3.2235 \(\int \frac{(f+g x) \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=186 \[ \frac{2 (e f-d g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt{d+e x}}-\frac{2 \sqrt{2 c d-b e} (e f-d g) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{e^2}-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 c e^2 (d+e x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.773768, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ \frac{2 (e f-d g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt{d+e x}}-\frac{2 \sqrt{2 c d-b e} (e f-d g) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{e^2}-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 c e^2 (d+e x)^{3/2}} \]

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])/(d + e*x)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 77.3358, size = 165, normalized size = 0.89 \[ \frac{2 \sqrt{b e - 2 c d} \left (d g - e f\right ) \operatorname{atan}{\left (\frac{\sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{\sqrt{d + e x} \sqrt{b e - 2 c d}} \right )}}{e^{2}} - \frac{2 \left (d g - e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{e^{2} \sqrt{d + e x}} - \frac{2 g \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{3 c e^{2} \left (d + e x\right )^{\frac{3}{2}}} \]

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)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.5439, size = 131, normalized size = 0.7 \[ \frac{2 \sqrt{(d+e x) (c (d-e x)-b e)} \left (\frac{3 \sqrt{2 c d-b e} (d g-e f) \tanh ^{-1}\left (\frac{\sqrt{-b e+c d-c e x}}{\sqrt{2 c d-b e}}\right )}{\sqrt{c (d-e x)-b e}}+e \left (\frac{b g}{c}+3 f+g x\right )-4 d g\right )}{3 e^2 \sqrt{d+e x}} \]

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])/(d + e*x)^(3/2),x]

[Out]

(2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(-4*d*g + e*(3*f + (b*g)/c + g*x) + (3
*Sqrt[2*c*d - b*e]*(-(e*f) + d*g)*ArcTanh[Sqrt[c*d - b*e - c*e*x]/Sqrt[2*c*d - b
*e]])/Sqrt[-(b*e) + c*(d - e*x)]))/(3*e^2*Sqrt[d + e*x])

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Maple [A]  time = 0.033, size = 330, normalized size = 1.8 \[{\frac{2}{3\,c{e}^{2}}\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}} \left ( 3\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) bcdeg-3\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) bc{e}^{2}f-6\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ){c}^{2}{d}^{2}g+6\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ){c}^{2}def+xceg\sqrt{-cex-be+cd}\sqrt{be-2\,cd}+beg\sqrt{-cex-be+cd}\sqrt{be-2\,cd}-4\,cdg\sqrt{-cex-be+cd}\sqrt{be-2\,cd}+3\,fce\sqrt{-cex-be+cd}\sqrt{be-2\,cd} \right ){\frac{1}{\sqrt{ex+d}}}{\frac{1}{\sqrt{-cex-be+cd}}}{\frac{1}{\sqrt{be-2\,cd}}}} \]

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)^(3/2),x)

[Out]

2/3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*(3*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e
-2*c*d)^(1/2))*b*c*d*e*g-3*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c*
e^2*f-6*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^2*d^2*g+6*arctan((-c*
e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^2*d*e*f+x*c*e*g*(-c*e*x-b*e+c*d)^(1/2)*(
b*e-2*c*d)^(1/2)+b*e*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-4*c*d*g*(-c*e*x-
b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+3*f*c*e*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2
))/(e*x+d)^(1/2)/(-c*e*x-b*e+c*d)^(1/2)/c/e^2/(b*e-2*c*d)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

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)/(e*x + d)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.297106, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \, c^{2} e^{3} g x^{3} + 3 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (c e f - c d g\right )} \sqrt{2 \, c d - b e} \sqrt{e x + d} \log \left (-\frac{c e^{2} x^{2} - 3 \, c d^{2} + 2 \, b d e - 2 \,{\left (c d e - b e^{2}\right )} x - 2 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{2 \, c d - b e} \sqrt{e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 2 \,{\left (3 \, c^{2} e^{3} f - 2 \,{\left (2 \, c^{2} d e^{2} - b c e^{3}\right )} g\right )} x^{2} - 6 \,{\left (c^{2} d^{2} e - b c d e^{2}\right )} f + 2 \,{\left (4 \, c^{2} d^{3} - 5 \, b c d^{2} e + b^{2} d e^{2}\right )} g + 2 \,{\left (3 \, b c e^{3} f -{\left (c^{2} d^{2} e + 3 \, b c d e^{2} - b^{2} e^{3}\right )} g\right )} x}{3 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{e x + d} c e^{2}}, -\frac{2 \,{\left (c^{2} e^{3} g x^{3} + 3 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (c e f - c d g\right )} \sqrt{-2 \, c d + b e} \sqrt{e x + d} \arctan \left (-\frac{\sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (2 \, c d - b e\right )} \sqrt{e x + d}}{{\left (c e^{2} x^{2} + b e^{2} x - c d^{2} + b d e\right )} \sqrt{-2 \, c d + b e}}\right ) +{\left (3 \, c^{2} e^{3} f - 2 \,{\left (2 \, c^{2} d e^{2} - b c e^{3}\right )} g\right )} x^{2} - 3 \,{\left (c^{2} d^{2} e - b c d e^{2}\right )} f +{\left (4 \, c^{2} d^{3} - 5 \, b c d^{2} e + b^{2} d e^{2}\right )} g +{\left (3 \, b c e^{3} f -{\left (c^{2} d^{2} e + 3 \, b c d e^{2} - b^{2} e^{3}\right )} g\right )} x\right )}}{3 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{e x + d} c e^{2}}\right ] \]

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)/(e*x + d)^(3/2),x, algorithm="fricas")

[Out]

[-1/3*(2*c^2*e^3*g*x^3 + 3*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(c*e*f - c
*d*g)*sqrt(2*c*d - b*e)*sqrt(e*x + d)*log(-(c*e^2*x^2 - 3*c*d^2 + 2*b*d*e - 2*(c
*d*e - b*e^2)*x - 2*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(2*c*d - b*e)
*sqrt(e*x + d))/(e^2*x^2 + 2*d*e*x + d^2)) + 2*(3*c^2*e^3*f - 2*(2*c^2*d*e^2 - b
*c*e^3)*g)*x^2 - 6*(c^2*d^2*e - b*c*d*e^2)*f + 2*(4*c^2*d^3 - 5*b*c*d^2*e + b^2*
d*e^2)*g + 2*(3*b*c*e^3*f - (c^2*d^2*e + 3*b*c*d*e^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*e^2), -2/3*(c^2*e^3*g*x^3 + 3*
sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(c*e*f - c*d*g)*sqrt(-2*c*d + b*e)*sq
rt(e*x + d)*arctan(-sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*c*d - b*e)*sqr
t(e*x + d)/((c*e^2*x^2 + b*e^2*x - c*d^2 + b*d*e)*sqrt(-2*c*d + b*e))) + (3*c^2*
e^3*f - 2*(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 + (4*c^2*
d^3 - 5*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 - 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*e^2)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- \left (d + e x\right ) \left (b e - c d + c e x\right )} \left (f + g x\right )}{\left (d + e x\right )^{\frac{3}{2}}}\, dx \]

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)**(3/2),x)

[Out]

Integral(sqrt(-(d + e*x)*(b*e - c*d + c*e*x))*(f + g*x)/(d + e*x)**(3/2), 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)/(e*x + d)^(3/2),x, algorithm="giac")

[Out]

Exception raised: AttributeError