Optimal. Leaf size=66 \[ \frac{36 b \sqrt [6]{a+b x}}{7 \sqrt [6]{c+d x} (b c-a d)^2}+\frac{6 \sqrt [6]{a+b x}}{7 (c+d x)^{7/6} (b c-a d)} \]
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Rubi [A] time = 0.0092258, antiderivative size = 66, 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 = {45, 37} \[ \frac{36 b \sqrt [6]{a+b x}}{7 \sqrt [6]{c+d x} (b c-a d)^2}+\frac{6 \sqrt [6]{a+b x}}{7 (c+d x)^{7/6} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{1}{(a+b x)^{5/6} (c+d x)^{13/6}} \, dx &=\frac{6 \sqrt [6]{a+b x}}{7 (b c-a d) (c+d x)^{7/6}}+\frac{(6 b) \int \frac{1}{(a+b x)^{5/6} (c+d x)^{7/6}} \, dx}{7 (b c-a d)}\\ &=\frac{6 \sqrt [6]{a+b x}}{7 (b c-a d) (c+d x)^{7/6}}+\frac{36 b \sqrt [6]{a+b x}}{7 (b c-a d)^2 \sqrt [6]{c+d x}}\\ \end{align*}
Mathematica [A] time = 0.0156081, size = 46, normalized size = 0.7 \[ \frac{6 \sqrt [6]{a+b x} (-a d+7 b c+6 b d x)}{7 (c+d x)^{7/6} (b c-a d)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 53, normalized size = 0.8 \begin{align*} -{\frac{-36\,bdx+6\,ad-42\,bc}{7\,{a}^{2}{d}^{2}-14\,abcd+7\,{b}^{2}{c}^{2}}\sqrt [6]{bx+a} \left ( dx+c \right ) ^{-{\frac{7}{6}}}} \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{5}{6}}{\left (d x + c\right )}^{\frac{13}{6}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.52248, size = 255, normalized size = 3.86 \begin{align*} \frac{6 \,{\left (6 \, b d x + 7 \, b c - a d\right )}{\left (b x + a\right )}^{\frac{1}{6}}{\left (d x + c\right )}^{\frac{5}{6}}}{7 \,{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} +{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2} + 2 \,{\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} 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(-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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