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

Optimal. Leaf size=209 \[ \frac{315 a^4 \sqrt{b} c \sqrt{c \left (a+b x^2\right )^3} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{128 \left (a+b x^2\right )^{3/2}}+\frac{315 a^3 b c x \sqrt{c \left (a+b x^2\right )^3}}{128 \left (a+b x^2\right )}+\frac{105}{64} a^2 b c x \sqrt{c \left (a+b x^2\right )^3}+\frac{21}{16} a b c x \left (a+b x^2\right ) \sqrt{c \left (a+b x^2\right )^3}-\frac{c \left (a+b x^2\right )^3 \sqrt{c \left (a+b x^2\right )^3}}{x}+\frac{9}{8} b c x \left (a+b x^2\right )^2 \sqrt{c \left (a+b x^2\right )^3} \]

[Out]

(105*a^2*b*c*x*Sqrt[c*(a + b*x^2)^3])/64 + (315*a^3*b*c*x*Sqrt[c*(a + b*x^2)^3])
/(128*(a + b*x^2)) + (21*a*b*c*x*(a + b*x^2)*Sqrt[c*(a + b*x^2)^3])/16 + (9*b*c*
x*(a + b*x^2)^2*Sqrt[c*(a + b*x^2)^3])/8 - (c*(a + b*x^2)^3*Sqrt[c*(a + b*x^2)^3
])/x + (315*a^4*Sqrt[b]*c*Sqrt[c*(a + b*x^2)^3]*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x
^2]])/(128*(a + b*x^2)^(3/2))

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Rubi [A]  time = 0.348095, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ \frac{315 a^4 \sqrt{b} c \sqrt{c \left (a+b x^2\right )^3} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{128 \left (a+b x^2\right )^{3/2}}+\frac{315 a^3 b c x \sqrt{c \left (a+b x^2\right )^3}}{128 \left (a+b x^2\right )}+\frac{105}{64} a^2 b c x \sqrt{c \left (a+b x^2\right )^3}+\frac{21}{16} a b c x \left (a+b x^2\right ) \sqrt{c \left (a+b x^2\right )^3}-\frac{c \left (a+b x^2\right )^3 \sqrt{c \left (a+b x^2\right )^3}}{x}+\frac{9}{8} b c x \left (a+b x^2\right )^2 \sqrt{c \left (a+b x^2\right )^3} \]

Antiderivative was successfully verified.

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

[Out]

(105*a^2*b*c*x*Sqrt[c*(a + b*x^2)^3])/64 + (315*a^3*b*c*x*Sqrt[c*(a + b*x^2)^3])
/(128*(a + b*x^2)) + (21*a*b*c*x*(a + b*x^2)*Sqrt[c*(a + b*x^2)^3])/16 + (9*b*c*
x*(a + b*x^2)^2*Sqrt[c*(a + b*x^2)^3])/8 - (c*(a + b*x^2)^3*Sqrt[c*(a + b*x^2)^3
])/x + (315*a^4*Sqrt[b]*c*Sqrt[c*(a + b*x^2)^3]*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x
^2]])/(128*(a + b*x^2)^(3/2))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c \left (a + b x^{2}\right )^{3}\right )^{\frac{3}{2}}}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*(b*x**2+a)**3)**(3/2)/x**2,x)

[Out]

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

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Mathematica [A]  time = 0.124534, size = 122, normalized size = 0.58 \[ \frac{\left (c \left (a+b x^2\right )^3\right )^{3/2} \left (315 a^4 \sqrt{b} x \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )+\sqrt{a+b x^2} \left (-128 a^4+325 a^3 b x^2+210 a^2 b^2 x^4+88 a b^3 x^6+16 b^4 x^8\right )\right )}{128 x \left (a+b x^2\right )^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

((c*(a + b*x^2)^3)^(3/2)*(Sqrt[a + b*x^2]*(-128*a^4 + 325*a^3*b*x^2 + 210*a^2*b^
2*x^4 + 88*a*b^3*x^6 + 16*b^4*x^8) + 315*a^4*Sqrt[b]*x*Log[b*x + Sqrt[b]*Sqrt[a
+ b*x^2]]))/(128*x*(a + b*x^2)^(9/2))

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Maple [A]  time = 0.017, size = 215, normalized size = 1. \[{\frac{1}{128\,c \left ( b{x}^{2}+a \right ) ^{3}x} \left ( c \left ( b{x}^{2}+a \right ) ^{3} \right ) ^{{\frac{3}{2}}} \left ( 16\,{b}^{2}{x}^{4} \left ( bc{x}^{2}+ac \right ) ^{5/2}\sqrt{bc}+315\,{a}^{4}b{c}^{3}\ln \left ({\frac{bcx+\sqrt{bc{x}^{2}+ac}\sqrt{bc}}{\sqrt{bc}}} \right ) x+56\,ba{x}^{2} \left ( bc{x}^{2}+ac \right ) ^{5/2}\sqrt{bc}+210\,{a}^{2}b{x}^{2} \left ( bc{x}^{2}+ac \right ) ^{3/2}c\sqrt{bc}+315\,{a}^{3}b{c}^{2}{x}^{2}\sqrt{bc{x}^{2}+ac}\sqrt{bc}-128\,{a}^{2} \left ( bc{x}^{2}+ac \right ) ^{5/2}\sqrt{bc} \right ) \left ( c \left ( b{x}^{2}+a \right ) \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{bc}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/128*(c*(b*x^2+a)^3)^(3/2)*(16*b^2*x^4*(b*c*x^2+a*c)^(5/2)*(b*c)^(1/2)+315*a^4*
b*c^3*ln((b*c*x+(b*c*x^2+a*c)^(1/2)*(b*c)^(1/2))/(b*c)^(1/2))*x+56*b*a*x^2*(b*c*
x^2+a*c)^(5/2)*(b*c)^(1/2)+210*a^2*b*x^2*(b*c*x^2+a*c)^(3/2)*c*(b*c)^(1/2)+315*a
^3*b*c^2*x^2*(b*c*x^2+a*c)^(1/2)*(b*c)^(1/2)-128*a^2*(b*c*x^2+a*c)^(5/2)*(b*c)^(
1/2))/(b*x^2+a)^3/(c*(b*x^2+a))^(3/2)/c/x/(b*c)^(1/2)

