Optimal. Leaf size=200 \[ -\frac{8 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g) \left (2 a e^2 g-c d (5 e f-3 d g)\right )}{105 c^3 d^3 e (d+e x)^{3/2}}+\frac{8 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)}{35 c^2 d^2 e \sqrt{d+e x}}+\frac{2 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{7 c d (d+e x)^{3/2}} \]
[Out]
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Rubi [A] time = 0.68347, antiderivative size = 200, 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 \[ -\frac{8 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g) \left (2 a e^2 g-c d (5 e f-3 d g)\right )}{105 c^3 d^3 e (d+e x)^{3/2}}+\frac{8 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)}{35 c^2 d^2 e \sqrt{d+e x}}+\frac{2 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{7 c d (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[((f + g*x)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/Sqrt[d + e*x],x]
[Out]
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Rubi in Sympy [A] time = 61.6233, size = 196, normalized size = 0.98 \[ \frac{2 \left (f + g x\right )^{2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{7 c d \left (d + e x\right )^{\frac{3}{2}}} - \frac{8 g \left (a e g - c d f\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{35 c^{2} d^{2} e \sqrt{d + e x}} + \frac{8 \left (a e g - c d f\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}} \left (2 a e^{2} g + 3 c d^{2} g - 5 c d e f\right )}{105 c^{3} d^{3} e \left (d + e x\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((g*x+f)**2*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/(e*x+d)**(1/2),x)
[Out]
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Mathematica [A] time = 0.135204, size = 90, normalized size = 0.45 \[ \frac{2 ((d+e x) (a e+c d x))^{3/2} \left (8 a^2 e^2 g^2-4 a c d e g (7 f+3 g x)+c^2 d^2 \left (35 f^2+42 f g x+15 g^2 x^2\right )\right )}{105 c^3 d^3 (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[((f + g*x)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/Sqrt[d + e*x],x]
[Out]
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Maple [A] time = 0.012, size = 116, normalized size = 0.6 \[{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 15\,{g}^{2}{x}^{2}{c}^{2}{d}^{2}-12\,acde{g}^{2}x+42\,{c}^{2}{d}^{2}fgx+8\,{a}^{2}{e}^{2}{g}^{2}-28\,acdefg+35\,{f}^{2}{c}^{2}{d}^{2} \right ) }{105\,{c}^{3}{d}^{3}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade}{\frac{1}{\sqrt{ex+d}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((g*x+f)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^(1/2),x)
[Out]
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Maxima [A] time = 0.733543, size = 180, normalized size = 0.9 \[ \frac{2 \,{\left (c d x + a e\right )}^{\frac{3}{2}} f^{2}}{3 \, c d} + \frac{4 \,{\left (3 \, c^{2} d^{2} x^{2} + a c d e x - 2 \, a^{2} e^{2}\right )} \sqrt{c d x + a e} f g}{15 \, c^{2} d^{2}} + \frac{2 \,{\left (15 \, c^{3} d^{3} x^{3} + 3 \, a c^{2} d^{2} e x^{2} - 4 \, a^{2} c d e^{2} x + 8 \, a^{3} e^{3}\right )} \sqrt{c d x + a e} g^{2}}{105 \, c^{3} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(g*x + f)^2/sqrt(e*x + d),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.270867, size = 522, normalized size = 2.61 \[ \frac{2 \,{\left (15 \, c^{4} d^{4} e g^{2} x^{5} + 35 \, a^{2} c^{2} d^{3} e^{2} f^{2} - 28 \, a^{3} c d^{2} e^{3} f g + 8 \, a^{4} d e^{4} g^{2} + 3 \,{\left (14 \, c^{4} d^{4} e f g +{\left (5 \, c^{4} d^{5} + 6 \, a c^{3} d^{3} e^{2}\right )} g^{2}\right )} x^{4} +{\left (35 \, c^{4} d^{4} e f^{2} + 14 \,{\left (3 \, c^{4} d^{5} + 4 \, a c^{3} d^{3} e^{2}\right )} f g +{\left (18 \, a c^{3} d^{4} e - a^{2} c^{2} d^{2} e^{3}\right )} g^{2}\right )} x^{3} +{\left (35 \,{\left (c^{4} d^{5} + 2 \, a c^{3} d^{3} e^{2}\right )} f^{2} + 14 \,{\left (4 \, a c^{3} d^{4} e - a^{2} c^{2} d^{2} e^{3}\right )} f g -{\left (a^{2} c^{2} d^{3} e^{2} - 4 \, a^{3} c d e^{4}\right )} g^{2}\right )} x^{2} +{\left (35 \,{\left (2 \, a c^{3} d^{4} e + a^{2} c^{2} d^{2} e^{3}\right )} f^{2} - 14 \,{\left (a^{2} c^{2} d^{3} e^{2} + 2 \, a^{3} c d e^{4}\right )} f g + 4 \,{\left (a^{3} c d^{2} e^{3} + 2 \, a^{4} e^{5}\right )} g^{2}\right )} x\right )}}{105 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} c^{3} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(g*x + f)^2/sqrt(e*x + d),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\left (d + e x\right ) \left (a e + c d x\right )} \left (f + g x\right )^{2}}{\sqrt{d + e x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((g*x+f)**2*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/(e*x+d)**(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(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(g*x + f)^2/sqrt(e*x + d),x, algorithm="giac")
[Out]