3.2209 \(\int \frac{(d+e x)^3 (f+g x)}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx\)

Optimal. Leaf size=340 \[ -\frac{5 (2 c d-b e)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+6 c d g+8 c e f)}{64 c^4 e^2}-\frac{5 (d+e x) (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+6 c d g+8 c e f)}{96 c^3 e^2}-\frac{(d+e x)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+6 c d g+8 c e f)}{24 c^2 e^2}+\frac{5 (2 c d-b e)^3 (-7 b e g+6 c d g+8 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{128 c^{9/2} e^2}-\frac{g (d+e x)^3 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c e^2} \]

[Out]

(-5*(2*c*d - b*e)^2*(8*c*e*f + 6*c*d*g - 7*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(64*c^4*e^2) - (5
*(2*c*d - b*e)*(8*c*e*f + 6*c*d*g - 7*b*e*g)*(d + e*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(96*c^3*e^2)
- ((8*c*e*f + 6*c*d*g - 7*b*e*g)*(d + e*x)^2*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(24*c^2*e^2) - (g*(d
+ e*x)^3*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(4*c*e^2) + (5*(2*c*d - b*e)^3*(8*c*e*f + 6*c*d*g - 7*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])])/(128*c^(9/2)*e^2)

________________________________________________________________________________________

Rubi [A]  time = 0.60936, antiderivative size = 340, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.114, Rules used = {794, 670, 640, 621, 204} \[ -\frac{5 (2 c d-b e)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+6 c d g+8 c e f)}{64 c^4 e^2}-\frac{5 (d+e x) (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+6 c d g+8 c e f)}{96 c^3 e^2}-\frac{(d+e x)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+6 c d g+8 c e f)}{24 c^2 e^2}+\frac{5 (2 c d-b e)^3 (-7 b e g+6 c d g+8 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{128 c^{9/2} e^2}-\frac{g (d+e x)^3 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*(2*c*d - b*e)^2*(8*c*e*f + 6*c*d*g - 7*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(64*c^4*e^2) - (5
*(2*c*d - b*e)*(8*c*e*f + 6*c*d*g - 7*b*e*g)*(d + e*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(96*c^3*e^2)
- ((8*c*e*f + 6*c*d*g - 7*b*e*g)*(d + e*x)^2*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(24*c^2*e^2) - (g*(d
+ e*x)^3*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(4*c*e^2) + (5*(2*c*d - b*e)^3*(8*c*e*f + 6*c*d*g - 7*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])])/(128*c^(9/2)*e^2)

Rule 794

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[(m*(g*(c*d - b*e) + c*e*f) + e*(p + 1)
*(2*c*f - b*g))/(c*e*(m + 2*p + 2)), Int[(d + e*x)^m*(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] && NeQ[m + 2*p + 2, 0] && (NeQ[m, 2] || Eq
Q[d, 0])

Rule 670

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

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)^3 (f+g x)}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx &=-\frac{g (d+e x)^3 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c e^2}-\frac{\left (\frac{1}{2} e \left (-2 c e^2 f+b e^2 g\right )+3 \left (-c e^3 f+\left (-c d e^2+b e^3\right ) g\right )\right ) \int \frac{(d+e x)^3}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{4 c e^3}\\ &=-\frac{(8 c e f+6 c d g-7 b e g) (d+e x)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{24 c^2 e^2}-\frac{g (d+e x)^3 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c e^2}+\frac{(5 (2 c d-b e) (8 c e f+6 c d g-7 b e g)) \int \frac{(d+e x)^2}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{48 c^2 e}\\ &=-\frac{5 (2 c d-b e) (8 c e f+6 c d g-7 b e g) (d+e x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{96 c^3 e^2}-\frac{(8 c e f+6 c d g-7 b e g) (d+e x)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{24 c^2 e^2}-\frac{g (d+e x)^3 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c e^2}+\frac{\left (5 (2 c d-b e)^2 (8 c e f+6 c d g-7 b e g)\right ) \int \frac{d+e x}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{64 c^3 e}\\ &=-\frac{5 (2 c d-b e)^2 (8 c e f+6 c d g-7 b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{64 c^4 e^2}-\frac{5 (2 c d-b e) (8 c e f+6 c d g-7 b e g) (d+e x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{96 c^3 e^2}-\frac{(8 c e f+6 c d g-7 b e g) (d+e x)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{24 c^2 e^2}-\frac{g (d+e x)^3 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c e^2}+\frac{\left (5 (2 c d-b e)^3 (8 c e f+6 c d g-7 b e g)\right ) \int \frac{1}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{128 c^4 e}\\ &=-\frac{5 (2 c d-b e)^2 (8 c e f+6 c d g-7 b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{64 c^4 e^2}-\frac{5 (2 c d-b e) (8 c e f+6 c d g-7 b e g) (d+e x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{96 c^3 e^2}-\frac{(8 c e f+6 c d g-7 b e g) (d+e x)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{24 c^2 e^2}-\frac{g (d+e x)^3 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c e^2}+\frac{\left (5 (2 c d-b e)^3 (8 c e f+6 c d g-7 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 )}{64 c^4 e}\\ &=-\frac{5 (2 c d-b e)^2 (8 c e f+6 c d g-7 b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{64 c^4 e^2}-\frac{5 (2 c d-b e) (8 c e f+6 c d g-7 b e g) (d+e x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{96 c^3 e^2}-\frac{(8 c e f+6 c d g-7 b e g) (d+e x)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{24 c^2 e^2}-\frac{g (d+e x)^3 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c e^2}+\frac{5 (2 c d-b e)^3 (8 c e f+6 c d g-7 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 )}{128 c^{9/2} e^2}\\ \end{align*}

