3.481 \(\int \frac{x^{5/2}}{\left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx\)

Optimal. Leaf size=628 \[ -\frac{a^{3/4} b^{5/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} (b c-a d)^3}+\frac{a^{3/4} b^{5/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} (b c-a d)^3}+\frac{a^{3/4} b^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} (b c-a d)^3}-\frac{a^{3/4} b^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} (b c-a d)^3}+\frac{\left (-3 a^2 d^2+30 a b c d+5 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{5/4} d^{3/4} (b c-a d)^3}-\frac{\left (-3 a^2 d^2+30 a b c d+5 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{5/4} d^{3/4} (b c-a d)^3}-\frac{\left (-3 a^2 d^2+30 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{5/4} d^{3/4} (b c-a d)^3}+\frac{\left (-3 a^2 d^2+30 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{5/4} d^{3/4} (b c-a d)^3}+\frac{x^{3/2} (3 a d+5 b c)}{16 c \left (c+d x^2\right ) (b c-a d)^2}+\frac{x^{3/2}}{4 \left (c+d x^2\right )^2 (b c-a d)} \]

[Out]

x^(3/2)/(4*(b*c - a*d)*(c + d*x^2)^2) + ((5*b*c + 3*a*d)*x^(3/2))/(16*c*(b*c - a
*d)^2*(c + d*x^2)) + (a^(3/4)*b^(5/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/
4)])/(Sqrt[2]*(b*c - a*d)^3) - (a^(3/4)*b^(5/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt
[x])/a^(1/4)])/(Sqrt[2]*(b*c - a*d)^3) - ((5*b^2*c^2 + 30*a*b*c*d - 3*a^2*d^2)*A
rcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(5/4)*d^(3/4)*(b*c -
a*d)^3) + ((5*b^2*c^2 + 30*a*b*c*d - 3*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqr
t[x])/c^(1/4)])/(32*Sqrt[2]*c^(5/4)*d^(3/4)*(b*c - a*d)^3) - (a^(3/4)*b^(5/4)*Lo
g[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*(b*c - a*d)
^3) + (a^(3/4)*b^(5/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x
])/(2*Sqrt[2]*(b*c - a*d)^3) + ((5*b^2*c^2 + 30*a*b*c*d - 3*a^2*d^2)*Log[Sqrt[c]
- Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(5/4)*d^(3/4)*(b*
c - a*d)^3) - ((5*b^2*c^2 + 30*a*b*c*d - 3*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4
)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(5/4)*d^(3/4)*(b*c - a*d)^3)

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Rubi [A]  time = 1.65786, antiderivative size = 628, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417 \[ -\frac{a^{3/4} b^{5/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} (b c-a d)^3}+\frac{a^{3/4} b^{5/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} (b c-a d)^3}+\frac{a^{3/4} b^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} (b c-a d)^3}-\frac{a^{3/4} b^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} (b c-a d)^3}+\frac{\left (-3 a^2 d^2+30 a b c d+5 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{5/4} d^{3/4} (b c-a d)^3}-\frac{\left (-3 a^2 d^2+30 a b c d+5 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{5/4} d^{3/4} (b c-a d)^3}-\frac{\left (-3 a^2 d^2+30 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{5/4} d^{3/4} (b c-a d)^3}+\frac{\left (-3 a^2 d^2+30 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{5/4} d^{3/4} (b c-a d)^3}+\frac{x^{3/2} (3 a d+5 b c)}{16 c \left (c+d x^2\right ) (b c-a d)^2}+\frac{x^{3/2}}{4 \left (c+d x^2\right )^2 (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[x^(5/2)/((a + b*x^2)*(c + d*x^2)^3),x]

