3.2185 \(\int (d+e x) (f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2} \, dx\)
Optimal. Leaf size=297 \[ \frac{(b+2 c x) (2 c d-b e)^3 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+2 c d g+12 c e f)}{512 c^4 e}+\frac{(b+2 c x) (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-7 b e g+2 c d g+12 c e f)}{192 c^3 e}+\frac{\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (7 b e g-12 c (d g+e f)-10 c e g x)}{60 c^2 e^2}+\frac{(2 c d-b e)^5 (-7 b e g+2 c d g+12 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{1024 c^{9/2} e^2} \]
[Out]
((2*c*d - b*e)^3*(12*c*e*f + 2*c*d*g - 7*b*e*g)*(b + 2*c*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(512*c^
4*e) + ((2*c*d - b*e)*(12*c*e*f + 2*c*d*g - 7*b*e*g)*(b + 2*c*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/
(192*c^3*e) + ((7*b*e*g - 12*c*(e*f + d*g) - 10*c*e*g*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(60*c^2*
e^2) + ((2*c*d - b*e)^5*(12*c*e*f + 2*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])])/(1024*c^(9/2)*e^2)
________________________________________________________________________________________
Rubi [A] time = 0.439888, antiderivative size = 297, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 4, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used =
{779, 612, 621, 204} \[ \frac{(b+2 c x) (2 c d-b e)^3 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+2 c d g+12 c e f)}{512 c^4 e}+\frac{(b+2 c x) (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-7 b e g+2 c d g+12 c e f)}{192 c^3 e}+\frac{\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (7 b e g-12 c (d g+e f)-10 c e g x)}{60 c^2 e^2}+\frac{(2 c d-b e)^5 (-7 b e g+2 c d g+12 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{1024 c^{9/2} e^2} \]
Antiderivative was successfully verified.
[In]
Int[(d + e*x)*(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)^3*(12*c*e*f + 2*c*d*g - 7*b*e*g)*(b + 2*c*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(512*c^
4*e) + ((2*c*d - b*e)*(12*c*e*f + 2*c*d*g - 7*b*e*g)*(b + 2*c*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/
(192*c^3*e) + ((7*b*e*g - 12*c*(e*f + d*g) - 10*c*e*g*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(60*c^2*
e^2) + ((2*c*d - b*e)^5*(12*c*e*f + 2*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])])/(1024*c^(9/2)*e^2)
Rule 779
Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((b
*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x)*(a + b*x + c*x^2)^(p + 1))/(2*c^2*(p + 1)*(2*p + 3
)), x] + Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)), Int[(a
+ b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] && !LeQ[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) (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx &=\frac{(7 b e g-12 c (e f+d g)-10 c e g x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{60 c^2 e^2}+\frac{((2 c d-b e) (12 c e f+2 c d g-7 b e g)) \int \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) (12 c e f+2 c d g-7 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}}{192 c^3 e}+\frac{(7 b e g-12 c (e f+d g)-10 c e g x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{60 c^2 e^2}+\frac{\left ((2 c d-b e)^3 (12 c e f+2 c d g-7 b e g)\right ) \int \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2} \, dx}{128 c^3 e}\\ &=\frac{(2 c d-b e)^3 (12 c e f+2 c d g-7 b e g) (b+2 c x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{512 c^4 e}+\frac{(2 c d-b e) (12 c e f+2 c d g-7 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}}{192 c^3 e}+\frac{(7 b e g-12 c (e f+d g)-10 c e g x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{60 c^2 e^2}+\frac{\left ((2 c d-b e)^5 (12 c e f+2 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}{1024 c^4 e}\\ &=\frac{(2 c d-b e)^3 (12 c e f+2 c d g-7 b e g) (b+2 c x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{512 c^4 e}+\frac{(2 c d-b e) (12 c e f+2 c d g-7 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}}{192 c^3 e}+\frac{(7 b e g-12 c (e f+d g)-10 c e g x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{60 c^2 e^2}+\frac{\left ((2 c d-b e)^5 (12 c e f+2 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 )}{512 c^4 e}\\ &=\frac{(2 c d-b e)^3 (12 c e f+2 c d g-7 b e g) (b+2 c x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{512 c^4 e}+\frac{(2 c d-b e) (12 c e f+2 c d g-7 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}}{192 c^3 e}+\frac{(7 b e g-12 c (e f+d g)-10 c e g x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{60 c^2 e^2}+\frac{(2 c d-b e)^5 (12 c e f+2 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 )}{1024 c^{9/2} e^2}\\ \end{align*}
Mathematica [B] time = 6.20173, size = 1240, normalized size = 4.18 \[ -\frac{(c d e+(c d-b e) e) \left (-6 c f e^2-\left (\frac{7}{2} e (c d-b e)-\frac{5 c d e}{2}\right ) g\right ) (d+e x)^2 ((d+e x) (c (d-e x)-b e))^{3/2} \left (\frac{21 (c d e+(c d-b e) e)^4 \left (-\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 )^4}{512 c^4 e^8 (d+e x)^4 \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{7}{10} \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}{8 \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 ) \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}}{21 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}}}-\frac{g (d+e x)^2 (c d-b e-c e x) ((d+e x) (c (d-e x)-b e))^{3/2}}{6 c e^2} \]
Antiderivative was successfully verified.