Mathematica [A]  time = 2.05945, size = 375, normalized size = 1.1 \[ \frac{\sqrt{(d+e x) (c (d-e x)-b e)} \left (-\frac{e^3 (-7 b e g+6 c d g+8 c e f) \left (8 c^3 e^6 (d+e x)^3 \sqrt{e (2 c d-b e)} \sqrt{\frac{b e-c d+c e x}{b e-2 c d}}-10 c^2 e^6 (d+e x)^2 \sqrt{e (2 c d-b e)} (b e-2 c d) \sqrt{\frac{b e-c d+c e x}{b e-2 c d}}+15 c e^6 (d+e x) \sqrt{e (2 c d-b e)} (b e-2 c d)^2 \sqrt{\frac{b e-c d+c e x}{b e-2 c d}}+15 \sqrt{c} e^{13/2} \sqrt{d+e x} (b e-2 c d)^3 \sin ^{-1}\left (\frac{\sqrt{c} \sqrt{e} \sqrt{d+e x}}{\sqrt{e (2 c d-b e)}}\right )\right )}{3 \sqrt{e (2 c d-b e)} \sqrt{\frac{b e-c d+c e x}{b e-2 c d}}}-16 c^4 e^9 g (d+e x)^4\right )}{64 c^5 e^{11} (d+e x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(-16*c^4*e^9*g*(d + e*x)^4 - (e^3*(8*c*e*f + 6*c*d*g - 7*b*e*g)*(15*c*
e^6*Sqrt[e*(2*c*d - b*e)]*(-2*c*d + b*e)^2*(d + e*x)*Sqrt[(-(c*d) + b*e + c*e*x)/(-2*c*d + b*e)] - 10*c^2*e^6*
Sqrt[e*(2*c*d - b*e)]*(-2*c*d + b*e)*(d + e*x)^2*Sqrt[(-(c*d) + b*e + c*e*x)/(-2*c*d + b*e)] + 8*c^3*e^6*Sqrt[
e*(2*c*d - b*e)]*(d + e*x)^3*Sqrt[(-(c*d) + b*e + c*e*x)/(-2*c*d + b*e)] + 15*Sqrt[c]*e^(13/2)*(-2*c*d + b*e)^
3*Sqrt[d + e*x]*ArcSin[(Sqrt[c]*Sqrt[e]*Sqrt[d + e*x])/Sqrt[e*(2*c*d - b*e)]]))/(3*Sqrt[e*(2*c*d - b*e)]*Sqrt[
(-(c*d) + b*e + c*e*x)/(-2*c*d + b*e)])))/(64*c^5*e^11*(d + e*x))

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Maple [B]  time = 0.018, size = 1208, normalized size = 3.6 \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)^3*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x)

[Out]

-x^2/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d*g-3/2*x/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d*f+5/2*d^3*f
/(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))+5/12*b/c^2*x*e*(-c*e^2
*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*f-5/16*b^3/c^3/(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))*e^3*f+7/24*e*g*b/c^2*x^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-35/96*e*g*b^2/c^3*x*(-
c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)+35/128*e^3*g*b^4/c^4/(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))+259/48/e*g*b/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^2+13/8*g/c^2*x*(-
c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b*d-15/8/e*g/c*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^2-5*g/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))*b*d^3+35/64*e*g*b^3/c^4*(-c*e^
2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-1/4*e*g*x^3/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-145/48*g*b^2/c^3*(-c*e^2
*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d+15/8/e*g/(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^4-1/3*x^2/c*e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*f-5/8*b^2/c^3*e*(-c*e^2*x^2-b*e^2*x
-b*d*e+c*d^2)^(1/2)*f+35/12/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b*d*f-3/e^2/c*(-c*e^2*x^2-b*e^2*x-b*d*e
+c*d^2)^(1/2)*d^3*g-11/3/e/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^2*f+15/8*b^2/c^2*e^2/(c*e^2)^(1/2)*arcta
n((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*f-15/4*b/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^2*f-15/8*e^2*g*b^3/c^3/(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+75/16*e*g*b^2/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))*d^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((e*x+d)^3*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 5.07512, size = 1797, normalized size = 5.29 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algorithm="fricas")

[Out]

