Optimal. Leaf size=72 \[ \frac{6 \sqrt [6]{a+b x} \left (\frac{b (c+d x)}{b c-a d}\right )^{5/6} \, _2F_1\left (\frac{1}{6},\frac{5}{6};\frac{7}{6};-\frac{d (a+b x)}{b c-a d}\right )}{b (c+d x)^{5/6}} \]
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Rubi [A] time = 0.0199112, 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} \[ \frac{6 \sqrt [6]{a+b x} \left (\frac{b (c+d x)}{b c-a d}\right )^{5/6} \, _2F_1\left (\frac{1}{6},\frac{5}{6};\frac{7}{6};-\frac{d (a+b x)}{b c-a d}\right )}{b (c+d x)^{5/6}} \]
Antiderivative was successfully verified.
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Rule 70
Rule 69
Rubi steps
\begin{align*} \int \frac{1}{(a+b x)^{5/6} (c+d x)^{5/6}} \, dx &=\frac{\left (\frac{b (c+d x)}{b c-a d}\right )^{5/6} \int \frac{1}{(a+b x)^{5/6} \left (\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}\right )^{5/6}} \, dx}{(c+d x)^{5/6}}\\ &=\frac{6 \sqrt [6]{a+b x} \left (\frac{b (c+d x)}{b c-a d}\right )^{5/6} \, _2F_1\left (\frac{1}{6},\frac{5}{6};\frac{7}{6};-\frac{d (a+b x)}{b c-a d}\right )}{b (c+d x)^{5/6}}\\ \end{align*}
Mathematica [A] time = 0.0229782, size = 71, normalized size = 0.99 \[ \frac{6 \sqrt [6]{a+b x} \left (\frac{b (c+d x)}{b c-a d}\right )^{5/6} \, _2F_1\left (\frac{1}{6},\frac{5}{6};\frac{7}{6};\frac{d (a+b x)}{a d-b c}\right )}{b (c+d x)^{5/6}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.034, size = 0, normalized size = 0. \begin{align*} \int{ \left ( bx+a \right ) ^{-{\frac{5}{6}}} \left ( dx+c \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{1}{{\left (b x + a\right )}^{\frac{5}{6}}{\left (d x + c\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 (b x + a\right )}^{\frac{1}{6}}{\left (d x + c\right )}^{\frac{1}{6}}}{b d x^{2} + a c +{\left (b c + a 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}{\left (a + b x\right )^{\frac{5}{6}} \left (c + d 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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