3.2218 \(\int \frac{(d+e x)^2 (f+g x)}{(c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=213 \[ \frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-3 b e g+4 c d g+2 c e f)}{c^2 e^2 (2 c d-b e)}-\frac{(-3 b e g+4 c d g+2 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{2 c^{5/2} e^2}+\frac{2 (d+e x)^2 (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}} \]

[Out]

(2*(c*e*f + c*d*g - b*e*g)*(d + e*x)^2)/(c*e^2*(2*c*d - b*e)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) + ((2*
c*e*f + 4*c*d*g - 3*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(c^2*e^2*(2*c*d - b*e)) - ((2*c*e*f + 4*
c*d*g - 3*b*e*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])])/(2*c^(5/2)*e^2
)

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Rubi [A]  time = 0.284045, antiderivative size = 213, 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 = {788, 640, 621, 204} \[ \frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-3 b e g+4 c d g+2 c e f)}{c^2 e^2 (2 c d-b e)}-\frac{(-3 b e g+4 c d g+2 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{2 c^{5/2} e^2}+\frac{2 (d+e x)^2 (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(c*e*f + c*d*g - b*e*g)*(d + e*x)^2)/(c*e^2*(2*c*d - b*e)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) + ((2*
c*e*f + 4*c*d*g - 3*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(c^2*e^2*(2*c*d - b*e)) - ((2*c*e*f + 4*
c*d*g - 3*b*e*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])])/(2*c^(5/2)*e^2
)

Rule 788

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

Rule 640

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

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{(d+e x)^2 (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx &=\frac{2 (c e f+c d g-b e g) (d+e x)^2}{c e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{(2 c e f+4 c d g-3 b e g) \int \frac{d+e x}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{c e (2 c d-b e)}\\ &=\frac{2 (c e f+c d g-b e g) (d+e x)^2}{c e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{(2 c e f+4 c d g-3 b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{c^2 e^2 (2 c d-b e)}-\frac{(2 c e f+4 c d g-3 b e g) \int \frac{1}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{2 c^2 e}\\ &=\frac{2 (c e f+c d g-b e g) (d+e x)^2}{c e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{(2 c e f+4 c d g-3 b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{c^2 e^2 (2 c d-b e)}-\frac{(2 c e f+4 c d g-3 b e 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 )}{c^2 e}\\ &=\frac{2 (c e f+c d g-b e g) (d+e x)^2}{c e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{(2 c e f+4 c d g-3 b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{c^2 e^2 (2 c d-b e)}-\frac{(2 c e f+4 c d g-3 b e 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 )}{2 c^{5/2} e^2}\\ \end{align*}

Mathematica [A]  time = 0.627517, size = 201, normalized size = 0.94 \[ \frac{\sqrt{c} \sqrt{e} (d+e x) \sqrt{e (2 c d-b e)} (c (3 d g+2 e f-e g x)-3 b e g)+e \sqrt{d+e x} (b e-2 c d) \sqrt{\frac{b e-c d+c e x}{b e-2 c d}} (-3 b e g+4 c d g+2 c e f) \sin ^{-1}\left (\frac{\sqrt{c} \sqrt{e} \sqrt{d+e x}}{\sqrt{e (2 c d-b e)}}\right )}{c^{5/2} e^{5/2} \sqrt{e (2 c d-b e)} \sqrt{(d+e x) (c (d-e x)-b e)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c]*Sqrt[e]*Sqrt[e*(2*c*d - b*e)]*(d + e*x)*(-3*b*e*g + c*(2*e*f + 3*d*g - e*g*x)) + e*(-2*c*d + b*e)*(2*
c*e*f + 4*c*d*g - 3*b*e*g)*Sqrt[d + e*x]*Sqrt[(-(c*d) + b*e + c*e*x)/(-2*c*d + b*e)]*ArcSin[(Sqrt[c]*Sqrt[e]*S
qrt[d + e*x])/Sqrt[e*(2*c*d - b*e)]])/(c^(5/2)*e^(5/2)*Sqrt[e*(2*c*d - b*e)]*Sqrt[(d + e*x)*(-(b*e) + c*(d - e
*x))])

