3.2184 \(\int (d+e x)^2 (f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2} \, dx\)
Optimal. Leaf size=413 \[ \frac{(b+2 c x) (2 c d-b e)^4 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-9 b e g+4 c d g+14 c e f)}{1024 c^5 e}+\frac{(b+2 c x) (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-9 b e g+4 c d g+14 c e f)}{384 c^4 e}-\frac{(2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-9 b e g+4 c d g+14 c e f)}{120 c^3 e^2}-\frac{(d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-9 b e g+4 c d g+14 c e f)}{84 c^2 e^2}+\frac{(2 c d-b e)^6 (-9 b e g+4 c d g+14 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{2048 c^{11/2} e^2}-\frac{g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e^2} \]
[Out]
((2*c*d - b*e)^4*(14*c*e*f + 4*c*d*g - 9*b*e*g)*(b + 2*c*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(1024*c
^5*e) + ((2*c*d - b*e)^2*(14*c*e*f + 4*c*d*g - 9*b*e*g)*(b + 2*c*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2
))/(384*c^4*e) - ((2*c*d - b*e)*(14*c*e*f + 4*c*d*g - 9*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(1
20*c^3*e^2) - ((14*c*e*f + 4*c*d*g - 9*b*e*g)*(d + e*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(84*c^2*e
^2) - (g*(d + e*x)^2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(7*c*e^2) + ((2*c*d - b*e)^6*(14*c*e*f + 4*c
*d*g - 9*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])])/(2048*c^(11/2)*
e^2)
________________________________________________________________________________________
Rubi [A] time = 0.60274, antiderivative size = 413, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 7, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.159, Rules used =
{1638, 12, 670, 640, 612, 621, 204} \[ \frac{(b+2 c x) (2 c d-b e)^4 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-9 b e g+4 c d g+14 c e f)}{1024 c^5 e}+\frac{(b+2 c x) (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-9 b e g+4 c d g+14 c e f)}{384 c^4 e}-\frac{(2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-9 b e g+4 c d g+14 c e f)}{120 c^3 e^2}-\frac{(d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-9 b e g+4 c d g+14 c e f)}{84 c^2 e^2}+\frac{(2 c d-b e)^6 (-9 b e g+4 c d g+14 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{2048 c^{11/2} e^2}-\frac{g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e^2} \]
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*d - b*e)^4*(14*c*e*f + 4*c*d*g - 9*b*e*g)*(b + 2*c*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(1024*c
^5*e) + ((2*c*d - b*e)^2*(14*c*e*f + 4*c*d*g - 9*b*e*g)*(b + 2*c*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2
))/(384*c^4*e) - ((2*c*d - b*e)*(14*c*e*f + 4*c*d*g - 9*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(1
20*c^3*e^2) - ((14*c*e*f + 4*c*d*g - 9*b*e*g)*(d + e*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(84*c^2*e
^2) - (g*(d + e*x)^2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(7*c*e^2) + ((2*c*d - b*e)^6*(14*c*e*f + 4*c
*d*g - 9*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])])/(2048*c^(11/2)*
e^2)
Rule 1638
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq
, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[(f*(d + e*x)^(m + q - 1)*(a + b*x + c*x^2)^(p + 1))/(c*e^(q - 1)*(
m + q + 2*p + 1)), x] + Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^
q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 1)*(d + e*x)^q + e*f*(m + p + q)*(d + e*x)^(q - 2)*(b*d - 2*a*e +
(2*c*d - b*e)*x), x], x], x] /; NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && PolyQ[Pq, x] &&
NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0]
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
