3.2188 \(\int \frac{(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2}}{(d+e x)^3} \, dx\)

Optimal. Leaf size=271 \[ -\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e^2 (d+e x)^3 (2 c d-b e)}-\frac{\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (b e g-6 c d g+4 c e f)}{2 e^2 (d+e x) (2 c d-b e)}-\frac{3 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (b e g-6 c d g+4 c e f)}{4 e^2}-\frac{3 (2 c d-b e) (b e g-6 c d g+4 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{8 \sqrt{c} e^2} \]

[Out]

(-3*(4*c*e*f - 6*c*d*g + b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(4*e^2) - ((4*c*e*f - 6*c*d*g + b*e
*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(2*e^2*(2*c*d - b*e)*(d + e*x)) - (2*(e*f - d*g)*(d*(c*d - b*
e) - b*e^2*x - c*e^2*x^2)^(5/2))/(e^2*(2*c*d - b*e)*(d + e*x)^3) - (3*(2*c*d - b*e)*(4*c*e*f - 6*c*d*g + 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])])/(8*Sqrt[c]*e^2)

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Rubi [A]  time = 0.467134, antiderivative size = 271, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {792, 664, 621, 204} \[ -\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e^2 (d+e x)^3 (2 c d-b e)}-\frac{\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (b e g-6 c d g+4 c e f)}{2 e^2 (d+e x) (2 c d-b e)}-\frac{3 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (b e g-6 c d g+4 c e f)}{4 e^2}-\frac{3 (2 c d-b e) (b e g-6 c d g+4 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{8 \sqrt{c} e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(4*c*e*f - 6*c*d*g + b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(4*e^2) - ((4*c*e*f - 6*c*d*g + b*e
*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(2*e^2*(2*c*d - b*e)*(d + e*x)) - (2*(e*f - d*g)*(d*(c*d - b*
e) - b*e^2*x - c*e^2*x^2)^(5/2))/(e^2*(2*c*d - b*e)*(d + e*x)^3) - (3*(2*c*d - b*e)*(4*c*e*f - 6*c*d*g + 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])])/(8*Sqrt[c]*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 664

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 + 2*p + 1)), x] - Dist[(p*(2*c*d - b*e))/(e^2*(m + 2*p + 1)), Int[(d + e*x)^(m + 1)*
(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] && (LeQ[-2, m, 0] || EqQ[m + p + 1, 0]) && NeQ[m + 2*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) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(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 )^{5/2}}{e^2 (2 c d-b e) (d+e x)^3}-\frac{(4 c e f-6 c d g+b e g) \int \frac{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^2} \, dx}{e (2 c d-b e)}\\ &=-\frac{(4 c e f-6 c d g+b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2 e^2 (2 c d-b e) (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 )^{5/2}}{e^2 (2 c d-b e) (d+e x)^3}-\frac{(3 (4 c e f-6 c d g+b e g)) \int \frac{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}{d+e x} \, dx}{4 e}\\ &=-\frac{3 (4 c e f-6 c d g+b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2}-\frac{(4 c e f-6 c d g+b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2 e^2 (2 c d-b e) (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 )^{5/2}}{e^2 (2 c d-b e) (d+e x)^3}-\frac{(3 (2 c d-b e) (4 c e f-6 c d g+b e g)) \int \frac{1}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{8 e}\\ &=-\frac{3 (4 c e f-6 c d g+b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2}-\frac{(4 c e f-6 c d g+b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2 e^2 (2 c d-b e) (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 )^{5/2}}{e^2 (2 c d-b e) (d+e x)^3}-\frac{(3 (2 c d-b e) (4 c e f-6 c d g+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 )}{4 e}\\ &=-\frac{3 (4 c e f-6 c d g+b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2}-\frac{(4 c e f-6 c d g+b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2 e^2 (2 c d-b e) (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 )^{5/2}}{e^2 (2 c d-b e) (d+e x)^3}-\frac{3 (2 c d-b e) (4 c e f-6 c d g+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 )}{8 \sqrt{c} e^2}\\ \end{align*}

