Optimal. Leaf size=101 \[ \frac{432 b^2 (a+b x)^{11/6}}{4301 (c+d x)^{11/6} (b c-a d)^3}+\frac{72 b (a+b x)^{11/6}}{391 (c+d x)^{17/6} (b c-a d)^2}+\frac{6 (a+b x)^{11/6}}{23 (c+d x)^{23/6} (b c-a d)} \]
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Rubi [A] time = 0.0197437, 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} \[ \frac{432 b^2 (a+b x)^{11/6}}{4301 (c+d x)^{11/6} (b c-a d)^3}+\frac{72 b (a+b x)^{11/6}}{391 (c+d x)^{17/6} (b c-a d)^2}+\frac{6 (a+b x)^{11/6}}{23 (c+d x)^{23/6} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{(a+b x)^{5/6}}{(c+d x)^{29/6}} \, dx &=\frac{6 (a+b x)^{11/6}}{23 (b c-a d) (c+d x)^{23/6}}+\frac{(12 b) \int \frac{(a+b x)^{5/6}}{(c+d x)^{23/6}} \, dx}{23 (b c-a d)}\\ &=\frac{6 (a+b x)^{11/6}}{23 (b c-a d) (c+d x)^{23/6}}+\frac{72 b (a+b x)^{11/6}}{391 (b c-a d)^2 (c+d x)^{17/6}}+\frac{\left (72 b^2\right ) \int \frac{(a+b x)^{5/6}}{(c+d x)^{17/6}} \, dx}{391 (b c-a d)^2}\\ &=\frac{6 (a+b x)^{11/6}}{23 (b c-a d) (c+d x)^{23/6}}+\frac{72 b (a+b x)^{11/6}}{391 (b c-a d)^2 (c+d x)^{17/6}}+\frac{432 b^2 (a+b x)^{11/6}}{4301 (b c-a d)^3 (c+d x)^{11/6}}\\ \end{align*}
Mathematica [A] time = 0.0431869, size = 77, normalized size = 0.76 \[ \frac{6 (a+b x)^{11/6} \left (187 a^2 d^2-22 a b d (23 c+6 d x)+b^2 \left (391 c^2+276 c d x+72 d^2 x^2\right )\right )}{4301 (c+d x)^{23/6} (b c-a d)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 105, normalized size = 1. \begin{align*} -{\frac{432\,{b}^{2}{d}^{2}{x}^{2}-792\,ab{d}^{2}x+1656\,{b}^{2}cdx+1122\,{a}^{2}{d}^{2}-3036\,abcd+2346\,{b}^{2}{c}^{2}}{4301\,{a}^{3}{d}^{3}-12903\,{a}^{2}cb{d}^{2}+12903\,a{b}^{2}{c}^{2}d-4301\,{b}^{3}{c}^{3}} \left ( bx+a \right ) ^{{\frac{11}{6}}} \left ( dx+c \right ) ^{-{\frac{23}{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{5}{6}}}{{\left (d x + c\right )}^{\frac{29}{6}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.90572, size = 702, normalized size = 6.95 \begin{align*} \frac{6 \,{\left (72 \, b^{3} d^{2} x^{3} + 391 \, a b^{2} c^{2} - 506 \, a^{2} b c d + 187 \, a^{3} d^{2} + 12 \,{\left (23 \, b^{3} c d - 5 \, a b^{2} d^{2}\right )} x^{2} +{\left (391 \, b^{3} c^{2} - 230 \, a b^{2} c d + 55 \, a^{2} b d^{2}\right )} x\right )}{\left (b x + a\right )}^{\frac{5}{6}}{\left (d x + c\right )}^{\frac{1}{6}}}{4301 \,{\left (b^{3} c^{7} - 3 \, a b^{2} c^{6} d + 3 \, a^{2} b c^{5} d^{2} - a^{3} c^{4} d^{3} +{\left (b^{3} c^{3} d^{4} - 3 \, a b^{2} c^{2} d^{5} + 3 \, a^{2} b c d^{6} - a^{3} d^{7}\right )} x^{4} + 4 \,{\left (b^{3} c^{4} d^{3} - 3 \, a b^{2} c^{3} d^{4} + 3 \, a^{2} b c^{2} d^{5} - a^{3} c d^{6}\right )} x^{3} + 6 \,{\left (b^{3} c^{5} d^{2} - 3 \, a b^{2} c^{4} d^{3} + 3 \, a^{2} b c^{3} d^{4} - a^{3} c^{2} d^{5}\right )} x^{2} + 4 \,{\left (b^{3} c^{6} d - 3 \, a b^{2} c^{5} d^{2} + 3 \, a^{2} b c^{4} d^{3} - a^{3} c^{3} 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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{\frac{5}{6}}}{{\left (d x + c\right )}^{\frac{29}{6}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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