[Out]

x^(3/2)/(4*(b*c - a*d)*(c + d*x^2)^2) + ((5*b*c + 3*a*d)*x^(3/2))/(16*c*(b*c - a
*d)^2*(c + d*x^2)) + (a^(3/4)*b^(5/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/
4)])/(Sqrt[2]*(b*c - a*d)^3) - (a^(3/4)*b^(5/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt
[x])/a^(1/4)])/(Sqrt[2]*(b*c - a*d)^3) - ((5*b^2*c^2 + 30*a*b*c*d - 3*a^2*d^2)*A
rcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(5/4)*d^(3/4)*(b*c -
a*d)^3) + ((5*b^2*c^2 + 30*a*b*c*d - 3*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqr
t[x])/c^(1/4)])/(32*Sqrt[2]*c^(5/4)*d^(3/4)*(b*c - a*d)^3) - (a^(3/4)*b^(5/4)*Lo
g[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*(b*c - a*d)
^3) + (a^(3/4)*b^(5/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x
])/(2*Sqrt[2]*(b*c - a*d)^3) + ((5*b^2*c^2 + 30*a*b*c*d - 3*a^2*d^2)*Log[Sqrt[c]
- Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(5/4)*d^(3/4)*(b*
c - a*d)^3) - ((5*b^2*c^2 + 30*a*b*c*d - 3*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4
)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(5/4)*d^(3/4)*(b*c - a*d)^3)

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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(x**(5/2)/(b*x**2+a)/(d*x**2+c)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 1.66167, size = 544, normalized size = 0.87 \[ \frac{-32 \sqrt{2} a^{3/4} b^{5/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+32 \sqrt{2} a^{3/4} b^{5/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+64 \sqrt{2} a^{3/4} b^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )-64 \sqrt{2} a^{3/4} b^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )+\frac{\sqrt{2} \left (-3 a^2 d^2+30 a b c d+5 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{5/4} d^{3/4}}-\frac{\sqrt{2} \left (-3 a^2 d^2+30 a b c d+5 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{5/4} d^{3/4}}-\frac{2 \sqrt{2} \left (-3 a^2 d^2+30 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{c^{5/4} d^{3/4}}+\frac{2 \sqrt{2} \left (-3 a^2 d^2+30 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{c^{5/4} d^{3/4}}+\frac{32 x^{3/2} (b c-a d)^2}{\left (c+d x^2\right )^2}+\frac{8 x^{3/2} (3 a d+5 b c) (b c-a d)}{c \left (c+d x^2\right )}}{128 (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(5/2)/((a + b*x^2)*(c + d*x^2)^3),x]

[Out]

((32*(b*c - a*d)^2*x^(3/2))/(c + d*x^2)^2 + (8*(b*c - a*d)*(5*b*c + 3*a*d)*x^(3/
2))/(c*(c + d*x^2)) + 64*Sqrt[2]*a^(3/4)*b^(5/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqr
t[x])/a^(1/4)] - 64*Sqrt[2]*a^(3/4)*b^(5/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])
/a^(1/4)] - (2*Sqrt[2]*(5*b^2*c^2 + 30*a*b*c*d - 3*a^2*d^2)*ArcTan[1 - (Sqrt[2]*
d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(5/4)*d^(3/4)) + (2*Sqrt[2]*(5*b^2*c^2 + 30*a*b*c*
d - 3*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(5/4)*d^(3/4))
- 32*Sqrt[2]*a^(3/4)*b^(5/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqr
t[b]*x] + 32*Sqrt[2]*a^(3/4)*b^(5/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[
x] + Sqrt[b]*x] + (Sqrt[2]*(5*b^2*c^2 + 30*a*b*c*d - 3*a^2*d^2)*Log[Sqrt[c] - Sq
rt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^(5/4)*d^(3/4)) - (Sqrt[2]*(5*b^2*
c^2 + 30*a*b*c*d - 3*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sq
rt[d]*x])/(c^(5/4)*d^(3/4)))/(128*(b*c - a*d)^3)

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Maple [A]  time = 0.028, size = 839, normalized size = 1.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)/(b*x^2+a)/(d*x^2+c)^3,x)

