Optimal. Leaf size=66 \[ -\frac {9 d \sqrt [3]{a+b x}}{2 \sqrt [3]{c+d x} (b c-a d)^2}-\frac {3}{2 (a+b x)^{2/3} \sqrt [3]{c+d x} (b c-a d)} \]
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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} \[ -\frac {9 d \sqrt [3]{a+b x}}{2 \sqrt [3]{c+d x} (b c-a d)^2}-\frac {3}{2 (a+b x)^{2/3} \sqrt [3]{c+d x} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rubi steps
\begin {align*} \int \frac {1}{(a+b x)^{5/3} (c+d x)^{4/3}} \, dx &=-\frac {3}{2 (b c-a d) (a+b x)^{2/3} \sqrt [3]{c+d x}}-\frac {(3 d) \int \frac {1}{(a+b x)^{2/3} (c+d x)^{4/3}} \, dx}{2 (b c-a d)}\\ &=-\frac {3}{2 (b c-a d) (a+b x)^{2/3} \sqrt [3]{c+d x}}-\frac {9 d \sqrt [3]{a+b x}}{2 (b c-a d)^2 \sqrt [3]{c+d x}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 45, normalized size = 0.68 \[ -\frac {3 (2 a d+b (c+3 d x))}{2 (a+b x)^{2/3} \sqrt [3]{c+d x} (b c-a d)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 126, normalized size = 1.91 \[ -\frac {3 \, {\left (3 \, b d x + b c + 2 \, a d\right )} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{2 \, {\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} + {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x\right )}} \]
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 \[ \int \frac {1}{{\left (b x + a\right )}^{\frac {5}{3}} {\left (d x + c\right )}^{\frac {4}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 53, normalized size = 0.80 \[ -\frac {3 \left (3 b d x +2 a d +b c \right )}{2 \left (b x +a \right )^{\frac {2}{3}} \left (d x +c \right )^{\frac {1}{3}} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )} \]
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 \[ \int \frac {1}{{\left (b x + a\right )}^{\frac {5}{3}} {\left (d x + c\right )}^{\frac {4}{3}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (a+b\,x\right )}^{5/3}\,{\left (c+d\,x\right )}^{4/3}} \,d x \]
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 \[ \int \frac {1}{\left (a + b x\right )^{\frac {5}{3}} \left (c + d x\right )^{\frac {4}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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