Optimal. Leaf size=66 \[ \frac{36 b (a+b x)^{13/6}}{247 (c+d x)^{13/6} (b c-a d)^2}+\frac{6 (a+b x)^{13/6}}{19 (c+d x)^{19/6} (b c-a d)} \]
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Rubi [A] time = 0.0092703, 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} \[ \frac{36 b (a+b x)^{13/6}}{247 (c+d x)^{13/6} (b c-a d)^2}+\frac{6 (a+b x)^{13/6}}{19 (c+d x)^{19/6} (b c-a d)} \]
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)^{25/6}} \, dx &=\frac{6 (a+b x)^{13/6}}{19 (b c-a d) (c+d x)^{19/6}}+\frac{(6 b) \int \frac{(a+b x)^{7/6}}{(c+d x)^{19/6}} \, dx}{19 (b c-a d)}\\ &=\frac{6 (a+b x)^{13/6}}{19 (b c-a d) (c+d x)^{19/6}}+\frac{36 b (a+b x)^{13/6}}{247 (b c-a d)^2 (c+d x)^{13/6}}\\ \end{align*}
Mathematica [A] time = 0.027283, size = 46, normalized size = 0.7 \[ \frac{6 (a+b x)^{13/6} (-13 a d+19 b c+6 b d x)}{247 (c+d x)^{19/6} (b c-a d)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 54, normalized size = 0.8 \begin{align*} -{\frac{-36\,bdx+78\,ad-114\,bc}{247\,{a}^{2}{d}^{2}-494\,abcd+247\,{b}^{2}{c}^{2}} \left ( bx+a \right ) ^{{\frac{13}{6}}} \left ( dx+c \right ) ^{-{\frac{19}{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{25}{6}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.92627, size = 497, normalized size = 7.53 \begin{align*} \frac{6 \,{\left (6 \, b^{3} d x^{3} + 19 \, a^{2} b c - 13 \, a^{3} d +{\left (19 \, b^{3} c - a b^{2} d\right )} x^{2} + 2 \,{\left (19 \, a b^{2} c - 10 \, a^{2} b d\right )} x\right )}{\left (b x + a\right )}^{\frac{1}{6}}{\left (d x + c\right )}^{\frac{5}{6}}}{247 \,{\left (b^{2} c^{6} - 2 \, a b c^{5} d + a^{2} c^{4} d^{2} +{\left (b^{2} c^{2} d^{4} - 2 \, a b c d^{5} + a^{2} d^{6}\right )} x^{4} + 4 \,{\left (b^{2} c^{3} d^{3} - 2 \, a b c^{2} d^{4} + a^{2} c d^{5}\right )} x^{3} + 6 \,{\left (b^{2} c^{4} d^{2} - 2 \, a b c^{3} d^{3} + a^{2} c^{2} d^{4}\right )} x^{2} + 4 \,{\left (b^{2} c^{5} d - 2 \, a b c^{4} d^{2} + a^{2} c^{3} 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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