[Out]

3/16/(a*d-b*c)^3/(d*x^2+c)^2*d^3/c*x^(7/2)*a^2+1/8/(a*d-b*c)^3/(d*x^2+c)^2*d^2*x
^(7/2)*a*b-5/16/(a*d-b*c)^3/(d*x^2+c)^2*d*c*x^(7/2)*b^2-1/16/(a*d-b*c)^3/(d*x^2+
c)^2*x^(3/2)*a^2*d^2+5/8/(a*d-b*c)^3/(d*x^2+c)^2*x^(3/2)*c*a*b*d-9/16/(a*d-b*c)^
3/(d*x^2+c)^2*x^(3/2)*b^2*c^2+3/64/(a*d-b*c)^3/c*d/(c/d)^(1/4)*2^(1/2)*arctan(2^
(1/2)/(c/d)^(1/4)*x^(1/2)+1)*a^2-15/32/(a*d-b*c)^3/(c/d)^(1/4)*2^(1/2)*arctan(2^
(1/2)/(c/d)^(1/4)*x^(1/2)+1)*a*b-5/64/(a*d-b*c)^3*c/d/(c/d)^(1/4)*2^(1/2)*arctan
(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*b^2+3/64/(a*d-b*c)^3/c*d/(c/d)^(1/4)*2^(1/2)*arc
tan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*a^2-15/32/(a*d-b*c)^3/(c/d)^(1/4)*2^(1/2)*arc
tan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*a*b-5/64/(a*d-b*c)^3*c/d/(c/d)^(1/4)*2^(1/2)*
arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*b^2+3/128/(a*d-b*c)^3/c*d/(c/d)^(1/4)*2^(1
/2)*ln((x-(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2))/(x+(c/d)^(1/4)*x^(1/2)*2^(1/2
)+(c/d)^(1/2)))*a^2-15/64/(a*d-b*c)^3/(c/d)^(1/4)*2^(1/2)*ln((x-(c/d)^(1/4)*x^(1
/2)*2^(1/2)+(c/d)^(1/2))/(x+(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2)))*a*b-5/128/
(a*d-b*c)^3*c/d/(c/d)^(1/4)*2^(1/2)*ln((x-(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2
))/(x+(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2)))*b^2+1/4*a*b/(a*d-b*c)^3/(a/b)^(1
/4)*2^(1/2)*ln((x-(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2))/(x+(a/b)^(1/4)*x^(1/2
)*2^(1/2)+(a/b)^(1/2)))+1/2*a*b/(a*d-b*c)^3/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(
a/b)^(1/4)*x^(1/2)+1)+1/2*a*b/(a*d-b*c)^3/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/
b)^(1/4)*x^(1/2)-1)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(5/2)/((b*x^2 + a)*(d*x^2 + c)^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 64.5506, size = 7329, normalized size = 11.67 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(5/2)/((b*x^2 + a)*(d*x^2 + c)^3),x, algorithm="fricas")

[Out]