[In]
Integrate[(d + e*x)*(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)^2*(c*d - b*e - c*e*x)*((d + e*x)*(-(b*e) + c*(d - e*x)))^(3/2))/(6*c*e^2) - ((c*d*e + e*(c*d - b
*e))*(-6*c*e^2*f - ((-5*c*d*e)/2 + (7*e*(c*d - b*e))/2)*g)*(d + e*x)^2*((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)*((7*(3/(8*(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)))/10 + (21*(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*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) + (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^4*e^8*(d + e*x)^4*(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)))/(21*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))])
________________________________________________________________________________________
Maple [B] time = 0.01, size = 2117, normalized size = 7.1 \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)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x)
[Out]
-15/32*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^2*f-
15/16*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^4*e*f+9/32*
b^2/c*e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d*f+3/8*e*g*b^2/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*
d^2+75/256*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^2-5/8*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
^3-11/64*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-9/128*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+15/128*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*f-1/16*b^2/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)
^(3/2)*e*f-3/128*b^4/c^3*e^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*f+1/48/e*g/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d
^2)^(3/2)*b*d^2-1/6*g/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*b*d-3/8*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^5+45/64*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^4+7/256*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-11/128*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+7/96*e*g*b^2/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e
+c*d^2)^(3/2)*x+7/1024*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))+1/16/e*g*c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^4+1/16/e*g*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^6-3/64*b^3/c^2*e^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d
^2)^(1/2)*x*f-9/16*b*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^2*e*f+9/64*b^3/c^2*e^2*(-c*e^2*x^2-b*e^2*x-b*d
*e+c*d^2)^(1/2)*d*f-9/32*b^2/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^2*e*f+3/16*e*g*b^3/c^2*(-c*e^2*x^2-b*e
^2*x-b*d*e+c*d^2)^(1/2)*d^2+15/16*b^2*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-3/256*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))*f-1/8*b/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*e*f+3/16*d^3*f*(-c*e^2*x^2-b*e^2*x-b*d*
e+c*d^2)^(1/2)*b+1/4*d*f*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x-1/5*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/c
/e*f-1/12*g/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*b^2*d+7/60/e*g*b/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(
5/2)+1/32/e*g*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b*d^4-1/6/e*g*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/c+
1/24/e*g*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*d^2+7/192*e*g*b^3/c^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)
+7/512*e^3*g*b^5/c^4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)+1/8*d*f/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*b
+3/8*d^3*f*c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x+3/8*d^5*f*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))-1/5*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/c/e^2*d*g-5/32*g/c*(-c
*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b^2*d^3-5/16*g*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*b*d^3
