3.2264 \(\int \frac{\sqrt{d+e x} (f+g x)}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx\)

Optimal. Leaf size=117 \[ -\frac{2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-2 b e g+c d g+3 c e f)}{3 c^2 e^2 \sqrt{d+e x}}-\frac{2 g \sqrt{d+e x} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{3 c e^2} \]

[Out]

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

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Rubi [A]  time = 0.455701, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043 \[ -\frac{2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-2 b e g+c d g+3 c e f)}{3 c^2 e^2 \sqrt{d+e x}}-\frac{2 g \sqrt{d+e x} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{3 c e^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 46.5025, size = 109, normalized size = 0.93 \[ - \frac{2 g \sqrt{d + e x} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{3 c e^{2}} + \frac{2 \left (2 b e g - c d g - 3 c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{3 c^{2} e^{2} \sqrt{d + e x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0989573, size = 63, normalized size = 0.54 \[ -\frac{2 \sqrt{(d+e x) (c (d-e x)-b e)} (c (2 d g+3 e f+e g x)-2 b e g)}{3 c^2 e^2 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 79, normalized size = 0.7 \[ -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( -cegx+2\,beg-2\,cdg-3\,fce \right ) }{3\,{c}^{2}{e}^{2}}\sqrt{ex+d}{\frac{1}{\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*(c*e*x+b*e-c*d)*(-c*e*g*x+2*b*e*g-2*c*d*g-3*c*e*f)*(e*x+d)^(1/2)/c^2/e^2/(-
c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)

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Maxima [A]  time = 0.730278, size = 149, normalized size = 1.27 \[ \frac{2 \,{\left (c e x - c d + b e\right )} f}{\sqrt{-c e x + c d - b e} c e} + \frac{2 \,{\left (c^{2} e^{2} x^{2} - 2 \, c^{2} d^{2} + 4 \, b c d e - 2 \, b^{2} e^{2} +{\left (c^{2} d e - b c e^{2}\right )} x\right )} g}{3 \, \sqrt{-c e x + c d - b e} c^{2} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(c*e*x - c*d + b*e)*f/(sqrt(-c*e*x + c*d - b*e)*c*e) + 2/3*(c^2*e^2*x^2 - 2*c^
2*d^2 + 4*b*c*d*e - 2*b^2*e^2 + (c^2*d*e - b*c*e^2)*x)*g/(sqrt(-c*e*x + c*d - b*
e)*c^2*e^2)

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Fricas [A]  time = 0.272447, size = 236, normalized size = 2.02 \[ \frac{2 \,{\left (c^{2} e^{3} g x^{3} +{\left (3 \, c^{2} e^{3} f +{\left (2 \, c^{2} d e^{2} - b c e^{3}\right )} g\right )} x^{2} - 3 \,{\left (c^{2} d^{2} e - b c d e^{2}\right )} f - 2 \,{\left (c^{2} d^{3} - 2 \, b c d^{2} e + b^{2} d e^{2}\right )} g +{\left (3 \, b c e^{3} f -{\left (c^{2} d^{2} e - 3 \, b c d e^{2} + 2 \, b^{2} e^{3}\right )} g\right )} x\right )}}{3 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{e x + d} c^{2} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(d + e*x)*(f + g*x)/sqrt(-(d + e*x)*(b*e - c*d + c*e*x)), x)

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GIAC/XCAS [A]  time = 22.8779, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x