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 = 0.319046, 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, Rules used = {792, 662, 621, 204} \[ -\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^3 (2 c d-b e)}-\frac{2 g \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (d+e x)}-\frac{\sqrt{c} g \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{e^2} \]

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

Rule 792

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d*g - e*f)*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/((2*c*d - b*e)*(m + p + 1)), x] + Dist[(m*(g*(c*d - b*e)
+ c*e*f) + e*(p + 1)*(2*c*f - b*g))/(e*(2*c*d - b*e)*(m + p + 1)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p,
x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ((L
tQ[m, -1] &&  !IGtQ[m + p + 1, 0]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p + 1, 0]

Rule 662

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
a + b*x + c*x^2)^p)/(e*(m + p + 1)), x] - Dist[(c*p)/(e^2*(m + p + 1)), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2
)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[
p, 0] && (LtQ[m, -2] || EqQ[m + 2*p + 1, 0]) && NeQ[m + p + 1, 0] && IntegerQ[2*p]

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \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 &=-\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 (2 c d-b e) (d+e x)^3}+\frac{g \int \frac{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^2} \, dx}{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{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 (2 c d-b e) (d+e x)^3}-\frac{(c g) \int \frac{1}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{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{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 (2 c d-b e) (d+e x)^3}-\frac{(2 c g) \operatorname{Subst}\left (\int \frac{1}{-4 c e^2-x^2} \, dx,x,\frac{-b e^2-2 c e^2 x}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}\right )}{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{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 (2 c d-b e) (d+e x)^3}-\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}\\ \end{align*}

Mathematica [C]  time = 0.234121, size = 146, normalized size = 0.87 \[ \frac{2 \sqrt{(d+e x) (c (d-e x)-b e)} \left ((b e-c d+c e x) (-b e g+c d g+c e f)+\frac{g (b e-2 c d)^2 \, _2F_1\left (-\frac{3}{2},-\frac{3}{2};-\frac{1}{2};\frac{c (d+e x)}{2 c d-b e}\right )}{\sqrt{\frac{b e-c d+c e x}{b e-2 c d}}}\right )}{3 c e^2 (d+e x)^2 (2 c d-b e)} \]

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]

(2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*((c*e*f + c*d*g - b*e*g)*(-(c*d) + b*e + c*e*x) + ((-2*c*d + b*e)^2*
g*Hypergeometric2F1[-3/2, -3/2, -1/2, (c*(d + e*x))/(2*c*d - b*e)])/Sqrt[(-(c*d) + b*e + c*e*x)/(-2*c*d + b*e)
]))/(3*c*e^2*(2*c*d - b*e)*(d + e*x)^2)

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Maple [B]  time = 0.013, size = 404, normalized size = 2.4 \begin{align*} -{\frac{-2\,dg+2\,ef}{3\,{e}^{4} \left ( -b{e}^{2}+2\,cde \right ) } \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}c{e}^{2}+ \left ( -b{e}^{2}+2\,cde \right ) \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{d}{e}} \right ) ^{-3}}-2\,{\frac{g}{{e}^{3} \left ( -b{e}^{2}+2\,cde \right ) } \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}c{e}^{2}+ \left ( -b{e}^{2}+2\,cde \right ) \left ( x+{\frac{d}{e}} \right ) \right ) ^{3/2} \left ( x+{\frac{d}{e}} \right ) ^{-2}}-2\,{\frac{gc}{e \left ( -b{e}^{2}+2\,cde \right ) }\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}c{e}^{2}+ \left ( -b{e}^{2}+2\,cde \right ) \left ( x+{\frac{d}{e}} \right ) }}+{\frac{gecb}{-b{e}^{2}+2\,cde}\arctan \left ({\sqrt{c{e}^{2}} \left ( x+{\frac{d}{e}}-{\frac{-b{e}^{2}+2\,cde}{2\,c{e}^{2}}} \right ){\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}c{e}^{2}+ \left ( -b{e}^{2}+2\,cde \right ) \left ( x+{\frac{d}{e}} \right ) }}}} \right ){\frac{1}{\sqrt{c{e}^{2}}}}}-2\,{\frac{g{c}^{2}d}{ \left ( -b{e}^{2}+2\,cde \right ) \sqrt{c{e}^{2}}}\arctan \left ({\sqrt{c{e}^{2}} \left ( x+{\frac{d}{e}}-1/2\,{\frac{-b{e}^{2}+2\,cde}{c{e}^{2}}} \right ){\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}c{e}^{2}+ \left ( -b{e}^{2}+2\,cde \right ) \left ( x+{\frac{d}{e}} \right ) }}}} \right ) } \end{align*}

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]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 10.0301, size = 1200, normalized size = 7.14 \begin{align*} \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{\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}}{2 \,{\left (c^{2} e^{2} x^{2} + b c e^{2} x - c^{2} d^{2} + b c d e\right )}}\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 ] \end{align*}

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, 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)*ar
ctan(1/2*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*c*e*x + b*e)*sqrt(c)/(c^2*e^2*x^2 + b*c*e^2*x - c^2*d^2
+ b*c*d*e)) - 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. \begin{align*} \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 \end{align*}

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 [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

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, algorithm="giac")

[Out]

Exception raised: NotImplementedError