3.2234 \(\int \frac{(f+g x) \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=118 \[ -\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-2 b e g-c d g+5 c e f)}{15 c^2 e^2 (d+e x)^{3/2}}-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{5 c e^2 \sqrt{d+e x}} \]

[Out]

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

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Rubi [A]  time = 0.454698, antiderivative size = 118, 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 \[ -\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-2 b e g-c d g+5 c e f)}{15 c^2 e^2 (d+e x)^{3/2}}-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{5 c e^2 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 43.3288, size = 109, normalized size = 0.92 \[ - \frac{2 g \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{5 c e^{2} \sqrt{d + e x}} + \frac{2 \left (2 b e g + c d g - 5 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{15 c^{2} e^{2} \left (d + e x\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.103689, size = 76, normalized size = 0.64 \[ \frac{2 (b e-c d+c e x) \sqrt{(d+e x) (c (d-e x)-b e)} (c (2 d g+5 e f+3 e g x)-2 b e g)}{15 c^2 e^2 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 79, normalized size = 0.7 \[ -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( -3\,cegx+2\,beg-2\,cdg-5\,cef \right ) }{15\,{c}^{2}{e}^{2}}\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}{\frac{1}{\sqrt{ex+d}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15*(c*e*x+b*e-c*d)*(-3*c*e*g*x+2*b*e*g-2*c*d*g-5*c*e*f)*(-c*e^2*x^2-b*e^2*x-b
*d*e+c*d^2)^(1/2)/c^2/e^2/(e*x+d)^(1/2)

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Maxima [A]  time = 0.722946, size = 151, normalized size = 1.28 \[ \frac{2 \,{\left (c e x - c d + b e\right )} \sqrt{-c e x + c d - b e} f}{3 \, c e} + \frac{2 \,{\left (3 \, c^{2} e^{2} x^{2} - 2 \, c^{2} d^{2} + 4 \, b c d e - 2 \, b^{2} e^{2} -{\left (c^{2} d e - b c e^{2}\right )} x\right )} \sqrt{-c e x + c d - b e} g}{15 \, c^{2} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(c*e*x - c*d + b*e)*sqrt(-c*e*x + c*d - b*e)*f/(c*e) + 2/15*(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)*sqrt(-c*e*x + c*d -
b*e)*g/(c^2*e^2)

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Fricas [A]  time = 0.295134, size = 398, normalized size = 3.37 \[ -\frac{2 \,{\left (3 \, c^{3} e^{4} g x^{4} +{\left (5 \, c^{3} e^{4} f -{\left (c^{3} d e^{3} - 4 \, b c^{2} e^{4}\right )} g\right )} x^{3} -{\left (5 \,{\left (c^{3} d e^{3} - 2 \, b c^{2} e^{4}\right )} f +{\left (5 \, c^{3} d^{2} e^{2} - 6 \, b c^{2} d e^{3} + b^{2} c e^{4}\right )} g\right )} x^{2} + 5 \,{\left (c^{3} d^{3} e - 2 \, b c^{2} d^{2} e^{2} + b^{2} c d e^{3}\right )} f + 2 \,{\left (c^{3} d^{4} - 3 \, b c^{2} d^{3} e + 3 \, b^{2} c d^{2} e^{2} - b^{3} d e^{3}\right )} g -{\left (5 \,{\left (c^{3} d^{2} e^{2} - b^{2} c e^{4}\right )} f -{\left (c^{3} d^{3} e - 4 \, b c^{2} d^{2} e^{2} + 5 \, b^{2} c d e^{3} - 2 \, b^{3} e^{4}\right )} g\right )} x\right )}}{15 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{e x + d} c^{2} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15*(3*c^3*e^4*g*x^4 + (5*c^3*e^4*f - (c^3*d*e^3 - 4*b*c^2*e^4)*g)*x^3 - (5*(c
^3*d*e^3 - 2*b*c^2*e^4)*f + (5*c^3*d^2*e^2 - 6*b*c^2*d*e^3 + b^2*c*e^4)*g)*x^2 +
5*(c^3*d^3*e - 2*b*c^2*d^2*e^2 + b^2*c*d*e^3)*f + 2*(c^3*d^4 - 3*b*c^2*d^3*e +
3*b^2*c*d^2*e^2 - b^3*d*e^3)*g - (5*(c^3*d^2*e^2 - b^2*c*e^4)*f - (c^3*d^3*e - 4
*b*c^2*d^2*e^2 + 5*b^2*c*d*e^3 - 2*b^3*e^4)*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. \[ \int \frac{\sqrt{- \left (d + e x\right ) \left (b e - c d + c e x\right )} \left (f + g x\right )}{\sqrt{d + e x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: AttributeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: AttributeError