Optimal. Leaf size=115 \[ \frac{3 a^4 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{128 b^{5/2}}-\frac{3 a^3 x \sqrt{a+b x^2}}{128 b^2}+\frac{a^2 x^3 \sqrt{a+b x^2}}{64 b}+\frac{1}{8} x^5 \left (a+b x^2\right )^{3/2}+\frac{1}{16} a x^5 \sqrt{a+b x^2} \]
[Out]
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Rubi [A] time = 0.136401, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{3 a^4 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{128 b^{5/2}}-\frac{3 a^3 x \sqrt{a+b x^2}}{128 b^2}+\frac{a^2 x^3 \sqrt{a+b x^2}}{64 b}+\frac{1}{8} x^5 \left (a+b x^2\right )^{3/2}+\frac{1}{16} a x^5 \sqrt{a+b x^2} \]
Antiderivative was successfully verified.
[In] Int[x^4*(a + b*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 16.9259, size = 104, normalized size = 0.9 \[ \frac{3 a^{4} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{128 b^{\frac{5}{2}}} - \frac{3 a^{3} x \sqrt{a + b x^{2}}}{128 b^{2}} + \frac{a^{2} x^{3} \sqrt{a + b x^{2}}}{64 b} + \frac{a x^{5} \sqrt{a + b x^{2}}}{16} + \frac{x^{5} \left (a + b x^{2}\right )^{\frac{3}{2}}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4*(b*x**2+a)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0731491, size = 88, normalized size = 0.77 \[ \frac{3 a^4 \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{128 b^{5/2}}+\sqrt{a+b x^2} \left (-\frac{3 a^3 x}{128 b^2}+\frac{a^2 x^3}{64 b}+\frac{3 a x^5}{16}+\frac{b x^7}{8}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x^4*(a + b*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.01, size = 95, normalized size = 0.8 \[{\frac{{x}^{3}}{8\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{ax}{16\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{{a}^{2}x}{64\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{a}^{3}x}{128\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{3\,{a}^{4}}{128}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4*(b*x^2+a)^(3/2),x)
[Out]
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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)^(3/2)*x^4,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.265435, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a^{4} \log \left (-2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right ) + 2 \,{\left (16 \, b^{3} x^{7} + 24 \, a b^{2} x^{5} + 2 \, a^{2} b x^{3} - 3 \, a^{3} x\right )} \sqrt{b x^{2} + a} \sqrt{b}}{256 \, b^{\frac{5}{2}}}, \frac{3 \, a^{4} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) +{\left (16 \, b^{3} x^{7} + 24 \, a b^{2} x^{5} + 2 \, a^{2} b x^{3} - 3 \, a^{3} x\right )} \sqrt{b x^{2} + a} \sqrt{-b}}{128 \, \sqrt{-b} b^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(3/2)*x^4,x, algorithm="fricas")
[Out]
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Sympy [A] time = 26.197, size = 148, normalized size = 1.29 \[ - \frac{3 a^{\frac{7}{2}} x}{128 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{a^{\frac{5}{2}} x^{3}}{128 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{13 a^{\frac{3}{2}} x^{5}}{64 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{5 \sqrt{a} b x^{7}}{16 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 a^{4} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{128 b^{\frac{5}{2}}} + \frac{b^{2} x^{9}}{8 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4*(b*x**2+a)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.215324, size = 103, normalized size = 0.9 \[ \frac{1}{128} \,{\left (2 \,{\left (4 \,{\left (2 \, b x^{2} + 3 \, a\right )} x^{2} + \frac{a^{2}}{b}\right )} x^{2} - \frac{3 \, a^{3}}{b^{2}}\right )} \sqrt{b x^{2} + a} x - \frac{3 \, a^{4}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{128 \, b^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(3/2)*x^4,x, algorithm="giac")
[Out]