Optimal. Leaf size=101 \[ -\frac{27 d^2 (c+d x)^{4/3}}{140 (a+b x)^{4/3} (b c-a d)^3}+\frac{9 d (c+d x)^{4/3}}{35 (a+b x)^{7/3} (b c-a d)^2}-\frac{3 (c+d x)^{4/3}}{10 (a+b x)^{10/3} (b c-a d)} \]
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Rubi [A] time = 0.0173515, 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} \[ -\frac{27 d^2 (c+d x)^{4/3}}{140 (a+b x)^{4/3} (b c-a d)^3}+\frac{9 d (c+d x)^{4/3}}{35 (a+b x)^{7/3} (b c-a d)^2}-\frac{3 (c+d x)^{4/3}}{10 (a+b x)^{10/3} (b c-a d)} \]
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)^{13/3}} \, dx &=-\frac{3 (c+d x)^{4/3}}{10 (b c-a d) (a+b x)^{10/3}}-\frac{(3 d) \int \frac{\sqrt [3]{c+d x}}{(a+b x)^{10/3}} \, dx}{5 (b c-a d)}\\ &=-\frac{3 (c+d x)^{4/3}}{10 (b c-a d) (a+b x)^{10/3}}+\frac{9 d (c+d x)^{4/3}}{35 (b c-a d)^2 (a+b x)^{7/3}}+\frac{\left (9 d^2\right ) \int \frac{\sqrt [3]{c+d x}}{(a+b x)^{7/3}} \, dx}{35 (b c-a d)^2}\\ &=-\frac{3 (c+d x)^{4/3}}{10 (b c-a d) (a+b x)^{10/3}}+\frac{9 d (c+d x)^{4/3}}{35 (b c-a d)^2 (a+b x)^{7/3}}-\frac{27 d^2 (c+d x)^{4/3}}{140 (b c-a d)^3 (a+b x)^{4/3}}\\ \end{align*}
Mathematica [A] time = 0.0400891, size = 77, normalized size = 0.76 \[ -\frac{3 (c+d x)^{4/3} \left (35 a^2 d^2+10 a b d (3 d x-4 c)+b^2 \left (14 c^2-12 c d x+9 d^2 x^2\right )\right )}{140 (a+b x)^{10/3} (b c-a d)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 105, normalized size = 1. \begin{align*}{\frac{27\,{b}^{2}{d}^{2}{x}^{2}+90\,ab{d}^{2}x-36\,{b}^{2}cdx+105\,{a}^{2}{d}^{2}-120\,abcd+42\,{b}^{2}{c}^{2}}{140\,{a}^{3}{d}^{3}-420\,{a}^{2}cb{d}^{2}+420\,a{b}^{2}{c}^{2}d-140\,{b}^{3}{c}^{3}} \left ( dx+c \right ) ^{{\frac{4}{3}}} \left ( bx+a \right ) ^{-{\frac{10}{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{13}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.75163, size = 689, normalized size = 6.82 \begin{align*} -\frac{3 \,{\left (9 \, b^{2} d^{3} x^{3} + 14 \, b^{2} c^{3} - 40 \, a b c^{2} d + 35 \, a^{2} c d^{2} - 3 \,{\left (b^{2} c d^{2} - 10 \, a b d^{3}\right )} x^{2} +{\left (2 \, b^{2} c^{2} d - 10 \, a b c d^{2} + 35 \, a^{2} d^{3}\right )} x\right )}{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}}}{140 \,{\left (a^{4} b^{3} c^{3} - 3 \, a^{5} b^{2} c^{2} d + 3 \, a^{6} b c d^{2} - a^{7} d^{3} +{\left (b^{7} c^{3} - 3 \, a b^{6} c^{2} d + 3 \, a^{2} b^{5} c d^{2} - a^{3} b^{4} d^{3}\right )} x^{4} + 4 \,{\left (a b^{6} c^{3} - 3 \, a^{2} b^{5} c^{2} d + 3 \, a^{3} b^{4} c d^{2} - a^{4} b^{3} d^{3}\right )} x^{3} + 6 \,{\left (a^{2} b^{5} c^{3} - 3 \, a^{3} b^{4} c^{2} d + 3 \, a^{4} b^{3} c d^{2} - a^{5} b^{2} d^{3}\right )} x^{2} + 4 \,{\left (a^{3} b^{4} c^{3} - 3 \, a^{4} b^{3} c^{2} d + 3 \, a^{5} b^{2} c d^{2} - a^{6} b 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] 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{13}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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