1/64*(128*(-a^3*b^5/(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a
^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6
- 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2
*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12))^(1/4)*(b^2*c^5 - 2*a*b*c^4*d + a^2*c^
3*d^2 + (b^2*c^3*d^2 - 2*a*b*c^2*d^3 + a^2*c*d^4)*x^4 + 2*(b^2*c^4*d - 2*a*b*c^3
*d^2 + a^2*c^2*d^3)*x^2)*arctan(-(b^9*c^9 - 9*a*b^8*c^8*d + 36*a^2*b^7*c^7*d^2 -
84*a^3*b^6*c^6*d^3 + 126*a^4*b^5*c^5*d^4 - 126*a^5*b^4*c^4*d^5 + 84*a^6*b^3*c^3
*d^6 - 36*a^7*b^2*c^2*d^7 + 9*a^8*b*c*d^8 - a^9*d^9)*(-a^3*b^5/(b^12*c^12 - 12*a
*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4
- 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*
c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d
^12))^(3/4)/(a^2*b^4*sqrt(x) + sqrt(a^4*b^8*x - (a^3*b^11*c^6 - 6*a^4*b^10*c^5*d
+ 15*a^5*b^9*c^4*d^2 - 20*a^6*b^8*c^3*d^3 + 15*a^7*b^7*c^2*d^4 - 6*a^8*b^6*c*d^
5 + a^9*b^5*d^6)*sqrt(-a^3*b^5/(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*
d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*
b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 +
66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12))))) + 4*(b^2*c^5 - 2*a*b*c^
4*d + a^2*c^3*d^2 + (b^2*c^3*d^2 - 2*a*b*c^2*d^3 + a^2*c*d^4)*x^4 + 2*(b^2*c^4*d
- 2*a*b*c^3*d^2 + a^2*c^2*d^3)*x^2)*(-(625*b^8*c^8 + 15000*a*b^7*c^7*d + 133500
*a^2*b^6*c^6*d^2 + 513000*a^3*b^5*c^5*d^3 + 649350*a^4*b^4*c^4*d^4 - 307800*a^5*
b^3*c^3*d^5 + 48060*a^6*b^2*c^2*d^6 - 3240*a^7*b*c*d^7 + 81*a^8*d^8)/(b^12*c^17*
d^3 - 12*a*b^11*c^16*d^4 + 66*a^2*b^10*c^15*d^5 - 220*a^3*b^9*c^14*d^6 + 495*a^4
*b^8*c^13*d^7 - 792*a^5*b^7*c^12*d^8 + 924*a^6*b^6*c^11*d^9 - 792*a^7*b^5*c^10*d
^10 + 495*a^8*b^4*c^9*d^11 - 220*a^9*b^3*c^8*d^12 + 66*a^10*b^2*c^7*d^13 - 12*a^
11*b*c^6*d^14 + a^12*c^5*d^15))^(1/4)*arctan((b^9*c^13*d^2 - 9*a*b^8*c^12*d^3 +
36*a^2*b^7*c^11*d^4 - 84*a^3*b^6*c^10*d^5 + 126*a^4*b^5*c^9*d^6 - 126*a^5*b^4*c^
8*d^7 + 84*a^6*b^3*c^7*d^8 - 36*a^7*b^2*c^6*d^9 + 9*a^8*b*c^5*d^10 - a^9*c^4*d^1
1)*(-(625*b^8*c^8 + 15000*a*b^7*c^7*d + 133500*a^2*b^6*c^6*d^2 + 513000*a^3*b^5*
c^5*d^3 + 649350*a^4*b^4*c^4*d^4 - 307800*a^5*b^3*c^3*d^5 + 48060*a^6*b^2*c^2*d^
