3.2172 \(\int (d+e x)^3 (f+g x) \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2} \, dx\)

Optimal. Leaf size=414 \[ \frac{7 (b+2 c x) (2 c d-b e)^3 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-3 b e g+2 c d g+4 c e f)}{512 c^5 e}-\frac{7 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-3 b e g+2 c d g+4 c e f)}{192 c^4 e^2}-\frac{7 (d+e x) (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-3 b e g+2 c d g+4 c e f)}{160 c^3 e^2}-\frac{(d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-3 b e g+2 c d g+4 c e f)}{20 c^2 e^2}+\frac{7 (2 c d-b e)^5 (-3 b e g+2 c d g+4 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{1024 c^{11/2} e^2}-\frac{g (d+e x)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{6 c e^2} \]

[Out]

(7*(2*c*d - b*e)^3*(4*c*e*f + 2*c*d*g - 3*b*e*g)*(b + 2*c*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(512*c
^5*e) - (7*(2*c*d - b*e)^2*(4*c*e*f + 2*c*d*g - 3*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(192*c^4
*e^2) - (7*(2*c*d - b*e)*(4*c*e*f + 2*c*d*g - 3*b*e*g)*(d + e*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/
(160*c^3*e^2) - ((4*c*e*f + 2*c*d*g - 3*b*e*g)*(d + e*x)^2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(20*c^
2*e^2) - (g*(d + e*x)^3*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(6*c*e^2) + (7*(2*c*d - b*e)^5*(4*c*e*f +
2*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])])/(1024*c^(11
/2)*e^2)

________________________________________________________________________________________

Rubi [A]  time = 0.830252, antiderivative size = 414, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {794, 670, 640, 612, 621, 204} \[ \frac{7 (b+2 c x) (2 c d-b e)^3 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-3 b e g+2 c d g+4 c e f)}{512 c^5 e}-\frac{7 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-3 b e g+2 c d g+4 c e f)}{192 c^4 e^2}-\frac{7 (d+e x) (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-3 b e g+2 c d g+4 c e f)}{160 c^3 e^2}-\frac{(d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-3 b e g+2 c d g+4 c e f)}{20 c^2 e^2}+\frac{7 (2 c d-b e)^5 (-3 b e g+2 c d g+4 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{1024 c^{11/2} e^2}-\frac{g (d+e x)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{6 c e^2} \]

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]

