Optimal. Leaf size=72 \[ \frac{6 \sqrt [6]{a+b x} \sqrt [6]{c+d x} \, _2F_1\left (-\frac{1}{6},\frac{1}{6};\frac{7}{6};-\frac{d (a+b x)}{b c-a d}\right )}{b \sqrt [6]{\frac{b (c+d x)}{b c-a d}}} \]
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Rubi [A] time = 0.0193598, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {70, 69} \[ \frac{6 \sqrt [6]{a+b x} \sqrt [6]{c+d x} \, _2F_1\left (-\frac{1}{6},\frac{1}{6};\frac{7}{6};-\frac{d (a+b x)}{b c-a d}\right )}{b \sqrt [6]{\frac{b (c+d x)}{b c-a d}}} \]
Antiderivative was successfully verified.
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Rule 70
Rule 69
Rubi steps
\begin{align*} \int \frac{\sqrt [6]{c+d x}}{(a+b x)^{5/6}} \, dx &=\frac{\sqrt [6]{c+d x} \int \frac{\sqrt [6]{\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}}}{(a+b x)^{5/6}} \, dx}{\sqrt [6]{\frac{b (c+d x)}{b c-a d}}}\\ &=\frac{6 \sqrt [6]{a+b x} \sqrt [6]{c+d x} \, _2F_1\left (-\frac{1}{6},\frac{1}{6};\frac{7}{6};-\frac{d (a+b x)}{b c-a d}\right )}{b \sqrt [6]{\frac{b (c+d x)}{b c-a d}}}\\ \end{align*}
Mathematica [A] time = 0.0238663, size = 71, normalized size = 0.99 \[ \frac{6 \sqrt [6]{a+b x} \sqrt [6]{c+d x} \, _2F_1\left (-\frac{1}{6},\frac{1}{6};\frac{7}{6};\frac{d (a+b x)}{a d-b c}\right )}{b \sqrt [6]{\frac{b (c+d x)}{b c-a d}}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.016, size = 0, normalized size = 0. \begin{align*} \int{\sqrt [6]{dx+c} \left ( bx+a \right ) ^{-{\frac{5}{6}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x + c\right )}^{\frac{1}{6}}}{{\left (b x + a\right )}^{\frac{5}{6}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (d x + c\right )}^{\frac{1}{6}}}{{\left (b x + a\right )}^{\frac{5}{6}}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt [6]{c + d x}}{\left (a + b x\right )^{\frac{5}{6}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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