Optimal. Leaf size=193 \[ -\frac{4 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-4 b e g+3 c d g+5 c e f)}{15 c^3 e^2 \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-4 b e g+3 c d g+5 c e f)}{15 c^2 e^2}-\frac{2 g (d+e x)^{3/2} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{5 c e^2} \]
[Out]
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Rubi [A] time = 0.750226, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ -\frac{4 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-4 b e g+3 c d g+5 c e f)}{15 c^3 e^2 \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-4 b e g+3 c d g+5 c e f)}{15 c^2 e^2}-\frac{2 g (d+e x)^{3/2} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{5 c e^2} \]
Antiderivative was successfully verified.
[In] Int[((d + e*x)^(3/2)*(f + g*x))/Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2],x]
[Out]
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Rubi in Sympy [A] time = 67.4585, size = 185, normalized size = 0.96 \[ - \frac{2 g \left (d + e x\right )^{\frac{3}{2}} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{5 c e^{2}} + \frac{2 \sqrt{d + e x} \left (4 b e g - 3 c d g - 5 c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{15 c^{2} e^{2}} - \frac{4 \left (b e - 2 c d\right ) \left (4 b e g - 3 c d g - 5 c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{15 c^{3} e^{2} \sqrt{d + e x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**(3/2)*(g*x+f)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.140751, size = 118, normalized size = 0.61 \[ \frac{2 \sqrt{d+e x} (b e-c d+c e x) \left (8 b^2 e^2 g-2 b c e (13 d g+5 e f+2 e g x)+c^2 \left (18 d^2 g+d e (25 f+9 g x)+e^2 x (5 f+3 g x)\right )\right )}{15 c^3 e^2 \sqrt{(d+e x) (c (d-e x)-b e)}} \]
Antiderivative was successfully verified.
[In] Integrate[((d + e*x)^(3/2)*(f + g*x))/Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2],x]
[Out]
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Maple [A] time = 0.008, size = 139, normalized size = 0.7 \[{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( 3\,g{x}^{2}{c}^{2}{e}^{2}-4\,bc{e}^{2}gx+9\,{c}^{2}degx+5\,{c}^{2}{e}^{2}fx+8\,{b}^{2}{e}^{2}g-26\,bcdeg-10\,bc{e}^{2}f+18\,{c}^{2}{d}^{2}g+25\,{c}^{2}def \right ) }{15\,{c}^{3}{e}^{2}}\sqrt{ex+d}{\frac{1}{\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(3/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x)
[Out]
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Maxima [A] time = 0.727456, size = 271, normalized size = 1.4 \[ \frac{2 \,{\left (c^{2} e^{2} x^{2} - 5 \, c^{2} d^{2} + 7 \, b c d e - 2 \, b^{2} e^{2} +{\left (4 \, c^{2} d e - b c e^{2}\right )} x\right )} f}{3 \, \sqrt{-c e x + c d - b e} c^{2} e} + \frac{2 \,{\left (3 \, c^{3} e^{3} x^{3} - 18 \, c^{3} d^{3} + 44 \, b c^{2} d^{2} e - 34 \, b^{2} c d e^{2} + 8 \, b^{3} e^{3} +{\left (6 \, c^{3} d e^{2} - b c^{2} e^{3}\right )} x^{2} +{\left (9 \, c^{3} d^{2} e - 13 \, b c^{2} d e^{2} + 4 \, b^{2} c e^{3}\right )} x\right )} g}{15 \, \sqrt{-c e x + c d - b e} c^{3} e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(3/2)*(g*x + f)/sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.275722, size = 417, normalized size = 2.16 \[ \frac{2 \,{\left (3 \, c^{3} e^{4} g x^{4} +{\left (5 \, c^{3} e^{4} f +{\left (9 \, c^{3} d e^{3} - b c^{2} e^{4}\right )} g\right )} x^{3} +{\left (5 \,{\left (5 \, c^{3} d e^{3} - b c^{2} e^{4}\right )} f +{\left (15 \, c^{3} d^{2} e^{2} - 14 \, b c^{2} d e^{3} + 4 \, b^{2} c e^{4}\right )} g\right )} x^{2} - 5 \,{\left (5 \, c^{3} d^{3} e - 7 \, b c^{2} d^{2} e^{2} + 2 \, b^{2} c d e^{3}\right )} f - 2 \,{\left (9 \, c^{3} d^{4} - 22 \, b c^{2} d^{3} e + 17 \, b^{2} c d^{2} e^{2} - 4 \, b^{3} d e^{3}\right )} g -{\left (5 \,{\left (c^{3} d^{2} e^{2} - 6 \, b c^{2} d e^{3} + 2 \, b^{2} c e^{4}\right )} f +{\left (9 \, c^{3} d^{3} e - 31 \, b c^{2} d^{2} e^{2} + 30 \, b^{2} c d e^{3} - 8 \, b^{3} e^{4}\right )} g\right )} x\right )}}{15 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{e x + d} c^{3} e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(3/2)*(g*x + f)/sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{\frac{3}{2}} \left (f + g x\right )}{\sqrt{- \left (d + e x\right ) \left (b e - c d + c e x\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**(3/2)*(g*x+f)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(1/2),x)
[Out]
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GIAC/XCAS [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(3/2)*(g*x + f)/sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e),x, algorithm="giac")
[Out]