(7*(2*c*d - b*e)^3*(4*c*e*f + 2*c*d*g - 3*b*e*g)*(b + 2*c*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(512*c
^5*e) - (7*(2*c*d - b*e)^2*(4*c*e*f + 2*c*d*g - 3*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(192*c^4
*e^2) - (7*(2*c*d - b*e)*(4*c*e*f + 2*c*d*g - 3*b*e*g)*(d + e*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/
(160*c^3*e^2) - ((4*c*e*f + 2*c*d*g - 3*b*e*g)*(d + e*x)^2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(20*c^
2*e^2) - (g*(d + e*x)^3*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(6*c*e^2) + (7*(2*c*d - b*e)^5*(4*c*e*f +
2*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])])/(1024*c^(11
/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 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 (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 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{6 c e^2}-\frac{\left (\frac{3}{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 (d+e x)^3 \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2} \, dx}{6 c e^3}\\ &=-\frac{(4 c e f+2 c d g-3 b e g) (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{20 c^2 e^2}-\frac{g (d+e x)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{6 c e^2}+\frac{(7 (2 c d-b e) (4 c e f+2 c d g-3 b e g)) \int (d+e x)^2 \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2} \, dx}{40 c^2 e}\\ &=-\frac{7 (2 c d-b e) (4 c e f+2 c d g-3 b e g) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{160 c^3 e^2}-\frac{(4 c e f+2 c d g-3 b e g) (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{20 c^2 e^2}-\frac{g (d+e x)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{6 c e^2}+\frac{\left (7 (2 c d-b e)^2 (4 c e f+2 c d g-3 b e g)\right ) \int (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{7 (2 c d-b e)^2 (4 c e f+2 c d g-3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{192 c^4 e^2}-\frac{7 (2 c d-b e) (4 c e f+2 c d g-3 b e g) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{160 c^3 e^2}-\frac{(4 c e f+2 c d g-3 b e g) (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{20 c^2 e^2}-\frac{g (d+e x)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{6 c e^2}+\frac{\left (7 (2 c d-b e)^3 (4 c e f+2 c d g-3 b e g)\right ) \int \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2} \, dx}{128 c^4 e}\\ &=\frac{7 (2 c d-b e)^3 (4 c e f+2 c d g-3 b e g) (b+2 c x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{512 c^5 e}-\frac{7 (2 c d-b e)^2 (4 c e f+2 c d g-3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{192 c^4 e^2}-\frac{7 (2 c d-b e) (4 c e f+2 c d g-3 b e g) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{160 c^3 e^2}-\frac{(4 c e f+2 c d g-3 b e g) (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{20 c^2 e^2}-\frac{g (d+e x)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{6 c e^2}+\frac{\left (7 (2 c d-b e)^5 (4 c e f+2 c d g-3 b e g)\right ) \int \frac{1}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{1024 c^5 e}\\ &=\frac{7 (2 c d-b e)^3 (4 c e f+2 c d g-3 b e g) (b+2 c x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{512 c^5 e}-\frac{7 (2 c d-b e)^2 (4 c e f+2 c d g-3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{192 c^4 e^2}-\frac{7 (2 c d-b e) (4 c e f+2 c d g-3 b e g) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{160 c^3 e^2}-\frac{(4 c e f+2 c d g-3 b e g) (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{20 c^2 e^2}-\frac{g (d+e x)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{6 c e^2}+\frac{\left (7 (2 c d-b e)^5 (4 c e f+2 c d g-3 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 )}{512 c^5 e}\\ &=\frac{7 (2 c d-b e)^3 (4 c e f+2 c d g-3 b e g) (b+2 c x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{512 c^5 e}-\frac{7 (2 c d-b e)^2 (4 c e f+2 c d g-3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{192 c^4 e^2}-\frac{7 (2 c d-b e) (4 c e f+2 c d g-3 b e g) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{160 c^3 e^2}-\frac{(4 c e f+2 c d g-3 b e g) (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{20 c^2 e^2}-\frac{g (d+e x)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{6 c e^2}+\frac{7 (2 c d-b e)^5 (4 c e f+2 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 )}{1024 c^{11/2} e^2}\\ \end{align*}

