Optimal. Leaf size=101 \[ \frac {27 d^2 \sqrt [3]{a+b x}}{5 \sqrt [3]{c+d x} (b c-a d)^3}+\frac {9 d}{5 (a+b x)^{2/3} \sqrt [3]{c+d x} (b c-a d)^2}-\frac {3}{5 (a+b x)^{5/3} \sqrt [3]{c+d x} (b c-a d)} \]
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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} \[ \frac {27 d^2 \sqrt [3]{a+b x}}{5 \sqrt [3]{c+d x} (b c-a d)^3}+\frac {9 d}{5 (a+b x)^{2/3} \sqrt [3]{c+d x} (b c-a d)^2}-\frac {3}{5 (a+b x)^{5/3} \sqrt [3]{c+d x} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rubi steps
\begin {align*} \int \frac {1}{(a+b x)^{8/3} (c+d x)^{4/3}} \, dx &=-\frac {3}{5 (b c-a d) (a+b x)^{5/3} \sqrt [3]{c+d x}}-\frac {(6 d) \int \frac {1}{(a+b x)^{5/3} (c+d x)^{4/3}} \, dx}{5 (b c-a d)}\\ &=-\frac {3}{5 (b c-a d) (a+b x)^{5/3} \sqrt [3]{c+d x}}+\frac {9 d}{5 (b c-a d)^2 (a+b x)^{2/3} \sqrt [3]{c+d x}}+\frac {\left (9 d^2\right ) \int \frac {1}{(a+b x)^{2/3} (c+d x)^{4/3}} \, dx}{5 (b c-a d)^2}\\ &=-\frac {3}{5 (b c-a d) (a+b x)^{5/3} \sqrt [3]{c+d x}}+\frac {9 d}{5 (b c-a d)^2 (a+b x)^{2/3} \sqrt [3]{c+d x}}+\frac {27 d^2 \sqrt [3]{a+b x}}{5 (b c-a d)^3 \sqrt [3]{c+d x}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 75, normalized size = 0.74 \[ \frac {3 \left (5 a^2 d^2+5 a b d (c+3 d x)+b^2 \left (-c^2+3 c d x+9 d^2 x^2\right )\right )}{5 (a+b x)^{5/3} \sqrt [3]{c+d x} (b c-a d)^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 273, normalized size = 2.70 \[ \frac {3 \, {\left (9 \, b^{2} d^{2} x^{2} - b^{2} c^{2} + 5 \, a b c d + 5 \, a^{2} d^{2} + 3 \, {\left (b^{2} c d + 5 \, a b d^{2}\right )} x\right )} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{5 \, {\left (a^{2} b^{3} c^{4} - 3 \, a^{3} b^{2} c^{3} d + 3 \, a^{4} b c^{2} d^{2} - a^{5} c d^{3} + {\left (b^{5} c^{3} d - 3 \, a b^{4} c^{2} d^{2} + 3 \, a^{2} b^{3} c d^{3} - a^{3} b^{2} d^{4}\right )} x^{3} + {\left (b^{5} c^{4} - a b^{4} c^{3} d - 3 \, a^{2} b^{3} c^{2} d^{2} + 5 \, a^{3} b^{2} c d^{3} - 2 \, a^{4} b d^{4}\right )} x^{2} + {\left (2 \, a b^{4} c^{4} - 5 \, a^{2} b^{3} c^{3} d + 3 \, a^{3} b^{2} c^{2} d^{2} + a^{4} b c d^{3} - a^{5} d^{4}\right )} x\right )}} \]
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 \[ \int \frac {1}{{\left (b x + a\right )}^{\frac {8}{3}} {\left (d x + c\right )}^{\frac {4}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 105, normalized size = 1.04 \[ -\frac {3 \left (9 b^{2} x^{2} d^{2}+15 a b \,d^{2} x +3 b^{2} c d x +5 a^{2} d^{2}+5 a b c d -b^{2} c^{2}\right )}{5 \left (b x +a \right )^{\frac {5}{3}} \left (d x +c \right )^{\frac {1}{3}} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )} \]
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 \[ \int \frac {1}{{\left (b x + a\right )}^{\frac {8}{3}} {\left (d x + c\right )}^{\frac {4}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (a+b\,x\right )}^{8/3}\,{\left (c+d\,x\right )}^{4/3}} \,d x \]
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 \[ \int \frac {1}{\left (a + b x\right )^{\frac {8}{3}} \left (c + d x\right )^{\frac {4}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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