Optimal. Leaf size=101 \[ \frac{432 b^2 (a+b x)^{13/6}}{6175 (c+d x)^{13/6} (b c-a d)^3}+\frac{72 b (a+b x)^{13/6}}{475 (c+d x)^{19/6} (b c-a d)^2}+\frac{6 (a+b x)^{13/6}}{25 (c+d x)^{25/6} (b c-a d)} \]
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Rubi [A] time = 0.0182365, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {45, 37} \[ \frac{432 b^2 (a+b x)^{13/6}}{6175 (c+d x)^{13/6} (b c-a d)^3}+\frac{72 b (a+b x)^{13/6}}{475 (c+d x)^{19/6} (b c-a d)^2}+\frac{6 (a+b x)^{13/6}}{25 (c+d x)^{25/6} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{(a+b x)^{7/6}}{(c+d x)^{31/6}} \, dx &=\frac{6 (a+b x)^{13/6}}{25 (b c-a d) (c+d x)^{25/6}}+\frac{(12 b) \int \frac{(a+b x)^{7/6}}{(c+d x)^{25/6}} \, dx}{25 (b c-a d)}\\ &=\frac{6 (a+b x)^{13/6}}{25 (b c-a d) (c+d x)^{25/6}}+\frac{72 b (a+b x)^{13/6}}{475 (b c-a d)^2 (c+d x)^{19/6}}+\frac{\left (72 b^2\right ) \int \frac{(a+b x)^{7/6}}{(c+d x)^{19/6}} \, dx}{475 (b c-a d)^2}\\ &=\frac{6 (a+b x)^{13/6}}{25 (b c-a d) (c+d x)^{25/6}}+\frac{72 b (a+b x)^{13/6}}{475 (b c-a d)^2 (c+d x)^{19/6}}+\frac{432 b^2 (a+b x)^{13/6}}{6175 (b c-a d)^3 (c+d x)^{13/6}}\\ \end{align*}
Mathematica [A] time = 0.0440663, size = 77, normalized size = 0.76 \[ \frac{6 (a+b x)^{13/6} \left (247 a^2 d^2-26 a b d (25 c+6 d x)+b^2 \left (475 c^2+300 c d x+72 d^2 x^2\right )\right )}{6175 (c+d x)^{25/6} (b c-a d)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 105, normalized size = 1. \begin{align*} -{\frac{432\,{b}^{2}{d}^{2}{x}^{2}-936\,ab{d}^{2}x+1800\,{b}^{2}cdx+1482\,{a}^{2}{d}^{2}-3900\,abcd+2850\,{b}^{2}{c}^{2}}{6175\,{a}^{3}{d}^{3}-18525\,{a}^{2}cb{d}^{2}+18525\,a{b}^{2}{c}^{2}d-6175\,{b}^{3}{c}^{3}} \left ( bx+a \right ) ^{{\frac{13}{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{7}{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 = 1.86186, size = 880, normalized size = 8.71 \begin{align*} \frac{6 \,{\left (72 \, b^{4} d^{2} x^{4} + 475 \, a^{2} b^{2} c^{2} - 650 \, a^{3} b c d + 247 \, a^{4} d^{2} + 12 \,{\left (25 \, b^{4} c d - a b^{3} d^{2}\right )} x^{3} +{\left (475 \, b^{4} c^{2} - 50 \, a b^{3} c d + 7 \, a^{2} b^{2} d^{2}\right )} x^{2} + 2 \,{\left (475 \, a b^{3} c^{2} - 500 \, a^{2} b^{2} c d + 169 \, a^{3} b d^{2}\right )} x\right )}{\left (b x + a\right )}^{\frac{1}{6}}{\left (d x + c\right )}^{\frac{5}{6}}}{6175 \,{\left (b^{3} c^{8} - 3 \, a b^{2} c^{7} d + 3 \, a^{2} b c^{6} d^{2} - a^{3} c^{5} d^{3} +{\left (b^{3} c^{3} d^{5} - 3 \, a b^{2} c^{2} d^{6} + 3 \, a^{2} b c d^{7} - a^{3} d^{8}\right )} x^{5} + 5 \,{\left (b^{3} c^{4} d^{4} - 3 \, a b^{2} c^{3} d^{5} + 3 \, a^{2} b c^{2} d^{6} - a^{3} c d^{7}\right )} x^{4} + 10 \,{\left (b^{3} c^{5} d^{3} - 3 \, a b^{2} c^{4} d^{4} + 3 \, a^{2} b c^{3} d^{5} - a^{3} c^{2} d^{6}\right )} x^{3} + 10 \,{\left (b^{3} c^{6} d^{2} - 3 \, a b^{2} c^{5} d^{3} + 3 \, a^{2} b c^{4} d^{4} - a^{3} c^{3} d^{5}\right )} x^{2} + 5 \,{\left (b^{3} c^{7} d - 3 \, a b^{2} c^{6} d^{2} + 3 \, a^{2} b c^{5} d^{3} - a^{3} c^{4} d^{4}\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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