Optimal. Leaf size=100 \[ \frac {2 (a+b x)^{3/2} \sqrt [3]{c+d x} F_1\left (\frac {3}{2};-\frac {1}{3},1;\frac {5}{2};-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{3 (b e-a f) \sqrt [3]{\frac {b (c+d x)}{b c-a d}}} \]
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Rubi [A]
time = 0.03, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {142, 141}
\begin {gather*} \frac {2 (a+b x)^{3/2} \sqrt [3]{c+d x} F_1\left (\frac {3}{2};-\frac {1}{3},1;\frac {5}{2};-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{3 (b e-a f) \sqrt [3]{\frac {b (c+d x)}{b c-a d}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 141
Rule 142
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x} \sqrt [3]{c+d x}}{e+f x} \, dx &=\frac {\sqrt [3]{c+d x} \int \frac {\sqrt {a+b x} \sqrt [3]{\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}}}{e+f x} \, dx}{\sqrt [3]{\frac {b (c+d x)}{b c-a d}}}\\ &=\frac {2 (a+b x)^{3/2} \sqrt [3]{c+d x} F_1\left (\frac {3}{2};-\frac {1}{3},1;\frac {5}{2};-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{3 (b e-a f) \sqrt [3]{\frac {b (c+d x)}{b c-a d}}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(202\) vs. \(2(100)=200\).
time = 20.43, size = 202, normalized size = 2.02 \begin {gather*} \frac {6 \sqrt {a+b x} \left (7 f (c+d x)-\frac {\left (\frac {b (c+d x)}{d (a+b x)}\right )^{2/3} \left (-7 (5 b d e-2 b c f-3 a d f) F_1\left (\frac {1}{6};\frac {2}{3},1;\frac {7}{6};\frac {-b c+a d}{d (a+b x)},\frac {-b e+a f}{f (a+b x)}\right )-\frac {3 (b c-a d) (b e-a f) F_1\left (\frac {7}{6};\frac {2}{3},1;\frac {13}{6};\frac {-b c+a d}{d (a+b x)},\frac {-b e+a f}{f (a+b x)}\right )}{a+b x}\right )}{b}\right )}{35 f^2 (c+d x)^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {b x +a}\, \left (d x +c \right )^{\frac {1}{3}}}{f x +e}\, dx\]
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a + b x} \sqrt [3]{c + d x}}{e + f x}\, dx \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 {\sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{1/3}}{e+f\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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