Optimal. Leaf size=100 \[ \frac {3 (a+b x)^{4/3} \sqrt {c+d x} F_1\left (\frac {4}{3};-\frac {1}{2},1;\frac {7}{3};-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{4 (b e-a f) \sqrt {\frac {b (c+d x)}{b c-a d}}} \]
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Rubi [A] time = 0.04, 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 = {137, 136} \[ \frac {3 (a+b x)^{4/3} \sqrt {c+d x} F_1\left (\frac {4}{3};-\frac {1}{2},1;\frac {7}{3};-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{4 (b e-a f) \sqrt {\frac {b (c+d x)}{b c-a d}}} \]
Antiderivative was successfully verified.
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Rule 136
Rule 137
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{a+b x} \sqrt {c+d x}}{e+f x} \, dx &=\frac {\sqrt {c+d x} \int \frac {\sqrt [3]{a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}}}{e+f x} \, dx}{\sqrt {\frac {b (c+d x)}{b c-a d}}}\\ &=\frac {3 (a+b x)^{4/3} \sqrt {c+d x} F_1\left (\frac {4}{3};-\frac {1}{2},1;\frac {7}{3};-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{4 (b e-a f) \sqrt {\frac {b (c+d x)}{b c-a d}}}\\ \end {align*}
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Mathematica [B] time = 0.58, size = 201, normalized size = 2.01 \[ \frac {6 \sqrt {c+d x} \left (\frac {\left (\frac {d (a+b x)}{b (c+d x)}\right )^{2/3} \left (7 (-2 a d f-3 b c f+5 b d e) F_1\left (\frac {1}{6};\frac {2}{3},1;\frac {7}{6};\frac {b c-a d}{b c+b d x},\frac {c f-d e}{f (c+d x)}\right )+\frac {3 (b c-a d) (c f-d e) F_1\left (\frac {7}{6};\frac {2}{3},1;\frac {13}{6};\frac {b c-a d}{b c+b d x},\frac {c f-d e}{f (c+d x)}\right )}{c+d x}\right )}{d}+7 f (a+b x)\right )}{35 f^2 (a+b x)^{2/3}} \]
Warning: Unable to verify antiderivative.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {{\left (b x + a\right )}^{\frac {1}{3}} \sqrt {d x + c}}{f x + e}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.28, size = 0, normalized size = 0.00 \[ \int \frac {\left (b x +a \right )^{\frac {1}{3}} \sqrt {d x +c}}{f x +e}\, dx \]
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 \[ \int \frac {{\left (b x + a\right )}^{\frac {1}{3}} \sqrt {d x + c}}{f x + e}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,x\right )}^{1/3}\,\sqrt {c+d\,x}}{e+f\,x} \,d x \]
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 \[ \int \frac {\sqrt [3]{a + b x} \sqrt {c + d x}}{e + f x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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