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

Optimal. Leaf size=125 \[ \frac{2 g \sqrt{d+e x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c d e}-\frac{2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (2 a e^2 g-c d (3 e f-d g)\right )}{3 c^2 d^2 e \sqrt{d+e x}} \]

[Out]

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

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Rubi [A]  time = 0.334967, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{2 g \sqrt{d+e x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c d e}-\frac{2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (2 a e^2 g-c d (3 e f-d g)\right )}{3 c^2 d^2 e \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 35.6517, size = 119, normalized size = 0.95 \[ \frac{2 g \sqrt{d + e x} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{3 c d e} - \frac{2 \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )} \left (2 a e^{2} g + c d^{2} g - 3 c d e f\right )}{3 c^{2} d^{2} e \sqrt{d + e x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((g*x+f)*(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*g*sqrt(d + e*x)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(3*c*d*e) - 2*s
qrt(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
)/(3*c**2*d**2*e*sqrt(d + e*x))

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Mathematica [A]  time = 0.0601866, size = 53, normalized size = 0.42 \[ \frac{2 \sqrt{(d+e x) (a e+c d x)} (c d (3 f+g x)-2 a e g)}{3 c^2 d^2 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[d + e*x]*(f + g*x))/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)]*(-2*a*e*g + c*d*(3*f + g*x)))/(3*c^2*d^2*Sqrt[d
+ e*x])

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Maple [A]  time = 0.008, size = 67, normalized size = 0.5 \[ -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -xcdg+2\,aeg-3\,cdf \right ) }{3\,{c}^{2}{d}^{2}}\sqrt{ex+d}{\frac{1}{\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*(c*d*x+a*e)*(-c*d*g*x+2*a*e*g-3*c*d*f)*(e*x+d)^(1/2)/c^2/d^2/(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 = 0.761707, size = 88, normalized size = 0.7 \[ \frac{2 \, \sqrt{c d x + a e} f}{c d} + \frac{2 \,{\left (c^{2} d^{2} x^{2} - a c d e x - 2 \, a^{2} e^{2}\right )} g}{3 \, \sqrt{c d x + a e} c^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(e*x + d)*(g*x + f)/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/(c*d) + 2/3*(c^2*d^2*x^2 - a*c*d*e*x - 2*a^2*e^2)*g/(sqrt(
c*d*x + a*e)*c^2*d^2)

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Fricas [A]  time = 0.293412, size = 198, normalized size = 1.58 \[ \frac{2 \,{\left (c^{2} d^{2} e g x^{3} + 3 \, a c d^{2} e f - 2 \, a^{2} d e^{2} g +{\left (3 \, c^{2} d^{2} e f +{\left (c^{2} d^{3} - a c d e^{2}\right )} g\right )} x^{2} +{\left (3 \,{\left (c^{2} d^{3} + a c d e^{2}\right )} f -{\left (a c d^{2} e + 2 \, a^{2} e^{3}\right )} g\right )} x\right )}}{3 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} c^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

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

Verification of antiderivative is not currently implemented for this CAS.

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