Optimal. Leaf size=296 \[ -\frac{3 \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{2 a^{3/4} b^{3/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{3 \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{a^{3/4} b^{3/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{3 \sqrt [3]{x} \left (a x^{2/3}+b\right )}{\sqrt{a} b \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{a x+b \sqrt [3]{x}}}+\frac{3 x^{2/3}}{b \sqrt{a x+b \sqrt [3]{x}}} \]
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Rubi [A] time = 0.53324, antiderivative size = 296, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.467 \[ -\frac{3 \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{2 a^{3/4} b^{3/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{3 \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{a^{3/4} b^{3/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{3 \sqrt [3]{x} \left (a x^{2/3}+b\right )}{\sqrt{a} b \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{a x+b \sqrt [3]{x}}}+\frac{3 x^{2/3}}{b \sqrt{a x+b \sqrt [3]{x}}} \]
Antiderivative was successfully verified.
[In] Int[(b*x^(1/3) + a*x)^(-3/2),x]
[Out]
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Rubi in Sympy [A] time = 46.2122, size = 270, normalized size = 0.91 \[ \frac{3 x^{\frac{2}{3}}}{b \sqrt{a x + b \sqrt [3]{x}}} - \frac{3 \sqrt{a x + b \sqrt [3]{x}}}{\sqrt{a} b \left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right )} + \frac{3 \sqrt{\frac{a x^{\frac{2}{3}} + b}{\left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right )^{2}}} \left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right ) \sqrt{a x + b \sqrt [3]{x}} E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{a^{\frac{3}{4}} b^{\frac{3}{4}} \sqrt [6]{x} \left (a x^{\frac{2}{3}} + b\right )} - \frac{3 \sqrt{\frac{a x^{\frac{2}{3}} + b}{\left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right )^{2}}} \left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right ) \sqrt{a x + b \sqrt [3]{x}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{2 a^{\frac{3}{4}} b^{\frac{3}{4}} \sqrt [6]{x} \left (a x^{\frac{2}{3}} + b\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x**(1/3)+a*x)**(3/2),x)
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Mathematica [C] time = 0.055254, size = 65, normalized size = 0.22 \[ -\frac{3 x^{2/3} \left (\sqrt{\frac{b}{a x^{2/3}}+1} \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{4};-\frac{b}{a x^{2/3}}\right )-1\right )}{b \sqrt{a x+b \sqrt [3]{x}}} \]
Antiderivative was successfully verified.
[In] Integrate[(b*x^(1/3) + a*x)^(-3/2),x]
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Maple [A] time = 0.007, size = 243, normalized size = 0.8 \[ -{\frac{3}{2\,ab} \left ( 2\,\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }\sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-2\,{\frac{a\sqrt [3]{x}-\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{a\sqrt [3]{x}}{\sqrt{-ab}}}}{\it EllipticE} \left ( \sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) b-\sqrt{\sqrt [3]{x} \left ( b+a{x}^{{\frac{2}{3}}} \right ) }\sqrt{{1 \left ( a\sqrt [3]{x}+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-2\,{\frac{a\sqrt [3]{x}-\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{a\sqrt [3]{x}{\frac{1}{\sqrt{-ab}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( a\sqrt [3]{x}+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ) b-2\,{x}^{2/3}\sqrt{b\sqrt [3]{x}+ax}a \right ){\frac{1}{\sqrt [3]{x}}} \left ( b+a{x}^{{\frac{2}{3}}} \right ) ^{-1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x^(1/3)+a*x)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a x + b x^{\frac{1}{3}}\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a*x + b*x^(1/3))^(-3/2),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (a x + b x^{\frac{1}{3}}\right )}^{\frac{3}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a*x + b*x^(1/3))^(-3/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a x + b \sqrt [3]{x}\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x**(1/3)+a*x)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a x + b x^{\frac{1}{3}}\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a*x + b*x^(1/3))^(-3/2),x, algorithm="giac")
[Out]