3.658 \(\int \frac{\sqrt{d+e x} (f+g x)^2}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx\)

Optimal. Leaf size=200 \[ -\frac{8 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g) \left (2 a e^2 g-c d (3 e f-d g)\right )}{15 c^3 d^3 e \sqrt{d+e x}}+\frac{8 g \sqrt{d+e x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}{15 c^2 d^2 e}+\frac{2 (f+g x)^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 c d \sqrt{d+e x}} \]

[Out]

(-8*(c*d*f - a*e*g)*(2*a*e^2*g - c*d*(3*e*f - d*g))*Sqrt[a*d*e + (c*d^2 + a*e^2)
*x + c*d*e*x^2])/(15*c^3*d^3*e*Sqrt[d + e*x]) + (8*g*(c*d*f - a*e*g)*Sqrt[d + e*
x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(15*c^2*d^2*e) + (2*(f + g*x)^2*
Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(5*c*d*Sqrt[d + e*x])

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Rubi [A]  time = 0.707216, 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 \[ -\frac{8 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g) \left (2 a e^2 g-c d (3 e f-d g)\right )}{15 c^3 d^3 e \sqrt{d+e x}}+\frac{8 g \sqrt{d+e x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}{15 c^2 d^2 e}+\frac{2 (f+g x)^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 c d \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[d + e*x]*(f + g*x)^2)/Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]

[Out]

(-8*(c*d*f - a*e*g)*(2*a*e^2*g - c*d*(3*e*f - d*g))*Sqrt[a*d*e + (c*d^2 + a*e^2)
*x + c*d*e*x^2])/(15*c^3*d^3*e*Sqrt[d + e*x]) + (8*g*(c*d*f - a*e*g)*Sqrt[d + e*
x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(15*c^2*d^2*e) + (2*(f + g*x)^2*
Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(5*c*d*Sqrt[d + e*x])

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Rubi in Sympy [A]  time = 61.8147, size = 194, normalized size = 0.97 \[ \frac{2 \left (f + g x\right )^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{5 c d \sqrt{d + e x}} - \frac{8 g \sqrt{d + e x} \left (a e g - c d f\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{15 c^{2} d^{2} e} + \frac{8 \left (a e g - c d f\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )} \left (2 a e^{2} g + c d^{2} g - 3 c d e f\right )}{15 c^{3} d^{3} e \sqrt{d + e x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((g*x+f)**2*(e*x+d)**(1/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)

[Out]

2*(f + g*x)**2*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(5*c*d*sqrt(d + e*
x)) - 8*g*sqrt(d + e*x)*(a*e*g - c*d*f)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*
d**2))/(15*c**2*d**2*e) + 8*(a*e*g - c*d*f)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2
+ c*d**2))*(2*a*e**2*g + c*d**2*g - 3*c*d*e*f)/(15*c**3*d**3*e*sqrt(d + e*x))

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Mathematica [A]  time = 0.125982, size = 89, normalized size = 0.44 \[ \frac{2 \sqrt{(d+e x) (a e+c d x)} \left (8 a^2 e^2 g^2-4 a c d e g (5 f+g x)+c^2 d^2 \left (15 f^2+10 f g x+3 g^2 x^2\right )\right )}{15 c^3 d^3 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[d + e*x]*(f + g*x)^2)/Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]

[Out]

(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(8*a^2*e^2*g^2 - 4*a*c*d*e*g*(5*f + g*x) + c^2*
d^2*(15*f^2 + 10*f*g*x + 3*g^2*x^2)))/(15*c^3*d^3*Sqrt[d + e*x])

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Maple [A]  time = 0.011, size = 116, normalized size = 0.6 \[{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 3\,{g}^{2}{x}^{2}{c}^{2}{d}^{2}-4\,acde{g}^{2}x+10\,{c}^{2}{d}^{2}fgx+8\,{a}^{2}{e}^{2}{g}^{2}-20\,acdefg+15\,{f}^{2}{c}^{2}{d}^{2} \right ) }{15\,{c}^{3}{d}^{3}}\sqrt{ex+d}{\frac{1}{\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((g*x+f)^2*(e*x+d)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x)

[Out]

