3.1965 \(\int \frac{(d+e x)^5}{(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx\)

Optimal. Leaf size=231 \[ -\frac{10 e (d+e x)^2}{3 c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{5 e^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c^3 d^3}+\frac{5 e^{3/2} \left (c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 c^{7/2} d^{7/2}}-\frac{2 (d+e x)^4}{3 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

[Out]

(-2*(d + e*x)^4)/(3*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (10*e*(d + e*x)^2)/(3*c^2*d^2*Sqrt[a*
d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (5*e^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(c^3*d^3) + (5*e^(
3/2)*(c*d^2 - a*e^2)*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^
2)*x + c*d*e*x^2])])/(2*c^(7/2)*d^(7/2))

________________________________________________________________________________________

Rubi [A]  time = 0.139867, antiderivative size = 231, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.108, Rules used = {668, 640, 621, 206} \[ -\frac{10 e (d+e x)^2}{3 c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{5 e^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c^3 d^3}+\frac{5 e^{3/2} \left (c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 c^{7/2} d^{7/2}}-\frac{2 (d+e x)^4}{3 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^5/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]

[Out]

(-2*(d + e*x)^4)/(3*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (10*e*(d + e*x)^2)/(3*c^2*d^2*Sqrt[a*
d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (5*e^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(c^3*d^3) + (5*e^(
3/2)*(c*d^2 - a*e^2)*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^
2)*x + c*d*e*x^2])])/(2*c^(7/2)*d^(7/2))

Rule 668

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*(p + 1)), x] - Dist[(e^2*(m + p))/(c*(p + 1)), Int[(d + e*x)^(m - 2)*(a + b*x +
c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &
& LtQ[p, -1] && GtQ[m, 1] && 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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{(d+e x)^5}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx &=-\frac{2 (d+e x)^4}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{(5 e) \int \frac{(d+e x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{3 c d}\\ &=-\frac{2 (d+e x)^4}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac{10 e (d+e x)^2}{3 c^2 d^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{\left (5 e^2\right ) \int \frac{d+e x}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{c^2 d^2}\\ &=-\frac{2 (d+e x)^4}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac{10 e (d+e x)^2}{3 c^2 d^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{5 e^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c^3 d^3}+\frac{\left (5 e^2 \left (c d^2-a e^2\right )\right ) \int \frac{1}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{2 c^3 d^3}\\ &=-\frac{2 (d+e x)^4}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac{10 e (d+e x)^2}{3 c^2 d^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{5 e^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c^3 d^3}+\frac{\left (5 e^2 \left (c d^2-a e^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 c d e-x^2} \, dx,x,\frac{c d^2+a e^2+2 c d e x}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{c^3 d^3}\\ &=-\frac{2 (d+e x)^4}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac{10 e (d+e x)^2}{3 c^2 d^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{5 e^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c^3 d^3}+\frac{5 e^{3/2} \left (c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{c d^2+a e^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{2 c^{7/2} d^{7/2}}\\ \end{align*}

Mathematica [C]  time = 0.0654721, size = 112, normalized size = 0.48 \[ -\frac{2 \left (c d^2-a e^2\right )^2 \sqrt{(d+e x) (a e+c d x)} \, _2F_1\left (-\frac{5}{2},-\frac{3}{2};-\frac{1}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right )}{3 c^3 d^3 (a e+c d x)^2 \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^5/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]

[Out]

(-2*(c*d^2 - a*e^2)^2*Sqrt[(a*e + c*d*x)*(d + e*x)]*Hypergeometric2F1[-5/2, -3/2, -1/2, (e*(a*e + c*d*x))/(-(c
*d^2) + a*e^2)])/(3*c^3*d^3*(a*e + c*d*x)^2*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])

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Maple [B]  time = 0.08, size = 3215, normalized size = 13.9 \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)^5/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x)

[Out]