6 - 3240*a^7*b*c*d^7 + 81*a^8*d^8)/(b^12*c^17*d^3 - 12*a*b^11*c^16*d^4 + 66*a^2*
b^10*c^15*d^5 - 220*a^3*b^9*c^14*d^6 + 495*a^4*b^8*c^13*d^7 - 792*a^5*b^7*c^12*d
^8 + 924*a^6*b^6*c^11*d^9 - 792*a^7*b^5*c^10*d^10 + 495*a^8*b^4*c^9*d^11 - 220*a
^9*b^3*c^8*d^12 + 66*a^10*b^2*c^7*d^13 - 12*a^11*b*c^6*d^14 + a^12*c^5*d^15))^(3
/4)/((125*b^6*c^6 + 2250*a*b^5*c^5*d + 13275*a^2*b^4*c^4*d^2 + 24300*a^3*b^3*c^3
*d^3 - 7965*a^4*b^2*c^2*d^4 + 810*a^5*b*c*d^5 - 27*a^6*d^6)*sqrt(x) - sqrt((1562
5*b^12*c^12 + 562500*a*b^11*c^11*d + 8381250*a^2*b^10*c^10*d^2 + 65812500*a^3*b^
9*c^9*d^3 + 283584375*a^4*b^8*c^8*d^4 + 609525000*a^5*b^7*c^7*d^5 + 382657500*a^
6*b^6*c^6*d^6 - 365715000*a^7*b^5*c^5*d^7 + 102090375*a^8*b^4*c^4*d^8 - 14215500
*a^9*b^3*c^3*d^9 + 1086210*a^10*b^2*c^2*d^10 - 43740*a^11*b*c*d^11 + 729*a^12*d^
12)*x - (625*b^14*c^17*d + 11250*a*b^13*c^16*d^2 + 52875*a^2*b^12*c^15*d^3 - 755
00*a^3*b^11*c^14*d^4 - 716775*a^4*b^10*c^13*d^5 + 1042350*a^5*b^9*c^12*d^6 + 328
8235*a^6*b^8*c^11*d^7 - 10986600*a^7*b^7*c^10*d^8 + 13692171*a^8*b^6*c^9*d^9 - 9
010386*a^9*b^5*c^8*d^10 + 3283065*a^10*b^4*c^7*d^11 - 646380*a^11*b^3*c^6*d^12 +
68715*a^12*b^2*c^5*d^13 - 3726*a^13*b*c^4*d^14 + 81*a^14*c^3*d^15)*sqrt(-(625*b
^8*c^8 + 15000*a*b^7*c^7*d + 133500*a^2*b^6*c^6*d^2 + 513000*a^3*b^5*c^5*d^3 + 6
49350*a^4*b^4*c^4*d^4 - 307800*a^5*b^3*c^3*d^5 + 48060*a^6*b^2*c^2*d^6 - 3240*a^
7*b*c*d^7 + 81*a^8*d^8)/(b^12*c^17*d^3 - 12*a*b^11*c^16*d^4 + 66*a^2*b^10*c^15*d
^5 - 220*a^3*b^9*c^14*d^6 + 495*a^4*b^8*c^13*d^7 - 792*a^5*b^7*c^12*d^8 + 924*a^
6*b^6*c^11*d^9 - 792*a^7*b^5*c^10*d^10 + 495*a^8*b^4*c^9*d^11 - 220*a^9*b^3*c^8*
d^12 + 66*a^10*b^2*c^7*d^13 - 12*a^11*b*c^6*d^14 + a^12*c^5*d^15))))) - 32*(-a^3
*b^5/(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3
+ 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*
c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*
a^11*b*c*d^11 + a^12*d^12))^(1/4)*(b^2*c^5 - 2*a*b*c^4*d + a^2*c^3*d^2 + (b^2*c^
3*d^2 - 2*a*b*c^2*d^3 + a^2*c*d^4)*x^4 + 2*(b^2*c^4*d - 2*a*b*c^3*d^2 + a^2*c^2*
d^3)*x^2)*log(a^2*b^4*sqrt(x) + (b^9*c^9 - 9*a*b^8*c^8*d + 36*a^2*b^7*c^7*d^2 -
84*a^3*b^6*c^6*d^3 + 126*a^4*b^5*c^5*d^4 - 126*a^5*b^4*c^4*d^5 + 84*a^6*b^3*c^3*
