3.477 \(\int \frac{1}{x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx\)

Optimal. Leaf size=570 \[ -\frac{b^{9/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{5/4} (b c-a d)^2}+\frac{b^{9/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{5/4} (b c-a d)^2}+\frac{b^{9/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{5/4} (b c-a d)^2}-\frac{b^{9/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{5/4} (b c-a d)^2}+\frac{d^{5/4} (9 b c-5 a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{9/4} (b c-a d)^2}-\frac{d^{5/4} (9 b c-5 a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{9/4} (b c-a d)^2}-\frac{d^{5/4} (9 b c-5 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{9/4} (b c-a d)^2}+\frac{d^{5/4} (9 b c-5 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} c^{9/4} (b c-a d)^2}-\frac{4 b c-5 a d}{2 a c^2 \sqrt{x} (b c-a d)}-\frac{d}{2 c \sqrt{x} \left (c+d x^2\right ) (b c-a d)} \]

[Out]

-(4*b*c - 5*a*d)/(2*a*c^2*(b*c - a*d)*Sqrt[x]) - d/(2*c*(b*c - a*d)*Sqrt[x]*(c +
d*x^2)) + (b^(9/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(5
/4)*(b*c - a*d)^2) - (b^(9/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sq
rt[2]*a^(5/4)*(b*c - a*d)^2) - (d^(5/4)*(9*b*c - 5*a*d)*ArcTan[1 - (Sqrt[2]*d^(1
/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c^(9/4)*(b*c - a*d)^2) + (d^(5/4)*(9*b*c - 5*a
*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c^(9/4)*(b*c - a*d
)^2) - (b^(9/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*S
qrt[2]*a^(5/4)*(b*c - a*d)^2) + (b^(9/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*S
qrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(5/4)*(b*c - a*d)^2) + (d^(5/4)*(9*b*c - 5*a*d
)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(9/4)
*(b*c - a*d)^2) - (d^(5/4)*(9*b*c - 5*a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)
*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(9/4)*(b*c - a*d)^2)

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Rubi [A]  time = 1.69739, antiderivative size = 570, 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 \[ -\frac{b^{9/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{5/4} (b c-a d)^2}+\frac{b^{9/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{5/4} (b c-a d)^2}+\frac{b^{9/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{5/4} (b c-a d)^2}-\frac{b^{9/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{5/4} (b c-a d)^2}+\frac{d^{5/4} (9 b c-5 a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{9/4} (b c-a d)^2}-\frac{d^{5/4} (9 b c-5 a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{9/4} (b c-a d)^2}-\frac{d^{5/4} (9 b c-5 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{9/4} (b c-a d)^2}+\frac{d^{5/4} (9 b c-5 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} c^{9/4} (b c-a d)^2}-\frac{4 b c-5 a d}{2 a c^2 \sqrt{x} (b c-a d)}-\frac{d}{2 c \sqrt{x} \left (c+d x^2\right ) (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(3/2)*(a + b*x^2)*(c + d*x^2)^2),x]

[Out]

-(4*b*c - 5*a*d)/(2*a*c^2*(b*c - a*d)*Sqrt[x]) - d/(2*c*(b*c - a*d)*Sqrt[x]*(c +
d*x^2)) + (b^(9/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(5
/4)*(b*c - a*d)^2) - (b^(9/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sq
rt[2]*a^(5/4)*(b*c - a*d)^2) - (d^(5/4)*(9*b*c - 5*a*d)*ArcTan[1 - (Sqrt[2]*d^(1
/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c^(9/4)*(b*c - a*d)^2) + (d^(5/4)*(9*b*c - 5*a
*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c^(9/4)*(b*c - a*d
)^2) - (b^(9/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*S
qrt[2]*a^(5/4)*(b*c - a*d)^2) + (b^(9/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*S
qrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(5/4)*(b*c - a*d)^2) + (d^(5/4)*(9*b*c - 5*a*d
)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(9/4)
*(b*c - a*d)^2) - (d^(5/4)*(9*b*c - 5*a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)
*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(9/4)*(b*c - a*d)^2)

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

[Out]

