*(3*c*d^2 - a*e^
2 + 2*Sqrt[c]*d*Sqrt[c*d^2 + a*e^2])*Log[Sqrt[c*d^2 + a*e^2] + Sqrt[2]*c^(1/4)*S
qrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]*Sqrt[d + e*x] + Sqrt[c]*(d + e*x)])/(2*Sqrt
[2]*(c*d^2 + a*e^2)^(5/2)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]])

_______________________________________________________________________________________

Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**(5/2)/(c*x**2+a),x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [C]  time = 0.697264, size = 229, normalized size = 0.31 \[ -\frac{2 e \left (a e^2+c d (7 d+6 e x)\right )}{3 (d+e x)^{3/2} \left (a e^2+c d^2\right )^2}-\frac{i c \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-i \sqrt{a} \sqrt{c} e}}\right )}{\sqrt{a} \left (\sqrt{c} d-i \sqrt{a} e\right )^2 \sqrt{c d-i \sqrt{a} \sqrt{c} e}}+\frac{i c \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d+i \sqrt{a} \sqrt{c} e}}\right )}{\sqrt{a} \left (\sqrt{c} d+i \sqrt{a} e\right )^2 \sqrt{c d+i \sqrt{a} \sqrt{c} e}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^(5/2)*(a + c*x^2)),x]

[Out]

(-2*e*(a*e^2 + c*d*(7*d + 6*e*x)))/(3*(c*d^2 + a*e^2)^2*(d + e*x)^(3/2)) - (I*c*
ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - I*Sqrt[a]*Sqrt[c]*e]])/(Sqrt[a]*(Sqrt
[c]*d - I*Sqrt[a]*e)^2*Sqrt[c*d - I*Sqrt[a]*Sqrt[c]*e]) + (I*c*ArcTanh[(Sqrt[c]*
Sqrt[d + e*x])/Sqrt[c*d + I*Sqrt[a]*Sqrt[c]*e]])/(Sqrt[a]*(Sqrt[c]*d + I*Sqrt[a]
*e)^2*Sqrt[c*d + I*Sqrt[a]*Sqrt[c]*e])

_______________________________________________________________________________________

Maple [B]  time = 0.106, size = 7258, normalized size = 9.9 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^(5/2)/(c*x^2+a),x)

[Out]

result too large to display

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} + a\right )}{\left (e x + d\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + a)*(e*x + d)^(5/2)),x, algorithm="maxima")

[Out]

integrate(1/((c*x^2 + a)*(e*x + d)^(5/2)), x)

_______________________________________________________________________________________

Fricas [A]  time = 0.324572, size = 6743, normalized size = 9.16 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + a)*(e*x + d)^(5/2)),x, algorithm="fricas")

[Out]

