3.2187 \(\int \frac{(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2}}{(d+e x)^2} \, dx\)

Optimal. Leaf size=278 \[ \frac{(2 c d-b e)^2 (-b e g-4 c d g+6 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{16 c^{3/2} e^2}+\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e^2 (d+e x)^2 (2 c d-b e)}+\frac{\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-b e g-4 c d g+6 c e f)}{3 e^2 (2 c d-b e)}+\frac{(b+2 c x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-b e g-4 c d g+6 c e f)}{8 c e} \]

[Out]

((6*c*e*f - 4*c*d*g - b*e*g)*(b + 2*c*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(8*c*e) + ((6*c*e*f - 4*c*
d*g - b*e*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)) + (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)^2) + ((2*c*d - b*e)^2*(6*c*e*f - 4*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])])/(16*c^(3/2)*e^2)

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Rubi [A]  time = 0.42014, antiderivative size = 278, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.114, Rules used = {792, 664, 612, 621, 204} \[ \frac{(2 c d-b e)^2 (-b e g-4 c d g+6 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{16 c^{3/2} e^2}+\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e^2 (d+e x)^2 (2 c d-b e)}+\frac{\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-b e g-4 c d g+6 c e f)}{3 e^2 (2 c d-b e)}+\frac{(b+2 c x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-b e g-4 c d g+6 c e f)}{8 c e} \]

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)^2,x]

[Out]

((6*c*e*f - 4*c*d*g - b*e*g)*(b + 2*c*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(8*c*e) + ((6*c*e*f - 4*c*
d*g - b*e*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)) + (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)^2) + ((2*c*d - b*e)^2*(6*c*e*f - 4*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])])/(16*c^(3/2)*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 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 612

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p +
1)), x] - Dist[(p*(b^2 - 4*a*c))/(2*c*(2*p + 1)), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*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)^2} \, 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)^2}+\frac{(6 c e f-4 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} \, dx}{e (2 c d-b e)}\\ &=\frac{(6 c e f-4 c d g-b e 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)}+\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)^2}-\frac{((-2 c d+b e) (6 c e f-4 c d g-b e g)) \int \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2} \, dx}{2 e (2 c d-b e)}\\ &=\frac{(6 c e f-4 c d g-b e g) (b+2 c x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{8 c e}+\frac{(6 c e f-4 c d g-b e 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)}+\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)^2}+\frac{\left ((2 c d-b e)^2 (6 c e f-4 c d g-b e g)\right ) \int \frac{1}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{16 c e}\\ &=\frac{(6 c e f-4 c d g-b e g) (b+2 c x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{8 c e}+\frac{(6 c e f-4 c d g-b e 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)}+\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)^2}+\frac{\left ((2 c d-b e)^2 (6 c e f-4 c d g-b e g)\right ) \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 )}{8 c e}\\ &=\frac{(6 c e f-4 c d g-b e g) (b+2 c x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{8 c e}+\frac{(6 c e f-4 c d g-b e 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)}+\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)^2}+\frac{(2 c d-b e)^2 (6 c e f-4 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 )}{16 c^{3/2} e^2}\\ \end{align*}

Mathematica [A]  time = 0.983889, size = 228, normalized size = 0.82 \[ \frac{\sqrt{(d+e x) (c (d-e x)-b e)} \left (-\sqrt{c} \sqrt{e} \left (3 b^2 e^2 g+2 b c e (-14 d g+15 e f+7 e g x)+4 c^2 \left (10 d^2 g-6 d e (2 f+g x)+e^2 x (3 f+2 g x)\right )\right )-\frac{3 \sqrt{e (2 c d-b e)} (b e-2 c d) (-b e g-4 c d g+6 c e f) \sin ^{-1}\left (\frac{\sqrt{c} \sqrt{e} \sqrt{d+e x}}{\sqrt{e (2 c d-b e)}}\right )}{\sqrt{d+e x} \sqrt{\frac{b e-c d+c e x}{b e-2 c d}}}\right )}{24 c^{3/2} e^{5/2}} \]

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)^2,x]

[Out]

(Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(-(Sqrt[c]*Sqrt[e]*(3*b^2*e^2*g + 2*b*c*e*(15*e*f - 14*d*g + 7*e*g*x)
+ 4*c^2*(10*d^2*g - 6*d*e*(2*f + g*x) + e^2*x*(3*f + 2*g*x)))) - (3*Sqrt[e*(2*c*d - b*e)]*(-2*c*d + b*e)*(6*c*
e*f - 4*c*d*g - b*e*g)*ArcSin[(Sqrt[c]*Sqrt[e]*Sqrt[d + e*x])/Sqrt[e*(2*c*d - b*e)]])/(Sqrt[d + e*x]*Sqrt[(-(c
*d) + b*e + c*e*x)/(-2*c*d + b*e)])))/(24*c^(3/2)*e^(5/2))

