Optimal. Leaf size=32 \[ \frac {3 (c+d x)^{2/3}}{2 d^2 \sqrt [3]{a d+b c+2 b d x}} \]
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Rubi [A] time = 0.00, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {74} \begin {gather*} \frac {3 (c+d x)^{2/3}}{2 d^2 \sqrt [3]{a d+b c+2 b d x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 74
Rubi steps
\begin {align*} \int \frac {a+b x}{\sqrt [3]{c+d x} (b c+a d+2 b d x)^{4/3}} \, dx &=\frac {3 (c+d x)^{2/3}}{2 d^2 \sqrt [3]{b c+a d+2 b d x}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 32, normalized size = 1.00 \begin {gather*} \frac {3 (c+d x)^{2/3}}{2 d^2 \sqrt [3]{a d+b (c+2 d x)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 138.74, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a+b x}{\sqrt [3]{c+d x} (b c+a d+2 b d x)^{4/3}} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.51, size = 44, normalized size = 1.38 \begin {gather*} \frac {3 \, {\left (2 \, b d x + b c + a d\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{2 \, {\left (2 \, b d^{3} x + b c d^{2} + a d^{3}\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 {b x + a}{{\left (2 \, b d x + b c + a d\right )}^{\frac {4}{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 = 27, normalized size = 0.84 \begin {gather*} \frac {3 \left (d x +c \right )^{\frac {2}{3}}}{2 \left (2 b d x +a d +b c \right )^{\frac {1}{3}} d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.68, size = 26, normalized size = 0.81 \begin {gather*} \frac {3 \, {\left (d x + c\right )}^{\frac {2}{3}}}{2 \, {\left (2 \, b d x + b c + a d\right )}^{\frac {1}{3}} d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.05, size = 75, normalized size = 2.34 \begin {gather*} \frac {\left (\frac {3\,c}{4\,b\,d^3}+\frac {3\,x}{4\,b\,d^2}\right )\,{\left (a\,d+b\,c+2\,b\,d\,x\right )}^{2/3}}{x\,{\left (c+d\,x\right )}^{1/3}+\frac {\left (2\,a\,d^3+2\,b\,c\,d^2\right )\,{\left (c+d\,x\right )}^{1/3}}{4\,b\,d^3}} \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 {a + b x}{\sqrt [3]{c + d x} \left (a d + b c + 2 b d x\right )^{\frac {4}{3}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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