Mathematica [A]  time = 0.980982, size = 208, normalized size = 0.77 \[ \frac{\sqrt{(d+e x) (c (d-e x)-b e)} \left (-\frac{\sqrt{e} \left (b e (13 d g-8 e f+5 e g x)+c \left (-28 d^2 g+10 d e (2 f-g x)+2 e^2 x (2 f+g x)\right )\right )}{d+e x}-\frac{3 \sqrt{e (2 c d-b e)} (b e g-6 c d g+4 c e f) \sin ^{-1}\left (\frac{\sqrt{c} \sqrt{e} \sqrt{d+e x}}{\sqrt{e (2 c d-b e)}}\right )}{\sqrt{c} \sqrt{d+e x} \sqrt{\frac{b e-c d+c e x}{b e-2 c d}}}\right )}{4 e^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(-((Sqrt[e]*(b*e*(-8*e*f + 13*d*g + 5*e*g*x) + c*(-28*d^2*g + 10*d*e*(
2*f - g*x) + 2*e^2*x*(2*f + g*x))))/(d + e*x)) - (3*Sqrt[e*(2*c*d - b*e)]*(4*c*e*f - 6*c*d*g + b*e*g)*ArcSin[(
Sqrt[c]*Sqrt[e]*Sqrt[d + e*x])/Sqrt[e*(2*c*d - b*e)]])/(Sqrt[c]*Sqrt[d + e*x]*Sqrt[(-(c*d) + b*e + c*e*x)/(-2*
c*d + b*e)])))/(4*e^(5/2))

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Maple [B]  time = 0.013, size = 2535, normalized size = 9.4 \begin{align*} \text{result too large to display} \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)^(3/2)/(e*x+d)^3,x)

[Out]

