Optimal. Leaf size=82 \[ -\frac{6 b^2 \sqrt [6]{\frac{b (c+d x)}{b c-a d}} \, _2F_1\left (-\frac{1}{6},\frac{19}{6};\frac{5}{6};-\frac{d (a+b x)}{b c-a d}\right )}{\sqrt [6]{a+b x} \sqrt [6]{c+d x} (b c-a d)^3} \]
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Rubi [A] time = 0.0223072, antiderivative size = 82, normalized size of antiderivative = 1., 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} \[ -\frac{6 b^2 \sqrt [6]{\frac{b (c+d x)}{b c-a d}} \, _2F_1\left (-\frac{1}{6},\frac{19}{6};\frac{5}{6};-\frac{d (a+b x)}{b c-a d}\right )}{\sqrt [6]{a+b x} \sqrt [6]{c+d x} (b c-a d)^3} \]
Antiderivative was successfully verified.
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Rule 70
Rule 69
Rubi steps
\begin{align*} \int \frac{1}{(a+b x)^{7/6} (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}{(a+b x)^{7/6} \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 \sqrt [6]{\frac{b (c+d x)}{b c-a d}} \, _2F_1\left (-\frac{1}{6},\frac{19}{6};\frac{5}{6};-\frac{d (a+b x)}{b c-a d}\right )}{(b c-a d)^3 \sqrt [6]{a+b x} \sqrt [6]{c+d x}}\\ \end{align*}
Mathematica [A] time = 0.0344366, size = 79, normalized size = 0.96 \[ -\frac{6 b \left (\frac{b (c+d x)}{b c-a d}\right )^{7/6} \, _2F_1\left (-\frac{1}{6},\frac{19}{6};\frac{5}{6};\frac{d (a+b x)}{a d-b c}\right )}{\sqrt [6]{a+b x} (c+d x)^{7/6} (b c-a d)^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.039, size = 0, normalized size = 0. \begin{align*} \int{ \left ( bx+a \right ) ^{-{\frac{7}{6}}} \left ( dx+c \right ) ^{-{\frac{19}{6}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x + a\right )}^{\frac{7}{6}}{\left (d x + c\right )}^{\frac{19}{6}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x + a\right )}^{\frac{5}{6}}{\left (d x + c\right )}^{\frac{5}{6}}}{b^{2} d^{4} x^{6} + a^{2} c^{4} + 2 \,{\left (2 \, b^{2} c d^{3} + a b d^{4}\right )} x^{5} +{\left (6 \, b^{2} c^{2} d^{2} + 8 \, a b c d^{3} + a^{2} d^{4}\right )} x^{4} + 4 \,{\left (b^{2} c^{3} d + 3 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x^{3} +{\left (b^{2} c^{4} + 8 \, a b c^{3} d + 6 \, a^{2} c^{2} d^{2}\right )} x^{2} + 2 \,{\left (a b c^{4} + 2 \, a^{2} c^{3} d\right )} x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x + a\right )}^{\frac{7}{6}}{\left (d x + c\right )}^{\frac{19}{6}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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