d^6 - 36*a^7*b^2*c^2*d^7 + 9*a^8*b*c*d^8 - a^9*d^9)*(-a^3*b^5/(b^12*c^12 - 12*a*
b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 -
792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c
^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^
12))^(3/4)) + 32*(-a^3*b^5/(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2
- 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*
c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a
^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12))^(1/4)*(b^2*c^5 - 2*a*b*c^4*d +
a^2*c^3*d^2 + (b^2*c^3*d^2 - 2*a*b*c^2*d^3 + a^2*c*d^4)*x^4 + 2*(b^2*c^4*d - 2*
a*b*c^3*d^2 + a^2*c^2*d^3)*x^2)*log(a^2*b^4*sqrt(x) - (b^9*c^9 - 9*a*b^8*c^8*d +
36*a^2*b^7*c^7*d^2 - 84*a^3*b^6*c^6*d^3 + 126*a^4*b^5*c^5*d^4 - 126*a^5*b^4*c^4
*d^5 + 84*a^6*b^3*c^3*d^6 - 36*a^7*b^2*c^2*d^7 + 9*a^8*b*c*d^8 - a^9*d^9)*(-a^3*
b^5/(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 +
495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c
^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a
^11*b*c*d^11 + a^12*d^12))^(3/4)) - (b^2*c^5 - 2*a*b*c^4*d + a^2*c^3*d^2 + (b^2*
c^3*d^2 - 2*a*b*c^2*d^3 + a^2*c*d^4)*x^4 + 2*(b^2*c^4*d - 2*a*b*c^3*d^2 + a^2*c^
2*d^3)*x^2)*(-(625*b^8*c^8 + 15000*a*b^7*c^7*d + 133500*a^2*b^6*c^6*d^2 + 513000
*a^3*b^5*c^5*d^3 + 649350*a^4*b^4*c^4*d^4 - 307800*a^5*b^3*c^3*d^5 + 48060*a^6*b
^2*c^2*d^6 - 3240*a^7*b*c*d^7 + 81*a^8*d^8)/(b^12*c^17*d^3 - 12*a*b^11*c^16*d^4
+ 66*a^2*b^10*c^15*d^5 - 220*a^3*b^9*c^14*d^6 + 495*a^4*b^8*c^13*d^7 - 792*a^5*b
^7*c^12*d^8 + 924*a^6*b^6*c^11*d^9 - 792*a^7*b^5*c^10*d^10 + 495*a^8*b^4*c^9*d^1
1 - 220*a^9*b^3*c^8*d^12 + 66*a^10*b^2*c^7*d^13 - 12*a^11*b*c^6*d^14 + a^12*c^5*
d^15))^(1/4)*log((b^9*c^13*d^2 - 9*a*b^8*c^12*d^3 + 36*a^2*b^7*c^11*d^4 - 84*a^3
*b^6*c^10*d^5 + 126*a^4*b^5*c^9*d^6 - 126*a^5*b^4*c^8*d^7 + 84*a^6*b^3*c^7*d^8 -
36*a^7*b^2*c^6*d^9 + 9*a^8*b*c^5*d^10 - a^9*c^4*d^11)*(-(625*b^8*c^8 + 15000*a*
b^7*c^7*d + 133500*a^2*b^6*c^6*d^2 + 513000*a^3*b^5*c^5*d^3 + 649350*a^4*b^4*c^4
*d^4 - 307800*a^5*b^3*c^3*d^5 + 48060*a^6*b^2*c^2*d^6 - 3240*a^7*b*c*d^7 + 81*a^
8*d^8)/(b^12*c^17*d^3 - 12*a*b^11*c^16*d^4 + 66*a^2*b^10*c^15*d^5 - 220*a^3*b^9*
c^14*d^6 + 495*a^4*b^8*c^13*d^7 - 792*a^5*b^7*c^12*d^8 + 924*a^6*b^6*c^11*d^9 -