-1/6*(24*c*d*e^2*x + 28*c*d^2*e + 4*a*e^3 - 3*(c^2*d^5 + 2*a*c*d^3*e^2 + a^2*d*e
^4 + (c^2*d^4*e + 2*a*c*d^2*e^3 + a^2*e^5)*x)*sqrt(e*x + d)*sqrt(-(c^4*d^5 - 10*
a*c^3*d^3*e^2 + 5*a^2*c^2*d*e^4 + (a*c^5*d^10 + 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d
^6*e^4 + 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 + a^6*e^10)*sqrt(-(25*c^7*d^8*e^2
- 100*a*c^6*d^6*e^4 + 110*a^2*c^5*d^4*e^6 - 20*a^3*c^4*d^2*e^8 + a^4*c^3*e^10)/(
a*c^10*d^20 + 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 + 120*a^4*c^7*d^14*e^6 +
210*a^5*c^6*d^12*e^8 + 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 + 120*a^8*c
^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 + 10*a^10*c*d^2*e^18 + a^11*e^20)))/(a*c^5*d^1
0 + 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 + 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^
8 + a^6*e^10))*log((5*c^4*d^4*e - 10*a*c^3*d^2*e^3 + a^2*c^2*e^5)*sqrt(e*x + d)
+ (15*a*c^4*d^6*e^2 - 35*a^2*c^3*d^4*e^4 + 13*a^3*c^2*d^2*e^6 - a^4*c*e^8 + (a*c
^6*d^13 + 2*a^2*c^5*d^11*e^2 - 5*a^3*c^4*d^9*e^4 - 20*a^4*c^3*d^7*e^6 - 25*a^5*c
^2*d^5*e^8 - 14*a^6*c*d^3*e^10 - 3*a^7*d*e^12)*sqrt(-(25*c^7*d^8*e^2 - 100*a*c^6
*d^6*e^4 + 110*a^2*c^5*d^4*e^6 - 20*a^3*c^4*d^2*e^8 + a^4*c^3*e^10)/(a*c^10*d^20
+ 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 + 120*a^4*c^7*d^14*e^6 + 210*a^5*c^
6*d^12*e^8 + 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 + 120*a^8*c^3*d^6*e^14
+ 45*a^9*c^2*d^4*e^16 + 10*a^10*c*d^2*e^18 + a^11*e^20)))*sqrt(-(c^4*d^5 - 10*a
*c^3*d^3*e^2 + 5*a^2*c^2*d*e^4 + (a*c^5*d^10 + 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^
6*e^4 + 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 + a^6*e^10)*sqrt(-(25*c^7*d^8*e^2 -
100*a*c^6*d^6*e^4 + 110*a^2*c^5*d^4*e^6 - 20*a^3*c^4*d^2*e^8 + a^4*c^3*e^10)/(a
*c^10*d^20 + 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 + 120*a^4*c^7*d^14*e^6 +
210*a^5*c^6*d^12*e^8 + 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 + 120*a^8*c^
3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 + 10*a^10*c*d^2*e^18 + a^11*e^20)))/(a*c^5*d^10
+ 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 + 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8
+ a^6*e^10))) + 3*(c^2*d^5 + 2*a*c*d^3*e^2 + a^2*d*e^4 + (c^2*d^4*e + 2*a*c*d^2
*e^3 + a^2*e^5)*x)*sqrt(e*x + d)*sqrt(-(c^4*d^5 - 10*a*c^3*d^3*e^2 + 5*a^2*c^2*d
*e^4 + (a*c^5*d^10 + 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 + 10*a^4*c^2*d^4*e^6
+ 5*a^5*c*d^2*e^8 + a^6*e^10)*sqrt(-(25*c^7*d^8*e^2 - 100*a*c^6*d^6*e^4 + 110*a
^2*c^5*d^4*e^6 - 20*a^3*c^4*d^2*e^8 + a^4*c^3*e^10)/(a*c^10*d^20 + 10*a^2*c^9*d^
18*e^2 + 45*a^3*c^8*d^16*e^4 + 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 + 252
*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 + 120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^
4*e^16 + 10*a^10*c*d^2*e^18 + a^11*e^20)))/(a*c^5*d^10 + 5*a^2*c^4*d^8*e^2 + 10*
a^3*c^3*d^6*e^4 + 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 + a^6*e^10))*log((5*c^4*d
