Optimal. Leaf size=74 \[ \frac {6 (a+b x)^{7/6} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/6} \, _2F_1\left (\frac {5}{6},\frac {7}{6};\frac {13}{6};-\frac {d (a+b x)}{b c-a d}\right )}{7 b (c+d x)^{5/6}} \]
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Rubi [A]
time = 0.01, antiderivative size = 74, 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)^{7/6} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/6} \, _2F_1\left (\frac {5}{6},\frac {7}{6};\frac {13}{6};-\frac {d (a+b x)}{b c-a d}\right )}{7 b (c+d x)^{5/6}} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 72
Rubi steps
\begin {align*} \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{5/6}} \, dx &=\frac {\left (\frac {b (c+d x)}{b c-a d}\right )^{5/6} \int \frac {\sqrt [6]{a+b x}}{\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 (a+b x)^{7/6} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/6} \, _2F_1\left (\frac {5}{6},\frac {7}{6};\frac {13}{6};-\frac {d (a+b x)}{b c-a d}\right )}{7 b (c+d x)^{5/6}}\\ \end {align*}
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Mathematica [A]
time = 10.04, size = 73, normalized size = 0.99 \begin {gather*} \frac {6 (a+b x)^{7/6} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/6} \, _2F_1\left (\frac {5}{6},\frac {7}{6};\frac {13}{6};\frac {d (a+b x)}{-b c+a d}\right )}{7 b (c+d x)^{5/6}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{\frac {1}{6}}}{\left (d x +c \right )^{\frac {5}{6}}}\, dx\]
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [6]{a + b x}}{\left (c + d x\right )^{\frac {5}{6}}}\, dx \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 \frac {{\left (a+b\,x\right )}^{1/6}}{{\left (c+d\,x\right )}^{5/6}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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