Optimal. Leaf size=66 \[ \frac {9 d \sqrt [3]{c+d x}}{4 \sqrt [3]{a+b x} (b c-a d)^2}-\frac {3 \sqrt [3]{c+d x}}{4 (a+b x)^{4/3} (b c-a d)} \]
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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} \[ \frac {9 d \sqrt [3]{c+d x}}{4 \sqrt [3]{a+b x} (b c-a d)^2}-\frac {3 \sqrt [3]{c+d x}}{4 (a+b x)^{4/3} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rubi steps
\begin {align*} \int \frac {1}{(a+b x)^{7/3} (c+d x)^{2/3}} \, dx &=-\frac {3 \sqrt [3]{c+d x}}{4 (b c-a d) (a+b x)^{4/3}}-\frac {(3 d) \int \frac {1}{(a+b x)^{4/3} (c+d x)^{2/3}} \, dx}{4 (b c-a d)}\\ &=-\frac {3 \sqrt [3]{c+d x}}{4 (b c-a d) (a+b x)^{4/3}}+\frac {9 d \sqrt [3]{c+d x}}{4 (b c-a d)^2 \sqrt [3]{a+b x}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 46, normalized size = 0.70 \[ \frac {3 \sqrt [3]{c+d x} (4 a d-b c+3 b d x)}{4 (a+b x)^{4/3} (b c-a d)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 118, normalized size = 1.79 \[ \frac {3 \, {\left (3 \, b d x - b c + 4 \, a d\right )} {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}}}{4 \, {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} + {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2} + 2 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x\right )}} \]
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 \[ \int \frac {1}{{\left (b x + a\right )}^{\frac {7}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}\,{d x} \]
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 \[ \frac {3 \left (d x +c \right )^{\frac {1}{3}} \left (3 b d x +4 a d -b c \right )}{4 \left (b x +a \right )^{\frac {4}{3}} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )} \]
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 \[ \int \frac {1}{{\left (b x + a\right )}^{\frac {7}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.98, size = 71, normalized size = 1.08 \[ \frac {\left (\frac {9\,d\,x}{4\,{\left (a\,d-b\,c\right )}^2}+\frac {12\,a\,d-3\,b\,c}{4\,b\,{\left (a\,d-b\,c\right )}^2}\right )\,{\left (c+d\,x\right )}^{1/3}}{x\,{\left (a+b\,x\right )}^{1/3}+\frac {a\,{\left (a+b\,x\right )}^{1/3}}{b}} \]
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 \[ \int \frac {1}{\left (a + b x\right )^{\frac {7}{3}} \left (c + d x\right )^{\frac {2}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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