2/15*(c*d*x+a*e)*(3*c^2*d^2*g^2*x^2-4*a*c*d*e*g^2*x+10*c^2*d^2*f*g*x+8*a^2*e^2*g
^2-20*a*c*d*e*f*g+15*c^2*d^2*f^2)*(e*x+d)^(1/2)/c^3/d^3/(c*d*e*x^2+a*e^2*x+c*d^2
*x+a*d*e)^(1/2)

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Maxima [A]  time = 0.764394, size = 180, normalized size = 0.9 \[ \frac{2 \, \sqrt{c d x + a e} f^{2}}{c d} + \frac{4 \,{\left (c^{2} d^{2} x^{2} - a c d e x - 2 \, a^{2} e^{2}\right )} f g}{3 \, \sqrt{c d x + a e} c^{2} d^{2}} + \frac{2 \,{\left (3 \, c^{3} d^{3} x^{3} - a c^{2} d^{2} e x^{2} + 4 \, a^{2} c d e^{2} x + 8 \, a^{3} e^{3}\right )} g^{2}}{15 \, \sqrt{c d x + a e} c^{3} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(e*x + d)*(g*x + f)^2/sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x),x, algorithm="maxima")

[Out]

2*sqrt(c*d*x + a*e)*f^2/(c*d) + 4/3*(c^2*d^2*x^2 - a*c*d*e*x - 2*a^2*e^2)*f*g/(s
qrt(c*d*x + a*e)*c^2*d^2) + 2/15*(3*c^3*d^3*x^3 - a*c^2*d^2*e*x^2 + 4*a^2*c*d*e^
2*x + 8*a^3*e^3)*g^2/(sqrt(c*d*x + a*e)*c^3*d^3)

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Fricas [A]  time = 0.287774, size = 378, normalized size = 1.89 \[ \frac{2 \,{\left (3 \, c^{3} d^{3} e g^{2} x^{4} + 15 \, a c^{2} d^{3} e f^{2} - 20 \, a^{2} c d^{2} e^{2} f g + 8 \, a^{3} d e^{3} g^{2} +{\left (10 \, c^{3} d^{3} e f g +{\left (3 \, c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} g^{2}\right )} x^{3} +{\left (15 \, c^{3} d^{3} e f^{2} + 10 \,{\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} f g -{\left (a c^{2} d^{3} e - 4 \, a^{2} c d e^{3}\right )} g^{2}\right )} x^{2} +{\left (15 \,{\left (c^{3} d^{4} + a c^{2} d^{2} e^{2}\right )} f^{2} - 10 \,{\left (a c^{2} d^{3} e + 2 \, a^{2} c d e^{3}\right )} f g + 4 \,{\left (a^{2} c d^{2} e^{2} + 2 \, a^{3} e^{4}\right )} g^{2}\right )} x\right )}}{15 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} c^{3} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(e*x + d)*(g*x + f)^2/sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x),x, algorithm="fricas")

[Out]

2/15*(3*c^3*d^3*e*g^2*x^4 + 15*a*c^2*d^3*e*f^2 - 20*a^2*c*d^2*e^2*f*g + 8*a^3*d*
e^3*g^2 + (10*c^3*d^3*e*f*g + (3*c^3*d^4 - a*c^2*d^2*e^2)*g^2)*x^3 + (15*c^3*d^3
*e*f^2 + 10*(c^3*d^4 - a*c^2*d^2*e^2)*f*g - (a*c^2*d^3*e - 4*a^2*c*d*e^3)*g^2)*x
^2 + (15*(c^3*d^4 + a*c^2*d^2*e^2)*f^2 - 10*(a*c^2*d^3*e + 2*a^2*c*d*e^3)*f*g +
4*(a^2*c*d^2*e^2 + 2*a^3*e^4)*g^2)*x)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*
x)*sqrt(e*x + d)*c^3*d^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d + e x} \left (f + g x\right )^{2}}{\sqrt{\left (d + e x\right ) \left (a e + c d x\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x+f)**2*(e*x+d)**(1/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)

[Out]

Integral(sqrt(d + e*x)*(f + g*x)**2/sqrt((d + e*x)*(a*e + c*d*x)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{e x + d}{\left (g x + f\right )}^{2}}{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(e*x + d)*(g*x + f)^2/sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x),x, algorithm="giac")

[Out]

integrate(sqrt(e*x + d)*(g*x + f)^2/sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x),
x)