Optimal. Leaf size=72 \[ -\frac{6 (c+d x)^{5/6} \, _2F_1\left (-\frac{5}{6},-\frac{1}{6};\frac{5}{6};-\frac{d (a+b x)}{b c-a d}\right )}{b \sqrt [6]{a+b x} \left (\frac{b (c+d x)}{b c-a d}\right )^{5/6}} \]
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Rubi [A] time = 0.020786, 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} \[ -\frac{6 (c+d x)^{5/6} \, _2F_1\left (-\frac{5}{6},-\frac{1}{6};\frac{5}{6};-\frac{d (a+b x)}{b c-a d}\right )}{b \sqrt [6]{a+b x} \left (\frac{b (c+d x)}{b c-a d}\right )^{5/6}} \]
Antiderivative was successfully verified.
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Rule 70
Rule 69
Rubi steps
\begin{align*} \int \frac{(c+d x)^{5/6}}{(a+b x)^{7/6}} \, dx &=\frac{(c+d x)^{5/6} \int \frac{\left (\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}\right )^{5/6}}{(a+b x)^{7/6}} \, dx}{\left (\frac{b (c+d x)}{b c-a d}\right )^{5/6}}\\ &=-\frac{6 (c+d x)^{5/6} \, _2F_1\left (-\frac{5}{6},-\frac{1}{6};\frac{5}{6};-\frac{d (a+b x)}{b c-a d}\right )}{b \sqrt [6]{a+b x} \left (\frac{b (c+d x)}{b c-a d}\right )^{5/6}}\\ \end{align*}
Mathematica [A] time = 0.0221944, size = 71, normalized size = 0.99 \[ -\frac{6 (c+d x)^{5/6} \, _2F_1\left (-\frac{5}{6},-\frac{1}{6};\frac{5}{6};\frac{d (a+b x)}{a d-b c}\right )}{b \sqrt [6]{a+b x} \left (\frac{b (c+d x)}{b c-a d}\right )^{5/6}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.031, size = 0, normalized size = 0. \begin{align*} \int{ \left ( dx+c \right ) ^{{\frac{5}{6}}} \left ( bx+a \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{{\left (d x + c\right )}^{\frac{5}{6}}}{{\left (b x + a\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^{2} x^{2} + 2 \, a b x + a^{2}}, 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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