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 = 0.07, 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 = {140, 139, 138} \[ \frac {2 \sqrt {a+b x} (c+d x)^{2/5} (e+f x)^{3/5} F_1\left (\frac {1}{2};-\frac {2}{5},-\frac {3}{5};\frac {3}{2};-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b \left (\frac {b (c+d x)}{b c-a d}\right )^{2/5} \left (\frac {b (e+f x)}{b e-a f}\right )^{3/5}} \]
Antiderivative was successfully verified.
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Rule 138
Rule 139
Rule 140
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] time = 2.91, size = 536, normalized size = 4.36 \[ \frac {2 \sqrt {a+b x} \left (15 b^2 (c+d x) (e+f x)-2 (a+b x) \left (\frac {9 \left (25 a^2 d^2 f^2-10 a b d f (2 c f+3 d e)+b^2 \left (-2 c^2 f^2+24 c d e f+3 d^2 e^2\right )\right ) F_1\left (\frac {1}{2};\frac {3}{5},\frac {2}{5};\frac {3}{2};\frac {a d-b c}{d (a+b x)},\frac {a f-b e}{f (a+b x)}\right )}{15 d f (a+b x) F_1\left (\frac {1}{2};\frac {3}{5},\frac {2}{5};\frac {3}{2};\frac {a d-b c}{d (a+b x)},\frac {a f-b e}{f (a+b x)}\right )+(4 a d f-4 b d e) F_1\left (\frac {3}{2};\frac {3}{5},\frac {7}{5};\frac {5}{2};\frac {a d-b c}{d (a+b x)},\frac {a f-b e}{f (a+b x)}\right )+6 f (a d-b c) F_1\left (\frac {3}{2};\frac {8}{5},\frac {2}{5};\frac {5}{2};\frac {a d-b c}{d (a+b x)},\frac {a f-b e}{f (a+b x)}\right )}+\frac {(b c-a d) (b e-a f) \left (\frac {b (c+d x)}{d (a+b x)}\right )^{3/5} \left (\frac {b (e+f x)}{f (a+b x)}\right )^{2/5} (-5 a d f+2 b c f+3 b d e) F_1\left (\frac {3}{2};\frac {3}{5},\frac {2}{5};\frac {5}{2};\frac {a d-b c}{d (a+b x)},\frac {a f-b e}{f (a+b x)}\right )}{d f (a+b x)^2}-\frac {3 b^2 (c+d x) (e+f x) (-5 a d f+2 b c f+3 b d e)}{d f (a+b x)^2}\right )\right )}{45 b^3 (c+d x)^{3/5} (e+f x)^{2/5}} \]
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 \[ \int \frac {{\left (d x + c\right )}^{\frac {2}{5}} {\left (f x + e\right )}^{\frac {3}{5}}}{\sqrt {b x + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.18, 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 \[ \int \frac {{\left (d x + c\right )}^{\frac {2}{5}} {\left (f x + e\right )}^{\frac {3}{5}}}{\sqrt {b x + a}}\,{d x} \]
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 \[ \int \frac {{\left (e+f\,x\right )}^{3/5}\,{\left (c+d\,x\right )}^{2/5}}{\sqrt {a+b\,x}} \,d x \]
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 \[ \int \frac {\left (c + d x\right )^{\frac {2}{5}} \left (e + f x\right )^{\frac {3}{5}}}{\sqrt {a + b x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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