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Maple [B]  time = 0.01, size = 1320, normalized size = 6.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 5.33575, size = 1057, normalized size = 4.96 \begin{align*} \left [-\frac{{\left (2 \,{\left (c^{2} d e - b c e^{2}\right )} f +{\left (4 \, c^{2} d^{2} - 7 \, b c d e + 3 \, b^{2} e^{2}\right )} g -{\left (2 \, c^{2} e^{2} f +{\left (4 \, c^{2} d e - 3 \, b c e^{2}\right )} g\right )} x\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 \,{\left (c^{2} e g x - 2 \, c^{2} e f - 3 \,{\left (c^{2} d - b c e\right )} g\right )} \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{4 \,{\left (c^{4} e^{3} x - c^{4} d e^{2} + b c^{3} e^{3}\right )}}, -\frac{{\left (2 \,{\left (c^{2} d e - b c e^{2}\right )} f +{\left (4 \, c^{2} d^{2} - 7 \, b c d e + 3 \, b^{2} e^{2}\right )} g -{\left (2 \, c^{2} e^{2} f +{\left (4 \, c^{2} d e - 3 \, b c e^{2}\right )} g\right )} x\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 \,{\left (c^{2} e g x - 2 \, c^{2} e f - 3 \,{\left (c^{2} d - b c e\right )} g\right )} \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{2 \,{\left (c^{4} e^{3} x - c^{4} d e^{2} + b c^{3} e^{3}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.31007, size = 576, normalized size = 2.7 \begin{align*} \frac{\sqrt{-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}{\left ({\left (\frac{{\left (4 \, c^{3} d^{2} g e^{3} - 4 \, b c^{2} d g e^{4} + b^{2} c g e^{5}\right )} x}{4 \, c^{4} d^{2} e^{3} - 4 \, b c^{3} d e^{4} + b^{2} c^{2} e^{5}} - \frac{8 \, c^{3} d^{3} g e^{2} + 8 \, c^{3} d^{2} f e^{3} - 20 \, b c^{2} d^{2} g e^{3} - 8 \, b c^{2} d f e^{4} + 14 \, b^{2} c d g e^{4} + 2 \, b^{2} c f e^{5} - 3 \, b^{3} g e^{5}}{4 \, c^{4} d^{2} e^{3} - 4 \, b c^{3} d e^{4} + b^{2} c^{2} e^{5}}\right )} x - \frac{12 \, c^{3} d^{4} g e + 8 \, c^{3} d^{3} f e^{2} - 24 \, b c^{2} d^{3} g e^{2} - 8 \, b c^{2} d^{2} f e^{3} + 15 \, b^{2} c d^{2} g e^{3} + 2 \, b^{2} c d f e^{4} - 3 \, b^{3} d g e^{4}}{4 \, c^{4} d^{2} e^{3} - 4 \, b c^{3} d e^{4} + b^{2} c^{2} e^{5}}\right )}}{c x^{2} e^{2} - c d^{2} + b x e^{2} + b d e} - \frac{{\left (4 \, c d g + 2 \, c f e - 3 \, b g e\right )} \sqrt{-c e^{2}} e^{\left (-3\right )} \log \left ({\left | -2 \,{\left (\sqrt{-c e^{2}} x - \sqrt{-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )} c - \sqrt{-c e^{2}} b \right |}\right )}{2 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e)*(((4*c^3*d^2*g*e^3 - 4*b*c^2*d*g*e^4 + b^2*c*g*e^5)*x/(4*c^4*d^2*e^
3 - 4*b*c^3*d*e^4 + b^2*c^2*e^5) - (8*c^3*d^3*g*e^2 + 8*c^3*d^2*f*e^3 - 20*b*c^2*d^2*g*e^3 - 8*b*c^2*d*f*e^4 +
14*b^2*c*d*g*e^4 + 2*b^2*c*f*e^5 - 3*b^3*g*e^5)/(4*c^4*d^2*e^3 - 4*b*c^3*d*e^4 + b^2*c^2*e^5))*x - (12*c^3*d^
4*g*e + 8*c^3*d^3*f*e^2 - 24*b*c^2*d^3*g*e^2 - 8*b*c^2*d^2*f*e^3 + 15*b^2*c*d^2*g*e^3 + 2*b^2*c*d*f*e^4 - 3*b^
3*d*g*e^4)/(4*c^4*d^2*e^3 - 4*b*c^3*d*e^4 + b^2*c^2*e^5))/(c*x^2*e^2 - c*d^2 + b*x*e^2 + b*d*e) - 1/2*(4*c*d*g
+ 2*c*f*e - 3*b*g*e)*sqrt(-c*e^2)*e^(-3)*log(abs(-2*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d
*e))*c - sqrt(-c*e^2)*b))/c^3