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

Optimal. Leaf size=627 \[ -\frac{\left (-3 a^2 d^2+14 a b c d+21 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{7/4} \sqrt [4]{d} (b c-a d)^3}+\frac{\left (-3 a^2 d^2+14 a b c d+21 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{7/4} \sqrt [4]{d} (b c-a d)^3}-\frac{\left (-3 a^2 d^2+14 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{7/4} \sqrt [4]{d} (b c-a d)^3}+\frac{\left (-3 a^2 d^2+14 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{7/4} \sqrt [4]{d} (b c-a d)^3}+\frac{\sqrt [4]{a} b^{7/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} (b c-a d)^3}-\frac{\sqrt [4]{a} b^{7/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} (b c-a d)^3}+\frac{\sqrt [4]{a} b^{7/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} (b c-a d)^3}-\frac{\sqrt [4]{a} b^{7/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} (b c-a d)^3}+\frac{\sqrt{x} (a d+7 b c)}{16 c \left (c+d x^2\right ) (b c-a d)^2}+\frac{\sqrt{x}}{4 \left (c+d x^2\right )^2 (b c-a d)} \]

[Out]

Sqrt[x]/(4*(b*c - a*d)*(c + d*x^2)^2) + ((7*b*c + a*d)*Sqrt[x])/(16*c*(b*c - a*d
)^2*(c + d*x^2)) + (a^(1/4)*b^(7/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)
])/(Sqrt[2]*(b*c - a*d)^3) - (a^(1/4)*b^(7/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x
])/a^(1/4)])/(Sqrt[2]*(b*c - a*d)^3) - ((21*b^2*c^2 + 14*a*b*c*d - 3*a^2*d^2)*Ar
cTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(7/4)*d^(1/4)*(b*c -
a*d)^3) + ((21*b^2*c^2 + 14*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^(7/4)*d^(1/4)*(b*c - a*d)^3) + (a^(1/4)*b^(7/4)*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^(1/4)*b^(7/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x
])/(2*Sqrt[2]*(b*c - a*d)^3) - ((21*b^2*c^2 + 14*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^(7/4)*d^(1/4)*(b
*c - a*d)^3) + ((21*b^2*c^2 + 14*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^(7/4)*d^(1/4)*(b*c - a*d)^3)

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Rubi [A]  time = 1.49367, antiderivative size = 627, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417 \[ -\frac{\left (-3 a^2 d^2+14 a b c d+21 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{7/4} \sqrt [4]{d} (b c-a d)^3}+\frac{\left (-3 a^2 d^2+14 a b c d+21 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{7/4} \sqrt [4]{d} (b c-a d)^3}-\frac{\left (-3 a^2 d^2+14 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{7/4} \sqrt [4]{d} (b c-a d)^3}+\frac{\left (-3 a^2 d^2+14 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{7/4} \sqrt [4]{d} (b c-a d)^3}+\frac{\sqrt [4]{a} b^{7/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} (b c-a d)^3}-\frac{\sqrt [4]{a} b^{7/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} (b c-a d)^3}+\frac{\sqrt [4]{a} b^{7/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} (b c-a d)^3}-\frac{\sqrt [4]{a} b^{7/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} (b c-a d)^3}+\frac{\sqrt{x} (a d+7 b c)}{16 c \left (c+d x^2\right ) (b c-a d)^2}+\frac{\sqrt{x}}{4 \left (c+d x^2\right )^2 (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[x]/(4*(b*c - a*d)*(c + d*x^2)^2) + ((7*b*c + a*d)*Sqrt[x])/(16*c*(b*c - a*d
)^2*(c + d*x^2)) + (a^(1/4)*b^(7/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)
])/(Sqrt[2]*(b*c - a*d)^3) - (a^(1/4)*b^(7/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x
])/a^(1/4)])/(Sqrt[2]*(b*c - a*d)^3) - ((21*b^2*c^2 + 14*a*b*c*d - 3*a^2*d^2)*Ar
cTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(7/4)*d^(1/4)*(b*c -
a*d)^3) + ((21*b^2*c^2 + 14*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^(7/4)*d^(1/4)*(b*c - a*d)^3) + (a^(1/4)*b^(7/4)*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^(1/4)*b^(7/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x
])/(2*Sqrt[2]*(b*c - a*d)^3) - ((21*b^2*c^2 + 14*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^(7/4)*d^(1/4)*(b
*c - a*d)^3) + ((21*b^2*c^2 + 14*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^(7/4)*d^(1/4)*(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**(3/2)/(b*x**2+a)/(d*x**2+c)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 1.66039, size = 543, normalized size = 0.87 \[ \frac{-\frac{\sqrt{2} \left (-3 a^2 d^2+14 a b c d+21 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{7/4} \sqrt [4]{d}}+\frac{\sqrt{2} \left (-3 a^2 d^2+14 a b c d+21 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{7/4} \sqrt [4]{d}}-\frac{2 \sqrt{2} \left (-3 a^2 d^2+14 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{c^{7/4} \sqrt [4]{d}}+\frac{2 \sqrt{2} \left (-3 a^2 d^2+14 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{c^{7/4} \sqrt [4]{d}}+32 \sqrt{2} \sqrt [4]{a} b^{7/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-32 \sqrt{2} \sqrt [4]{a} b^{7/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+64 \sqrt{2} \sqrt [4]{a} b^{7/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )-64 \sqrt{2} \sqrt [4]{a} b^{7/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )+\frac{32 \sqrt{x} (b c-a d)^2}{\left (c+d x^2\right )^2}+\frac{8 \sqrt{x} (a d+7 b c) (b c-a d)}{c \left (c+d x^2\right )}}{128 (b c-a d)^3} \]