19/96*d^7*c/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-5/6*e^3/c*x^3/(a*d*e+(a*e
^2+c*d^2)*x+c*d*e*x^2)^(3/2)+e^4*x^4/d/c/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+5/96*e^8/d^5/c^5/(a*d*e+(a*e^
2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^4+5/4*e/c^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-15/32*d^3/c/(a*d*e+(a*e^2+c*
d^2)*x+c*d*e*x^2)^(3/2)+5/4*e*d^4/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-71/
16*e^2*d/c^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a-5/2*e^2/d/c^2*x/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)
-5/4*e^5/d^4/c^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^2+5/2*e^2/d/c^2*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d
*e*c)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(d*e*c)^(1/2)+19/12*e*d^8*c^2/(-a^2*e^4+2*a*c*d^2*e^2-c^2
*d^4)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+11/4*e^4/d/c^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^2-35/
4*e^2*d/c*x^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-115/16*e*d^2/c*x/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)
-43/48*e^6/d^3/c^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^3+1/12*e^2*d^5/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(
a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a-19/16*e^3/c^2*x/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a+5/2*e^2*d^3
/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x+73/12*e^9/c^2/(-a^2*e^4+2*a*c*d^2*
e^2-c^2*d^4)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^4-97/12*e^5*d^4/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(a
*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^2-5/2*e^8/d^3/c^3/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2
)*x+c*d*e*x^2)^(1/2)*x*a^3-5/2*e^6/d/c^2/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1
/2)*x*a^2-61/6*e^10/d/c^2/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^4-11/
6*e^4*d^5*c/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a+5/2*e^4*d/c/(-a^2*e
^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a+67/3*e^8*d/c/(-a^2*e^4+2*a*c*d^2*e^2-c^2
*d^4)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^3+5/48*e^11/d^4/c^4/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*
e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*a^5-61/48*e^9/d^2/c^3/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^
2)*x+c*d*e*x^2)^(3/2)*x*a^4+5/6*e^12/d^3/c^3/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x
^2)^(1/2)*x*a^5-43/24*e^5*d^2/c/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*a^2
-5/2*e^7/d^2/c^3/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^3+11/16*e^5/d^2/c^
3*x/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^2+1/2*e^6*d/c^2/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c
*d^2)*x+c*d*e*x^2)^(3/2)*a^3-14/3*e^11/d^2/c^3/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e
*x^2)^(1/2)*a^5-5/4*e^6/d^3/c^3*x^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^2+19/6*e^2*d^7*c^2/(-a^2*e^4+2*a
*c*d^2*e^2-c^2*d^4)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x+19/48*e*d^6*c/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)
/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x+5/96*e^12/d^5/c^5/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*
d^2)*x+c*d*e*x^2)^(3/2)*a^6-7/12*e^10/d^3/c^4/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^
2)^(3/2)*a^5+73/96*e^8/d/c^3/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^4+5/12
*e^13/d^4/c^4/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^6-5/16*e^7/d^4/c^4*
x/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^3+5/6*e^5/d^2/c^2*x^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a-43
/3*e^6*d^3/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^2+67/24*e^7/c^2/(-a^
2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*a^3-11/48*e^3*d^4/(-a^2*e^4+2*a*c*d^2*e
^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*a+5/2*e^4/d^3/c^3*x/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1
/2)*a-5/2*e^4/d^3/c^3*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/
(d*e*c)^(1/2)*a+2/3*e^3*d^6*c/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a+4*e
^4/d/c^2*x^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a-97/96*e^4*d^3/c/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e
+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^2+4*e^7*d^2/c/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(a*d*e+(a*e^2+c*d^2)*x+c*
d*e*x^2)^(1/2)*a^3+5/2*e^3*d^2/c/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a-5/
4*e^9/d^4/c^4/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^4

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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)^5/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 12.2847, size = 1311, normalized size = 5.68 \begin{align*} \left [\frac{15 \,{\left (a^{2} c d^{2} e^{3} - a^{3} e^{5} +{\left (c^{3} d^{4} e - a c^{2} d^{2} e^{3}\right )} x^{2} + 2 \,{\left (a c^{2} d^{3} e^{2} - a^{2} c d e^{4}\right )} x\right )} \sqrt{\frac{e}{c d}} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} + 8 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x + 4 \,{\left (2 \, c^{2} d^{2} e x + c^{2} d^{3} + a c d e^{2}\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{\frac{e}{c d}}\right ) + 4 \,{\left (3 \, c^{2} d^{2} e^{2} x^{2} - 2 \, c^{2} d^{4} - 10 \, a c d^{2} e^{2} + 15 \, a^{2} e^{4} - 2 \,{\left (7 \, c^{2} d^{3} e - 10 \, a c d e^{3}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{12 \,{\left (c^{5} d^{5} x^{2} + 2 \, a c^{4} d^{4} e x + a^{2} c^{3} d^{3} e^{2}\right )}}, -\frac{15 \,{\left (a^{2} c d^{2} e^{3} - a^{3} e^{5} +{\left (c^{3} d^{4} e - a c^{2} d^{2} e^{3}\right )} x^{2} + 2 \,{\left (a c^{2} d^{3} e^{2} - a^{2} c d e^{4}\right )} x\right )} \sqrt{-\frac{e}{c d}} \arctan \left (\frac{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt{-\frac{e}{c d}}}{2 \,{\left (c d e^{2} x^{2} + a d e^{2} +{\left (c d^{2} e + a e^{3}\right )} x\right )}}\right ) - 2 \,{\left (3 \, c^{2} d^{2} e^{2} x^{2} - 2 \, c^{2} d^{4} - 10 \, a c d^{2} e^{2} + 15 \, a^{2} e^{4} - 2 \,{\left (7 \, c^{2} d^{3} e - 10 \, a c d e^{3}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{6 \,{\left (c^{5} d^{5} x^{2} + 2 \, a c^{4} d^{4} e x + a^{2} c^{3} d^{3} e^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="fricas")