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)^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{g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e^2}-\frac{\int -\frac{1}{2} e^2 (14 c e f+4 c d g-9 b e g) (d+e x)^2 \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx}{7 c e^3}\\ &=-\frac{g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e^2}+\frac{(14 c e f+4 c d g-9 b e g) \int (d+e x)^2 \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx}{14 c e}\\ &=-\frac{(14 c e f+4 c d g-9 b e g) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{84 c^2 e^2}-\frac{g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e^2}+\frac{((2 c d-b e) (14 c e f+4 c d g-9 b e g)) \int (d+e x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx}{24 c^2 e}\\ &=-\frac{(2 c d-b e) (14 c e f+4 c d g-9 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{120 c^3 e^2}-\frac{(14 c e f+4 c d g-9 b e g) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{84 c^2 e^2}-\frac{g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e^2}+\frac{\left ((2 c d-b e)^2 (14 c e f+4 c d g-9 b e g)\right ) \int \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx}{48 c^3 e}\\ &=\frac{(2 c d-b e)^2 (14 c e f+4 c d g-9 b e g) (b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{384 c^4 e}-\frac{(2 c d-b e) (14 c e f+4 c d g-9 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{120 c^3 e^2}-\frac{(14 c e f+4 c d g-9 b e g) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{84 c^2 e^2}-\frac{g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e^2}+\frac{\left ((2 c d-b e)^4 (14 c e f+4 c d g-9 b e g)\right ) \int \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2} \, dx}{256 c^4 e}\\ &=\frac{(2 c d-b e)^4 (14 c e f+4 c d g-9 b e g) (b+2 c x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{1024 c^5 e}+\frac{(2 c d-b e)^2 (14 c e f+4 c d g-9 b e g) (b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{384 c^4 e}-\frac{(2 c d-b e) (14 c e f+4 c d g-9 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{120 c^3 e^2}-\frac{(14 c e f+4 c d g-9 b e g) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{84 c^2 e^2}-\frac{g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e^2}+\frac{\left ((2 c d-b e)^6 (14 c e f+4 c d g-9 b e g)\right ) \int \frac{1}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{2048 c^5 e}\\ &=\frac{(2 c d-b e)^4 (14 c e f+4 c d g-9 b e g) (b+2 c x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{1024 c^5 e}+\frac{(2 c d-b e)^2 (14 c e f+4 c d g-9 b e g) (b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{384 c^4 e}-\frac{(2 c d-b e) (14 c e f+4 c d g-9 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{120 c^3 e^2}-\frac{(14 c e f+4 c d g-9 b e g) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{84 c^2 e^2}-\frac{g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e^2}+\frac{\left ((2 c d-b e)^6 (14 c e f+4 c d g-9 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 )}{1024 c^5 e}\\ &=\frac{(2 c d-b e)^4 (14 c e f+4 c d g-9 b e g) (b+2 c x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{1024 c^5 e}+\frac{(2 c d-b e)^2 (14 c e f+4 c d g-9 b e g) (b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{384 c^4 e}-\frac{(2 c d-b e) (14 c e f+4 c d g-9 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{120 c^3 e^2}-\frac{(14 c e f+4 c d g-9 b e g) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{84 c^2 e^2}-\frac{g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e^2}+\frac{(2 c d-b e)^6 (14 c e f+4 c d g-9 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 )}{2048 c^{11/2} e^2}\\ \end{align*}
