Optimal. Leaf size=101 \[ -\frac {27 d^2 (c+d x)^{2/3}}{40 (a+b x)^{2/3} (b c-a d)^3}+\frac {9 d (c+d x)^{2/3}}{20 (a+b x)^{5/3} (b c-a d)^2}-\frac {3 (c+d x)^{2/3}}{8 (a+b x)^{8/3} (b c-a d)} \]
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Rubi [A] time = 0.02, antiderivative size = 101, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {27 d^2 (c+d x)^{2/3}}{40 (a+b x)^{2/3} (b c-a d)^3}+\frac {9 d (c+d x)^{2/3}}{20 (a+b x)^{5/3} (b c-a d)^2}-\frac {3 (c+d x)^{2/3}}{8 (a+b x)^{8/3} (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)^{11/3} \sqrt [3]{c+d x}} \, dx &=-\frac {3 (c+d x)^{2/3}}{8 (b c-a d) (a+b x)^{8/3}}-\frac {(3 d) \int \frac {1}{(a+b x)^{8/3} \sqrt [3]{c+d x}} \, dx}{4 (b c-a d)}\\ &=-\frac {3 (c+d x)^{2/3}}{8 (b c-a d) (a+b x)^{8/3}}+\frac {9 d (c+d x)^{2/3}}{20 (b c-a d)^2 (a+b x)^{5/3}}+\frac {\left (9 d^2\right ) \int \frac {1}{(a+b x)^{5/3} \sqrt [3]{c+d x}} \, dx}{20 (b c-a d)^2}\\ &=-\frac {3 (c+d x)^{2/3}}{8 (b c-a d) (a+b x)^{8/3}}+\frac {9 d (c+d x)^{2/3}}{20 (b c-a d)^2 (a+b x)^{5/3}}-\frac {27 d^2 (c+d x)^{2/3}}{40 (b c-a d)^3 (a+b x)^{2/3}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 77, normalized size = 0.76 \begin {gather*} -\frac {3 (c+d x)^{2/3} \left (20 a^2 d^2+8 a b d (3 d x-2 c)+b^2 \left (5 c^2-6 c d x+9 d^2 x^2\right )\right )}{40 (a+b x)^{8/3} (b c-a d)^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.17, size = 73, normalized size = 0.72 \begin {gather*} -\frac {3 (c+d x)^{8/3} \left (\frac {20 d^2 (a+b x)^2}{(c+d x)^2}-\frac {16 b d (a+b x)}{c+d x}+5 b^2\right )}{40 (a+b x)^{8/3} (b c-a d)^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.59, size = 251, normalized size = 2.49 \begin {gather*} -\frac {3 \, {\left (9 \, b^{2} d^{2} x^{2} + 5 \, b^{2} c^{2} - 16 \, a b c d + 20 \, a^{2} d^{2} - 6 \, {\left (b^{2} c d - 4 \, a b d^{2}\right )} x\right )} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{40 \, {\left (a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3} + {\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} x^{3} + 3 \, {\left (a b^{5} c^{3} - 3 \, a^{2} b^{4} c^{2} d + 3 \, a^{3} b^{3} c d^{2} - a^{4} b^{2} d^{3}\right )} x^{2} + 3 \, {\left (a^{2} b^{4} c^{3} - 3 \, a^{3} b^{3} c^{2} d + 3 \, a^{4} b^{2} c d^{2} - a^{5} b d^{3}\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 {11}{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 = 0.01, size = 105, normalized size = 1.04 \begin {gather*} \frac {3 \left (d x +c \right )^{\frac {2}{3}} \left (9 b^{2} x^{2} d^{2}+24 a b \,d^{2} x -6 b^{2} c d x +20 a^{2} d^{2}-16 a b c d +5 b^{2} c^{2}\right )}{40 \left (b x +a \right )^{\frac {8}{3}} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\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 {11}{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 = -0.01 \begin {gather*} \int \frac {1}{{\left (a+b\,x\right )}^{11/3}\,{\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 {11}{3}} \sqrt [3]{c + d x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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