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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)*(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 = 9.37025, size = 3206, normalized size = 10.79 \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)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algorithm="fricas")
[Out]
[-1/30720*(15*(12*(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 - 384*b*c^5*d^5*e + 720*b^2*c^4*d^4*e^2 - 640*b^3*c^3*d^3*e^3 + 300*b^4*c^2*d^2
*e^4 - 72*b^5*c*d*e^5 + 7*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 + (12*c^6*d*e^4 + 13*b*c^5*e^5)*g)*x^4 + 16*(12*(10*c^6*d*e^4 + 11*b*c^5*e^5)*f - (140*c^6*d^2*e^3 -
272*b*c^5*d*e^4 - 3*b^2*c^4*e^5)*g)*x^3 - 8*(12*(32*c^6*d^2*e^3 - 62*b*c^5*d*e^4 - b^2*c^4*e^5)*f + (384*c^6*
d^3*e^2 - 348*b*c^5*d^2*e^3 - 48*b^2*c^4*d*e^4 + 7*b^3*c^3*e^5)*g)*x^2 + 12*(128*c^6*d^4*e - 456*b*c^5*d^3*e^2
+ 428*b^2*c^4*d^2*e^3 - 130*b^3*c^3*d*e^4 + 15*b^4*c^2*e^5)*f + (1536*c^6*d^5 - 4368*b*c^5*d^4*e + 5328*b^2*c
^4*d^3*e^2 - 3256*b^3*c^3*d^2*e^3 + 940*b^4*c^2*d*e^4 - 105*b^5*c*e^5)*g - 2*(12*(200*c^6*d^3*e^2 - 172*b*c^5*
d^2*e^3 - 38*b^2*c^4*d*e^4 + 5*b^3*c^3*e^5)*f - (240*c^6*d^4*e - 816*b*c^5*d^3*e^2 + 792*b^2*c^4*d^2*e^3 - 276
*b^3*c^3*d*e^4 + 35*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^5*e^2), -1/15360*(15*(12
*(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 - 384*b*c^5*d^5*e + 720*b^2*c^4*d^4*e^2 - 640*b^3*c^3*d^3*e^3 + 300*b^4*c^2*d^2*e^4 - 72*b^5*c*d
*e^5 + 7*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)*sqrt(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 + (12*c^6*d*e^4 + 13*b
*c^5*e^5)*g)*x^4 + 16*(12*(10*c^6*d*e^4 + 11*b*c^5*e^5)*f - (140*c^6*d^2*e^3 - 272*b*c^5*d*e^4 - 3*b^2*c^4*e^5
)*g)*x^3 - 8*(12*(32*c^6*d^2*e^3 - 62*b*c^5*d*e^4 - b^2*c^4*e^5)*f + (384*c^6*d^3*e^2 - 348*b*c^5*d^2*e^3 - 48
*b^2*c^4*d*e^4 + 7*b^3*c^3*e^5)*g)*x^2 + 12*(128*c^6*d^4*e - 456*b*c^5*d^3*e^2 + 428*b^2*c^4*d^2*e^3 - 130*b^3
*c^3*d*e^4 + 15*b^4*c^2*e^5)*f + (1536*c^6*d^5 - 4368*b*c^5*d^4*e + 5328*b^2*c^4*d^3*e^2 - 3256*b^3*c^3*d^2*e^
3 + 940*b^4*c^2*d*e^4 - 105*b^5*c*e^5)*g - 2*(12*(200*c^6*d^3*e^2 - 172*b*c^5*d^2*e^3 - 38*b^2*c^4*d*e^4 + 5*b
^3*c^3*e^5)*f - (240*c^6*d^4*e - 816*b*c^5*d^3*e^2 + 792*b^2*c^4*d^2*e^3 - 276*b^3*c^3*d*e^4 + 35*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^5*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 ) \left (f + g x\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)*(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)*(f + g*x), x)
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Giac [B] time = 1.21483, size = 956, normalized size = 3.22 \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)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algorithm="giac")
[Out]
-1/7680*sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e)*(2*(4*(2*(8*(10*c*g*x*e^3 + (12*c^6*d*g*e^10 + 12*c^6*f*e^1
1 + 13*b*c^5*g*e^11)*e^(-8)/c^5)*x - (140*c^6*d^2*g*e^9 - 120*c^6*d*f*e^10 - 272*b*c^5*d*g*e^10 - 132*b*c^5*f*
e^11 - 3*b^2*c^4*g*e^11)*e^(-8)/c^5)*x - (384*c^6*d^3*g*e^8 + 384*c^6*d^2*f*e^9 - 348*b*c^5*d^2*g*e^9 - 744*b*
c^5*d*f*e^10 - 48*b^2*c^4*d*g*e^10 - 12*b^2*c^4*f*e^11 + 7*b^3*c^3*g*e^11)*e^(-8)/c^5)*x + (240*c^6*d^4*g*e^7
- 2400*c^6*d^3*f*e^8 - 816*b*c^5*d^3*g*e^8 + 2064*b*c^5*d^2*f*e^9 + 792*b^2*c^4*d^2*g*e^9 + 456*b^2*c^4*d*f*e^
10 - 276*b^3*c^3*d*g*e^10 - 60*b^3*c^3*f*e^11 + 35*b^4*c^2*g*e^11)*e^(-8)/c^5)*x + (1536*c^6*d^5*g*e^6 + 1536*
c^6*d^4*f*e^7 - 4368*b*c^5*d^4*g*e^7 - 5472*b*c^5*d^3*f*e^8 + 5328*b^2*c^4*d^3*g*e^8 + 5136*b^2*c^4*d^2*f*e^9
- 3256*b^3*c^3*d^2*g*e^9 - 1560*b^3*c^3*d*f*e^10 + 940*b^4*c^2*d*g*e^10 + 180*b^4*c^2*f*e^11 - 105*b^5*c*g*e^1
1)*e^(-8)/c^5) + 1/1024*(64*c^6*d^6*g + 384*c^6*d^5*f*e - 384*b*c^5*d^5*g*e - 960*b*c^5*d^4*f*e^2 + 720*b^2*c^
4*d^4*g*e^2 + 960*b^2*c^4*d^3*f*e^3 - 640*b^3*c^3*d^3*g*e^3 - 480*b^3*c^3*d^2*f*e^4 + 300*b^4*c^2*d^2*g*e^4 +
120*b^4*c^2*d*f*e^5 - 72*b^5*c*d*g*e^5 - 12*b^5*c*f*e^6 + 7*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^5