Timed out

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Mathematica [A]  time = 1.68955, size = 540, normalized size = 0.95 \[ \frac{1}{16} \left (-\frac{4 \sqrt{2} b^{9/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{5/4} (b c-a d)^2}+\frac{4 \sqrt{2} b^{9/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{5/4} (b c-a d)^2}+\frac{8 \sqrt{2} b^{9/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{a^{5/4} (b c-a d)^2}-\frac{8 \sqrt{2} b^{9/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{a^{5/4} (b c-a d)^2}+\frac{\sqrt{2} d^{5/4} (9 b c-5 a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{9/4} (b c-a d)^2}+\frac{\sqrt{2} d^{5/4} (5 a d-9 b c) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{9/4} (b c-a d)^2}+\frac{2 \sqrt{2} d^{5/4} (5 a d-9 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{c^{9/4} (b c-a d)^2}+\frac{2 \sqrt{2} d^{5/4} (9 b c-5 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{c^{9/4} (b c-a d)^2}+\frac{8 d^2 x^{3/2}}{c^2 \left (c+d x^2\right ) (b c-a d)}-\frac{32}{a c^2 \sqrt{x}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^(3/2)*(a + b*x^2)*(c + d*x^2)^2),x]

[Out]

(-32/(a*c^2*Sqrt[x]) + (8*d^2*x^(3/2))/(c^2*(b*c - a*d)*(c + d*x^2)) + (8*Sqrt[2
]*b^(9/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(5/4)*(b*c - a*d)^2)
- (8*Sqrt[2]*b^(9/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(5/4)*(b
*c - a*d)^2) + (2*Sqrt[2]*d^(5/4)*(-9*b*c + 5*a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*S
qrt[x])/c^(1/4)])/(c^(9/4)*(b*c - a*d)^2) + (2*Sqrt[2]*d^(5/4)*(9*b*c - 5*a*d)*A
rcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(9/4)*(b*c - a*d)^2) - (4*Sqrt[
2]*b^(9/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(5/4)*
(b*c - a*d)^2) + (4*Sqrt[2]*b^(9/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x
] + Sqrt[b]*x])/(a^(5/4)*(b*c - a*d)^2) + (Sqrt[2]*d^(5/4)*(9*b*c - 5*a*d)*Log[S
qrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^(9/4)*(b*c - a*d)^2) +
(Sqrt[2]*d^(5/4)*(-9*b*c + 5*a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x]
+ Sqrt[d]*x])/(c^(9/4)*(b*c - a*d)^2))/16

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Maple [A]  time = 0.029, size = 582, normalized size = 1. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(3/2)/(b*x^2+a)/(d*x^2+c)^2,x)

[Out]

-1/2*d^3/c^2/(a*d-b*c)^2*x^(3/2)/(d*x^2+c)*a+1/2*d^2/c/(a*d-b*c)^2*x^(3/2)/(d*x^
2+c)*b-5/16*d^2/c^2/(a*d-b*c)^2/(c/d)^(1/4)*2^(1/2)*a*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)))-5/8*d^2/c^2/(a
*d-b*c)^2/(c/d)^(1/4)*2^(1/2)*a*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)-5/8*d^2/c^
2/(a*d-b*c)^2/(c/d)^(1/4)*2^(1/2)*a*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)+9/16*d
/c/(a*d-b*c)^2/(c/d)^(1/4)*2^(1/2)*b*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)))+9/8*d/c/(a*d-b*c)^2/(c/d)^(1/4)
*2^(1/2)*b*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)+9/8*d/c/(a*d-b*c)^2/(c/d)^(1/4)
*2^(1/2)*b*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)-2/a/c^2/x^(1/2)-1/4*b^2/a/(a*d-
b*c)^2/(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*b^2/a/(a*d-b*c)^2/(a/b)^(1/4)*2^(1/2)
*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)-1/2*b^2/a/(a*d-b*c)^2/(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(1/((b*x^2 + a)*(d*x^2 + c)^2*x^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 21.8001, size = 4498, normalized size = 7.89 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)*(d*x^2 + c)^2*x^(3/2)),x, algorithm="fricas")

[Out]

