Optimal. Leaf size=66 \[ \frac {9 d (c+d x)^{2/3}}{10 (a+b x)^{2/3} (b c-a d)^2}-\frac {3 (c+d x)^{2/3}}{5 (a+b x)^{5/3} (b c-a d)} \]
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Rubi [A] time = 0.01, antiderivative size = 66, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {9 d (c+d x)^{2/3}}{10 (a+b x)^{2/3} (b c-a d)^2}-\frac {3 (c+d x)^{2/3}}{5 (a+b x)^{5/3} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rubi steps
\begin {align*} \int \frac {1}{(a+b x)^{8/3} \sqrt [3]{c+d x}} \, dx &=-\frac {3 (c+d x)^{2/3}}{5 (b c-a d) (a+b x)^{5/3}}-\frac {(3 d) \int \frac {1}{(a+b x)^{5/3} \sqrt [3]{c+d x}} \, dx}{5 (b c-a d)}\\ &=-\frac {3 (c+d x)^{2/3}}{5 (b c-a d) (a+b x)^{5/3}}+\frac {9 d (c+d x)^{2/3}}{10 (b c-a d)^2 (a+b x)^{2/3}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 46, normalized size = 0.70 \begin {gather*} \frac {3 (c+d x)^{2/3} (5 a d-2 b c+3 b d x)}{10 (a+b x)^{5/3} (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.16, size = 51, normalized size = 0.77 \begin {gather*} \frac {3 (c+d x)^{5/3} \left (\frac {5 d (a+b x)}{c+d x}-2 b\right )}{10 (a+b x)^{5/3} (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.40, size = 118, normalized size = 1.79 \begin {gather*} \frac {3 \, {\left (3 \, b d x - 2 \, b c + 5 \, a d\right )} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{10 \, {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} + {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2} + 2 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (b x + a\right )}^{\frac {8}{3}} {\left (d x + c\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 54, normalized size = 0.82 \begin {gather*} \frac {3 \left (d x +c \right )^{\frac {2}{3}} \left (3 b d x +5 a d -2 b c \right )}{10 \left (b x +a \right )^{\frac {5}{3}} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (b x + a\right )}^{\frac {8}{3}} {\left (d x + c\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{{\left (a+b\,x\right )}^{8/3}\,{\left (c+d\,x\right )}^{1/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b x\right )^{\frac {8}{3}} \sqrt [3]{c + d x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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