3.2225 \(\int \frac{(d+e x)^3 (f+g x)}{(c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}} \, dx\)
Optimal. Leaf size=177 \[ -\frac{2 g (d+e x)}{c^2 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{g \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{c^{5/2} e^2}+\frac{2 (d+e x)^3 (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]
[Out]
(2*(c*e*f + c*d*g - b*e*g)*(d + e*x)^3)/(3*c*e^2*(2*c*d - b*e)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2)) -
(2*g*(d + e*x))/(c^2*e^2*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) + (g*ArcTan[(e*(b + 2*c*x))/(2*Sqrt[c]*Sqr
t[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])])/(c^(5/2)*e^2)
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Rubi [A] time = 0.301974, antiderivative size = 177, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 4, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used =
{788, 652, 621, 204} \[ -\frac{2 g (d+e x)}{c^2 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{g \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{c^{5/2} e^2}+\frac{2 (d+e x)^3 (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
Int[((d + e*x)^3*(f + g*x))/(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2),x]
[Out]
(2*(c*e*f + c*d*g - b*e*g)*(d + e*x)^3)/(3*c*e^2*(2*c*d - b*e)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2)) -
(2*g*(d + e*x))/(c^2*e^2*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) + (g*ArcTan[(e*(b + 2*c*x))/(2*Sqrt[c]*Sqr
t[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])])/(c^(5/2)*e^2)
Rule 788
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((g*(c*d - b*e) + c*e*f)*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(c*(p + 1)*(2*c*d - b*e)), x] - Dist[(e*(m*(g
*(c*d - b*e) + c*e*f) + e*(p + 1)*(2*c*f - b*g)))/(c*(p + 1)*(2*c*d - b*e)), Int[(d + e*x)^(m - 1)*(a + b*x +
c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2,
0] && LtQ[p, -1] && GtQ[m, 0]
Rule 652
Int[((d_.) + (e_.)*(x_))^2*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)*(a + b*x +
c*x^2)^(p + 1))/(c*(p + 1)), x] - Dist[(e^2*(p + 2))/(c*(p + 1)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; Fr
eeQ[{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] && !IntegerQ[p] && LtQ[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)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx &=\frac{2 (c e f+c d g-b e g) (d+e x)^3}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{g \int \frac{(d+e x)^2}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{c e}\\ &=\frac{2 (c e f+c d g-b e g) (d+e x)^3}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{2 g (d+e x)}{c^2 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{g \int \frac{1}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{c^2 e}\\ &=\frac{2 (c e f+c d g-b e g) (d+e x)^3}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{2 g (d+e x)}{c^2 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{(2 g) \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 )}{c^2 e}\\ &=\frac{2 (c e f+c d g-b e g) (d+e x)^3}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{2 g (d+e x)}{c^2 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{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 )}{c^{5/2} e^2}\\ \end{align*}
Mathematica [A] time = 1.33697, size = 228, normalized size = 1.29 \[ \frac{2 \left (-\frac{\sqrt{c} (d+e x) \left (3 b^2 e^2 g+4 b c e g (e x-2 d)+c^2 \left (5 d^2 g-d e (f+7 g x)-e^2 f x\right )\right )}{b e-c d+c e x}-\frac{3 \sqrt{e} g \sqrt{d+e x} (b e-2 c d)^2 \sqrt{\frac{b e-c d+c e x}{b e-2 c d}} \sin ^{-1}\left (\frac{\sqrt{c} \sqrt{e} \sqrt{d+e x}}{\sqrt{e (2 c d-b e)}}\right )}{\sqrt{e (2 c d-b e)}}\right )}{3 c^{5/2} e^2 (b e-2 c d) \sqrt{(d+e x) (c (d-e x)-b e)}} \]
Antiderivative was successfully verified.