Mathematica [B]  time = 6.25705, size = 1207, normalized size = 2.92 \[ -\frac{\left (-6 c f e^2-\left (\frac{9}{2} e (c d-b e)-\frac{3 c d e}{2}\right ) g\right ) \sqrt{(d+e x) (c (d-e x)-b e)} \left (1-\frac{c e^2 (d+e x)}{(c d e+(c d-b e) e) \left (\frac{c d e^2}{c d e+(c d-b e) e}+\frac{(c d-b e) e^2}{c d e+(c d-b e) e}\right )}\right )^{3/2} \left (\frac{63 (c d e+(c d-b e) e)^5 \left (-\frac{32 c^4 (d+e x)^4 e^8}{35 (c d e+(c d-b e) e)^4 \left (\frac{c d e^2}{c d e+(c d-b e) e}+\frac{(c d-b e) e^2}{c d e+(c d-b e) e}\right )^4}-\frac{16 c^3 (d+e x)^3 e^6}{15 (c d e+(c d-b e) e)^3 \left (\frac{c d e^2}{c d e+(c d-b e) e}+\frac{(c d-b e) e^2}{c d e+(c d-b e) e}\right )^3}-\frac{4 c^2 (d+e x)^2 e^4}{3 (c d e+(c d-b e) e)^2 \left (\frac{c d e^2}{c d e+(c d-b e) e}+\frac{(c d-b e) e^2}{c d e+(c d-b e) e}\right )^2}-\frac{2 c (d+e x) e^2}{(c d e+(c d-b e) e) \left (\frac{c d e^2}{c d e+(c d-b e) e}+\frac{(c d-b e) e^2}{c d e+(c d-b e) e}\right )}+\frac{2 \sqrt{c} \sqrt{d+e x} \sin ^{-1}\left (\frac{\sqrt{c} e \sqrt{d+e x}}{\sqrt{c d e+(c d-b e) e} \sqrt{\frac{c d e^2}{c d e+(c d-b e) e}+\frac{(c d-b e) e^2}{c d e+(c d-b e) e}}}\right ) e}{\sqrt{c d e+(c d-b e) e} \sqrt{\frac{c d e^2}{c d e+(c d-b e) e}+\frac{(c d-b e) e^2}{c d e+(c d-b e) e}} \sqrt{1-\frac{c e^2 (d+e x)}{(c d e+(c d-b e) e) \left (\frac{c d e^2}{c d e+(c d-b e) e}+\frac{(c d-b e) e^2}{c d e+(c d-b e) e}\right )}}}\right ) \left (\frac{c d e^2}{c d e+(c d-b e) e}+\frac{(c d-b e) e^2}{c d e+(c d-b e) e}\right )^5}{512 c^5 e^{10} (d+e x)^5 \left (1-\frac{c e^2 (d+e x)}{(c d e+(c d-b e) e) \left (\frac{c d e^2}{c d e+(c d-b e) e}+\frac{(c d-b e) e^2}{c d e+(c d-b e) e}\right )}\right )}+\frac{9}{10 \left (1-\frac{c e^2 (d+e x)}{(c d e+(c d-b e) e) \left (\frac{c d e^2}{c d e+(c d-b e) e}+\frac{(c d-b e) e^2}{c d e+(c d-b e) e}\right )}\right )}\right ) (d+e x)^4}{27 c e^3 \sqrt{\frac{e}{\frac{c d e^2}{c d e+(c d-b e) e}+\frac{(c d-b e) e^2}{c d e+(c d-b e) e}}} \sqrt{\frac{e (c d-b e-c e x)}{c d e+(c d-b e) e}}}-\frac{g (c d-b e-c e x) \sqrt{(d+e x) (c (d-e x)-b e)} (d+e x)^4}{6 c e^2} \]

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]