-1/8*(16*b*c^2 - 16*a*c*d + 4*(4*b*c*d - 5*a*d^2)*x^2 + 16*(-b^9/(a^5*b^8*c^8 -
8*a^6*b^7*c^7*d + 28*a^7*b^6*c^6*d^2 - 56*a^8*b^5*c^5*d^3 + 70*a^9*b^4*c^4*d^4 -
56*a^10*b^3*c^3*d^5 + 28*a^11*b^2*c^2*d^6 - 8*a^12*b*c*d^7 + a^13*d^8))^(1/4)*(
a*b*c^4 - a^2*c^3*d + (a*b*c^3*d - a^2*c^2*d^2)*x^2)*sqrt(x)*arctan((a^4*b^6*c^6
- 6*a^5*b^5*c^5*d + 15*a^6*b^4*c^4*d^2 - 20*a^7*b^3*c^3*d^3 + 15*a^8*b^2*c^2*d^
4 - 6*a^9*b*c*d^5 + a^10*d^6)*(-b^9/(a^5*b^8*c^8 - 8*a^6*b^7*c^7*d + 28*a^7*b^6*
c^6*d^2 - 56*a^8*b^5*c^5*d^3 + 70*a^9*b^4*c^4*d^4 - 56*a^10*b^3*c^3*d^5 + 28*a^1
1*b^2*c^2*d^6 - 8*a^12*b*c*d^7 + a^13*d^8))^(3/4)/(b^7*sqrt(x) + sqrt(b^14*x - (
a^3*b^13*c^4 - 4*a^4*b^12*c^3*d + 6*a^5*b^11*c^2*d^2 - 4*a^6*b^10*c*d^3 + a^7*b^
9*d^4)*sqrt(-b^9/(a^5*b^8*c^8 - 8*a^6*b^7*c^7*d + 28*a^7*b^6*c^6*d^2 - 56*a^8*b^
5*c^5*d^3 + 70*a^9*b^4*c^4*d^4 - 56*a^10*b^3*c^3*d^5 + 28*a^11*b^2*c^2*d^6 - 8*a
^12*b*c*d^7 + a^13*d^8))))) + 4*(a*b*c^4 - a^2*c^3*d + (a*b*c^3*d - a^2*c^2*d^2)
*x^2)*sqrt(x)*(-(6561*b^4*c^4*d^5 - 14580*a*b^3*c^3*d^6 + 12150*a^2*b^2*c^2*d^7
- 4500*a^3*b*c*d^8 + 625*a^4*d^9)/(b^8*c^17 - 8*a*b^7*c^16*d + 28*a^2*b^6*c^15*d
^2 - 56*a^3*b^5*c^14*d^3 + 70*a^4*b^4*c^13*d^4 - 56*a^5*b^3*c^12*d^5 + 28*a^6*b^
2*c^11*d^6 - 8*a^7*b*c^10*d^7 + a^8*c^9*d^8))^(1/4)*arctan(-(b^6*c^13 - 6*a*b^5*
c^12*d + 15*a^2*b^4*c^11*d^2 - 20*a^3*b^3*c^10*d^3 + 15*a^4*b^2*c^9*d^4 - 6*a^5*
b*c^8*d^5 + a^6*c^7*d^6)*(-(6561*b^4*c^4*d^5 - 14580*a*b^3*c^3*d^6 + 12150*a^2*b
^2*c^2*d^7 - 4500*a^3*b*c*d^8 + 625*a^4*d^9)/(b^8*c^17 - 8*a*b^7*c^16*d + 28*a^2
*b^6*c^15*d^2 - 56*a^3*b^5*c^14*d^3 + 70*a^4*b^4*c^13*d^4 - 56*a^5*b^3*c^12*d^5
+ 28*a^6*b^2*c^11*d^6 - 8*a^7*b*c^10*d^7 + a^8*c^9*d^8))^(3/4)/((729*b^3*c^3*d^4
- 1215*a*b^2*c^2*d^5 + 675*a^2*b*c*d^6 - 125*a^3*d^7)*sqrt(x) - sqrt((531441*b^
6*c^6*d^8 - 1771470*a*b^5*c^5*d^9 + 2460375*a^2*b^4*c^4*d^10 - 1822500*a^3*b^3*c
^3*d^11 + 759375*a^4*b^2*c^2*d^12 - 168750*a^5*b*c*d^13 + 15625*a^6*d^14)*x - (6
561*b^8*c^13*d^5 - 40824*a*b^7*c^12*d^6 + 109836*a^2*b^6*c^11*d^7 - 166824*a^3*b
^5*c^10*d^8 + 156406*a^4*b^4*c^9*d^9 - 92680*a^5*b^3*c^8*d^10 + 33900*a^6*b^2*c^
7*d^11 - 7000*a^7*b*c^6*d^12 + 625*a^8*c^5*d^13)*sqrt(-(6561*b^4*c^4*d^5 - 14580
*a*b^3*c^3*d^6 + 12150*a^2*b^2*c^2*d^7 - 4500*a^3*b*c*d^8 + 625*a^4*d^9)/(b^8*c^
17 - 8*a*b^7*c^16*d + 28*a^2*b^6*c^15*d^2 - 56*a^3*b^5*c^14*d^3 + 70*a^4*b^4*c^1
3*d^4 - 56*a^5*b^3*c^12*d^5 + 28*a^6*b^2*c^11*d^6 - 8*a^7*b*c^10*d^7 + a^8*c^9*d
^8))))) + 4*(-b^9/(a^5*b^8*c^8 - 8*a^6*b^7*c^7*d + 28*a^7*b^6*c^6*d^2 - 56*a^8*b
^5*c^5*d^3 + 70*a^9*b^4*c^4*d^4 - 56*a^10*b^3*c^3*d^5 + 28*a^11*b^2*c^2*d^6 - 8*
a^12*b*c*d^7 + a^13*d^8))^(1/4)*(a*b*c^4 - a^2*c^3*d + (a*b*c^3*d - a^2*c^2*d^2)
