Optimal. Leaf size=79 \[ \frac {6 \sqrt [6]{a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/6} \, _2F_1\left (\frac {1}{6},\frac {11}{6};\frac {7}{6};-\frac {d (a+b x)}{b c-a d}\right )}{(c+d x)^{5/6} (b c-a d)} \]
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Rubi [A] time = 0.02, antiderivative size = 79, 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} \[ \frac {6 \sqrt [6]{a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/6} \, _2F_1\left (\frac {1}{6},\frac {11}{6};\frac {7}{6};-\frac {d (a+b x)}{b c-a d}\right )}{(c+d x)^{5/6} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rubi steps
\begin {align*} \int \frac {1}{(a+b x)^{5/6} (c+d x)^{11/6}} \, dx &=\frac {\left (b \left (\frac {b (c+d x)}{b c-a d}\right )^{5/6}\right ) \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 )^{11/6}} \, dx}{(b c-a d) (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 {11}{6};\frac {7}{6};-\frac {d (a+b x)}{b c-a d}\right )}{(b c-a d) (c+d x)^{5/6}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 71, normalized size = 0.90 \[ \frac {6 \sqrt [6]{a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{11/6} \, _2F_1\left (\frac {1}{6},\frac {11}{6};\frac {7}{6};\frac {d (a+b x)}{a d-b c}\right )}{b (c+d x)^{11/6}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.02, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {1}{6}}}{b d^{2} x^{3} + a c^{2} + {\left (2 \, b c d + a d^{2}\right )} x^{2} + {\left (b c^{2} + 2 \, a c d\right )} x}, x\right ) \]
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 \[ \int \frac {1}{{\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {11}{6}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b x +a \right )^{\frac {5}{6}} \left (d x +c \right )^{\frac {11}{6}}}\, dx \]
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 \[ \int \frac {1}{{\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {11}{6}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (a+b\,x\right )}^{5/6}\,{\left (c+d\,x\right )}^{11/6}} \,d x \]
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 \[ \int \frac {1}{\left (a + b x\right )^{\frac {5}{6}} \left (c + d x\right )^{\frac {11}{6}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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