Optimal. Leaf size=81 \[ \frac{6 (a+b x)^{5/6} \sqrt [6]{\frac{b (c+d x)}{b c-a d}} \, _2F_1\left (\frac{5}{6},\frac{7}{6};\frac{11}{6};-\frac{d (a+b x)}{b c-a d}\right )}{5 \sqrt [6]{c+d x} (b c-a d)} \]
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Rubi [A] time = 0.0204251, antiderivative size = 81, 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} \[ \frac{6 (a+b x)^{5/6} \sqrt [6]{\frac{b (c+d x)}{b c-a d}} \, _2F_1\left (\frac{5}{6},\frac{7}{6};\frac{11}{6};-\frac{d (a+b x)}{b c-a d}\right )}{5 \sqrt [6]{c+d x} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 70
Rule 69
Rubi steps
\begin{align*} \int \frac{1}{\sqrt [6]{a+b x} (c+d x)^{7/6}} \, dx &=\frac{\left (b \sqrt [6]{\frac{b (c+d x)}{b c-a d}}\right ) \int \frac{1}{\sqrt [6]{a+b x} \left (\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}\right )^{7/6}} \, dx}{(b c-a d) \sqrt [6]{c+d x}}\\ &=\frac{6 (a+b x)^{5/6} \sqrt [6]{\frac{b (c+d x)}{b c-a d}} \, _2F_1\left (\frac{5}{6},\frac{7}{6};\frac{11}{6};-\frac{d (a+b x)}{b c-a d}\right )}{5 (b c-a d) \sqrt [6]{c+d x}}\\ \end{align*}
Mathematica [A] time = 0.0337345, size = 73, normalized size = 0.9 \[ \frac{6 (a+b x)^{5/6} \left (\frac{b (c+d x)}{b c-a d}\right )^{7/6} \, _2F_1\left (\frac{5}{6},\frac{7}{6};\frac{11}{6};\frac{d (a+b x)}{a d-b c}\right )}{5 b (c+d x)^{7/6}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.036, size = 0, normalized size = 0. \begin{align*} \int{{\frac{1}{\sqrt [6]{bx+a}}} \left ( dx+c \right ) ^{-{\frac{7}{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{1}{{\left (b x + a\right )}^{\frac{1}{6}}{\left (d x + c\right )}^{\frac{7}{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 (b x + a\right )}^{\frac{5}{6}}{\left (d x + c\right )}^{\frac{5}{6}}}{b d^{2} x^{3} + a c^{2} +{\left (2 \, b c d + a d^{2}\right )} x^{2} +{\left (b c^{2} + 2 \, a c d\right )} x}, 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{1}{\sqrt [6]{a + b x} \left (c + d x\right )^{\frac{7}{6}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x + a\right )}^{\frac{1}{6}}{\left (d x + c\right )}^{\frac{7}{6}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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