Optimal. Leaf size=60 \[ \frac{6 \sqrt [3]{x}}{b \sqrt{a x+b x^{2/3}}}-\frac{6 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )}{b^{3/2}} \]
[Out]
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Rubi [A] time = 0.103153, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{6 \sqrt [3]{x}}{b \sqrt{a x+b x^{2/3}}}-\frac{6 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )}{b^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(b*x^(2/3) + a*x)^(-3/2),x]
[Out]
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Rubi in Sympy [A] time = 8.95854, size = 53, normalized size = 0.88 \[ \frac{6 \sqrt [3]{x}}{b \sqrt{a x + b x^{\frac{2}{3}}}} - \frac{6 \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x + b x^{\frac{2}{3}}}} \right )}}{b^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x**(2/3)+a*x)**(3/2),x)
[Out]
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Mathematica [A] time = 0.107749, size = 71, normalized size = 1.18 \[ \frac{6 \sqrt{a x+b x^{2/3}}}{b \sqrt [3]{x} \left (a \sqrt [3]{x}+b\right )}-\frac{6 \tanh ^{-1}\left (\frac{\sqrt{a x+b x^{2/3}}}{\sqrt{b} \sqrt [3]{x}}\right )}{b^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(b*x^(2/3) + a*x)^(-3/2),x]
[Out]
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Maple [A] time = 0.005, size = 55, normalized size = 0.9 \[ 6\,{\frac{x \left ( b+a\sqrt [3]{x} \right ) }{ \left ( b{x}^{2/3}+ax \right ) ^{3/2}{b}^{5/2}} \left ({b}^{3/2}-{\it Artanh} \left ({\frac{\sqrt{b+a\sqrt [3]{x}}}{\sqrt{b}}} \right ) b\sqrt{b+a\sqrt [3]{x}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x^(2/3)+a*x)^(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((a*x + b*x^(2/3))^(-3/2),x, algorithm="maxima")
[Out]
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a*x + b*x^(2/3))^(-3/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a x + b x^{\frac{2}{3}}\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x**(2/3)+a*x)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.226018, size = 117, normalized size = 1.95 \[ -\frac{6 \,{\left (\sqrt{b} \arctan \left (\frac{\sqrt{b}}{\sqrt{-b}}\right ) + \sqrt{-b}\right )}{\rm sign}\left (x^{\frac{1}{3}}\right )}{\sqrt{-b} b^{\frac{3}{2}}} + \frac{6 \, \arctan \left (\frac{\sqrt{a x^{\frac{1}{3}} + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b{\rm sign}\left (x^{\frac{1}{3}}\right )} + \frac{6}{\sqrt{a x^{\frac{1}{3}} + b} b{\rm sign}\left (x^{\frac{1}{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a*x + b*x^(2/3))^(-3/2),x, algorithm="giac")
[Out]