Mathematica [B] time = 6.35967, size = 1329, normalized size = 3.22 \[ -\frac{g (c d-b e-c e x) ((d+e x) (c (d-e x)-b e))^{3/2} (d+e x)^3}{7 c e^2}-\frac{2 (c d e+(c d-b e) e) \left (-7 c f e^2-\left (\frac{9}{2} e (c d-b e)-\frac{5 c d e}{2}\right ) g\right ) ((d+e x) (c (d-e x)-b e))^{3/2} \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 )^{5/2} \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}{2048 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 )^2}+\frac{3}{4} \left (\frac{1}{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 )}}+\frac{3}{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 )^2}\right )\right ) (d+e x)^3}{63 c e^4 \left (\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}}\right )^{3/2} (c d-b e-c e x) \sqrt{\frac{e (c d-b e-c e x)}{c d e+(c d-b e) 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]
-(g*(d + e*x)^3*(c*d - b*e - c*e*x)*((d + e*x)*(-(b*e) + c*(d - e*x)))^(3/2))/(7*c*e^2) - (2*(c*d*e + e*(c*d -
b*e))*(-7*c*e^2*f - ((-5*c*d*e)/2 + (9*e*(c*d - b*e))/2)*g)*(d + e*x)^3*((d + e*x)*(-(b*e) + c*(d - e*x)))^(3
/2)*(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)))))^(5/2)*((3*(3/(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)))))^2) + (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)))))^(-1)))/4 + (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))))])))/(2048*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)))))^2)))/(63*c*e^4*(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)*(c*d - b*e - c*e*x)*Sqrt[(e*(c*d - b*e - c*e*x))/(c*d*e + e*(c*d - b*e))])
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Maple [B] time = 0.011, size = 2799, normalized size = 6.8 \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]
105/256*b^4/c^2*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^2
*f-31/64*e^2*g*b^3/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^2+7/128*e^5*g*b^6/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-147/512*e^4*g*b^5/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^2-175/128*e^2*g*b^3/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+5/24*e*g*b^2/c^2*(-c*e^2*x^2-b*e^2*x-b*
d*e+c*d^2)^(3/2)*x*d+105/128*e^3*g*b^4/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+19/128*e^3*g*b^4/c^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d+3/4*e*g*b^2/c*(-c*e^2*
x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^3-35/32*b^3/c*e^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^3*f-7/32*b^3/c^2*e^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d*f+21/32*b^2/c*
(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^2*e^2*f-7/24/c*e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*b*d*f-21/
16*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))*b*d^5*f-21/256*b
^5/c^3*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))*d*f-2/5*(-c*
e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/c/e*d*f+1/12/e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*d^3*g+21/16*e*g*b^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^5+19/256*e^3*g*b^5/c
^4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d-9/2048*e^6*g*b^7/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))-9/512*e^4*g*b^5/c^4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x-13/48*g
*b/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*d^2-21/32*g*b*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+3/8*e*g*b^3/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^3+5/48*e*g
*b^3/c^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d-3/64*e^2*g*b^3/c^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x+
105/64*b^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^4*e^2*f+1/
8*c^2/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^7*g+1/24/c/e*
(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*b*d^3*g+1/8*c/e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^5*g-7/16/c*e