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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(((b*x^2 + a)^3*c)^(3/2)/x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.344913, size = 1, normalized size = 0. \[ \left [\frac{315 \,{\left (a^{4} b c x^{3} + a^{5} c x\right )} \sqrt{b c} \log \left (-\frac{2 \, b^{2} c x^{4} + 3 \, a b c x^{2} + a^{2} c + 2 \, \sqrt{b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c} \sqrt{b c} x}{b x^{2} + a}\right ) + 2 \,{\left (16 \, b^{4} c x^{8} + 88 \, a b^{3} c x^{6} + 210 \, a^{2} b^{2} c x^{4} + 325 \, a^{3} b c x^{2} - 128 \, a^{4} c\right )} \sqrt{b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c}}{256 \,{\left (b x^{3} + a x\right )}}, \frac{315 \,{\left (a^{4} b c x^{3} + a^{5} c x\right )} \sqrt{-b c} \arctan \left (\frac{b^{2} c x^{3} + a b c x}{\sqrt{b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c} \sqrt{-b c}}\right ) +{\left (16 \, b^{4} c x^{8} + 88 \, a b^{3} c x^{6} + 210 \, a^{2} b^{2} c x^{4} + 325 \, a^{3} b c x^{2} - 128 \, a^{4} c\right )} \sqrt{b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c}}{128 \,{\left (b x^{3} + a x\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/256*(315*(a^4*b*c*x^3 + a^5*c*x)*sqrt(b*c)*log(-(2*b^2*c*x^4 + 3*a*b*c*x^2 +
a^2*c + 2*sqrt(b^3*c*x^6 + 3*a*b^2*c*x^4 + 3*a^2*b*c*x^2 + a^3*c)*sqrt(b*c)*x)/(
b*x^2 + a)) + 2*(16*b^4*c*x^8 + 88*a*b^3*c*x^6 + 210*a^2*b^2*c*x^4 + 325*a^3*b*c
*x^2 - 128*a^4*c)*sqrt(b^3*c*x^6 + 3*a*b^2*c*x^4 + 3*a^2*b*c*x^2 + a^3*c))/(b*x^
3 + a*x), 1/128*(315*(a^4*b*c*x^3 + a^5*c*x)*sqrt(-b*c)*arctan((b^2*c*x^3 + a*b*
c*x)/(sqrt(b^3*c*x^6 + 3*a*b^2*c*x^4 + 3*a^2*b*c*x^2 + a^3*c)*sqrt(-b*c))) + (16
*b^4*c*x^8 + 88*a*b^3*c*x^6 + 210*a^2*b^2*c*x^4 + 325*a^3*b*c*x^2 - 128*a^4*c)*s
qrt(b^3*c*x^6 + 3*a*b^2*c*x^4 + 3*a^2*b*c*x^2 + a^3*c))/(b*x^3 + a*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((c*(b*x**2+a)**3)**(3/2)/x**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.278366, size = 250, normalized size = 1.2 \[ \frac{1}{256} \,{\left (\frac{512 \, \sqrt{b c} a^{5} c{\rm sign}\left (b x^{2} + a\right )}{{\left (\sqrt{b c} x - \sqrt{b c x^{2} + a c}\right )}^{2} - a c} - 315 \, \sqrt{b c} a^{4}{\rm ln}\left ({\left (\sqrt{b c} x - \sqrt{b c x^{2} + a c}\right )}^{2}\right ){\rm sign}\left (b x^{2} + a\right ) + 2 \,{\left (325 \, a^{3} b{\rm sign}\left (b x^{2} + a\right ) + 2 \,{\left (105 \, a^{2} b^{2}{\rm sign}\left (b x^{2} + a\right ) + 4 \,{\left (2 \, b^{4} x^{2}{\rm sign}\left (b x^{2} + a\right ) + 11 \, a b^{3}{\rm sign}\left (b x^{2} + a\right )\right )} x^{2}\right )} x^{2}\right )} \sqrt{b c x^{2} + a c} x\right )} c \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/256*(512*sqrt(b*c)*a^5*c*sign(b*x^2 + a)/((sqrt(b*c)*x - sqrt(b*c*x^2 + a*c))^
2 - a*c) - 315*sqrt(b*c)*a^4*ln((sqrt(b*c)*x - sqrt(b*c*x^2 + a*c))^2)*sign(b*x^
2 + a) + 2*(325*a^3*b*sign(b*x^2 + a) + 2*(105*a^2*b^2*sign(b*x^2 + a) + 4*(2*b^
4*x^2*sign(b*x^2 + a) + 11*a*b^3*sign(b*x^2 + a))*x^2)*x^2)*sqrt(b*c*x^2 + a*c)*
x)*c