Optimal. Leaf size=82 \[ \frac {6 (a+b x)^{7/6} \sqrt [6]{c+d x} (b c-a d) \, _2F_1\left (-\frac {7}{6},\frac {7}{6};\frac {13}{6};-\frac {d (a+b x)}{b c-a d}\right )}{7 b^2 \sqrt [6]{\frac {b (c+d x)}{b c-a d}}} \]
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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 = {70, 69} \[ \frac {6 (a+b x)^{7/6} \sqrt [6]{c+d x} (b c-a d) \, _2F_1\left (-\frac {7}{6},\frac {7}{6};\frac {13}{6};-\frac {d (a+b x)}{b c-a d}\right )}{7 b^2 \sqrt [6]{\frac {b (c+d x)}{b c-a d}}} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rubi steps
\begin {align*} \int \sqrt [6]{a+b x} (c+d x)^{7/6} \, dx &=\frac {\left ((b c-a d) \sqrt [6]{c+d x}\right ) \int \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 \sqrt [6]{\frac {b (c+d x)}{b c-a d}}}\\ &=\frac {6 (b c-a d) (a+b x)^{7/6} \sqrt [6]{c+d x} \, _2F_1\left (-\frac {7}{6},\frac {7}{6};\frac {13}{6};-\frac {d (a+b x)}{b c-a d}\right )}{7 b^2 \sqrt [6]{\frac {b (c+d x)}{b c-a d}}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 73, normalized size = 0.89 \[ \frac {6 (a+b x)^{7/6} (c+d x)^{7/6} \, _2F_1\left (-\frac {7}{6},\frac {7}{6};\frac {13}{6};\frac {d (a+b x)}{a d-b c}\right )}{7 b \left (\frac {b (c+d x)}{b c-a d}\right )^{7/6}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {7}{6}}, x\right ) \]
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 \[ \int {\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {7}{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \left (b x +a \right )^{\frac {1}{6}} \left (d x +c \right )^{\frac {7}{6}}\, dx \]
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 \[ \int {\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {7}{6}}\,{d x} \]
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 \[ \int {\left (a+b\,x\right )}^{1/6}\,{\left (c+d\,x\right )}^{7/6} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt [6]{a + b x} \left (c + d x\right )^{\frac {7}{6}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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