Antiderivative was successfully verified.

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

[Out]

((32*(b*c - a*d)^2*Sqrt[x])/(c + d*x^2)^2 + (8*(b*c - a*d)*(7*b*c + a*d)*Sqrt[x]
)/(c*(c + d*x^2)) + 64*Sqrt[2]*a^(1/4)*b^(7/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[
x])/a^(1/4)] - 64*Sqrt[2]*a^(1/4)*b^(7/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a
^(1/4)] - (2*Sqrt[2]*(21*b^2*c^2 + 14*a*b*c*d - 3*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d
^(1/4)*Sqrt[x])/c^(1/4)])/(c^(7/4)*d^(1/4)) + (2*Sqrt[2]*(21*b^2*c^2 + 14*a*b*c*
d - 3*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(7/4)*d^(1/4))
+ 32*Sqrt[2]*a^(1/4)*b^(7/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqr
t[b]*x] - 32*Sqrt[2]*a^(1/4)*b^(7/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[
x] + Sqrt[b]*x] - (Sqrt[2]*(21*b^2*c^2 + 14*a*b*c*d - 3*a^2*d^2)*Log[Sqrt[c] - S
qrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^(7/4)*d^(1/4)) + (Sqrt[2]*(21*b^
2*c^2 + 14*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])/(c^(7/4)*d^(1/4)))/(128*(b*c - a*d)^3)

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Maple [A]  time = 0.027, size = 848, normalized size = 1.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16/(a*d-b*c)^3/(d*x^2+c)^2*d^3/c*x^(5/2)*a^2+3/8/(a*d-b*c)^3/(d*x^2+c)^2*d^2*x
^(5/2)*a*b-7/16/(a*d-b*c)^3/(d*x^2+c)^2*d*c*x^(5/2)*b^2+7/8/(a*d-b*c)^3/(d*x^2+c
)^2*x^(1/2)*c*a*b*d-11/16/(a*d-b*c)^3/(d*x^2+c)^2*x^(1/2)*b^2*c^2-3/16/(a*d-b*c)
^3/(d*x^2+c)^2*x^(1/2)*a^2*d^2+3/64/(a*d-b*c)^3/c^2*(c/d)^(1/4)*2^(1/2)*arctan(2
^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*a^2*d^2-7/32/(a*d-b*c)^3/c*(c/d)^(1/4)*2^(1/2)*arc
tan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*a*b*d-21/64/(a*d-b*c)^3*(c/d)^(1/4)*2^(1/2)*a
rctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*b^2+3/64/(a*d-b*c)^3/c^2*(c/d)^(1/4)*2^(1/2
)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*a^2*d^2-7/32/(a*d-b*c)^3/c*(c/d)^(1/4)*2
^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*a*b*d-21/64/(a*d-b*c)^3*(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^2*(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*d^2-7/64/(a*d-b*c)^3/c*(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*d-21/128/(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)))*b^2+1/4*b^2/(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*b^2/(a*d-b*c)^3*(a/b)^(1/4)*2^(1/2)*arc
tan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+1/2*b^2/(a*d-b*c)^3*(a/b)^(1/4)*2^(1/2)*arcta
n(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^(3/2)/((b*x^2 + a)*(d*x^2 + c)^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [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^(3/2)/((b*x^2 + a)*(d*x^2 + c)^3),x, algorithm="fricas")

[Out]

Timed out

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.398723, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done