Optimal. Leaf size=213 \[ -\frac{6 b^{15/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{77 a^{13/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{12 b^3 \sqrt{a x+b \sqrt [3]{x}}}{77 a^3}-\frac{36 b^2 x^{2/3} \sqrt{a x+b \sqrt [3]{x}}}{385 a^2}+\frac{4 b x^{4/3} \sqrt{a x+b \sqrt [3]{x}}}{55 a}+\frac{2}{5} x^2 \sqrt{a x+b \sqrt [3]{x}} \]
[Out]
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Rubi [A] time = 0.552869, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353 \[ -\frac{6 b^{15/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{77 a^{13/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{12 b^3 \sqrt{a x+b \sqrt [3]{x}}}{77 a^3}-\frac{36 b^2 x^{2/3} \sqrt{a x+b \sqrt [3]{x}}}{385 a^2}+\frac{4 b x^{4/3} \sqrt{a x+b \sqrt [3]{x}}}{55 a}+\frac{2}{5} x^2 \sqrt{a x+b \sqrt [3]{x}} \]
Antiderivative was successfully verified.
[In] Int[x*Sqrt[b*x^(1/3) + a*x],x]
[Out]
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Rubi in Sympy [A] time = 47.4334, size = 204, normalized size = 0.96 \[ \frac{2 x^{2} \sqrt{a x + b \sqrt [3]{x}}}{5} + \frac{4 b x^{\frac{4}{3}} \sqrt{a x + b \sqrt [3]{x}}}{55 a} - \frac{36 b^{2} x^{\frac{2}{3}} \sqrt{a x + b \sqrt [3]{x}}}{385 a^{2}} + \frac{12 b^{3} \sqrt{a x + b \sqrt [3]{x}}}{77 a^{3}} - \frac{6 b^{\frac{15}{4}} \sqrt{\frac{a x^{\frac{2}{3}} + b}{\left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right )^{2}}} \left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right ) \sqrt{a x + b \sqrt [3]{x}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{77 a^{\frac{13}{4}} \sqrt [6]{x} \left (a x^{\frac{2}{3}} + b\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(b*x**(1/3)+a*x)**(1/2),x)
[Out]
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Mathematica [C] time = 0.0839891, size = 118, normalized size = 0.55 \[ \frac{2 \sqrt [3]{x} \left (77 a^4 x^{8/3}+91 a^3 b x^2-4 a^2 b^2 x^{4/3}+30 b^4 \sqrt{\frac{b}{a x^{2/3}}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b}{a x^{2/3}}\right )+12 a b^3 x^{2/3}+30 b^4\right )}{385 a^3 \sqrt{a x+b \sqrt [3]{x}}} \]
Antiderivative was successfully verified.
[In] Integrate[x*Sqrt[b*x^(1/3) + a*x],x]
[Out]
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Maple [A] time = 0.023, size = 198, normalized size = 0.9 \[{\frac{2\,{x}^{2}}{5}\sqrt{b\sqrt [3]{x}+ax}}+{\frac{4\,b}{55\,a}{x}^{{\frac{4}{3}}}\sqrt{b\sqrt [3]{x}+ax}}-{\frac{36\,{b}^{2}}{385\,{a}^{2}}{x}^{{\frac{2}{3}}}\sqrt{b\sqrt [3]{x}+ax}}+{\frac{12\,{b}^{3}}{77\,{a}^{3}}\sqrt{b\sqrt [3]{x}+ax}}-{\frac{6\,{b}^{4}}{77\,{a}^{4}}\sqrt{-ab}\sqrt{{a \left ( \sqrt [3]{x}+{\frac{1}{a}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-2\,{\frac{a}{\sqrt{-ab}} \left ( \sqrt [3]{x}-{\frac{\sqrt{-ab}}{a}} \right ) }}\sqrt{-{a\sqrt [3]{x}{\frac{1}{\sqrt{-ab}}}}}{\it EllipticF} \left ( \sqrt{{a \left ( \sqrt [3]{x}+{\frac{1}{a}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ){\frac{1}{\sqrt{b\sqrt [3]{x}+ax}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(b*x^(1/3)+a*x)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{a x + b x^{\frac{1}{3}}} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a*x + b*x^(1/3))*x,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\sqrt{a x + b x^{\frac{1}{3}}} x, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a*x + b*x^(1/3))*x,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int x \sqrt{a x + b \sqrt [3]{x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(b*x**(1/3)+a*x)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{a x + b x^{\frac{1}{3}}} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a*x + b*x^(1/3))*x,x, algorithm="giac")
[Out]