^4*e - 10*a*c^3*d^2*e^3 + a^2*c^2*e^5)*sqrt(e*x + d) - (15*a*c^4*d^6*e^2 - 35*a^
2*c^3*d^4*e^4 + 13*a^3*c^2*d^2*e^6 - a^4*c*e^8 + (a*c^6*d^13 + 2*a^2*c^5*d^11*e^
2 - 5*a^3*c^4*d^9*e^4 - 20*a^4*c^3*d^7*e^6 - 25*a^5*c^2*d^5*e^8 - 14*a^6*c*d^3*e
^10 - 3*a^7*d*e^12)*sqrt(-(25*c^7*d^8*e^2 - 100*a*c^6*d^6*e^4 + 110*a^2*c^5*d^4*
e^6 - 20*a^3*c^4*d^2*e^8 + a^4*c^3*e^10)/(a*c^10*d^20 + 10*a^2*c^9*d^18*e^2 + 45
*a^3*c^8*d^16*e^4 + 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 + 252*a^6*c^5*d^
10*e^10 + 210*a^7*c^4*d^8*e^12 + 120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 + 10
*a^10*c*d^2*e^18 + a^11*e^20)))*sqrt(-(c^4*d^5 - 10*a*c^3*d^3*e^2 + 5*a^2*c^2*d*
e^4 + (a*c^5*d^10 + 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 + 10*a^4*c^2*d^4*e^6
+ 5*a^5*c*d^2*e^8 + a^6*e^10)*sqrt(-(25*c^7*d^8*e^2 - 100*a*c^6*d^6*e^4 + 110*a^
2*c^5*d^4*e^6 - 20*a^3*c^4*d^2*e^8 + a^4*c^3*e^10)/(a*c^10*d^20 + 10*a^2*c^9*d^1
8*e^2 + 45*a^3*c^8*d^16*e^4 + 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 + 252*
a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 + 120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4
*e^16 + 10*a^10*c*d^2*e^18 + a^11*e^20)))/(a*c^5*d^10 + 5*a^2*c^4*d^8*e^2 + 10*a
^3*c^3*d^6*e^4 + 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 + a^6*e^10))) - 3*(c^2*d^5
+ 2*a*c*d^3*e^2 + a^2*d*e^4 + (c^2*d^4*e + 2*a*c*d^2*e^3 + a^2*e^5)*x)*sqrt(e*x
+ d)*sqrt(-(c^4*d^5 - 10*a*c^3*d^3*e^2 + 5*a^2*c^2*d*e^4 - (a*c^5*d^10 + 5*a^2*
c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 + 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 + a^6*e^
10)*sqrt(-(25*c^7*d^8*e^2 - 100*a*c^6*d^6*e^4 + 110*a^2*c^5*d^4*e^6 - 20*a^3*c^4
*d^2*e^8 + a^4*c^3*e^10)/(a*c^10*d^20 + 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^
4 + 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 + 252*a^6*c^5*d^10*e^10 + 210*a^
7*c^4*d^8*e^12 + 120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 + 10*a^10*c*d^2*e^18
+ a^11*e^20)))/(a*c^5*d^10 + 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 + 10*a^4*c^
2*d^4*e^6 + 5*a^5*c*d^2*e^8 + a^6*e^10))*log((5*c^4*d^4*e - 10*a*c^3*d^2*e^3 + a
^2*c^2*e^5)*sqrt(e*x + d) + (15*a*c^4*d^6*e^2 - 35*a^2*c^3*d^4*e^4 + 13*a^3*c^2*
d^2*e^6 - a^4*c*e^8 - (a*c^6*d^13 + 2*a^2*c^5*d^11*e^2 - 5*a^3*c^4*d^9*e^4 - 20*
a^4*c^3*d^7*e^6 - 25*a^5*c^2*d^5*e^8 - 14*a^6*c*d^3*e^10 - 3*a^7*d*e^12)*sqrt(-(
25*c^7*d^8*e^2 - 100*a*c^6*d^6*e^4 + 110*a^2*c^5*d^4*e^6 - 20*a^3*c^4*d^2*e^8 +
a^4*c^3*e^10)/(a*c^10*d^20 + 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 + 120*a^4
*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 + 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e
^12 + 120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 + 10*a^10*c*d^2*e^18 + a^11*e^2
0)))*sqrt(-(c^4*d^5 - 10*a*c^3*d^3*e^2 + 5*a^2*c^2*d*e^4 - (a*c^5*d^10 + 5*a^2*c
^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 + 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 + a^6*e^1
0)*sqrt(-(25*c^7*d^8*e^2 - 100*a*c^6*d^6*e^4 + 110*a^2*c^5*d^4*e^6 - 20*a^3*c^4*
