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

Optimal. Leaf size=681 \[ -\frac{b^{13/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{5/4} (b c-a d)^3}+\frac{b^{13/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{5/4} (b c-a d)^3}+\frac{b^{13/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{5/4} (b c-a d)^3}-\frac{b^{13/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{5/4} (b c-a d)^3}+\frac{d^{5/4} \left (45 a^2 d^2-130 a b c d+117 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{13/4} (b c-a d)^3}-\frac{d^{5/4} \left (45 a^2 d^2-130 a b c d+117 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{13/4} (b c-a d)^3}-\frac{d^{5/4} \left (45 a^2 d^2-130 a b c d+117 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{13/4} (b c-a d)^3}+\frac{d^{5/4} \left (45 a^2 d^2-130 a b c d+117 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{13/4} (b c-a d)^3}-\frac{45 a^2 d^2-85 a b c d+32 b^2 c^2}{16 a c^3 \sqrt{x} (b c-a d)^2}-\frac{d (17 b c-9 a d)}{16 c^2 \sqrt{x} \left (c+d x^2\right ) (b c-a d)^2}-\frac{d}{4 c \sqrt{x} \left (c+d x^2\right )^2 (b c-a d)} \]

[Out]

-(32*b^2*c^2 - 85*a*b*c*d + 45*a^2*d^2)/(16*a*c^3*(b*c - a*d)^2*Sqrt[x]) - d/(4*
c*(b*c - a*d)*Sqrt[x]*(c + d*x^2)^2) - (d*(17*b*c - 9*a*d))/(16*c^2*(b*c - a*d)^
2*Sqrt[x]*(c + d*x^2)) + (b^(13/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)]
)/(Sqrt[2]*a^(5/4)*(b*c - a*d)^3) - (b^(13/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x
])/a^(1/4)])/(Sqrt[2]*a^(5/4)*(b*c - a*d)^3) - (d^(5/4)*(117*b^2*c^2 - 130*a*b*c
*d + 45*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(1
3/4)*(b*c - a*d)^3) + (d^(5/4)*(117*b^2*c^2 - 130*a*b*c*d + 45*a^2*d^2)*ArcTan[1
+ (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(13/4)*(b*c - a*d)^3) - (b^
(13/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^
(5/4)*(b*c - a*d)^3) + (b^(13/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] +
Sqrt[b]*x])/(2*Sqrt[2]*a^(5/4)*(b*c - a*d)^3) + (d^(5/4)*(117*b^2*c^2 - 130*a*b
*c*d + 45*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(
64*Sqrt[2]*c^(13/4)*(b*c - a*d)^3) - (d^(5/4)*(117*b^2*c^2 - 130*a*b*c*d + 45*a^
2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c
^(13/4)*(b*c - a*d)^3)

_______________________________________________________________________________________

Rubi [A]  time = 2.27686, antiderivative size = 681, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 11, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.458 \[ -\frac{b^{13/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{5/4} (b c-a d)^3}+\frac{b^{13/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{5/4} (b c-a d)^3}+\frac{b^{13/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{5/4} (b c-a d)^3}-\frac{b^{13/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{5/4} (b c-a d)^3}+\frac{d^{5/4} \left (45 a^2 d^2-130 a b c d+117 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{13/4} (b c-a d)^3}-\frac{d^{5/4} \left (45 a^2 d^2-130 a b c d+117 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{13/4} (b c-a d)^3}-\frac{d^{5/4} \left (45 a^2 d^2-130 a b c d+117 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{13/4} (b c-a d)^3}+\frac{d^{5/4} \left (45 a^2 d^2-130 a b c d+117 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{13/4} (b c-a d)^3}-\frac{45 a^2 d^2-85 a b c d+32 b^2 c^2}{16 a c^3 \sqrt{x} (b c-a d)^2}-\frac{d (17 b c-9 a d)}{16 c^2 \sqrt{x} \left (c+d x^2\right ) (b c-a d)^2}-\frac{d}{4 c \sqrt{x} \left (c+d x^2\right )^2 (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

-(32*b^2*c^2 - 85*a*b*c*d + 45*a^2*d^2)/(16*a*c^3*(b*c - a*d)^2*Sqrt[x]) - d/(4*
c*(b*c - a*d)*Sqrt[x]*(c + d*x^2)^2) - (d*(17*b*c - 9*a*d))/(16*c^2*(b*c - a*d)^
2*Sqrt[x]*(c + d*x^2)) + (b^(13/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)]
)/(Sqrt[2]*a^(5/4)*(b*c - a*d)^3) - (b^(13/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x
])/a^(1/4)])/(Sqrt[2]*a^(5/4)*(b*c - a*d)^3) - (d^(5/4)*(117*b^2*c^2 - 130*a*b*c
*d + 45*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(1
3/4)*(b*c - a*d)^3) + (d^(5/4)*(117*b^2*c^2 - 130*a*b*c*d + 45*a^2*d^2)*ArcTan[1
+ (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(13/4)*(b*c - a*d)^3) - (b^
(13/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^
(5/4)*(b*c - a*d)^3) + (b^(13/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] +
Sqrt[b]*x])/(2*Sqrt[2]*a^(5/4)*(b*c - a*d)^3) + (d^(5/4)*(117*b^2*c^2 - 130*a*b
*c*d + 45*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(
64*Sqrt[2]*c^(13/4)*(b*c - a*d)^3) - (d^(5/4)*(117*b^2*c^2 - 130*a*b*c*d + 45*a^
2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c
^(13/4)*(b*c - a*d)^3)

