Optimal. Leaf size=136 \[ \frac{243 d^3 (c+d x)^{4/3}}{1820 (a+b x)^{4/3} (b c-a d)^4}-\frac{81 d^2 (c+d x)^{4/3}}{455 (a+b x)^{7/3} (b c-a d)^3}+\frac{27 d (c+d x)^{4/3}}{130 (a+b x)^{10/3} (b c-a d)^2}-\frac{3 (c+d x)^{4/3}}{13 (a+b x)^{13/3} (b c-a d)} \]
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Rubi [A] time = 0.0283204, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {45, 37} \[ \frac{243 d^3 (c+d x)^{4/3}}{1820 (a+b x)^{4/3} (b c-a d)^4}-\frac{81 d^2 (c+d x)^{4/3}}{455 (a+b x)^{7/3} (b c-a d)^3}+\frac{27 d (c+d x)^{4/3}}{130 (a+b x)^{10/3} (b c-a d)^2}-\frac{3 (c+d x)^{4/3}}{13 (a+b x)^{13/3} (b c-a d)} \]
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)^{16/3}} \, dx &=-\frac{3 (c+d x)^{4/3}}{13 (b c-a d) (a+b x)^{13/3}}-\frac{(9 d) \int \frac{\sqrt [3]{c+d x}}{(a+b x)^{13/3}} \, dx}{13 (b c-a d)}\\ &=-\frac{3 (c+d x)^{4/3}}{13 (b c-a d) (a+b x)^{13/3}}+\frac{27 d (c+d x)^{4/3}}{130 (b c-a d)^2 (a+b x)^{10/3}}+\frac{\left (27 d^2\right ) \int \frac{\sqrt [3]{c+d x}}{(a+b x)^{10/3}} \, dx}{65 (b c-a d)^2}\\ &=-\frac{3 (c+d x)^{4/3}}{13 (b c-a d) (a+b x)^{13/3}}+\frac{27 d (c+d x)^{4/3}}{130 (b c-a d)^2 (a+b x)^{10/3}}-\frac{81 d^2 (c+d x)^{4/3}}{455 (b c-a d)^3 (a+b x)^{7/3}}-\frac{\left (81 d^3\right ) \int \frac{\sqrt [3]{c+d x}}{(a+b x)^{7/3}} \, dx}{455 (b c-a d)^3}\\ &=-\frac{3 (c+d x)^{4/3}}{13 (b c-a d) (a+b x)^{13/3}}+\frac{27 d (c+d x)^{4/3}}{130 (b c-a d)^2 (a+b x)^{10/3}}-\frac{81 d^2 (c+d x)^{4/3}}{455 (b c-a d)^3 (a+b x)^{7/3}}+\frac{243 d^3 (c+d x)^{4/3}}{1820 (b c-a d)^4 (a+b x)^{4/3}}\\ \end{align*}
Mathematica [A] time = 0.0594977, size = 118, normalized size = 0.87 \[ \frac{3 (c+d x)^{4/3} \left (195 a^2 b d^2 (3 d x-4 c)+455 a^3 d^3+39 a b^2 d \left (14 c^2-12 c d x+9 d^2 x^2\right )+b^3 \left (126 c^2 d x-140 c^3-108 c d^2 x^2+81 d^3 x^3\right )\right )}{1820 (a+b x)^{13/3} (b c-a d)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 171, normalized size = 1.3 \begin{align*}{\frac{243\,{b}^{3}{d}^{3}{x}^{3}+1053\,a{b}^{2}{d}^{3}{x}^{2}-324\,{b}^{3}c{d}^{2}{x}^{2}+1755\,{a}^{2}b{d}^{3}x-1404\,a{b}^{2}c{d}^{2}x+378\,{b}^{3}{c}^{2}dx+1365\,{a}^{3}{d}^{3}-2340\,{a}^{2}cb{d}^{2}+1638\,a{b}^{2}{c}^{2}d-420\,{b}^{3}{c}^{3}}{1820\,{d}^{4}{a}^{4}-7280\,b{d}^{3}c{a}^{3}+10920\,{b}^{2}{d}^{2}{c}^{2}{a}^{2}-7280\,{b}^{3}d{c}^{3}a+1820\,{b}^{4}{c}^{4}} \left ( dx+c \right ) ^{{\frac{4}{3}}} \left ( bx+a \right ) ^{-{\frac{13}{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{16}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.77431, size = 1098, normalized size = 8.07 \begin{align*} \frac{3 \,{\left (81 \, b^{3} d^{4} x^{4} - 140 \, b^{3} c^{4} + 546 \, a b^{2} c^{3} d - 780 \, a^{2} b c^{2} d^{2} + 455 \, a^{3} c d^{3} - 27 \,{\left (b^{3} c d^{3} - 13 \, a b^{2} d^{4}\right )} x^{3} + 9 \,{\left (2 \, b^{3} c^{2} d^{2} - 13 \, a b^{2} c d^{3} + 65 \, a^{2} b d^{4}\right )} x^{2} -{\left (14 \, b^{3} c^{3} d - 78 \, a b^{2} c^{2} d^{2} + 195 \, a^{2} b c d^{3} - 455 \, a^{3} d^{4}\right )} x\right )}{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}}}{1820 \,{\left (a^{5} b^{4} c^{4} - 4 \, a^{6} b^{3} c^{3} d + 6 \, a^{7} b^{2} c^{2} d^{2} - 4 \, a^{8} b c d^{3} + a^{9} d^{4} +{\left (b^{9} c^{4} - 4 \, a b^{8} c^{3} d + 6 \, a^{2} b^{7} c^{2} d^{2} - 4 \, a^{3} b^{6} c d^{3} + a^{4} b^{5} d^{4}\right )} x^{5} + 5 \,{\left (a b^{8} c^{4} - 4 \, a^{2} b^{7} c^{3} d + 6 \, a^{3} b^{6} c^{2} d^{2} - 4 \, a^{4} b^{5} c d^{3} + a^{5} b^{4} d^{4}\right )} x^{4} + 10 \,{\left (a^{2} b^{7} c^{4} - 4 \, a^{3} b^{6} c^{3} d + 6 \, a^{4} b^{5} c^{2} d^{2} - 4 \, a^{5} b^{4} c d^{3} + a^{6} b^{3} d^{4}\right )} x^{3} + 10 \,{\left (a^{3} b^{6} c^{4} - 4 \, a^{4} b^{5} c^{3} d + 6 \, a^{5} b^{4} c^{2} d^{2} - 4 \, a^{6} b^{3} c d^{3} + a^{7} b^{2} d^{4}\right )} x^{2} + 5 \,{\left (a^{4} b^{5} c^{4} - 4 \, a^{5} b^{4} c^{3} d + 6 \, a^{6} b^{3} c^{2} d^{2} - 4 \, a^{7} b^{2} c d^{3} + a^{8} b 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 (d x + c\right )}^{\frac{1}{3}}}{{\left (b x + a\right )}^{\frac{16}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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