Optimal. Leaf size=350 \[ \frac{4 a^{9/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 b^{3/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{8 a^{9/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 b^{3/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{8 a^{5/2} \sqrt [3]{x} \left (a x^{2/3}+b\right )}{5 b \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{a x+b \sqrt [3]{x}}}-\frac{8 a^2 \sqrt{a x+b \sqrt [3]{x}}}{5 b \sqrt [3]{x}}-\frac{2 \left (a x+b \sqrt [3]{x}\right )^{3/2}}{3 x^2}-\frac{4 a \sqrt{a x+b \sqrt [3]{x}}}{5 x} \]
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Rubi [A] time = 0.796356, antiderivative size = 350, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421 \[ \frac{4 a^{9/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 b^{3/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{8 a^{9/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 b^{3/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{8 a^{5/2} \sqrt [3]{x} \left (a x^{2/3}+b\right )}{5 b \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{a x+b \sqrt [3]{x}}}-\frac{8 a^2 \sqrt{a x+b \sqrt [3]{x}}}{5 b \sqrt [3]{x}}-\frac{2 \left (a x+b \sqrt [3]{x}\right )^{3/2}}{3 x^2}-\frac{4 a \sqrt{a x+b \sqrt [3]{x}}}{5 x} \]
Antiderivative was successfully verified.
[In] Int[(b*x^(1/3) + a*x)^(3/2)/x^3,x]
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Rubi in Sympy [A] time = 72.9508, size = 320, normalized size = 0.91 \[ - \frac{8 a^{\frac{9}{4}} \sqrt{\frac{a x^{\frac{2}{3}} + b}{\left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right )^{2}}} \left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right ) \sqrt{a x + b \sqrt [3]{x}} E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{5 b^{\frac{3}{4}} \sqrt [6]{x} \left (a x^{\frac{2}{3}} + b\right )} + \frac{4 a^{\frac{9}{4}} \sqrt{\frac{a x^{\frac{2}{3}} + b}{\left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right )^{2}}} \left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right ) \sqrt{a x + b \sqrt [3]{x}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{5 b^{\frac{3}{4}} \sqrt [6]{x} \left (a x^{\frac{2}{3}} + b\right )} + \frac{8 a^{\frac{5}{2}} \sqrt{a x + b \sqrt [3]{x}}}{5 b \left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right )} - \frac{8 a^{2} \sqrt{a x + b \sqrt [3]{x}}}{5 b \sqrt [3]{x}} - \frac{4 a \sqrt{a x + b \sqrt [3]{x}}}{5 x} - \frac{2 \left (a x + b \sqrt [3]{x}\right )^{\frac{3}{2}}}{3 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**(1/3)+a*x)**(3/2)/x**3,x)
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Mathematica [C] time = 0.0869826, size = 108, normalized size = 0.31 \[ -\frac{2 \left (-12 a^3 x^2 \sqrt{\frac{b}{a x^{2/3}}+1} \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{4};-\frac{b}{a x^{2/3}}\right )+12 a^3 x^2+23 a^2 b x^{4/3}+16 a b^2 x^{2/3}+5 b^3\right )}{15 b x^{4/3} \sqrt{a x+b \sqrt [3]{x}}} \]
Antiderivative was successfully verified.
[In] Integrate[(b*x^(1/3) + a*x)^(3/2)/x^3,x]
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Maple [A] time = 0.038, size = 339, normalized size = 1. \[{\frac{2}{15\,b{x}^{3}} \left ( 12\,{a}^{2}b\sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-2\,{\frac{a\sqrt [3]{x}-\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{a\sqrt [3]{x}}{\sqrt{-ab}}}}{x}^{8/3}\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{\it EllipticE} \left ( \sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) -6\,{a}^{2}b\sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-2\,{\frac{a\sqrt [3]{x}-\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{a\sqrt [3]{x}}{\sqrt{-ab}}}}{x}^{8/3}\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{\it EllipticF} \left ( \sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) -12\,\sqrt{b\sqrt [3]{x}+ax}{x}^{10/3}{a}^{3}-12\,\sqrt{b\sqrt [3]{x}+ax}{x}^{8/3}{a}^{2}b-16\,\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{x}^{2}a{b}^{2}-11\,\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{x}^{8/3}{a}^{2}b-5\,\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{x}^{4/3}{b}^{3} \right ) \left ( b+a{x}^{{\frac{2}{3}}} \right ) ^{-1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^(1/3)+a*x)^(3/2)/x^3,x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (a x + b x^{\frac{1}{3}}\right )}^{\frac{3}{2}}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a*x + b*x^(1/3))^(3/2)/x^3,x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (a x + b x^{\frac{1}{3}}\right )}^{\frac{3}{2}}}{x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a*x + b*x^(1/3))^(3/2)/x^3,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((b*x**(1/3)+a*x)**(3/2)/x**3,x)
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a*x + b*x^(1/3))^(3/2)/x^3,x, algorithm="giac")
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