_______________________________________________________________________________________

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)**3,x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [A]  time = 2.81875, size = 637, normalized size = 0.94 \[ \frac{1}{128} \left (\frac{32 \sqrt{2} b^{13/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{5/4} (a d-b c)^3}+\frac{32 \sqrt{2} b^{13/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{5/4} (b c-a d)^3}-\frac{64 \sqrt{2} b^{13/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{a^{5/4} (a d-b c)^3}+\frac{64 \sqrt{2} b^{13/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{a^{5/4} (a d-b c)^3}+\frac{\sqrt{2} d^{5/4} \left (45 a^2 d^2-130 a b c d+117 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{13/4} (b c-a d)^3}+\frac{\sqrt{2} d^{5/4} \left (45 a^2 d^2-130 a b c d+117 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{13/4} (a d-b c)^3}-\frac{2 \sqrt{2} d^{5/4} \left (45 a^2 d^2-130 a b c d+117 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{c^{13/4} (b c-a d)^3}+\frac{2 \sqrt{2} d^{5/4} \left (45 a^2 d^2-130 a b c d+117 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{c^{13/4} (b c-a d)^3}+\frac{8 d^2 x^{3/2} (21 b c-13 a d)}{c^3 \left (c+d x^2\right ) (b c-a d)^2}+\frac{32 d^2 x^{3/2}}{c^2 \left (c+d x^2\right )^2 (b c-a d)}-\frac{256}{a c^3 \sqrt{x}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-256/(a*c^3*Sqrt[x]) + (32*d^2*x^(3/2))/(c^2*(b*c - a*d)*(c + d*x^2)^2) + (8*d^
2*(21*b*c - 13*a*d)*x^(3/2))/(c^3*(b*c - a*d)^2*(c + d*x^2)) - (64*Sqrt[2]*b^(13
/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(5/4)*(-(b*c) + a*d)^3) +
(64*Sqrt[2]*b^(13/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(5/4)*(-(
b*c) + a*d)^3) - (2*Sqrt[2]*d^(5/4)*(117*b^2*c^2 - 130*a*b*c*d + 45*a^2*d^2)*Arc
Tan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(13/4)*(b*c - a*d)^3) + (2*Sqrt[2
]*d^(5/4)*(117*b^2*c^2 - 130*a*b*c*d + 45*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*S
qrt[x])/c^(1/4)])/(c^(13/4)*(b*c - a*d)^3) + (32*Sqrt[2]*b^(13/4)*Log[Sqrt[a] -
Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(5/4)*(-(b*c) + a*d)^3) + (32*S
qrt[2]*b^(13/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(
5/4)*(b*c - a*d)^3) + (Sqrt[2]*d^(5/4)*(117*b^2*c^2 - 130*a*b*c*d + 45*a^2*d^2)*
Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^(13/4)*(b*c - a*d
)^3) + (Sqrt[2]*d^(5/4)*(117*b^2*c^2 - 130*a*b*c*d + 45*a^2*d^2)*Log[Sqrt[c] + S
qrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^(13/4)*(-(b*c) + a*d)^3))/128

_______________________________________________________________________________________

Maple [A]  time = 0.033, size = 900, normalized size = 1.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-13/16*d^5/c^3/(a*d-b*c)^3/(d*x^2+c)^2*x^(7/2)*a^2+17/8*d^4/c^2/(a*d-b*c)^3/(d*x
^2+c)^2*x^(7/2)*a*b-21/16*d^3/c/(a*d-b*c)^3/(d*x^2+c)^2*x^(7/2)*b^2-17/16*d^4/c^
2/(a*d-b*c)^3/(d*x^2+c)^2*x^(3/2)*a^2+21/8*d^3/c/(a*d-b*c)^3/(d*x^2+c)^2*x^(3/2)
*a*b-25/16*d^2/(a*d-b*c)^3/(d*x^2+c)^2*x^(3/2)*b^2-45/128*d^3/c^3/(a*d-b*c)^3/(c
/d)^(1/4)*2^(1/2)*a^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)))-45/64*d^3/c^3/(a*d-b*c)^3/(c/d)^(1/4)*2^(1/2)*
a^2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)-45/64*d^3/c^3/(a*d-b*c)^3/(c/d)^(1/4)*
2^(1/2)*a^2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)+65/64*d^2/c^2/(a*d-b*c)^3/(c/d
)^(1/4)*2^(1/2)*a*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)))+65/32*d^2/c^2/(a*d-b*c)^3/(c/d)^(1/4)*2^(1/2)*a*
b*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)+65/32*d^2/c^2/(a*d-b*c)^3/(c/d)^(1/4)*2^
(1/2)*a*b*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)-117/128*d/c/(a*d-b*c)^3/(c/d)^(1
/4)*2^(1/2)*b^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)))-117/64*d/c/(a*d-b*c)^3/(c/d)^(1/4)*2^(1/2)*b^2*arcta
n(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)-117/64*d/c/(a*d-b*c)^3/(c/d)^(1/4)*2^(1/2)*b^2*
arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)-2/a/c^3/x^(1/2)+1/4*b^3/a/(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^3/a/(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*b^3/a/(a*d-b*c)^3/(a/b)^(1/4)*2^(1/2)*arctan(2^(
1/2)/(a/b)^(1/4)*x^(1/2)-1)

_______________________________________________________________________________________

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)^3*x^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.420483, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done