3.2177 \(\int \frac{(f+g x) \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^3} \, dx\)

Optimal. Leaf size=168 \[ -\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^3 (2 c d-b e)}-\frac{2 g \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (d+e x)}-\frac{\sqrt{c} g \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{e^2} \]

[Out]

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

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Rubi [A]  time = 0.685287, antiderivative size = 168, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^3 (2 c d-b e)}-\frac{2 g \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (d+e x)}-\frac{\sqrt{c} g \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{e^2} \]

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,x]

[Out]

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

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Rubi in Sympy [A]  time = 54.8414, size = 153, normalized size = 0.91 \[ - \frac{\sqrt{c} g \operatorname{atan}{\left (- \frac{e \left (- b - 2 c x\right )}{2 \sqrt{c} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} \right )}}{e^{2}} - \frac{2 g \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{e^{2} \left (d + e x\right )} - \frac{2 \left (d g - e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{3 e^{2} \left (d + e x\right )^{3} \left (b e - 2 c d\right )} \]

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,x)

[Out]

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

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Mathematica [C]  time = 0.585235, size = 164, normalized size = 0.98 \[ \frac{\sqrt{(d+e x) (c (d-e x)-b e)} \left (-\frac{2 (3 b e g-7 c d g+c e f)}{(d+e x) (b e-2 c d)}-\frac{3 i \sqrt{c} g \log \left (2 \sqrt{d+e x} \sqrt{c (d-e x)-b e}-\frac{i e (b+2 c x)}{\sqrt{c}}\right )}{\sqrt{d+e x} \sqrt{c (d-e x)-b e}}+\frac{2 (d g-e f)}{(d+e x)^2}\right )}{3 e^2} \]

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,x]

[Out]

(Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*((2*(-(e*f) + d*g))/(d + e*x)^2 - (2*(c*
e*f - 7*c*d*g + 3*b*e*g))/((-2*c*d + b*e)*(d + e*x)) - ((3*I)*Sqrt[c]*g*Log[((-I
)*e*(b + 2*c*x))/Sqrt[c] + 2*Sqrt[d + e*x]*Sqrt[-(b*e) + c*(d - e*x)]])/(Sqrt[d
+ e*x]*Sqrt[-(b*e) + c*(d - e*x)])))/(3*e^2)

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Maple [B]  time = 0.022, size = 404, normalized size = 2.4 \[ -2\,{\frac{g}{{e}^{3} \left ( -b{e}^{2}+2\,dec \right ) } \left ( -c \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+ \left ( -b{e}^{2}+2\,dec \right ) \left ({\frac{d}{e}}+x \right ) \right ) ^{3/2} \left ({\frac{d}{e}}+x \right ) ^{-2}}-2\,{\frac{cg}{e \left ( -b{e}^{2}+2\,dec \right ) }\sqrt{-c \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+ \left ( -b{e}^{2}+2\,dec \right ) \left ({\frac{d}{e}}+x \right ) }}+{\frac{bceg}{-b{e}^{2}+2\,dec}\arctan \left ({1\sqrt{c{e}^{2}} \left ( x+{\frac{d}{e}}-{\frac{-b{e}^{2}+2\,dec}{2\,c{e}^{2}}} \right ){\frac{1}{\sqrt{-c \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+ \left ( -b{e}^{2}+2\,dec \right ) \left ({\frac{d}{e}}+x \right ) }}}} \right ){\frac{1}{\sqrt{c{e}^{2}}}}}-2\,{\frac{g{c}^{2}d}{ \left ( -b{e}^{2}+2\,dec \right ) \sqrt{c{e}^{2}}}\arctan \left ({\sqrt{c{e}^{2}} \left ( x+{\frac{d}{e}}-1/2\,{\frac{-b{e}^{2}+2\,dec}{c{e}^{2}}} \right ){\frac{1}{\sqrt{-c \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+ \left ( -b{e}^{2}+2\,dec \right ) \left ({\frac{d}{e}}+x \right ) }}}} \right ) }-{\frac{-2\,dg+2\,ef}{3\,{e}^{4} \left ( -b{e}^{2}+2\,dec \right ) } \left ( -c \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+ \left ( -b{e}^{2}+2\,dec \right ) \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{3}{2}}} \left ({\frac{d}{e}}+x \right ) ^{-3}} \]