-(g*(d + e*x)^4*(c*d - b*e - c*e*x)*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))])/(6*c*e^2) - ((-6*c*e^2*f - ((-3*c*
d*e)/2 + (9*e*(c*d - b*e))/2)*g)*(d + e*x)^4*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(1 - (c*e^2*(d + e*x))/((c
*d*e + e*(c*d - b*e))*((c*d*e^2)/(c*d*e + e*(c*d - b*e)) + (e^2*(c*d - b*e))/(c*d*e + e*(c*d - b*e)))))^(3/2)*
(9/(10*(1 - (c*e^2*(d + e*x))/((c*d*e + e*(c*d - b*e))*((c*d*e^2)/(c*d*e + e*(c*d - b*e)) + (e^2*(c*d - b*e))/
(c*d*e + e*(c*d - b*e)))))) + (63*(c*d*e + e*(c*d - b*e))^5*((c*d*e^2)/(c*d*e + e*(c*d - b*e)) + (e^2*(c*d - b
*e))/(c*d*e + e*(c*d - b*e)))^5*((-2*c*e^2*(d + e*x))/((c*d*e + e*(c*d - b*e))*((c*d*e^2)/(c*d*e + e*(c*d - b*
e)) + (e^2*(c*d - b*e))/(c*d*e + e*(c*d - b*e)))) - (4*c^2*e^4*(d + e*x)^2)/(3*(c*d*e + e*(c*d - b*e))^2*((c*d
*e^2)/(c*d*e + e*(c*d - b*e)) + (e^2*(c*d - b*e))/(c*d*e + e*(c*d - b*e)))^2) - (16*c^3*e^6*(d + e*x)^3)/(15*(
c*d*e + e*(c*d - b*e))^3*((c*d*e^2)/(c*d*e + e*(c*d - b*e)) + (e^2*(c*d - b*e))/(c*d*e + e*(c*d - b*e)))^3) -
(32*c^4*e^8*(d + e*x)^4)/(35*(c*d*e + e*(c*d - b*e))^4*((c*d*e^2)/(c*d*e + e*(c*d - b*e)) + (e^2*(c*d - b*e))/
(c*d*e + e*(c*d - b*e)))^4) + (2*Sqrt[c]*e*Sqrt[d + e*x]*ArcSin[(Sqrt[c]*e*Sqrt[d + e*x])/(Sqrt[c*d*e + e*(c*d
- b*e)]*Sqrt[(c*d*e^2)/(c*d*e + e*(c*d - b*e)) + (e^2*(c*d - b*e))/(c*d*e + e*(c*d - b*e))])])/(Sqrt[c*d*e +
e*(c*d - b*e)]*Sqrt[(c*d*e^2)/(c*d*e + e*(c*d - b*e)) + (e^2*(c*d - b*e))/(c*d*e + e*(c*d - b*e))]*Sqrt[1 - (c
*e^2*(d + e*x))/((c*d*e + e*(c*d - b*e))*((c*d*e^2)/(c*d*e + e*(c*d - b*e)) + (e^2*(c*d - b*e))/(c*d*e + e*(c*
d - b*e))))])))/(512*c^5*e^10*(d + e*x)^5*(1 - (c*e^2*(d + e*x))/((c*d*e + e*(c*d - b*e))*((c*d*e^2)/(c*d*e +
e*(c*d - b*e)) + (e^2*(c*d - b*e))/(c*d*e + e*(c*d - b*e))))))))/(27*c*e^3*Sqrt[e/((c*d*e^2)/(c*d*e + e*(c*d -
b*e)) + (e^2*(c*d - b*e))/(c*d*e + e*(c*d - b*e)))]*Sqrt[(e*(c*d - b*e - c*e*x))/(c*d*e + e*(c*d - b*e))])

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Maple [B]  time = 0.027, size = 2217, normalized size = 5.4 \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]

35/16*b^2/c*e^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^3*f-2
1/16*b/c*e*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^2*f+21/32*b^2/c^2*e^2*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)
^(1/2)*d*f-35/32*b^3/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*e^3*f+35/128*b^4/c^3*e^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))*d*f-35/16*e^2*g*b^3/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^3+175/64*e*g*b^2/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))*d^4-7/32*e^4*g*b^5/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))*d+245/256*e^3*g*b^4/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^2-35/64*e^2*g*b^3/c^3*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d+21/16*e*g*b^2/c^2*x*(-c*e^2*x^2-b*e
^2*x-b*d*e+c*d^2)^(1/2)*d^2+7/40*b/c^2*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*e*f-21/160*e*g*b^2/c^3*x*(-c*e
^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)+21/1024*e^5*g*b^6/c^5/(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))+21/256*e^3*g*b^4/c^4*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)+7/32/e*g/c*(-c*
e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b*d^4-21/16*g/c*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b*d^3+7/16/e*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))*d^6-7/64*b^3/c^3*x*(-c*e
^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*e^3*f-7/256*b^5/c^4*e^5/(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-7/8/e*g/c*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d^2+21/32*e*g*b^3/c^3*
(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^2+59/48/e*g*b/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d^2-35/128*e
^2*g*b^4/c^4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d+3/20*e*g*b/c^2*x^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2
)+13/20*g/c^2*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*b*d+21/64*b^3/c^3*e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^
(1/2)*d*f-21/32*b^2/c^2*e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^2*f-35/16*b*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^4*f+7/8*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d
^3*f-11/15/e^2/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d^3*g-17/15/e/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)
*d^2*f-1/6*e*g*x^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/c+7/64*e*g*b^3/c^4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^
(3/2)+21/512*e^3*g*b^5/c^5*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)+7/16/e*g*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^
(1/2)*d^4-21/32*g/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b^2*d^3-77/120*g*b^2/c^3*(-c*e^2*x^2-b*e^2*x-b*d*
e+c*d^2)^(3/2)*d-7/4*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))*
b*d^5-3/4*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/c*d*f+7/16/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b*d^3*f
+7/8*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))*d^5*f+91/120/c^2
*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*b*d*f-3/5*x^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/c*d*g-1/5*x^2*(-c
*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/c*e*f-7/48*b^2/c^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*e*f-7/128*b^4/c^
4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*e^3*f

