Optimal. Leaf size=116 \[ \frac {3 d^2 (c+d x)^{2/3} \sqrt [3]{-\frac {b c+a d+2 b d x}{b c-a d}} F_1\left (\frac {2}{3};\frac {4}{3},3;\frac {5}{3};\frac {2 b (c+d x)}{b c-a d},\frac {b (c+d x)}{b c-a d}\right )}{2 (b c-a d)^4 \sqrt [3]{b c+a d+2 b d x}} \]
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Rubi [A]
time = 0.04, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {142, 141}
\begin {gather*} \frac {3 d^2 (c+d x)^{2/3} \sqrt [3]{-\frac {a d+b c+2 b d x}{b c-a d}} F_1\left (\frac {2}{3};\frac {4}{3},3;\frac {5}{3};\frac {2 b (c+d x)}{b c-a d},\frac {b (c+d x)}{b c-a d}\right )}{2 (b c-a d)^4 \sqrt [3]{a d+b c+2 b d x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 141
Rule 142
Rubi steps
\begin {align*} \int \frac {1}{(a+b x)^3 \sqrt [3]{c+d x} (b c+a d+2 b d x)^{4/3}} \, dx &=\frac {\left (d \sqrt [3]{\frac {d (b c+a d+2 b d x)}{-2 b c d+d (b c+a d)}}\right ) \int \frac {1}{(a+b x)^3 \sqrt [3]{c+d x} \left (\frac {d (b c+a d)}{-2 b c d+d (b c+a d)}+\frac {2 b d^2 x}{-2 b c d+d (b c+a d)}\right )^{4/3}} \, dx}{(-2 b c d+d (b c+a d)) \sqrt [3]{b c+a d+2 b d x}}\\ &=\frac {3 d^2 (c+d x)^{2/3} \sqrt [3]{-\frac {b c+a d+2 b d x}{b c-a d}} F_1\left (\frac {2}{3};\frac {4}{3},3;\frac {5}{3};\frac {2 b (c+d x)}{b c-a d},\frac {b (c+d x)}{b c-a d}\right )}{2 (b c-a d)^4 \sqrt [3]{b c+a d+2 b d x}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(324\) vs. \(2(116)=232\).
time = 20.95, size = 324, normalized size = 2.79 \begin {gather*} \frac {(c+d x)^{2/3} \left (\frac {5 \left (57 a^2 d^2+2 a b d (4 c+61 d x)+b^2 \left (-c^2+6 c d x+64 d^2 x^2\right )\right )}{(a+b x)^2}-\frac {d^2 \left (160 b (c+d x) (a d+b (c+2 d x))+95\ 2^{2/3} b (b c-a d) (c+d x) \sqrt [3]{\frac {b c+a d+2 b d x}{b c+b d x}} F_1\left (\frac {2}{3};\frac {1}{3},1;\frac {5}{3};\frac {b c-a d}{2 b c+2 b d x},\frac {b c-a d}{b c+b d x}\right )-16\ 2^{2/3} (b c-a d)^2 \sqrt [3]{\frac {b c+a d+2 b d x}{b c+b d x}} F_1\left (\frac {5}{3};\frac {1}{3},1;\frac {8}{3};\frac {b c-a d}{2 b c+2 b d x},\frac {b c-a d}{b c+b d x}\right )\right )}{b^2 (c+d x)^2}\right )}{10 (b c-a d)^4 \sqrt [3]{a d+b (c+2 d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (b x +a \right )^{3} \left (d x +c \right )^{\frac {1}{3}} \left (2 b d x +a d +b c \right )^{\frac {4}{3}}}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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*} \text {could not integrate} \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 )}^3\,{\left (c+d\,x\right )}^{1/3}\,{\left (a\,d+b\,c+2\,b\,d\,x\right )}^{4/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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