Optimal. Leaf size=123 \[ \frac {2 \sqrt {a+b x} (c+d x)^{2/5} (e+f x)^{3/5} F_1\left (\frac {1}{2};-\frac {2}{5},-\frac {3}{5};\frac {3}{2};-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b \left (\frac {b (c+d x)}{b c-a d}\right )^{2/5} \left (\frac {b (e+f x)}{b e-a f}\right )^{3/5}} \]
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Rubi [A]
time = 0.05, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {145, 144, 143}
\begin {gather*} \frac {2 \sqrt {a+b x} (c+d x)^{2/5} (e+f x)^{3/5} F_1\left (\frac {1}{2};-\frac {2}{5},-\frac {3}{5};\frac {3}{2};-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b \left (\frac {b (c+d x)}{b c-a d}\right )^{2/5} \left (\frac {b (e+f x)}{b e-a f}\right )^{3/5}} \end {gather*}
Antiderivative was successfully verified.
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Rule 143
Rule 144
Rule 145
Rubi steps
\begin {align*} \int \frac {(c+d x)^{2/5} (e+f x)^{3/5}}{\sqrt {a+b x}} \, dx &=\frac {(c+d x)^{2/5} \int \frac {\left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{2/5} (e+f x)^{3/5}}{\sqrt {a+b x}} \, dx}{\left (\frac {b (c+d x)}{b c-a d}\right )^{2/5}}\\ &=\frac {\left ((c+d x)^{2/5} (e+f x)^{3/5}\right ) \int \frac {\left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{2/5} \left (\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}\right )^{3/5}}{\sqrt {a+b x}} \, dx}{\left (\frac {b (c+d x)}{b c-a d}\right )^{2/5} \left (\frac {b (e+f x)}{b e-a f}\right )^{3/5}}\\ &=\frac {2 \sqrt {a+b x} (c+d x)^{2/5} (e+f x)^{3/5} F_1\left (\frac {1}{2};-\frac {2}{5},-\frac {3}{5};\frac {3}{2};-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b \left (\frac {b (c+d x)}{b c-a d}\right )^{2/5} \left (\frac {b (e+f x)}{b e-a f}\right )^{3/5}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(536\) vs. \(2(123)=246\).
time = 22.00, size = 536, normalized size = 4.36 \begin {gather*} \frac {2 \sqrt {a+b x} \left (15 b^2 (c+d x) (e+f x)-2 (a+b x) \left (-\frac {3 b^2 (3 b d e+2 b c f-5 a d f) (c+d x) (e+f x)}{d f (a+b x)^2}+\frac {(b c-a d) (b e-a f) (3 b d e+2 b c f-5 a d f) \left (\frac {b (c+d x)}{d (a+b x)}\right )^{3/5} \left (\frac {b (e+f x)}{f (a+b x)}\right )^{2/5} F_1\left (\frac {3}{2};\frac {3}{5},\frac {2}{5};\frac {5}{2};\frac {-b c+a d}{d (a+b x)},\frac {-b e+a f}{f (a+b x)}\right )}{d f (a+b x)^2}+\frac {9 \left (25 a^2 d^2 f^2-10 a b d f (3 d e+2 c f)+b^2 \left (3 d^2 e^2+24 c d e f-2 c^2 f^2\right )\right ) F_1\left (\frac {1}{2};\frac {3}{5},\frac {2}{5};\frac {3}{2};\frac {-b c+a d}{d (a+b x)},\frac {-b e+a f}{f (a+b x)}\right )}{15 d f (a+b x) F_1\left (\frac {1}{2};\frac {3}{5},\frac {2}{5};\frac {3}{2};\frac {-b c+a d}{d (a+b x)},\frac {-b e+a f}{f (a+b x)}\right )+(-4 b d e+4 a d f) F_1\left (\frac {3}{2};\frac {3}{5},\frac {7}{5};\frac {5}{2};\frac {-b c+a d}{d (a+b x)},\frac {-b e+a f}{f (a+b x)}\right )+6 (-b c+a d) f F_1\left (\frac {3}{2};\frac {8}{5},\frac {2}{5};\frac {5}{2};\frac {-b c+a d}{d (a+b x)},\frac {-b e+a f}{f (a+b x)}\right )}\right )\right )}{45 b^3 (c+d x)^{3/5} (e+f x)^{2/5}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (d x +c \right )^{\frac {2}{5}} \left (f x +e \right )^{\frac {3}{5}}}{\sqrt {b x +a}}\, dx\]
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 (c + d x\right )^{\frac {2}{5}} \left (e + f x\right )^{\frac {3}{5}}}{\sqrt {a + b 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 {{\left (e+f\,x\right )}^{3/5}\,{\left (c+d\,x\right )}^{2/5}}{\sqrt {a+b\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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