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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 = 8.56239, size = 3231, 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]

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]

[-1/30720*(105*(4*(32*c^6*d^5*e - 80*b*c^5*d^4*e^2 + 80*b^2*c^4*d^3*e^3 - 40*b^3*c^3*d^2*e^4 + 10*b^4*c^2*d*e^
5 - b^5*c*e^6)*f + (64*c^6*d^6 - 256*b*c^5*d^5*e + 400*b^2*c^4*d^4*e^2 - 320*b^3*c^3*d^3*e^3 + 140*b^4*c^2*d^2
*e^4 - 32*b^5*c*d*e^5 + 3*b^6*e^6)*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*(1280*c^6*e^5*g*x^5 + 128*(12*
c^6*e^5*f + (36*c^6*d*e^4 + b*c^5*e^5)*g)*x^4 + 16*(12*(30*c^6*d*e^4 + b*c^5*e^5)*f + (340*c^6*d^2*e^3 + 56*b*
c^5*d*e^4 - 9*b^2*c^4*e^5)*g)*x^3 + 8*(4*(224*c^6*d^2*e^3 + 46*b*c^5*d*e^4 - 7*b^2*c^4*e^5)*f + (128*c^6*d^3*e
^2 + 380*b*c^5*d^2*e^3 - 152*b^2*c^4*d*e^4 + 21*b^3*c^3*e^5)*g)*x^2 - 4*(2176*c^6*d^4*e - 4472*b*c^5*d^3*e^2 +
2996*b^2*c^4*d^2*e^3 - 910*b^3*c^3*d*e^4 + 105*b^4*c^2*e^5)*f - (5632*c^6*d^5 - 16752*b*c^5*d^4*e + 19408*b^2
*c^4*d^3*e^2 - 10808*b^3*c^3*d^2*e^3 + 2940*b^4*c^2*d*e^4 - 315*b^5*c*e^5)*g + 2*(4*(120*c^6*d^3*e^2 + 716*b*c
^5*d^2*e^3 - 266*b^2*c^4*d*e^4 + 35*b^3*c^3*e^5)*f - (1680*c^6*d^4*e - 3632*b*c^5*d^3*e^2 + 2680*b^2*c^4*d^2*e
^3 - 868*b^3*c^3*d*e^4 + 105*b^4*c^2*e^5)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(c^6*e^2), -1/1536
0*(105*(4*(32*c^6*d^5*e - 80*b*c^5*d^4*e^2 + 80*b^2*c^4*d^3*e^3 - 40*b^3*c^3*d^2*e^4 + 10*b^4*c^2*d*e^5 - b^5*
c*e^6)*f + (64*c^6*d^6 - 256*b*c^5*d^5*e + 400*b^2*c^4*d^4*e^2 - 320*b^3*c^3*d^3*e^3 + 140*b^4*c^2*d^2*e^4 - 3
2*b^5*c*d*e^5 + 3*b^6*e^6)*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)*sq
rt(c)/(c^2*e^2*x^2 + b*c*e^2*x - c^2*d^2 + b*c*d*e)) - 2*(1280*c^6*e^5*g*x^5 + 128*(12*c^6*e^5*f + (36*c^6*d*e
^4 + b*c^5*e^5)*g)*x^4 + 16*(12*(30*c^6*d*e^4 + b*c^5*e^5)*f + (340*c^6*d^2*e^3 + 56*b*c^5*d*e^4 - 9*b^2*c^4*e
^5)*g)*x^3 + 8*(4*(224*c^6*d^2*e^3 + 46*b*c^5*d*e^4 - 7*b^2*c^4*e^5)*f + (128*c^6*d^3*e^2 + 380*b*c^5*d^2*e^3
- 152*b^2*c^4*d*e^4 + 21*b^3*c^3*e^5)*g)*x^2 - 4*(2176*c^6*d^4*e - 4472*b*c^5*d^3*e^2 + 2996*b^2*c^4*d^2*e^3 -
910*b^3*c^3*d*e^4 + 105*b^4*c^2*e^5)*f - (5632*c^6*d^5 - 16752*b*c^5*d^4*e + 19408*b^2*c^4*d^3*e^2 - 10808*b^
3*c^3*d^2*e^3 + 2940*b^4*c^2*d*e^4 - 315*b^5*c*e^5)*g + 2*(4*(120*c^6*d^3*e^2 + 716*b*c^5*d^2*e^3 - 266*b^2*c^
4*d*e^4 + 35*b^3*c^3*e^5)*f - (1680*c^6*d^4*e - 3632*b*c^5*d^3*e^2 + 2680*b^2*c^4*d^2*e^3 - 868*b^3*c^3*d*e^4
+ 105*b^4*c^2*e^5)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(c^6*e^2)]

