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

Optimal. Leaf size=703 \[ -\frac{b^{7/4} (11 a d+b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{3/4} (b c-a d)^4}+\frac{b^{7/4} (11 a d+b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{3/4} (b c-a d)^4}-\frac{b^{7/4} (11 a d+b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{3/4} (b c-a d)^4}+\frac{b^{7/4} (11 a d+b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{3/4} (b c-a d)^4}+\frac{d^{3/4} \left (-3 a^2 d^2+22 a b c d+77 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{7/4} (b c-a d)^4}-\frac{d^{3/4} \left (-3 a^2 d^2+22 a b c d+77 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{7/4} (b c-a d)^4}+\frac{d^{3/4} \left (-3 a^2 d^2+22 a b c d+77 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{7/4} (b c-a d)^4}-\frac{d^{3/4} \left (-3 a^2 d^2+22 a b c d+77 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{7/4} (b c-a d)^4}-\frac{d \sqrt{x} (a d+23 b c)}{16 c \left (c+d x^2\right ) (b c-a d)^3}-\frac{3 d \sqrt{x}}{4 \left (c+d x^2\right )^2 (b c-a d)^2}-\frac{\sqrt{x}}{2 \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)} \]

[Out]

(-3*d*Sqrt[x])/(4*(b*c - a*d)^2*(c + d*x^2)^2) - Sqrt[x]/(2*(b*c - a*d)*(a + b*x
^2)*(c + d*x^2)^2) - (d*(23*b*c + a*d)*Sqrt[x])/(16*c*(b*c - a*d)^3*(c + d*x^2))
- (b^(7/4)*(b*c + 11*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqr
t[2]*a^(3/4)*(b*c - a*d)^4) + (b^(7/4)*(b*c + 11*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4
)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(3/4)*(b*c - a*d)^4) + (d^(3/4)*(77*b^2*c^2 +
22*a*b*c*d - 3*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[
2]*c^(7/4)*(b*c - a*d)^4) - (d^(3/4)*(77*b^2*c^2 + 22*a*b*c*d - 3*a^2*d^2)*ArcTa
n[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(7/4)*(b*c - a*d)^4) - (
b^(7/4)*(b*c + 11*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x
])/(8*Sqrt[2]*a^(3/4)*(b*c - a*d)^4) + (b^(7/4)*(b*c + 11*a*d)*Log[Sqrt[a] + Sqr
t[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(3/4)*(b*c - a*d)^4) + (
d^(3/4)*(77*b^2*c^2 + 22*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)*(b*c - a*d)^4) - (d^(3/4)*(77*b^2*
c^2 + 22*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])/(64*Sqrt[2]*c^(7/4)*(b*c - a*d)^4)

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Rubi [A]  time = 2.08094, antiderivative size = 703, 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 \[ -\frac{b^{7/4} (11 a d+b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{3/4} (b c-a d)^4}+\frac{b^{7/4} (11 a d+b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{3/4} (b c-a d)^4}-\frac{b^{7/4} (11 a d+b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{3/4} (b c-a d)^4}+\frac{b^{7/4} (11 a d+b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{3/4} (b c-a d)^4}+\frac{d^{3/4} \left (-3 a^2 d^2+22 a b c d+77 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{7/4} (b c-a d)^4}-\frac{d^{3/4} \left (-3 a^2 d^2+22 a b c d+77 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{7/4} (b c-a d)^4}+\frac{d^{3/4} \left (-3 a^2 d^2+22 a b c d+77 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{7/4} (b c-a d)^4}-\frac{d^{3/4} \left (-3 a^2 d^2+22 a b c d+77 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{7/4} (b c-a d)^4}-\frac{d \sqrt{x} (a d+23 b c)}{16 c \left (c+d x^2\right ) (b c-a d)^3}-\frac{3 d \sqrt{x}}{4 \left (c+d x^2\right )^2 (b c-a d)^2}-\frac{\sqrt{x}}{2 \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

(-3*d*Sqrt[x])/(4*(b*c - a*d)^2*(c + d*x^2)^2) - Sqrt[x]/(2*(b*c - a*d)*(a + b*x
^2)*(c + d*x^2)^2) - (d*(23*b*c + a*d)*Sqrt[x])/(16*c*(b*c - a*d)^3*(c + d*x^2))
- (b^(7/4)*(b*c + 11*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqr
t[2]*a^(3/4)*(b*c - a*d)^4) + (b^(7/4)*(b*c + 11*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4
)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(3/4)*(b*c - a*d)^4) + (d^(3/4)*(77*b^2*c^2 +
22*a*b*c*d - 3*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[
2]*c^(7/4)*(b*c - a*d)^4) - (d^(3/4)*(77*b^2*c^2 + 22*a*b*c*d - 3*a^2*d^2)*ArcTa
n[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(7/4)*(b*c - a*d)^4) - (
b^(7/4)*(b*c + 11*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x
])/(8*Sqrt[2]*a^(3/4)*(b*c - a*d)^4) + (b^(7/4)*(b*c + 11*a*d)*Log[Sqrt[a] + Sqr
t[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(3/4)*(b*c - a*d)^4) + (
d^(3/4)*(77*b^2*c^2 + 22*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)*(b*c - a*d)^4) - (d^(3/4)*(77*b^2*
c^2 + 22*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])/(64*Sqrt[2]*c^(7/4)*(b*c - a*d)^4)

