Optimal. Leaf size=274 \[ \frac{12 a^{5/4} \sqrt [4]{b} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 \sqrt{a x+b x^3}}-\frac{24 a^{5/4} \sqrt [4]{b} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 \sqrt{a x+b x^3}}+\frac{12}{5} b x \sqrt{a x+b x^3}-\frac{2 \left (a x+b x^3\right )^{3/2}}{x^2}+\frac{24 a \sqrt{b} x \left (a+b x^2\right )}{5 \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{a x+b x^3}} \]
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Rubi [A] time = 0.45754, antiderivative size = 274, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412 \[ \frac{12 a^{5/4} \sqrt [4]{b} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 \sqrt{a x+b x^3}}-\frac{24 a^{5/4} \sqrt [4]{b} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 \sqrt{a x+b x^3}}+\frac{12}{5} b x \sqrt{a x+b x^3}-\frac{2 \left (a x+b x^3\right )^{3/2}}{x^2}+\frac{24 a \sqrt{b} x \left (a+b x^2\right )}{5 \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{a x+b x^3}} \]
Antiderivative was successfully verified.
[In] Int[(a*x + b*x^3)^(3/2)/x^3,x]
[Out]
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Rubi in Sympy [A] time = 45.1036, size = 260, normalized size = 0.95 \[ - \frac{24 a^{\frac{5}{4}} \sqrt [4]{b} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \sqrt{a x + b x^{3}} E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{5 \sqrt{x} \left (a + b x^{2}\right )} + \frac{12 a^{\frac{5}{4}} \sqrt [4]{b} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \sqrt{a x + b x^{3}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{5 \sqrt{x} \left (a + b x^{2}\right )} + \frac{24 a \sqrt{b} \sqrt{a x + b x^{3}}}{5 \left (\sqrt{a} + \sqrt{b} x\right )} + \frac{12 b x \sqrt{a x + b x^{3}}}{5} - \frac{2 \left (a x + b x^{3}\right )^{\frac{3}{2}}}{x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**3+a*x)**(3/2)/x**3,x)
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Mathematica [C] time = 0.33325, size = 183, normalized size = 0.67 \[ \frac{2 \left (-12 a^{3/2} \sqrt{b} x \sqrt{\frac{b x^2}{a}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}}\right )\right |-1\right )+12 a^{3/2} \sqrt{b} x \sqrt{\frac{b x^2}{a}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}}\right )\right |-1\right )+\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}} \left (-5 a^2-4 a b x^2+b^2 x^4\right )\right )}{5 \sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}} \sqrt{x \left (a+b x^2\right )}} \]
Antiderivative was successfully verified.
[In] Integrate[(a*x + b*x^3)^(3/2)/x^3,x]
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Maple [A] time = 0.023, size = 194, normalized size = 0.7 \[ -2\,{\frac{ \left ( b{x}^{2}+a \right ) a}{\sqrt{x \left ( b{x}^{2}+a \right ) }}}+{\frac{2\,bx}{5}\sqrt{b{x}^{3}+ax}}+{\frac{12\,a}{5}\sqrt{-ab}\sqrt{{b \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-2\,{\frac{b}{\sqrt{-ab}} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }}\sqrt{-{bx{\frac{1}{\sqrt{-ab}}}}} \left ( -2\,{\frac{\sqrt{-ab}}{b}{\it EllipticE} \left ( \sqrt{{\frac{b}{\sqrt{-ab}} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }},1/2\,\sqrt{2} \right ) }+{\frac{1}{b}\sqrt{-ab}{\it EllipticF} \left ( \sqrt{{b \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ) } \right ){\frac{1}{\sqrt{b{x}^{3}+ax}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^3+a*x)^(3/2)/x^3,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{3} + a x\right )}^{\frac{3}{2}}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^3 + a*x)^(3/2)/x^3,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{b x^{3} + a x}{\left (b x^{2} + a\right )}}{x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^3 + a*x)^(3/2)/x^3,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x \left (a + b x^{2}\right )\right )^{\frac{3}{2}}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**3+a*x)**(3/2)/x**3,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{3} + a x\right )}^{\frac{3}{2}}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^3 + a*x)^(3/2)/x^3,x, algorithm="giac")
[Out]