-9*e^4*c^2/(-b*e^2+2*c*d*e)^2*b^2/(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*f-18*e^2*c^3/(-b*e^2+2*c*d*e)^2*b/(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^3*g+18*e^3*c^3/(
-b*e^2+2*c*d*e)^2*b/(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^2*f-6*e^2*c^2/(-b*e^2+2*c*d*e)^2*b*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e)
)^(1/2)*x*d*g-3/2*e^4*c/(-b*e^2+2*c*d*e)^2*b^3/(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*g+9*e^3*c^2/(-b*e^2+2*c*d*e)^2*b^2/(c*e^2)^(1/2)*a
rctan((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^2*
g+9/4*g*e^2*c/(-b*e^2+2*c*d*e)*b^2/(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-9/2*g*e*c^2/(-b*e^2+2*c*d*e)*b/(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^2+3/2*e^5*c/(-b*e^
2+2*c*d*e)^2*b^3/(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))*f+12*e*c^3/(-b*e^2+2*c*d*e)^2*d^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2
)*x*g-12*e^2*c^3/(-b*e^2+2*c*d*e)^2*d*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*x*f+6*e*c^2/(-b*e^2+2*
c*d*e)^2*d^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*b*g-6*e^2*c^2/(-b*e^2+2*c*d*e)^2*d*(-(x+d/e)^2*
c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*b*f+12*e*c^4/(-b*e^2+2*c*d*e)^2*d^4/(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))*g-12*e^2*c^4/(-b*e^2+2*c*
d*e)^2*d^3/(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))*f-3/2*g*e*c/(-b*e^2+2*c*d*e)*b*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*x+8/e^2
*c/(-b*e^2+2*c*d*e)^2/(x+d/e)^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(5/2)*d*g+6*e^3*c^2/(-b*e^2+2*c*d*
e)^2*b*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*x*f-3*e^2*c/(-b*e^2+2*c*d*e)^2*b^2*(-(x+d/e)^2*c*e^2+
(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*d*g+3*g*c^3/(-b*e^2+2*c*d*e)*d^3/(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))-3/8*g*e^3/(-b*e^2+2*c*d*e)*b^3/(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))+3*g*c^2/(-b*e^2+2*c*d*e)*d*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*x+3/2*g*c/(-b*e^2+2*c*d*e
)*d*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*b+2/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))^(5/2)*d*g-8/e*c/(-b*e^2+2*c*d*e)^2/(x+d/e)^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e)
)^(5/2)*f+3*e^3*c/(-b*e^2+2*c*d*e)^2*b^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*f-3/4*g*e/(-b*e^2+2
*c*d*e)*b^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)-2/e^3/(-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))^(5/2)*f+8*c^2/(-b*e^2+2*c*d*e)^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(3/2
)*d*g-8*e*c^2/(-b*e^2+2*c*d*e)^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(3/2)*f+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))^(5/2)+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))^(3/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 8.82789, size = 1337, normalized size = 4.93 \begin{align*} \left [\frac{3 \,{\left (4 \,{\left (2 \, c^{2} d^{2} e - b c d e^{2}\right )} f -{\left (12 \, c^{2} d^{3} - 8 \, b c d^{2} e + b^{2} d e^{2}\right )} g +{\left (4 \,{\left (2 \, c^{2} d e^{2} - b c e^{3}\right )} f -{\left (12 \, c^{2} d^{2} e - 8 \, b c d e^{2} + b^{2} e^{3}\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 (2 \, c^{2} e^{2} g x^{2} + 4 \,{\left (5 \, c^{2} d e - 2 \, b c e^{2}\right )} f -{\left (28 \, c^{2} d^{2} - 13 \, b c d e\right )} g +{\left (4 \, c^{2} e^{2} f - 5 \,{\left (2 \, c^{2} d e - b c e^{2}\right )} g\right )} x\right )} \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{16 \,{\left (c e^{3} x + c d e^{2}\right )}}, \frac{3 \,{\left (4 \,{\left (2 \, c^{2} d^{2} e - b c d e^{2}\right )} f -{\left (12 \, c^{2} d^{3} - 8 \, b c d^{2} e + b^{2} d e^{2}\right )} g +{\left (4 \,{\left (2 \, c^{2} d e^{2} - b c e^{3}\right )} f -{\left (12 \, c^{2} d^{2} e - 8 \, b c d e^{2} + b^{2} e^{3}\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 (2 \, c^{2} e^{2} g x^{2} + 4 \,{\left (5 \, c^{2} d e - 2 \, b c e^{2}\right )} f -{\left (28 \, c^{2} d^{2} - 13 \, b c d e\right )} g +{\left (4 \, c^{2} e^{2} f - 5 \,{\left (2 \, c^{2} d e - b c e^{2}\right )} g\right )} x\right )} \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{8 \,{\left (c e^{3} x + c d e^{2}\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)^(3/2)/(e*x+d)^3,x, algorithm="fricas")

[Out]

[1/16*(3*(4*(2*c^2*d^2*e - b*c*d*e^2)*f - (12*c^2*d^3 - 8*b*c*d^2*e + b^2*d*e^2)*g + (4*(2*c^2*d*e^2 - b*c*e^3
)*f - (12*c^2*d^2*e - 8*b*c*d*e^2 + b^2*e^3)*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*(2*c^2*e^2*g*x^2
+ 4*(5*c^2*d*e - 2*b*c*e^2)*f - (28*c^2*d^2 - 13*b*c*d*e)*g + (4*c^2*e^2*f - 5*(2*c^2*d*e - b*c*e^2)*g)*x)*sqr
t(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(c*e^3*x + c*d*e^2), 1/8*(3*(4*(2*c^2*d^2*e - b*c*d*e^2)*f - (12*c^2*
d^3 - 8*b*c*d^2*e + b^2*d*e^2)*g + (4*(2*c^2*d*e^2 - b*c*e^3)*f - (12*c^2*d^2*e - 8*b*c*d*e^2 + b^2*e^3)*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*(2*c^2*e^2*g*x^2 + 4*(5*c^2*d*e - 2*b*c*e^2)*f - (28*c^2*d^2 - 13*b*c*d*e)*g + (4*
c^2*e^2*f - 5*(2*c^2*d*e - b*c*e^2)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(c*e^3*x + c*d*e^2)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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)**(3/2)/(e*x+d)**3,x)

[Out]

Timed out

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

[Out]

Exception raised: NotImplementedError