*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b^2*d^3*f-7/48/c^2*e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*b^2*d*f+21
/64*b^3/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^2*e^2*f-7/8*e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*b*
d^3*f-1/3*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/c/e*d*g+7/96*b^2/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)
*x*e^2*f-31/128*e^2*g*b^4/c^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^2+61/210/e*g/c^2*(-c*e^2*x^2-b*e^2*x-b*
d*e+c*d^2)^(5/2)*b*d+7/256*b^4/c^3*e^4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*f-7/64*b^4/c^3*e^3*(-c*e^2*x^2
-b*e^2*x-b*d*e+c*d^2)^(1/2)*d*f+7/1024*b^6/c^4*e^6/(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+7/60*b/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)*f+7/24*(-c*e^2*x^2-b*e^2*x-b*d
*e+c*d^2)^(3/2)*x*d^2*f+7/32*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b*d^4*f-1/6*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*
d^2)^(5/2)/c*f-1/7*g*x^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/c-3/40*g*b^2/c^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d
^2)^(5/2)+7/512*b^5/c^4*e^4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*f+7/192*b^3/c^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c
*d^2)^(3/2)*e^2*f+7/16*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^6*f+7/48/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*b*d^2*f-9/35/e^2*g/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^
(5/2)*d^2-17/64*g*b^2/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^4-17/32*g*b*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^
(1/2)*x*d^4+3/28*g*b/c^2*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)-13/96*g*b^2/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*
d^2)^(3/2)*d^2-9/1024*e^4*g*b^6/c^5*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-3/128*e^2*g*b^4/c^4*(-c*e^2*x^2-b*e
^2*x-b*d*e+c*d^2)^(3/2)+1/16/e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b*d^5*g+7/16*c*(-c*e^2*x^2-b*e^2*x-b*d*e
+c*d^2)^(1/2)*x*d^4*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)^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 [B] time = 18.6842, size = 4220, normalized size = 10.22 \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)^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]
[1/430080*(105*(14*(64*c^7*d^6*e - 192*b*c^6*d^5*e^2 + 240*b^2*c^5*d^4*e^3 - 160*b^3*c^4*d^3*e^4 + 60*b^4*c^3*
d^2*e^5 - 12*b^5*c^2*d*e^6 + b^6*c*e^7)*f + (256*c^7*d^7 - 1344*b*c^6*d^6*e + 2688*b^2*c^5*d^5*e^2 - 2800*b^3*
c^4*d^4*e^3 + 1680*b^4*c^3*d^3*e^4 - 588*b^5*c^2*d^2*e^5 + 112*b^6*c*d*e^6 - 9*b^7*e^7)*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*(15360*c^7*e^6*g*x^6 + 1280*(14*c^7*e^6*f + (28*c^7*d*e^5 + 15*b*c^6*e^6)*g)*x^5 + 128*
(14*(24*c^7*d*e^5 + 13*b*c^6*e^6)*f - (24*c^7*d^2*e^4 - 556*b*c^6*d*e^5 - 3*b^2*c^5*e^6)*g)*x^4 - 16*(14*(20*c
^7*d^2*e^4 - 404*b*c^6*d*e^5 - 3*b^2*c^5*e^6)*f + (3920*c^7*d^3*e^3 - 5636*b*c^6*d^2*e^4 - 216*b^2*c^5*d*e^5 +
27*b^3*c^4*e^6)*g)*x^3 - 8*(14*(768*c^7*d^3*e^3 - 1092*b*c^6*d^2*e^4 - 60*b^2*c^5*d*e^5 + 7*b^3*c^4*e^6)*f +
(4992*c^7*d^4*e^2 - 3600*b*c^6*d^3*e^3 - 1932*b^2*c^5*d^2*e^4 + 568*b^3*c^4*d*e^5 - 63*b^4*c^3*e^6)*g)*x^2 + 1
4*(3072*c^7*d^5*e - 9840*b*c^6*d^4*e^2 + 10464*b^2*c^5*d^3*e^3 - 4816*b^3*c^4*d^2*e^4 + 1120*b^4*c^3*d*e^5 - 1
05*b^5*c^2*e^6)*f + (27648*c^7*d^6 - 97728*b*c^6*d^5*e + 145776*b^2*c^5*d^4*e^2 - 113440*b^3*c^4*d^3*e^3 + 478
24*b^4*c^3*d^2*e^4 - 10500*b^5*c^2*d*e^5 + 945*b^6*c*e^6)*g - 2*(14*(2160*c^7*d^4*e^2 - 1248*b*c^6*d^3*e^3 - 1
248*b^2*c^5*d^2*e^4 + 336*b^3*c^4*d*e^5 - 35*b^4*c^3*e^6)*f - (6720*c^7*d^5*e - 21648*b*c^6*d^4*e^2 + 24480*b^
