Optimal. Leaf size=136 \[ -\frac {243 d^3 \sqrt [3]{a+b x}}{40 \sqrt [3]{c+d x} (b c-a d)^4}-\frac {81 d^2}{40 (a+b x)^{2/3} \sqrt [3]{c+d x} (b c-a d)^3}+\frac {27 d}{40 (a+b x)^{5/3} \sqrt [3]{c+d x} (b c-a d)^2}-\frac {3}{8 (a+b x)^{8/3} \sqrt [3]{c+d x} (b c-a d)} \]
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Rubi [A] time = 0.03, antiderivative size = 136, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {243 d^3 \sqrt [3]{a+b x}}{40 \sqrt [3]{c+d x} (b c-a d)^4}-\frac {81 d^2}{40 (a+b x)^{2/3} \sqrt [3]{c+d x} (b c-a d)^3}+\frac {27 d}{40 (a+b x)^{5/3} \sqrt [3]{c+d x} (b c-a d)^2}-\frac {3}{8 (a+b x)^{8/3} \sqrt [3]{c+d x} (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} (c+d x)^{4/3}} \, dx &=-\frac {3}{8 (b c-a d) (a+b x)^{8/3} \sqrt [3]{c+d x}}-\frac {(9 d) \int \frac {1}{(a+b x)^{8/3} (c+d x)^{4/3}} \, dx}{8 (b c-a d)}\\ &=-\frac {3}{8 (b c-a d) (a+b x)^{8/3} \sqrt [3]{c+d x}}+\frac {27 d}{40 (b c-a d)^2 (a+b x)^{5/3} \sqrt [3]{c+d x}}+\frac {\left (27 d^2\right ) \int \frac {1}{(a+b x)^{5/3} (c+d x)^{4/3}} \, dx}{20 (b c-a d)^2}\\ &=-\frac {3}{8 (b c-a d) (a+b x)^{8/3} \sqrt [3]{c+d x}}+\frac {27 d}{40 (b c-a d)^2 (a+b x)^{5/3} \sqrt [3]{c+d x}}-\frac {81 d^2}{40 (b c-a d)^3 (a+b x)^{2/3} \sqrt [3]{c+d x}}-\frac {\left (81 d^3\right ) \int \frac {1}{(a+b x)^{2/3} (c+d x)^{4/3}} \, dx}{40 (b c-a d)^3}\\ &=-\frac {3}{8 (b c-a d) (a+b x)^{8/3} \sqrt [3]{c+d x}}+\frac {27 d}{40 (b c-a d)^2 (a+b x)^{5/3} \sqrt [3]{c+d x}}-\frac {81 d^2}{40 (b c-a d)^3 (a+b x)^{2/3} \sqrt [3]{c+d x}}-\frac {243 d^3 \sqrt [3]{a+b x}}{40 (b c-a d)^4 \sqrt [3]{c+d x}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 116, normalized size = 0.85 \begin {gather*} -\frac {3 \left (40 a^3 d^3+60 a^2 b d^2 (c+3 d x)+24 a b^2 d \left (-c^2+3 c d x+9 d^2 x^2\right )+b^3 \left (5 c^3-9 c^2 d x+27 c d^2 x^2+81 d^3 x^3\right )\right )}{40 (a+b x)^{8/3} \sqrt [3]{c+d x} (b c-a d)^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.13, size = 95, normalized size = 0.70 \begin {gather*} -\frac {3 (c+d x)^{8/3} \left (-\frac {24 b^2 d (a+b x)}{c+d x}+\frac {40 d^3 (a+b x)^3}{(c+d x)^3}+\frac {60 b d^2 (a+b x)^2}{(c+d x)^2}+5 b^3\right )}{40 (a+b x)^{8/3} (b c-a d)^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.52, size = 456, normalized size = 3.35 \begin {gather*} -\frac {3 \, {\left (81 \, b^{3} d^{3} x^{3} + 5 \, b^{3} c^{3} - 24 \, a b^{2} c^{2} d + 60 \, a^{2} b c d^{2} + 40 \, a^{3} d^{3} + 27 \, {\left (b^{3} c d^{2} + 8 \, a b^{2} d^{3}\right )} x^{2} - 9 \, {\left (b^{3} c^{2} d - 8 \, a b^{2} c d^{2} - 20 \, a^{2} b d^{3}\right )} x\right )} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{40 \, {\left (a^{3} b^{4} c^{5} - 4 \, a^{4} b^{3} c^{4} d + 6 \, a^{5} b^{2} c^{3} d^{2} - 4 \, a^{6} b c^{2} d^{3} + a^{7} c d^{4} + {\left (b^{7} c^{4} d - 4 \, a b^{6} c^{3} d^{2} + 6 \, a^{2} b^{5} c^{2} d^{3} - 4 \, a^{3} b^{4} c d^{4} + a^{4} b^{3} d^{5}\right )} x^{4} + {\left (b^{7} c^{5} - a b^{6} c^{4} d - 6 \, a^{2} b^{5} c^{3} d^{2} + 14 \, a^{3} b^{4} c^{2} d^{3} - 11 \, a^{4} b^{3} c d^{4} + 3 \, a^{5} b^{2} d^{5}\right )} x^{3} + 3 \, {\left (a b^{6} c^{5} - 3 \, a^{2} b^{5} c^{4} d + 2 \, a^{3} b^{4} c^{3} d^{2} + 2 \, a^{4} b^{3} c^{2} d^{3} - 3 \, a^{5} b^{2} c d^{4} + a^{6} b d^{5}\right )} x^{2} + {\left (3 \, a^{2} b^{5} c^{5} - 11 \, a^{3} b^{4} c^{4} d + 14 \, a^{4} b^{3} c^{3} d^{2} - 6 \, a^{5} b^{2} c^{2} d^{3} - a^{6} b c d^{4} + a^{7} d^{5}\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 {4}{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 = 171, normalized size = 1.26 \begin {gather*} -\frac {3 \left (81 b^{3} d^{3} x^{3}+216 a \,b^{2} d^{3} x^{2}+27 b^{3} c \,d^{2} x^{2}+180 a^{2} b \,d^{3} x +72 a \,b^{2} c \,d^{2} x -9 b^{3} c^{2} d x +40 a^{3} d^{3}+60 a^{2} b c \,d^{2}-24 a \,b^{2} c^{2} d +5 b^{3} c^{3}\right )}{40 \left (b x +a \right )^{\frac {8}{3}} \left (d x +c \right )^{\frac {1}{3}} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\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 {4}{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 )}^{4/3}} \,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}} \left (c + d x\right )^{\frac {4}{3}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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