d^2*e^8 + a^4*c^3*e^10)/(a*c^10*d^20 + 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4
+ 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 + 252*a^6*c^5*d^10*e^10 + 210*a^7
*c^4*d^8*e^12 + 120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 + 10*a^10*c*d^2*e^18
+ a^11*e^20)))/(a*c^5*d^10 + 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 + 10*a^4*c^2
*d^4*e^6 + 5*a^5*c*d^2*e^8 + a^6*e^10))) + 3*(c^2*d^5 + 2*a*c*d^3*e^2 + a^2*d*e^
4 + (c^2*d^4*e + 2*a*c*d^2*e^3 + a^2*e^5)*x)*sqrt(e*x + d)*sqrt(-(c^4*d^5 - 10*a
*c^3*d^3*e^2 + 5*a^2*c^2*d*e^4 - (a*c^5*d^10 + 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^
6*e^4 + 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 + a^6*e^10)*sqrt(-(25*c^7*d^8*e^2 -
100*a*c^6*d^6*e^4 + 110*a^2*c^5*d^4*e^6 - 20*a^3*c^4*d^2*e^8 + a^4*c^3*e^10)/(a
*c^10*d^20 + 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 + 120*a^4*c^7*d^14*e^6 +
210*a^5*c^6*d^12*e^8 + 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 + 120*a^8*c^
3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 + 10*a^10*c*d^2*e^18 + a^11*e^20)))/(a*c^5*d^10
+ 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 + 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8
+ a^6*e^10))*log((5*c^4*d^4*e - 10*a*c^3*d^2*e^3 + a^2*c^2*e^5)*sqrt(e*x + d) -
(15*a*c^4*d^6*e^2 - 35*a^2*c^3*d^4*e^4 + 13*a^3*c^2*d^2*e^6 - a^4*c*e^8 - (a*c^
6*d^13 + 2*a^2*c^5*d^11*e^2 - 5*a^3*c^4*d^9*e^4 - 20*a^4*c^3*d^7*e^6 - 25*a^5*c^
2*d^5*e^8 - 14*a^6*c*d^3*e^10 - 3*a^7*d*e^12)*sqrt(-(25*c^7*d^8*e^2 - 100*a*c^6*
d^6*e^4 + 110*a^2*c^5*d^4*e^6 - 20*a^3*c^4*d^2*e^8 + a^4*c^3*e^10)/(a*c^10*d^20
+ 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 + 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6
*d^12*e^8 + 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 + 120*a^8*c^3*d^6*e^14
+ 45*a^9*c^2*d^4*e^16 + 10*a^10*c*d^2*e^18 + a^11*e^20)))*sqrt(-(c^4*d^5 - 10*a*
c^3*d^3*e^2 + 5*a^2*c^2*d*e^4 - (a*c^5*d^10 + 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6
*e^4 + 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 + a^6*e^10)*sqrt(-(25*c^7*d^8*e^2 -
100*a*c^6*d^6*e^4 + 110*a^2*c^5*d^4*e^6 - 20*a^3*c^4*d^2*e^8 + a^4*c^3*e^10)/(a*
c^10*d^20 + 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 + 120*a^4*c^7*d^14*e^6 + 2
10*a^5*c^6*d^12*e^8 + 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 + 120*a^8*c^3
*d^6*e^14 + 45*a^9*c^2*d^4*e^16 + 10*a^10*c*d^2*e^18 + a^11*e^20)))/(a*c^5*d^10
+ 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 + 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8
+ a^6*e^10))))/((c^2*d^5 + 2*a*c*d^3*e^2 + a^2*d*e^4 + (c^2*d^4*e + 2*a*c*d^2*e^
3 + a^2*e^5)*x)*sqrt(e*x + d))

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + c x^{2}\right ) \left (d + e x\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**(5/2)/(c*x**2+a),x)

[Out]

Integral(1/((a + c*x**2)*(d + e*x)**(5/2)), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 91.5866, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + a)*(e*x + d)^(5/2)),x, algorithm="giac")

[Out]

Done