2*c^5*d^3*e^3 - 12576*b^3*c^4*d^2*e^4 + 3164*b^4*c^3*d*e^5 - 315*b^5*c^2*e^6)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x
+ c*d^2 - b*d*e))/(c^6*e^2), -1/215040*(105*(14*(64*c^7*d^6*e - 192*b*c^6*d^5*e^2 + 240*b^2*c^5*d^4*e^3 - 160*
b^3*c^4*d^3*e^4 + 60*b^4*c^3*d^2*e^5 - 12*b^5*c^2*d*e^6 + b^6*c*e^7)*f + (256*c^7*d^7 - 1344*b*c^6*d^6*e + 268
8*b^2*c^5*d^5*e^2 - 2800*b^3*c^4*d^4*e^3 + 1680*b^4*c^3*d^3*e^4 - 588*b^5*c^2*d^2*e^5 + 112*b^6*c*d*e^6 - 9*b^
7*e^7)*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*(15360*c^7*e^6*g*x^6 + 1280*(14*c^7*e^6*f + (28*c^7*d*e^5 + 15*b*c^6*e^6)
*g)*x^5 + 128*(14*(24*c^7*d*e^5 + 13*b*c^6*e^6)*f - (24*c^7*d^2*e^4 - 556*b*c^6*d*e^5 - 3*b^2*c^5*e^6)*g)*x^4
- 16*(14*(20*c^7*d^2*e^4 - 404*b*c^6*d*e^5 - 3*b^2*c^5*e^6)*f + (3920*c^7*d^3*e^3 - 5636*b*c^6*d^2*e^4 - 216*b
^2*c^5*d*e^5 + 27*b^3*c^4*e^6)*g)*x^3 - 8*(14*(768*c^7*d^3*e^3 - 1092*b*c^6*d^2*e^4 - 60*b^2*c^5*d*e^5 + 7*b^3
*c^4*e^6)*f + (4992*c^7*d^4*e^2 - 3600*b*c^6*d^3*e^3 - 1932*b^2*c^5*d^2*e^4 + 568*b^3*c^4*d*e^5 - 63*b^4*c^3*e
^6)*g)*x^2 + 14*(3072*c^7*d^5*e - 9840*b*c^6*d^4*e^2 + 10464*b^2*c^5*d^3*e^3 - 4816*b^3*c^4*d^2*e^4 + 1120*b^4
*c^3*d*e^5 - 105*b^5*c^2*e^6)*f + (27648*c^7*d^6 - 97728*b*c^6*d^5*e + 145776*b^2*c^5*d^4*e^2 - 113440*b^3*c^4
*d^3*e^3 + 47824*b^4*c^3*d^2*e^4 - 10500*b^5*c^2*d*e^5 + 945*b^6*c*e^6)*g - 2*(14*(2160*c^7*d^4*e^2 - 1248*b*c
^6*d^3*e^3 - 1248*b^2*c^5*d^2*e^4 + 336*b^3*c^4*d*e^5 - 35*b^4*c^3*e^6)*f - (6720*c^7*d^5*e - 21648*b*c^6*d^4*
e^2 + 24480*b^2*c^5*d^3*e^3 - 12576*b^3*c^4*d^2*e^4 + 3164*b^4*c^3*d*e^5 - 315*b^5*c^2*e^6)*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 \left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac{3}{2}} \left (d + e x\right )^{2} \left (f + g x\right )\, 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)*(b*e - c*d + c*e*x))**(3/2)*(d + e*x)**2*(f + g*x), x)
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Giac [B] time = 1.25695, size = 1229, normalized size = 2.98 \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)^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]
-1/107520*sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e)*(2*(4*(2*(8*(10*(12*c*g*x*e^4 + (28*c^7*d*g*e^13 + 14*c^7
*f*e^14 + 15*b*c^6*g*e^14)*e^(-10)/c^6)*x - (24*c^7*d^2*g*e^12 - 336*c^7*d*f*e^13 - 556*b*c^6*d*g*e^13 - 182*b
*c^6*f*e^14 - 3*b^2*c^5*g*e^14)*e^(-10)/c^6)*x - (3920*c^7*d^3*g*e^11 + 280*c^7*d^2*f*e^12 - 5636*b*c^6*d^2*g*
e^12 - 5656*b*c^6*d*f*e^13 - 216*b^2*c^5*d*g*e^13 - 42*b^2*c^5*f*e^14 + 27*b^3*c^4*g*e^14)*e^(-10)/c^6)*x - (4
992*c^7*d^4*g*e^10 + 10752*c^7*d^3*f*e^11 - 3600*b*c^6*d^3*g*e^11 - 15288*b*c^6*d^2*f*e^12 - 1932*b^2*c^5*d^2*
g*e^12 - 840*b^2*c^5*d*f*e^13 + 568*b^3*c^4*d*g*e^13 + 98*b^3*c^4*f*e^14 - 63*b^4*c^3*g*e^14)*e^(-10)/c^6)*x +
(6720*c^7*d^5*g*e^9 - 30240*c^7*d^4*f*e^10 - 21648*b*c^6*d^4*g*e^10 + 17472*b*c^6*d^3*f*e^11 + 24480*b^2*c^5*
d^3*g*e^11 + 17472*b^2*c^5*d^2*f*e^12 - 12576*b^3*c^4*d^2*g*e^12 - 4704*b^3*c^4*d*f*e^13 + 3164*b^4*c^3*d*g*e^
13 + 490*b^4*c^3*f*e^14 - 315*b^5*c^2*g*e^14)*e^(-10)/c^6)*x + (27648*c^7*d^6*g*e^8 + 43008*c^7*d^5*f*e^9 - 97
728*b*c^6*d^5*g*e^9 - 137760*b*c^6*d^4*f*e^10 + 145776*b^2*c^5*d^4*g*e^10 + 146496*b^2*c^5*d^3*f*e^11 - 113440
*b^3*c^4*d^3*g*e^11 - 67424*b^3*c^4*d^2*f*e^12 + 47824*b^4*c^3*d^2*g*e^12 + 15680*b^4*c^3*d*f*e^13 - 10500*b^5
*c^2*d*g*e^13 - 1470*b^5*c^2*f*e^14 + 945*b^6*c*g*e^14)*e^(-10)/c^6) + 1/2048*(256*c^7*d^7*g + 896*c^7*d^6*f*e
- 1344*b*c^6*d^6*g*e - 2688*b*c^6*d^5*f*e^2 + 2688*b^2*c^5*d^5*g*e^2 + 3360*b^2*c^5*d^4*f*e^3 - 2800*b^3*c^4*
d^4*g*e^3 - 2240*b^3*c^4*d^3*f*e^4 + 1680*b^4*c^3*d^3*g*e^4 + 840*b^4*c^3*d^2*f*e^5 - 588*b^5*c^2*d^2*g*e^5 -
168*b^5*c^2*d*f*e^6 + 112*b^6*c*d*g*e^6 + 14*b^6*c*f*e^7 - 9*b^7*g*e^7)*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