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

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [A]  time = 4.44656, size = 603, normalized size = 0.86 \[ \frac{-\frac{8 \sqrt{2} b^{7/4} (11 a d+b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{3/4}}+\frac{8 \sqrt{2} b^{7/4} (11 a d+b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{3/4}}-\frac{16 \sqrt{2} b^{7/4} (11 a d+b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{a^{3/4}}+\frac{16 \sqrt{2} b^{7/4} (11 a d+b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{a^{3/4}}+\frac{\sqrt{2} d^{3/4} \left (-3 a^2 d^2+22 a b c d+77 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{7/4}}+\frac{\sqrt{2} d^{3/4} \left (3 a^2 d^2-22 a b c d-77 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{7/4}}+\frac{2 \sqrt{2} d^{3/4} \left (-3 a^2 d^2+22 a b c d+77 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{c^{7/4}}-\frac{2 \sqrt{2} d^{3/4} \left (-3 a^2 d^2+22 a b c d+77 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{c^{7/4}}-\frac{64 b^2 \sqrt{x} (b c-a d)}{a+b x^2}+\frac{8 d \sqrt{x} (a d-b c) (a d+15 b c)}{c \left (c+d x^2\right )}-\frac{32 d \sqrt{x} (b c-a d)^2}{\left (c+d x^2\right )^2}}{128 (b c-a d)^4} \]

Antiderivative was successfully verified.

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

[Out]

((-64*b^2*(b*c - a*d)*Sqrt[x])/(a + b*x^2) - (32*d*(b*c - a*d)^2*Sqrt[x])/(c + d
*x^2)^2 + (8*d*(-(b*c) + a*d)*(15*b*c + a*d)*Sqrt[x])/(c*(c + d*x^2)) - (16*Sqrt
[2]*b^(7/4)*(b*c + 11*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/a^(3/4
) + (16*Sqrt[2]*b^(7/4)*(b*c + 11*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1
/4)])/a^(3/4) + (2*Sqrt[2]*d^(3/4)*(77*b^2*c^2 + 22*a*b*c*d - 3*a^2*d^2)*ArcTan[
1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/c^(7/4) - (2*Sqrt[2]*d^(3/4)*(77*b^2*c^2
+ 22*a*b*c*d - 3*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/c^(7/4
) - (8*Sqrt[2]*b^(7/4)*(b*c + 11*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt
[x] + Sqrt[b]*x])/a^(3/4) + (8*Sqrt[2]*b^(7/4)*(b*c + 11*a*d)*Log[Sqrt[a] + Sqrt
[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(3/4) + (Sqrt[2]*d^(3/4)*(77*b^2*c^2
+ 22*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) + (Sqrt[2]*d^(3/4)*(-77*b^2*c^2 - 22*a*b*c*d + 3*a^2*d^2)*Log[Sqr
t[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/c^(7/4))/(128*(b*c - a*d)^4
)

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Maple [A]  time = 0.034, size = 1094, normalized size = 1.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*d^4/(a*d-b*c)^4/(d*x^2+c)^2/c*x^(5/2)*a^2+7/8*d^3/(a*d-b*c)^4/(d*x^2+c)^2*x
^(5/2)*a*b-15/16*d^2/(a*d-b*c)^4/(d*x^2+c)^2*c*x^(5/2)*b^2+11/8*d^2/(a*d-b*c)^4/
(d*x^2+c)^2*x^(1/2)*c*a*b-19/16*d/(a*d-b*c)^4/(d*x^2+c)^2*x^(1/2)*b^2*c^2-3/16*d
^3/(a*d-b*c)^4/(d*x^2+c)^2*x^(1/2)*a^2+3/64*d^3/(a*d-b*c)^4/c^2*(c/d)^(1/4)*2^(1
/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*a^2-11/32*d^2/(a*d-b*c)^4/c*(c/d)^(1/4
)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*a*b-77/64*d/(a*d-b*c)^4*(c/d)^(1
/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*b^2+3/64*d^3/(a*d-b*c)^4/c^2*(
c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*a^2-11/32*d^2/(a*d-b*c)
^4/c*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*a*b-77/64*d/(a*d-
b*c)^4*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*b^2+3/128*d^3/(
a*d-b*c)^4/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^2-11/64*d^2/(a*d-b*c)^4/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-77/128*d/(a*d-b*c)^4*(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+1/2*b^2/(a*d-b*c)^4*x^(1/2)/(b*x^2+a)*a*d-1/2*b^3/(a*d-b*c)^4*x^(1/2)/(b
*x^2+a)*c+11/16*b^2/(a*d-b*c)^4*(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)))*d+1/16*b^3/(a*d-
b*c)^4*(a/b)^(1/4)/a*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)))*c+11/8*b^2/(a*d-b*c)^4*(a/b)^(1/4)*2^(1
/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)*d+1/8*b^3/(a*d-b*c)^4*(a/b)^(1/4)/a*2^
(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)*c+11/8*b^2/(a*d-b*c)^4*(a/b)^(1/4)*2
^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)*d+1/8*b^3/(a*d-b*c)^4*(a/b)^(1/4)/a
*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)*c

_______________________________________________________________________________________

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)^2*(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)^2*(d*x^2 + c)^3),x, algorithm="fricas")

[Out]

Timed out

_______________________________________________________________________________________

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.561003, 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)^2*(d*x^2 + c)^3),x, algorithm="giac")

[Out]

Done