Optimal. Leaf size=82 \[ \frac{6 (a+b x)^{13/6} \sqrt [6]{c+d x} (b c-a d) \, _2F_1\left (-\frac{7}{6},\frac{13}{6};\frac{19}{6};-\frac{d (a+b x)}{b c-a d}\right )}{13 b^2 \sqrt [6]{\frac{b (c+d x)}{b c-a d}}} \]
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Rubi [A] time = 0.0222514, antiderivative size = 82, 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} \[ \frac{6 (a+b x)^{13/6} \sqrt [6]{c+d x} (b c-a d) \, _2F_1\left (-\frac{7}{6},\frac{13}{6};\frac{19}{6};-\frac{d (a+b x)}{b c-a d}\right )}{13 b^2 \sqrt [6]{\frac{b (c+d x)}{b c-a d}}} \]
Antiderivative was successfully verified.
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Rule 70
Rule 69
Rubi steps
\begin{align*} \int (a+b x)^{7/6} (c+d x)^{7/6} \, dx &=\frac{\left ((b c-a d) \sqrt [6]{c+d x}\right ) \int (a+b x)^{7/6} \left (\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}\right )^{7/6} \, dx}{b \sqrt [6]{\frac{b (c+d x)}{b c-a d}}}\\ &=\frac{6 (b c-a d) (a+b x)^{13/6} \sqrt [6]{c+d x} \, _2F_1\left (-\frac{7}{6},\frac{13}{6};\frac{19}{6};-\frac{d (a+b x)}{b c-a d}\right )}{13 b^2 \sqrt [6]{\frac{b (c+d x)}{b c-a d}}}\\ \end{align*}
Mathematica [A] time = 0.0566748, size = 73, normalized size = 0.89 \[ \frac{6 (a+b x)^{13/6} (c+d x)^{7/6} \, _2F_1\left (-\frac{7}{6},\frac{13}{6};\frac{19}{6};\frac{d (a+b x)}{a d-b c}\right )}{13 b \left (\frac{b (c+d x)}{b c-a d}\right )^{7/6}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.018, size = 0, normalized size = 0. \begin{align*} \int \left ( bx+a \right ) ^{{\frac{7}{6}}} \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{\left (b x + a\right )}^{\frac{7}{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 ({\left (b d x^{2} + a c +{\left (b c + a d\right )} x\right )}{\left (b x + a\right )}^{\frac{1}{6}}{\left (d x + c\right )}^{\frac{1}{6}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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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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