792*a^7*b^5*c^10*d^10 + 495*a^8*b^4*c^9*d^11 - 220*a^9*b^3*c^8*d^12 + 66*a^10*b^
2*c^7*d^13 - 12*a^11*b*c^6*d^14 + a^12*c^5*d^15))^(3/4) - (125*b^6*c^6 + 2250*a*
b^5*c^5*d + 13275*a^2*b^4*c^4*d^2 + 24300*a^3*b^3*c^3*d^3 - 7965*a^4*b^2*c^2*d^4
+ 810*a^5*b*c*d^5 - 27*a^6*d^6)*sqrt(x)) + (b^2*c^5 - 2*a*b*c^4*d + a^2*c^3*d^2
+ (b^2*c^3*d^2 - 2*a*b*c^2*d^3 + a^2*c*d^4)*x^4 + 2*(b^2*c^4*d - 2*a*b*c^3*d^2
+ a^2*c^2*d^3)*x^2)*(-(625*b^8*c^8 + 15000*a*b^7*c^7*d + 133500*a^2*b^6*c^6*d^2
+ 513000*a^3*b^5*c^5*d^3 + 649350*a^4*b^4*c^4*d^4 - 307800*a^5*b^3*c^3*d^5 + 480
60*a^6*b^2*c^2*d^6 - 3240*a^7*b*c*d^7 + 81*a^8*d^8)/(b^12*c^17*d^3 - 12*a*b^11*c
^16*d^4 + 66*a^2*b^10*c^15*d^5 - 220*a^3*b^9*c^14*d^6 + 495*a^4*b^8*c^13*d^7 - 7
92*a^5*b^7*c^12*d^8 + 924*a^6*b^6*c^11*d^9 - 792*a^7*b^5*c^10*d^10 + 495*a^8*b^4
*c^9*d^11 - 220*a^9*b^3*c^8*d^12 + 66*a^10*b^2*c^7*d^13 - 12*a^11*b*c^6*d^14 + a
^12*c^5*d^15))^(1/4)*log(-(b^9*c^13*d^2 - 9*a*b^8*c^12*d^3 + 36*a^2*b^7*c^11*d^4
- 84*a^3*b^6*c^10*d^5 + 126*a^4*b^5*c^9*d^6 - 126*a^5*b^4*c^8*d^7 + 84*a^6*b^3*
c^7*d^8 - 36*a^7*b^2*c^6*d^9 + 9*a^8*b*c^5*d^10 - a^9*c^4*d^11)*(-(625*b^8*c^8 +
15000*a*b^7*c^7*d + 133500*a^2*b^6*c^6*d^2 + 513000*a^3*b^5*c^5*d^3 + 649350*a^
4*b^4*c^4*d^4 - 307800*a^5*b^3*c^3*d^5 + 48060*a^6*b^2*c^2*d^6 - 3240*a^7*b*c*d^
7 + 81*a^8*d^8)/(b^12*c^17*d^3 - 12*a*b^11*c^16*d^4 + 66*a^2*b^10*c^15*d^5 - 220
*a^3*b^9*c^14*d^6 + 495*a^4*b^8*c^13*d^7 - 792*a^5*b^7*c^12*d^8 + 924*a^6*b^6*c^
11*d^9 - 792*a^7*b^5*c^10*d^10 + 495*a^8*b^4*c^9*d^11 - 220*a^9*b^3*c^8*d^12 + 6
6*a^10*b^2*c^7*d^13 - 12*a^11*b*c^6*d^14 + a^12*c^5*d^15))^(3/4) - (125*b^6*c^6
+ 2250*a*b^5*c^5*d + 13275*a^2*b^4*c^4*d^2 + 24300*a^3*b^3*c^3*d^3 - 7965*a^4*b^
2*c^2*d^4 + 810*a^5*b*c*d^5 - 27*a^6*d^6)*sqrt(x)) + 4*((5*b*c*d + 3*a*d^2)*x^3
+ (9*b*c^2 - a*c*d)*x)*sqrt(x))/(b^2*c^5 - 2*a*b*c^4*d + a^2*c^3*d^2 + (b^2*c^3*
d^2 - 2*a*b*c^2*d^3 + a^2*c*d^4)*x^4 + 2*(b^2*c^4*d - 2*a*b*c^3*d^2 + a^2*c^2*d^
3)*x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(5/2)/(b*x**2+a)/(d*x**2+c)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.443891, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(5/2)/((b*x^2 + a)*(d*x^2 + c)^3),x, algorithm="giac")

[Out]

Done