*x^2)*sqrt(x)*log(b^7*sqrt(x) + (a^4*b^6*c^6 - 6*a^5*b^5*c^5*d + 15*a^6*b^4*c^4*
d^2 - 20*a^7*b^3*c^3*d^3 + 15*a^8*b^2*c^2*d^4 - 6*a^9*b*c*d^5 + a^10*d^6)*(-b^9/
(a^5*b^8*c^8 - 8*a^6*b^7*c^7*d + 28*a^7*b^6*c^6*d^2 - 56*a^8*b^5*c^5*d^3 + 70*a^
9*b^4*c^4*d^4 - 56*a^10*b^3*c^3*d^5 + 28*a^11*b^2*c^2*d^6 - 8*a^12*b*c*d^7 + a^1
3*d^8))^(3/4)) - 4*(-b^9/(a^5*b^8*c^8 - 8*a^6*b^7*c^7*d + 28*a^7*b^6*c^6*d^2 - 5
6*a^8*b^5*c^5*d^3 + 70*a^9*b^4*c^4*d^4 - 56*a^10*b^3*c^3*d^5 + 28*a^11*b^2*c^2*d
^6 - 8*a^12*b*c*d^7 + a^13*d^8))^(1/4)*(a*b*c^4 - a^2*c^3*d + (a*b*c^3*d - a^2*c
^2*d^2)*x^2)*sqrt(x)*log(b^7*sqrt(x) - (a^4*b^6*c^6 - 6*a^5*b^5*c^5*d + 15*a^6*b
^4*c^4*d^2 - 20*a^7*b^3*c^3*d^3 + 15*a^8*b^2*c^2*d^4 - 6*a^9*b*c*d^5 + a^10*d^6)
*(-b^9/(a^5*b^8*c^8 - 8*a^6*b^7*c^7*d + 28*a^7*b^6*c^6*d^2 - 56*a^8*b^5*c^5*d^3
+ 70*a^9*b^4*c^4*d^4 - 56*a^10*b^3*c^3*d^5 + 28*a^11*b^2*c^2*d^6 - 8*a^12*b*c*d^
7 + a^13*d^8))^(3/4)) + (a*b*c^4 - a^2*c^3*d + (a*b*c^3*d - a^2*c^2*d^2)*x^2)*sq
rt(x)*(-(6561*b^4*c^4*d^5 - 14580*a*b^3*c^3*d^6 + 12150*a^2*b^2*c^2*d^7 - 4500*a
^3*b*c*d^8 + 625*a^4*d^9)/(b^8*c^17 - 8*a*b^7*c^16*d + 28*a^2*b^6*c^15*d^2 - 56*
a^3*b^5*c^14*d^3 + 70*a^4*b^4*c^13*d^4 - 56*a^5*b^3*c^12*d^5 + 28*a^6*b^2*c^11*d
^6 - 8*a^7*b*c^10*d^7 + a^8*c^9*d^8))^(1/4)*log((b^6*c^13 - 6*a*b^5*c^12*d + 15*
a^2*b^4*c^11*d^2 - 20*a^3*b^3*c^10*d^3 + 15*a^4*b^2*c^9*d^4 - 6*a^5*b*c^8*d^5 +
a^6*c^7*d^6)*(-(6561*b^4*c^4*d^5 - 14580*a*b^3*c^3*d^6 + 12150*a^2*b^2*c^2*d^7 -
4500*a^3*b*c*d^8 + 625*a^4*d^9)/(b^8*c^17 - 8*a*b^7*c^16*d + 28*a^2*b^6*c^15*d^
2 - 56*a^3*b^5*c^14*d^3 + 70*a^4*b^4*c^13*d^4 - 56*a^5*b^3*c^12*d^5 + 28*a^6*b^2
*c^11*d^6 - 8*a^7*b*c^10*d^7 + a^8*c^9*d^8))^(3/4) - (729*b^3*c^3*d^4 - 1215*a*b
^2*c^2*d^5 + 675*a^2*b*c*d^6 - 125*a^3*d^7)*sqrt(x)) - (a*b*c^4 - a^2*c^3*d + (a
*b*c^3*d - a^2*c^2*d^2)*x^2)*sqrt(x)*(-(6561*b^4*c^4*d^5 - 14580*a*b^3*c^3*d^6 +
12150*a^2*b^2*c^2*d^7 - 4500*a^3*b*c*d^8 + 625*a^4*d^9)/(b^8*c^17 - 8*a*b^7*c^1
6*d + 28*a^2*b^6*c^15*d^2 - 56*a^3*b^5*c^14*d^3 + 70*a^4*b^4*c^13*d^4 - 56*a^5*b
^3*c^12*d^5 + 28*a^6*b^2*c^11*d^6 - 8*a^7*b*c^10*d^7 + a^8*c^9*d^8))^(1/4)*log(-
(b^6*c^13 - 6*a*b^5*c^12*d + 15*a^2*b^4*c^11*d^2 - 20*a^3*b^3*c^10*d^3 + 15*a^4*
b^2*c^9*d^4 - 6*a^5*b*c^8*d^5 + a^6*c^7*d^6)*(-(6561*b^4*c^4*d^5 - 14580*a*b^3*c
^3*d^6 + 12150*a^2*b^2*c^2*d^7 - 4500*a^3*b*c*d^8 + 625*a^4*d^9)/(b^8*c^17 - 8*a
*b^7*c^16*d + 28*a^2*b^6*c^15*d^2 - 56*a^3*b^5*c^14*d^3 + 70*a^4*b^4*c^13*d^4 -
56*a^5*b^3*c^12*d^5 + 28*a^6*b^2*c^11*d^6 - 8*a^7*b*c^10*d^7 + a^8*c^9*d^8))^(3/
4) - (729*b^3*c^3*d^4 - 1215*a*b^2*c^2*d^5 + 675*a^2*b*c*d^6 - 125*a^3*d^7)*sqrt
(x)))/((a*b*c^4 - a^2*c^3*d + (a*b*c^3*d - a^2*c^2*d^2)*x^2)*sqrt(x))

