Optimal. Leaf size=66 \[ \frac{9 d (c+d x)^{4/3}}{28 (a+b x)^{4/3} (b c-a d)^2}-\frac{3 (c+d x)^{4/3}}{7 (a+b x)^{7/3} (b c-a d)} \]
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Rubi [A] time = 0.0086674, 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} \[ \frac{9 d (c+d x)^{4/3}}{28 (a+b x)^{4/3} (b c-a d)^2}-\frac{3 (c+d x)^{4/3}}{7 (a+b x)^{7/3} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{\sqrt [3]{c+d x}}{(a+b x)^{10/3}} \, dx &=-\frac{3 (c+d x)^{4/3}}{7 (b c-a d) (a+b x)^{7/3}}-\frac{(3 d) \int \frac{\sqrt [3]{c+d x}}{(a+b x)^{7/3}} \, dx}{7 (b c-a d)}\\ &=-\frac{3 (c+d x)^{4/3}}{7 (b c-a d) (a+b x)^{7/3}}+\frac{9 d (c+d x)^{4/3}}{28 (b c-a d)^2 (a+b x)^{4/3}}\\ \end{align*}
Mathematica [A] time = 0.0229644, size = 46, normalized size = 0.7 \[ \frac{3 (c+d x)^{4/3} (7 a d-4 b c+3 b d x)}{28 (a+b x)^{7/3} (b c-a d)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 54, normalized size = 0.8 \begin{align*}{\frac{9\,bdx+21\,ad-12\,bc}{28\,{a}^{2}{d}^{2}-56\,abcd+28\,{b}^{2}{c}^{2}} \left ( dx+c \right ) ^{{\frac{4}{3}}} \left ( bx+a \right ) ^{-{\frac{7}{3}}}} \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 (d x + c\right )}^{\frac{1}{3}}}{{\left (b x + a\right )}^{\frac{10}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.71984, size = 370, normalized size = 5.61 \begin{align*} \frac{3 \,{\left (3 \, b d^{2} x^{2} - 4 \, b c^{2} + 7 \, a c d -{\left (b c d - 7 \, a d^{2}\right )} x\right )}{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}}}{28 \,{\left (a^{3} b^{2} c^{2} - 2 \, a^{4} b c d + a^{5} d^{2} +{\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} x^{3} + 3 \,{\left (a b^{4} c^{2} - 2 \, a^{2} b^{3} c d + a^{3} b^{2} d^{2}\right )} x^{2} + 3 \,{\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2}\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 (d x + c\right )}^{\frac{1}{3}}}{{\left (b x + a\right )}^{\frac{10}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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