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

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Giac [A]  time = 1.28546, size = 944, normalized size = 2.28 \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="giac")

[Out]

1/7680*sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e)*(2*(4*(2*(8*(10*g*x*e^3 + (36*c^5*d*g*e^10 + 12*c^5*f*e^11 +
b*c^4*g*e^11)*e^(-8)/c^5)*x + (340*c^5*d^2*g*e^9 + 360*c^5*d*f*e^10 + 56*b*c^4*d*g*e^10 + 12*b*c^4*f*e^11 - 9
*b^2*c^3*g*e^11)*e^(-8)/c^5)*x + (128*c^5*d^3*g*e^8 + 896*c^5*d^2*f*e^9 + 380*b*c^4*d^2*g*e^9 + 184*b*c^4*d*f*
e^10 - 152*b^2*c^3*d*g*e^10 - 28*b^2*c^3*f*e^11 + 21*b^3*c^2*g*e^11)*e^(-8)/c^5)*x - (1680*c^5*d^4*g*e^7 - 480
*c^5*d^3*f*e^8 - 3632*b*c^4*d^3*g*e^8 - 2864*b*c^4*d^2*f*e^9 + 2680*b^2*c^3*d^2*g*e^9 + 1064*b^2*c^3*d*f*e^10
- 868*b^3*c^2*d*g*e^10 - 140*b^3*c^2*f*e^11 + 105*b^4*c*g*e^11)*e^(-8)/c^5)*x - (5632*c^5*d^5*g*e^6 + 8704*c^5
*d^4*f*e^7 - 16752*b*c^4*d^4*g*e^7 - 17888*b*c^4*d^3*f*e^8 + 19408*b^2*c^3*d^3*g*e^8 + 11984*b^2*c^3*d^2*f*e^9
- 10808*b^3*c^2*d^2*g*e^9 - 3640*b^3*c^2*d*f*e^10 + 2940*b^4*c*d*g*e^10 + 420*b^4*c*f*e^11 - 315*b^5*g*e^11)*
e^(-8)/c^5) + 7/1024*(64*c^6*d^6*g + 128*c^6*d^5*f*e - 256*b*c^5*d^5*g*e - 320*b*c^5*d^4*f*e^2 + 400*b^2*c^4*d
^4*g*e^2 + 320*b^2*c^4*d^3*f*e^3 - 320*b^3*c^3*d^3*g*e^3 - 160*b^3*c^3*d^2*f*e^4 + 140*b^4*c^2*d^2*g*e^4 + 40*
b^4*c^2*d*f*e^5 - 32*b^5*c*d*g*e^5 - 4*b^5*c*f*e^6 + 3*b^6*g*e^6)*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^6