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 = 0.501148, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107 \[ \frac{2 \sqrt{a+b x} (c+d x)^{2/5} (e+f x)^{3/5} F_1\left (\frac{1}{2};-\frac{2}{5},-\frac{3}{5};\frac{3}{2};-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{b \left (\frac{b (c+d x)}{b c-a d}\right )^{2/5} \left (\frac{b (e+f x)}{b e-a f}\right )^{3/5}} \]
Antiderivative was successfully verified.
[In] Int[((c + d*x)^(2/5)*(e + f*x)^(3/5))/Sqrt[a + b*x],x]
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Rubi in Sympy [A] time = 63.4715, size = 102, normalized size = 0.83 \[ \frac{2 \sqrt{a + b x} \left (c + d x\right )^{\frac{2}{5}} \left (e + f x\right )^{\frac{3}{5}} \operatorname{appellf_{1}}{\left (\frac{1}{2},- \frac{3}{5},- \frac{2}{5},\frac{3}{2},\frac{f \left (a + b x\right )}{a f - b e},\frac{d \left (a + b x\right )}{a d - b c} \right )}}{b \left (\frac{b \left (- c - d x\right )}{a d - b c}\right )^{\frac{2}{5}} \left (\frac{b \left (- e - f x\right )}{a f - b e}\right )^{\frac{3}{5}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)**(2/5)*(f*x+e)**(3/5)/(b*x+a)**(1/2),x)
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Mathematica [B] time = 10.7698, size = 661, normalized size = 5.37 \[ \frac{2 \sqrt{a+b x} \left (15 b^2 (c+d x) (e+f x)-2 (a+b x) \left (\frac{9 \left (25 a^2 d^2 f^2-10 a b d f (2 c f+3 d e)+b^2 \left (-2 c^2 f^2+24 c d e f+3 d^2 e^2\right )\right ) F_1\left (\frac{1}{2};\frac{3}{5},\frac{2}{5};\frac{3}{2};\frac{a d-b c}{d (a+b x)},\frac{a f-b e}{f (a+b x)}\right )}{15 d f (a+b x) F_1\left (\frac{1}{2};\frac{3}{5},\frac{2}{5};\frac{3}{2};\frac{a d-b c}{d (a+b x)},\frac{a f-b e}{f (a+b x)}\right )+(4 a d f-4 b d e) F_1\left (\frac{3}{2};\frac{3}{5},\frac{7}{5};\frac{5}{2};\frac{a d-b c}{d (a+b x)},\frac{a f-b e}{f (a+b x)}\right )+6 f (a d-b c) F_1\left (\frac{3}{2};\frac{8}{5},\frac{2}{5};\frac{5}{2};\frac{a d-b c}{d (a+b x)},\frac{a f-b e}{f (a+b x)}\right )}+\frac{(-5 a d f+2 b c f+3 b d e) \left (\frac{25 (a+b x) (b c-a d) (b e-a f) F_1\left (\frac{3}{2};\frac{3}{5},\frac{2}{5};\frac{5}{2};\frac{a d-b c}{d (a+b x)},\frac{a f-b e}{f (a+b x)}\right )}{25 d f (a+b x) F_1\left (\frac{3}{2};\frac{3}{5},\frac{2}{5};\frac{5}{2};\frac{a d-b c}{d (a+b x)},\frac{a f-b e}{f (a+b x)}\right )+(4 a d f-4 b d e) F_1\left (\frac{5}{2};\frac{3}{5},\frac{7}{5};\frac{7}{2};\frac{a d-b c}{d (a+b x)},\frac{a f-b e}{f (a+b x)}\right )+6 f (a d-b c) F_1\left (\frac{5}{2};\frac{8}{5},\frac{2}{5};\frac{7}{2};\frac{a d-b c}{d (a+b x)},\frac{a f-b e}{f (a+b x)}\right )}-\frac{3 b^2 (c+d x) (e+f x)}{d f}\right )}{(a+b x)^2}\right )\right )}{45 b^3 (c+d x)^{3/5} (e+f x)^{2/5}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[((c + d*x)^(2/5)*(e + f*x)^(3/5))/Sqrt[a + b*x],x]
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Maple [F] time = 0.116, size = 0, normalized size = 0. \[ \int{1 \left ( dx+c \right ) ^{{\frac{2}{5}}} \left ( fx+e \right ) ^{{\frac{3}{5}}}{\frac{1}{\sqrt{bx+a}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+c)^(2/5)*(f*x+e)^(3/5)/(b*x+a)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}^{\frac{2}{5}}{\left (f x + e\right )}^{\frac{3}{5}}}{\sqrt{b x + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^(2/5)*(f*x + e)^(3/5)/sqrt(b*x + a),x, algorithm="maxima")
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^(2/5)*(f*x + e)^(3/5)/sqrt(b*x + a),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x+c)**(2/5)*(f*x+e)**(3/5)/(b*x+a)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}^{\frac{2}{5}}{\left (f x + e\right )}^{\frac{3}{5}}}{\sqrt{b x + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^(2/5)*(f*x + e)^(3/5)/sqrt(b*x + a),x, algorithm="giac")
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