[-1/768*(15*(8*(8*c^4*d^3*e - 12*b*c^3*d^2*e^2 + 6*b^2*c^2*d*e^3 - b^3*c*e^4)*f + (48*c^4*d^4 - 128*b*c^3*d^3*
e + 120*b^2*c^2*d^2*e^2 - 48*b^3*c*d*e^3 + 7*b^4*e^4)*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*(48*c^4*e^3
*g*x^3 + 8*(8*c^4*e^3*f + (24*c^4*d*e^2 - 7*b*c^3*e^3)*g)*x^2 + 8*(88*c^4*d^2*e - 70*b*c^3*d*e^2 + 15*b^2*c^2*
e^3)*f + (576*c^4*d^3 - 1036*b*c^3*d^2*e + 580*b^2*c^2*d*e^2 - 105*b^3*c*e^3)*g + 2*(8*(18*c^4*d*e^2 - 5*b*c^3
*e^3)*f + (180*c^4*d^2*e - 156*b*c^3*d*e^2 + 35*b^2*c^2*e^3)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))
/(c^5*e^2), -1/384*(15*(8*(8*c^4*d^3*e - 12*b*c^3*d^2*e^2 + 6*b^2*c^2*d*e^3 - b^3*c*e^4)*f + (48*c^4*d^4 - 128
*b*c^3*d^3*e + 120*b^2*c^2*d^2*e^2 - 48*b^3*c*d*e^3 + 7*b^4*e^4)*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*(48*c^4*e^3*g*x
^3 + 8*(8*c^4*e^3*f + (24*c^4*d*e^2 - 7*b*c^3*e^3)*g)*x^2 + 8*(88*c^4*d^2*e - 70*b*c^3*d*e^2 + 15*b^2*c^2*e^3)
*f + (576*c^4*d^3 - 1036*b*c^3*d^2*e + 580*b^2*c^2*d*e^2 - 105*b^3*c*e^3)*g + 2*(8*(18*c^4*d*e^2 - 5*b*c^3*e^3
)*f + (180*c^4*d^2*e - 156*b*c^3*d*e^2 + 35*b^2*c^2*e^3)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(c^
5*e^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.26752, size = 517, normalized size = 1.52 \begin{align*} -\frac{1}{192} \, \sqrt{-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}{\left (2 \,{\left (4 \,{\left (\frac{6 \, g x e}{c} + \frac{{\left (24 \, c^{3} d g e^{4} + 8 \, c^{3} f e^{5} - 7 \, b c^{2} g e^{5}\right )} e^{\left (-4\right )}}{c^{4}}\right )} x + \frac{{\left (180 \, c^{3} d^{2} g e^{3} + 144 \, c^{3} d f e^{4} - 156 \, b c^{2} d g e^{4} - 40 \, b c^{2} f e^{5} + 35 \, b^{2} c g e^{5}\right )} e^{\left (-4\right )}}{c^{4}}\right )} x + \frac{{\left (576 \, c^{3} d^{3} g e^{2} + 704 \, c^{3} d^{2} f e^{3} - 1036 \, b c^{2} d^{2} g e^{3} - 560 \, b c^{2} d f e^{4} + 580 \, b^{2} c d g e^{4} + 120 \, b^{2} c f e^{5} - 105 \, b^{3} g e^{5}\right )} e^{\left (-4\right )}}{c^{4}}\right )} + \frac{5 \,{\left (48 \, c^{4} d^{4} g + 64 \, c^{4} d^{3} f e - 128 \, b c^{3} d^{3} g e - 96 \, b c^{3} d^{2} f e^{2} + 120 \, b^{2} c^{2} d^{2} g e^{2} + 48 \, b^{2} c^{2} d f e^{3} - 48 \, b^{3} c d g e^{3} - 8 \, b^{3} c f e^{4} + 7 \, b^{4} g e^{4}\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 )}{128 \, c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algorithm="giac")

[Out]

-1/192*sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e)*(2*(4*(6*g*x*e/c + (24*c^3*d*g*e^4 + 8*c^3*f*e^5 - 7*b*c^2*g
*e^5)*e^(-4)/c^4)*x + (180*c^3*d^2*g*e^3 + 144*c^3*d*f*e^4 - 156*b*c^2*d*g*e^4 - 40*b*c^2*f*e^5 + 35*b^2*c*g*e
^5)*e^(-4)/c^4)*x + (576*c^3*d^3*g*e^2 + 704*c^3*d^2*f*e^3 - 1036*b*c^2*d^2*g*e^3 - 560*b*c^2*d*f*e^4 + 580*b^
2*c*d*g*e^4 + 120*b^2*c*f*e^5 - 105*b^3*g*e^5)*e^(-4)/c^4) + 5/128*(48*c^4*d^4*g + 64*c^4*d^3*f*e - 128*b*c^3*
d^3*g*e - 96*b*c^3*d^2*f*e^2 + 120*b^2*c^2*d^2*g*e^2 + 48*b^2*c^2*d*f*e^3 - 48*b^3*c*d*g*e^3 - 8*b^3*c*f*e^4 +
7*b^4*g*e^4)*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^5