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.357802, size = 979, normalized size = 1.72 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)*(d*x^2 + c)^2*x^(3/2)),x, algorithm="giac")

[Out]

1/4*(9*(c*d^3)^(3/4)*b*c - 5*(c*d^3)^(3/4)*a*d)*arctan(1/2*sqrt(2)*(sqrt(2)*(c/d
)^(1/4) + 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b^2*c^5*d - 2*sqrt(2)*a*b*c^4*d^2 + s
qrt(2)*a^2*c^3*d^3) + 1/4*(9*(c*d^3)^(3/4)*b*c - 5*(c*d^3)^(3/4)*a*d)*arctan(-1/
2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b^2*c^5*d - 2*
sqrt(2)*a*b*c^4*d^2 + sqrt(2)*a^2*c^3*d^3) - 1/8*(9*(c*d^3)^(3/4)*b*c - 5*(c*d^3
)^(3/4)*a*d)*ln(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)*b^2*c^5*d
- 2*sqrt(2)*a*b*c^4*d^2 + sqrt(2)*a^2*c^3*d^3) + 1/8*(9*(c*d^3)^(3/4)*b*c - 5*(c
*d^3)^(3/4)*a*d)*ln(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)*b^2*c
^5*d - 2*sqrt(2)*a*b*c^4*d^2 + sqrt(2)*a^2*c^3*d^3) - (a*b^3)^(3/4)*arctan(1/2*s
qrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*a^2*b^2*c^2 - 2*s
qrt(2)*a^3*b*c*d + sqrt(2)*a^4*d^2) - (a*b^3)^(3/4)*arctan(-1/2*sqrt(2)*(sqrt(2)
*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*a^2*b^2*c^2 - 2*sqrt(2)*a^3*b*c*
d + sqrt(2)*a^4*d^2) + 1/2*(a*b^3)^(3/4)*ln(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sq
rt(a/b))/(sqrt(2)*a^2*b^2*c^2 - 2*sqrt(2)*a^3*b*c*d + sqrt(2)*a^4*d^2) - 1/2*(a*
b^3)^(3/4)*ln(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*a^2*b^2*c^2
- 2*sqrt(2)*a^3*b*c*d + sqrt(2)*a^4*d^2) - 1/2*(4*b*c*d*x^2 - 5*a*d^2*x^2 + 4*b
*c^2 - 4*a*c*d)/((a*b*c^3 - a^2*c^2*d)*(d*x^(5/2) + c*sqrt(x)))