Optimal. Leaf size=136 \[ \frac{7776 b^3 (a+b x)^{7/6}}{43225 (c+d x)^{7/6} (b c-a d)^4}+\frac{1296 b^2 (a+b x)^{7/6}}{6175 (c+d x)^{13/6} (b c-a d)^3}+\frac{108 b (a+b x)^{7/6}}{475 (c+d x)^{19/6} (b c-a d)^2}+\frac{6 (a+b x)^{7/6}}{25 (c+d x)^{25/6} (b c-a d)} \]
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Rubi [A] time = 0.0339152, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {45, 37} \[ \frac{7776 b^3 (a+b x)^{7/6}}{43225 (c+d x)^{7/6} (b c-a d)^4}+\frac{1296 b^2 (a+b x)^{7/6}}{6175 (c+d x)^{13/6} (b c-a d)^3}+\frac{108 b (a+b x)^{7/6}}{475 (c+d x)^{19/6} (b c-a d)^2}+\frac{6 (a+b x)^{7/6}}{25 (c+d x)^{25/6} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{\sqrt [6]{a+b x}}{(c+d x)^{31/6}} \, dx &=\frac{6 (a+b x)^{7/6}}{25 (b c-a d) (c+d x)^{25/6}}+\frac{(18 b) \int \frac{\sqrt [6]{a+b x}}{(c+d x)^{25/6}} \, dx}{25 (b c-a d)}\\ &=\frac{6 (a+b x)^{7/6}}{25 (b c-a d) (c+d x)^{25/6}}+\frac{108 b (a+b x)^{7/6}}{475 (b c-a d)^2 (c+d x)^{19/6}}+\frac{\left (216 b^2\right ) \int \frac{\sqrt [6]{a+b x}}{(c+d x)^{19/6}} \, dx}{475 (b c-a d)^2}\\ &=\frac{6 (a+b x)^{7/6}}{25 (b c-a d) (c+d x)^{25/6}}+\frac{108 b (a+b x)^{7/6}}{475 (b c-a d)^2 (c+d x)^{19/6}}+\frac{1296 b^2 (a+b x)^{7/6}}{6175 (b c-a d)^3 (c+d x)^{13/6}}+\frac{\left (1296 b^3\right ) \int \frac{\sqrt [6]{a+b x}}{(c+d x)^{13/6}} \, dx}{6175 (b c-a d)^3}\\ &=\frac{6 (a+b x)^{7/6}}{25 (b c-a d) (c+d x)^{25/6}}+\frac{108 b (a+b x)^{7/6}}{475 (b c-a d)^2 (c+d x)^{19/6}}+\frac{1296 b^2 (a+b x)^{7/6}}{6175 (b c-a d)^3 (c+d x)^{13/6}}+\frac{7776 b^3 (a+b x)^{7/6}}{43225 (b c-a d)^4 (c+d x)^{7/6}}\\ \end{align*}
Mathematica [A] time = 0.0604482, size = 118, normalized size = 0.87 \[ \frac{6 (a+b x)^{7/6} \left (273 a^2 b d^2 (25 c+6 d x)-1729 a^3 d^3-21 a b^2 d \left (475 c^2+300 c d x+72 d^2 x^2\right )+b^3 \left (8550 c^2 d x+6175 c^3+5400 c d^2 x^2+1296 d^3 x^3\right )\right )}{43225 (c+d x)^{25/6} (b c-a d)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 171, normalized size = 1.3 \begin{align*} -{\frac{-7776\,{x}^{3}{b}^{3}{d}^{3}+9072\,a{b}^{2}{d}^{3}{x}^{2}-32400\,{b}^{3}c{d}^{2}{x}^{2}-9828\,{a}^{2}b{d}^{3}x+37800\,a{b}^{2}c{d}^{2}x-51300\,{b}^{3}{c}^{2}dx+10374\,{a}^{3}{d}^{3}-40950\,{a}^{2}cb{d}^{2}+59850\,a{b}^{2}{c}^{2}d-37050\,{b}^{3}{c}^{3}}{43225\,{a}^{4}{d}^{4}-172900\,{a}^{3}bc{d}^{3}+259350\,{a}^{2}{c}^{2}{b}^{2}{d}^{2}-172900\,{b}^{3}d{c}^{3}a+43225\,{b}^{4}{c}^{4}} \left ( bx+a \right ) ^{{\frac{7}{6}}} \left ( dx+c \right ) ^{-{\frac{25}{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{{\left (b x + a\right )}^{\frac{1}{6}}}{{\left (d x + c\right )}^{\frac{31}{6}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.64057, size = 1115, normalized size = 8.2 \begin{align*} \frac{6 \,{\left (1296 \, b^{4} d^{3} x^{4} + 6175 \, a b^{3} c^{3} - 9975 \, a^{2} b^{2} c^{2} d + 6825 \, a^{3} b c d^{2} - 1729 \, a^{4} d^{3} + 216 \,{\left (25 \, b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{3} + 18 \,{\left (475 \, b^{4} c^{2} d - 50 \, a b^{3} c d^{2} + 7 \, a^{2} b^{2} d^{3}\right )} x^{2} +{\left (6175 \, b^{4} c^{3} - 1425 \, a b^{3} c^{2} d + 525 \, a^{2} b^{2} c d^{2} - 91 \, a^{3} b d^{3}\right )} x\right )}{\left (b x + a\right )}^{\frac{1}{6}}{\left (d x + c\right )}^{\frac{5}{6}}}{43225 \,{\left (b^{4} c^{9} - 4 \, a b^{3} c^{8} d + 6 \, a^{2} b^{2} c^{7} d^{2} - 4 \, a^{3} b c^{6} d^{3} + a^{4} c^{5} d^{4} +{\left (b^{4} c^{4} d^{5} - 4 \, a b^{3} c^{3} d^{6} + 6 \, a^{2} b^{2} c^{2} d^{7} - 4 \, a^{3} b c d^{8} + a^{4} d^{9}\right )} x^{5} + 5 \,{\left (b^{4} c^{5} d^{4} - 4 \, a b^{3} c^{4} d^{5} + 6 \, a^{2} b^{2} c^{3} d^{6} - 4 \, a^{3} b c^{2} d^{7} + a^{4} c d^{8}\right )} x^{4} + 10 \,{\left (b^{4} c^{6} d^{3} - 4 \, a b^{3} c^{5} d^{4} + 6 \, a^{2} b^{2} c^{4} d^{5} - 4 \, a^{3} b c^{3} d^{6} + a^{4} c^{2} d^{7}\right )} x^{3} + 10 \,{\left (b^{4} c^{7} d^{2} - 4 \, a b^{3} c^{6} d^{3} + 6 \, a^{2} b^{2} c^{5} d^{4} - 4 \, a^{3} b c^{4} d^{5} + a^{4} c^{3} d^{6}\right )} x^{2} + 5 \,{\left (b^{4} c^{8} d - 4 \, a b^{3} c^{7} d^{2} + 6 \, a^{2} b^{2} c^{6} d^{3} - 4 \, a^{3} b c^{5} d^{4} + a^{4} c^{4} d^{5}\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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