Optimal. Leaf size=100 \[ \frac {3 (a+b x)^{7/3} \sqrt {\frac {b (c+d x)}{b c-a d}} F_1\left (\frac {7}{3};\frac {1}{2},1;\frac {10}{3};-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{7 (b e-a f) \sqrt {c+d x}} \]
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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 {3 (a+b x)^{7/3} \sqrt {\frac {b (c+d x)}{b c-a d}} F_1\left (\frac {7}{3};\frac {1}{2},1;\frac {10}{3};-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{7 \sqrt {c+d x} (b e-a f)} \end {gather*}
Antiderivative was successfully verified.
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Rule 141
Rule 142
Rubi steps
\begin {align*} \int \frac {(a+b x)^{4/3}}{\sqrt {c+d x} (e+f x)} \, dx &=\frac {\sqrt {\frac {b (c+d x)}{b c-a d}} \int \frac {(a+b x)^{4/3}}{\sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}} (e+f x)} \, dx}{\sqrt {c+d x}}\\ &=\frac {3 (a+b x)^{7/3} \sqrt {\frac {b (c+d x)}{b c-a d}} F_1\left (\frac {7}{3};\frac {1}{2},1;\frac {10}{3};-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{7 (b e-a f) \sqrt {c+d x}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(212\) vs. \(2(100)=200\).
time = 20.54, size = 212, normalized size = 2.12 \begin {gather*} \frac {6 \sqrt {c+d x} \left (7 b d f (a+b x)+\left (\frac {d (a+b x)}{b (c+d x)}\right )^{2/3} \left (7 b (5 b d e+2 b c f-7 a d f) F_1\left (\frac {1}{6};\frac {2}{3},1;\frac {7}{6};\frac {b c-a d}{b c+b d x},\frac {-d e+c f}{f (c+d x)}\right )-\frac {(b c-a d) (3 b d e+2 b c f-5 a d f) F_1\left (\frac {7}{6};\frac {2}{3},1;\frac {13}{6};\frac {b c-a d}{b c+b d x},\frac {-d e+c f}{f (c+d x)}\right )}{c+d x}\right )\right )}{35 d^2 f^2 (a+b x)^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{\frac {4}{3}}}{\left (f x +e \right ) \sqrt {d x +c}}\, dx\]
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 {\left (a + b x\right )^{\frac {4}{3}}}{\sqrt {c + d x} \left (e + f x\right )}\, 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 {{\left (a+b\,x\right )}^{4/3}}{\left (e+f\,x\right )\,\sqrt {c+d\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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