Optimal. Leaf size=82 \[ \frac {6 (b c-a d) (a+b x)^{13/6} \sqrt [6]{c+d x} \, _2F_1\left (-\frac {7}{6},\frac {13}{6};\frac {19}{6};-\frac {d (a+b x)}{b c-a d}\right )}{13 b^2 \sqrt [6]{\frac {b (c+d x)}{b c-a d}}} \]
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Rubi [A]
time = 0.02, antiderivative size = 82, normalized size of antiderivative = 1.00, 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 = {72, 71}
\begin {gather*} \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}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 72
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*}
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Mathematica [A]
time = 10.05, size = 73, normalized size = 0.89 \begin {gather*} \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)}{-b c+a d}\right )}{13 b \left (\frac {b (c+d x)}{b c-a d}\right )^{7/6}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \left (b x +a \right )^{\frac {7}{6}} \left (d x +c \right )^{\frac {7}{6}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (a+b\,x\right )}^{7/6}\,{\left (c+d\,x\right )}^{7/6} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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