[In]
Integrate[((d + e*x)^3*(f + g*x))/(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2),x]
[Out]
(2*(-((Sqrt[c]*(d + e*x)*(3*b^2*e^2*g + 4*b*c*e*g*(-2*d + e*x) + c^2*(5*d^2*g - e^2*f*x - d*e*(f + 7*g*x))))/(
-(c*d) + b*e + c*e*x)) - (3*Sqrt[e]*(-2*c*d + b*e)^2*g*Sqrt[d + e*x]*Sqrt[(-(c*d) + b*e + c*e*x)/(-2*c*d + b*e
)]*ArcSin[(Sqrt[c]*Sqrt[e]*Sqrt[d + e*x])/Sqrt[e*(2*c*d - b*e)]])/Sqrt[e*(2*c*d - b*e)]))/(3*c^(5/2)*e^2*(-2*c
*d + b*e)*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))])
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Maple [B] time = 0.011, size = 3485, normalized size = 19.7 \begin{align*} \text{output 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)^(5/2),x)
[Out]
1/3/c/e/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d^2*f-1/24*b^2/c^3*e/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*f-2
4*e^5*g*b^2/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^2-12*e^5*g*b^3/c
/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^2+3/2*e^3*g*b^3/c^2/(-b^2*e^4
+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d^2+3*e^6*g*b^4/c^2/(-b^2*e^4+4*b*c*d*e^3-4
*c^2*d^2*e^2)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d+1/24*e^5*g*b^4/c^3/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^
2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x+104/3*b*c*e^4/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)^2/(-c*e^2*x^2-b
*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^3*g+8*b*c*e^5/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c
*d^2)^(1/2)*x*d^2*f+1/2*b^2/c*e^4/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*
x*d*f+6*e^6*g*b^3/c/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d-3/4*e^4*
g*b^3/c^2/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*d+3*e^3*g*b^2/c/(-b^2*
e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*d^2+1/48*e^5*g*b^5/c^4/(-b^2*e^4+4*b*c
*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)-1/6*e^7*g*b^5/c^3/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2
*e^2)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)+25/12/e*g*b/c^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d^2+1/2*
e^3*g/c^3*b^3/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)+3/4*b/c^2*x/(-c*e^2*
x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d*g-1/2*e*g*b/c^2*x^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)-1/8*e*g*b^2/c^3*x/
(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)+4*b^2*e^5/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)^2/(-c*e^2*x^2-b*e^2*x-b*
d*e+c*d^2)^(1/2)*d^2*f+1/4*b/c^2*x*e/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*f-1/24*b^4/c^3/(-b^2*e^4+4*b*c*d*e
^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*e^5*f+1/3*b^4/c^2*e^7/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2
*e^2)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*f+3/2*x/c/e/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d^2*g+52/3*b
^2*e^4/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^3*g+1/(-b^2*e^4+4*b*c*d
*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*b*d^4*e*g+1/3/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/
(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*b*d^3*e^2*f-13/3*b/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e
^2*x-b*d*e+c*d^2)^(3/2)*x*d^3*e^2*g-1/3*e^7*g*b^4/c^2/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)^2/(-c*e^2*x^2-b*e^2
*x-b*d*e+c*d^2)^(1/2)*x-3/8*e^4*g*b^4/c^3/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2
)^(3/2)*d+e^3*g/c^2*b^2/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x+1/2/e*g/
c^3*b/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)+1/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))+3/2*x/c/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d*f+1/48*e*g*b^3/c^4/(-c*e^2*x^
2-b*e^2*x-b*d*e+c*d^2)^(3/2)-1/e*g/c^2*x/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-11/24*g*b^2/c^3/(-c*e^2*x^2-b*
e^2*x-b*d*e+c*d^2)^(3/2)*d+x^2/c*e/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*f-5/3/c/e^2/(-c*e^2*x^2-b*e^2*x-b*d*
e+c*d^2)^(3/2)*d^3*g+3*x^2/c/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d*g+5/12/c^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d
^2)^(3/2)*b*d*f+1/3*e*g*x^3/c/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)+2/3*b^3/c*e^7/(-b^2*e^4+4*b*c*d*e^3-4*c^2
