Optimal. Leaf size=84 \[ \frac {6 b^2 (a+b x)^{5/6} \sqrt [6]{\frac {b (c+d x)}{b c-a d}} \, _2F_1\left (\frac {5}{6},\frac {19}{6};\frac {11}{6};-\frac {d (a+b x)}{b c-a d}\right )}{5 \sqrt [6]{c+d x} (b c-a d)^3} \]
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Rubi [A] time = 0.02, antiderivative size = 84, 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} \[ \frac {6 b^2 (a+b x)^{5/6} \sqrt [6]{\frac {b (c+d x)}{b c-a d}} \, _2F_1\left (\frac {5}{6},\frac {19}{6};\frac {11}{6};-\frac {d (a+b x)}{b c-a d}\right )}{5 \sqrt [6]{c+d x} (b c-a d)^3} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{19/6}} \, dx &=\frac {\left (b^3 \sqrt [6]{\frac {b (c+d x)}{b c-a d}}\right ) \int \frac {1}{\sqrt [6]{a+b x} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{19/6}} \, dx}{(b c-a d)^3 \sqrt [6]{c+d x}}\\ &=\frac {6 b^2 (a+b x)^{5/6} \sqrt [6]{\frac {b (c+d x)}{b c-a d}} \, _2F_1\left (\frac {5}{6},\frac {19}{6};\frac {11}{6};-\frac {d (a+b x)}{b c-a d}\right )}{5 (b c-a d)^3 \sqrt [6]{c+d x}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 81, normalized size = 0.96 \[ \frac {6 b (a+b x)^{5/6} \left (\frac {b (c+d x)}{b c-a d}\right )^{7/6} \, _2F_1\left (\frac {5}{6},\frac {19}{6};\frac {11}{6};\frac {d (a+b x)}{a d-b c}\right )}{5 (c+d x)^{7/6} (b c-a d)^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {5}{6}}}{b d^{4} x^{5} + a c^{4} + {\left (4 \, b c d^{3} + a d^{4}\right )} x^{4} + 2 \, {\left (3 \, b c^{2} d^{2} + 2 \, a c d^{3}\right )} x^{3} + 2 \, {\left (2 \, b c^{3} d + 3 \, a c^{2} d^{2}\right )} x^{2} + {\left (b c^{4} + 4 \, a c^{3} d\right )} x}, x\right ) \]
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 \[ \int \frac {1}{{\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {19}{6}}}\,{d x} \]
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 \[ \int \frac {1}{\left (b x +a \right )^{\frac {1}{6}} \left (d x +c \right )^{\frac {19}{6}}}\, dx \]
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 \[ \int \frac {1}{{\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {19}{6}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (a+b\,x\right )}^{1/6}\,{\left (c+d\,x\right )}^{19/6}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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