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