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,x)

[Out]

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

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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,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.732268, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left ({\left (2 \, c d e^{2} - b e^{3}\right )} g x^{2} + 2 \,{\left (2 \, c d^{2} e - b d e^{2}\right )} g x +{\left (2 \, c d^{3} - b d^{2} e\right )} g\right )} \sqrt{-c} \log \left (8 \, c^{2} e^{2} x^{2} + 8 \, b c e^{2} x - 4 \, c^{2} d^{2} + 4 \, b c d e + b^{2} e^{2} - 4 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (2 \, c e x + b e\right )} \sqrt{-c}\right ) - 4 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left ({\left (c d e - b e^{2}\right )} f +{\left (5 \, c d^{2} - 2 \, b d e\right )} g -{\left (c e^{2} f -{\left (7 \, c d e - 3 \, b e^{2}\right )} g\right )} x\right )}}{6 \,{\left (2 \, c d^{3} e^{2} - b d^{2} e^{3} +{\left (2 \, c d e^{4} - b e^{5}\right )} x^{2} + 2 \,{\left (2 \, c d^{2} e^{3} - b d e^{4}\right )} x\right )}}, -\frac{3 \,{\left ({\left (2 \, c d e^{2} - b e^{3}\right )} g x^{2} + 2 \,{\left (2 \, c d^{2} e - b d e^{2}\right )} g x +{\left (2 \, c d^{3} - b d^{2} e\right )} g\right )} \sqrt{c} \arctan \left (\frac{2 \, c e x + b e}{2 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{c}}\right ) + 2 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left ({\left (c d e - b e^{2}\right )} f +{\left (5 \, c d^{2} - 2 \, b d e\right )} g -{\left (c e^{2} f -{\left (7 \, c d e - 3 \, b e^{2}\right )} g\right )} x\right )}}{3 \,{\left (2 \, c d^{3} e^{2} - b d^{2} e^{3} +{\left (2 \, c d e^{4} - b e^{5}\right )} x^{2} + 2 \,{\left (2 \, c d^{2} e^{3} - b d e^{4}\right )} x\right )}}\right ] \]

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,x, algorithm="fricas")

[Out]

[1/6*(3*((2*c*d*e^2 - b*e^3)*g*x^2 + 2*(2*c*d^2*e - b*d*e^2)*g*x + (2*c*d^3 - b*
d^2*e)*g)*sqrt(-c)*log(8*c^2*e^2*x^2 + 8*b*c*e^2*x - 4*c^2*d^2 + 4*b*c*d*e + b^2
*e^2 - 4*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*c*e*x + b*e)*sqrt(-c)) -
4*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*((c*d*e - b*e^2)*f + (5*c*d^2 - 2*b
*d*e)*g - (c*e^2*f - (7*c*d*e - 3*b*e^2)*g)*x))/(2*c*d^3*e^2 - b*d^2*e^3 + (2*c*
d*e^4 - b*e^5)*x^2 + 2*(2*c*d^2*e^3 - b*d*e^4)*x), -1/3*(3*((2*c*d*e^2 - b*e^3)*
g*x^2 + 2*(2*c*d^2*e - b*d*e^2)*g*x + (2*c*d^3 - b*d^2*e)*g)*sqrt(c)*arctan(1/2*
(2*c*e*x + b*e)/(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(c))) + 2*sqrt(-
c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*((c*d*e - b*e^2)*f + (5*c*d^2 - 2*b*d*e)*g
- (c*e^2*f - (7*c*d*e - 3*b*e^2)*g)*x))/(2*c*d^3*e^2 - b*d^2*e^3 + (2*c*d*e^4 -
b*e^5)*x^2 + 2*(2*c*d^2*e^3 - b*d*e^4)*x)]

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

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,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.777763, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

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,x, algorithm="giac")

[Out]

sage0*x