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Maple [B]  time = 0.013, size = 2174, normalized size = 7.8 \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)^2,x)

[Out]

-9/4*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^2*g+9/4*e^3*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*f+9/2*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^3*g-9/2*e^2*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*f+3/2*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*d*g-3/4*e^2/(-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)*f+2/e^2/(-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)*f-3/
2*e^2*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*f+3/8*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))*d*g+3*e*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*f+3/2*e*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*f+3*e*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))*f-3
*c^3/(-b*e^2+2*c*d*e)*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-3*c^2/(-b*e^2+2*c*d*e)*d^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^
(1/2)*x*g-3/2*c/(-b*e^2+2*c*d*e)*d^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*b*g-2/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)*d*g-2/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)*d*g+3/4*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
)*d*g-3/8*e^4/(-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))*f-1/16*g*e^2*b^3/c/(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*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-3/4*g*b*c
/(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+1/2*g/e*c*d*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*x+1/2*g/e*c^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))+2*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)*f-1/4*g*b*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*
e)*(x+d/e))^(1/2)*x-1/8*g*b^2/c*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)+1/3*g/e^2*(-(x+d/e)^2*c*e^2+
(-b*e^2+2*c*d*e)*(x+d/e))^(3/2)+1/4*g/e*d*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*b

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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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.36274, size = 1212, normalized size = 4.36 \begin{align*} \left [\frac{3 \,{\left (6 \,{\left (4 \, c^{3} d^{2} e - 4 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} f -{\left (16 \, c^{3} d^{3} - 12 \, b c^{2} d^{2} e + b^{3} e^{3}\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 \,{\left (8 \, c^{3} e^{2} g x^{2} - 6 \,{\left (8 \, c^{3} d e - 5 \, b c^{2} e^{2}\right )} f +{\left (40 \, c^{3} d^{2} - 28 \, b c^{2} d e + 3 \, b^{2} c e^{2}\right )} g + 2 \,{\left (6 \, c^{3} e^{2} f -{\left (12 \, c^{3} d e - 7 \, b c^{2} e^{2}\right )} g\right )} x\right )} \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{96 \, c^{2} e^{2}}, -\frac{3 \,{\left (6 \,{\left (4 \, c^{3} d^{2} e - 4 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} f -{\left (16 \, c^{3} d^{3} - 12 \, b c^{2} d^{2} e + b^{3} e^{3}\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 \,{\left (8 \, c^{3} e^{2} g x^{2} - 6 \,{\left (8 \, c^{3} d e - 5 \, b c^{2} e^{2}\right )} f +{\left (40 \, c^{3} d^{2} - 28 \, b c^{2} d e + 3 \, b^{2} c e^{2}\right )} g + 2 \,{\left (6 \, c^{3} e^{2} f -{\left (12 \, c^{3} d e - 7 \, b c^{2} e^{2}\right )} g\right )} x\right )} \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{48 \, c^{2} e^{2}}\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)^2,x, algorithm="fricas")

[Out]

[1/96*(3*(6*(4*c^3*d^2*e - 4*b*c^2*d*e^2 + b^2*c*e^3)*f - (16*c^3*d^3 - 12*b*c^2*d^2*e + b^3*e^3)*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*(8*c^3*e^2*g*x^2 - 6*(8*c^3*d*e - 5*b*c^2*e^2)*f + (40*c^3*d^2 - 28*b*c^2*d*e
+ 3*b^2*c*e^2)*g + 2*(6*c^3*e^2*f - (12*c^3*d*e - 7*b*c^2*e^2)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*
e))/(c^2*e^2), -1/48*(3*(6*(4*c^3*d^2*e - 4*b*c^2*d*e^2 + b^2*c*e^3)*f - (16*c^3*d^3 - 12*b*c^2*d^2*e + b^3*e^
3)*g)*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*(8*c^3*e^2*g*x^2 - 6*(8*c^3*d*e - 5*b*c^2*e^2)*f + (40*c^3*d^2 - 28*b*c^2*d*e
+ 3*b^2*c*e^2)*g + 2*(6*c^3*e^2*f - (12*c^3*d*e - 7*b*c^2*e^2)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*
e))/(c^2*e^2)]

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

[Out]

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

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Giac [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)^2,x, algorithm="giac")

[Out]

Timed out