*d^2*e^2)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*f+1/4*b^3/c^2*e^4/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-
c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d*f-16/3*c^2*e^4/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)^2/(-c*e^2*x^2-b*e^2
*x-b*d*e+c*d^2)^(1/2)*x*d^3*f-8*c*e^3/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^
(1/2)*b*d^4*g-8/3*c*e^4/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b*d^3*f+
2/3*c/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*d^3*e^2*f+2*c/(-b^2*e^4+4*
b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*d^4*e*g-16*c^2*e^3/(-b^2*e^4+4*b*c*d*e^3-4*c
^2*d^2*e^2)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^4*g-b/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^
2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*d^2*e^3*f-13/6*b^2/c/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b
*d*e+c*d^2)^(3/2)*d^3*e^2*g-1/2*b^2/c/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3
/2)*d^2*e^3*f-4*b^2*e^6/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d*f-2*
b^3/c*e^6/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d*f-1/12*b^3/c^2/(-b^2
*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*e^5*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)^(5/2),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 17.6471, size = 1602, normalized size = 9.05 \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)^(5/2),x, algorithm="fricas")
[Out]
[-1/6*(3*((2*c^3*d*e^2 - b*c^2*e^3)*g*x^2 - 2*(2*c^3*d^2*e - 3*b*c^2*d*e^2 + b^2*c*e^3)*g*x + (2*c^3*d^3 - 5*b
*c^2*d^2*e + 4*b^2*c*d*e^2 - 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*(c^3*d*e*f - (5*c^3*d^2 - 8
*b*c^2*d*e + 3*b^2*c*e^2)*g + (c^3*e^2*f + (7*c^3*d*e - 4*b*c^2*e^2)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 -
b*d*e))/(2*c^6*d^3*e^2 - 5*b*c^5*d^2*e^3 + 4*b^2*c^4*d*e^4 - b^3*c^3*e^5 + (2*c^6*d*e^4 - b*c^5*e^5)*x^2 - 2*
(2*c^6*d^2*e^3 - 3*b*c^5*d*e^4 + b^2*c^4*e^5)*x), -1/3*(3*((2*c^3*d*e^2 - b*c^2*e^3)*g*x^2 - 2*(2*c^3*d^2*e -
3*b*c^2*d*e^2 + b^2*c*e^3)*g*x + (2*c^3*d^3 - 5*b*c^2*d^2*e + 4*b^2*c*d*e^2 - b^3*e^3)*g)*sqrt(c)*arctan(1/2*s
qrt(-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*(c^3*d*e*f - (5*c^3*d^2 - 8*b*c^2*d*e + 3*b^2*c*e^2)*g + (c^3*e^2*f + (7*c^3*d*e - 4*b*c^2*e^2)*g)*x)*s
qrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(2*c^6*d^3*e^2 - 5*b*c^5*d^2*e^3 + 4*b^2*c^4*d*e^4 - b^3*c^3*e^5 +
(2*c^6*d*e^4 - b*c^5*e^5)*x^2 - 2*(2*c^6*d^2*e^3 - 3*b*c^5*d*e^4 + b^2*c^4*e^5)*x)]
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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 )}{\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac{5}{2}}}\, 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)**(5/2),x)
[Out]
Integral((d + e*x)**3*(f + g*x)/(-(d + e*x)*(b*e - c*d + c*e*x))**(5/2), x)
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Giac [B] time = 1.29808, size = 1135, normalized size = 6.41 \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)^(5/2),x, algorithm="giac")
[Out]
sqrt(-c*e^2)*g*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^3 + 2/3*sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e)*((((56*c^5*d^4*g*e^4 + 8*c^5*d^3*f*e^5 - 116*b*c^4*
d^3*g*e^5 - 12*b*c^4*d^2*f*e^6 + 90*b^2*c^3*d^2*g*e^6 + 6*b^2*c^3*d*f*e^7 - 31*b^3*c^2*d*g*e^7 - b^3*c^2*f*e^8
+ 4*b^4*c*g*e^8)*x/(16*c^6*d^4*e^3 - 32*b*c^5*d^3*e^4 + 24*b^2*c^4*d^2*e^5 - 8*b^3*c^3*d*e^6 + b^4*c^2*e^7) +
3*(24*c^5*d^5*g*e^3 + 8*c^5*d^4*f*e^4 - 36*b*c^4*d^4*g*e^4 - 12*b*c^4*d^3*f*e^5 + 10*b^2*c^3*d^3*g*e^5 + 6*b^
2*c^3*d^2*f*e^6 + 9*b^3*c^2*d^2*g*e^6 - b^3*c^2*d*f*e^7 - 6*b^4*c*d*g*e^7 + b^5*g*e^8)/(16*c^6*d^4*e^3 - 32*b*
c^5*d^3*e^4 + 24*b^2*c^4*d^2*e^5 - 8*b^3*c^3*d*e^6 + b^4*c^2*e^7))*x - 3*(8*c^5*d^6*g*e^2 - 8*c^5*d^5*f*e^3 -
44*b*c^4*d^5*g*e^3 + 12*b*c^4*d^4*f*e^4 + 70*b^2*c^3*d^4*g*e^4 - 6*b^2*c^3*d^3*f*e^5 - 49*b^3*c^2*d^3*g*e^5 +
b^3*c^2*d^2*f*e^6 + 16*b^4*c*d^2*g*e^6 - 2*b^5*d*g*e^7)/(16*c^6*d^4*e^3 - 32*b*c^5*d^3*e^4 + 24*b^2*c^4*d^2*e^
5 - 8*b^3*c^3*d*e^6 + b^4*c^2*e^7))*x - (40*c^5*d^7*g*e - 8*c^5*d^6*f*e^2 - 124*b*c^4*d^6*g*e^2 + 12*b*c^4*d^5
*f*e^3 + 150*b^2*c^3*d^5*g*e^3 - 6*b^2*c^3*d^4*f*e^4 - 89*b^3*c^2*d^4*g*e^4 + b^3*c^2*d^3*f*e^5 + 26*b^4*c*d^3
*g*e^5 - 3*b^5*d^2*g*e^6)/(16*c^6*d^4*e^3 - 32*b*c^5*d^3*e^4 + 24*b^2*c^4*d^2*e^5 - 8*b^3*c^3*d*e^6 + b^4*c^2*
e^7))/(c*x^2*e^2 - c*d^2 + b*x*e^2 + b*d*e)^2