[Out]

[1/12*(15*(a^2*c*d^2*e^3 - a^3*e^5 + (c^3*d^4*e - a*c^2*d^2*e^3)*x^2 + 2*(a*c^2*d^3*e^2 - a^2*c*d*e^4)*x)*sqrt
(e/(c*d))*log(8*c^2*d^2*e^2*x^2 + c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4 + 8*(c^2*d^3*e + a*c*d*e^3)*x + 4*(2*c^2*d
^2*e*x + c^2*d^3 + a*c*d*e^2)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e/(c*d))) + 4*(3*c^2*d^2*e^2*x^
2 - 2*c^2*d^4 - 10*a*c*d^2*e^2 + 15*a^2*e^4 - 2*(7*c^2*d^3*e - 10*a*c*d*e^3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^
2 + a*e^2)*x))/(c^5*d^5*x^2 + 2*a*c^4*d^4*e*x + a^2*c^3*d^3*e^2), -1/6*(15*(a^2*c*d^2*e^3 - a^3*e^5 + (c^3*d^4
*e - a*c^2*d^2*e^3)*x^2 + 2*(a*c^2*d^3*e^2 - a^2*c*d*e^4)*x)*sqrt(-e/(c*d))*arctan(1/2*sqrt(c*d*e*x^2 + a*d*e
+ (c*d^2 + a*e^2)*x)*(2*c*d*e*x + c*d^2 + a*e^2)*sqrt(-e/(c*d))/(c*d*e^2*x^2 + a*d*e^2 + (c*d^2*e + a*e^3)*x))
- 2*(3*c^2*d^2*e^2*x^2 - 2*c^2*d^4 - 10*a*c*d^2*e^2 + 15*a^2*e^4 - 2*(7*c^2*d^3*e - 10*a*c*d*e^3)*x)*sqrt(c*d
*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^5*d^5*x^2 + 2*a*c^4*d^4*e*x + a^2*c^3*d^3*e^2)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**5/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)

[Out]

Timed out

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Giac [B]  time = 1.43827, size = 1116, normalized size = 4.83 \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)^5/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="giac")

[Out]

1/3*((((3*(c^6*d^10*e^6 - 4*a*c^5*d^8*e^8 + 6*a^2*c^4*d^6*e^10 - 4*a^3*c^3*d^4*e^12 + a^4*c^2*d^2*e^14)*x/(c^7
*d^11*e^2 - 4*a*c^6*d^9*e^4 + 6*a^2*c^5*d^7*e^6 - 4*a^3*c^4*d^5*e^8 + a^4*c^3*d^3*e^10) - 4*(2*c^6*d^11*e^5 -
13*a*c^5*d^9*e^7 + 32*a^2*c^4*d^7*e^9 - 38*a^3*c^3*d^5*e^11 + 22*a^4*c^2*d^3*e^13 - 5*a^5*c*d*e^15)/(c^7*d^11*
e^2 - 4*a*c^6*d^9*e^4 + 6*a^2*c^5*d^7*e^6 - 4*a^3*c^4*d^5*e^8 + a^4*c^3*d^3*e^10))*x - 3*(9*c^6*d^12*e^4 - 46*
a*c^5*d^10*e^6 + 89*a^2*c^4*d^8*e^8 - 76*a^3*c^3*d^6*e^10 + 19*a^4*c^2*d^4*e^12 + 10*a^5*c*d^2*e^14 - 5*a^6*e^
16)/(c^7*d^11*e^2 - 4*a*c^6*d^9*e^4 + 6*a^2*c^5*d^7*e^6 - 4*a^3*c^4*d^5*e^8 + a^4*c^3*d^3*e^10))*x - 6*(3*c^6*
d^13*e^3 - 12*a*c^5*d^11*e^5 + 13*a^2*c^4*d^9*e^7 + 8*a^3*c^3*d^7*e^9 - 27*a^4*c^2*d^5*e^11 + 20*a^5*c*d^3*e^1
3 - 5*a^6*d*e^15)/(c^7*d^11*e^2 - 4*a*c^6*d^9*e^4 + 6*a^2*c^5*d^7*e^6 - 4*a^3*c^4*d^5*e^8 + a^4*c^3*d^3*e^10))
*x - (2*c^6*d^14*e^2 + 2*a*c^5*d^12*e^4 - 43*a^2*c^4*d^10*e^6 + 112*a^3*c^3*d^8*e^8 - 128*a^4*c^2*d^6*e^10 + 7
0*a^5*c*d^4*e^12 - 15*a^6*d^2*e^14)/(c^7*d^11*e^2 - 4*a*c^6*d^9*e^4 + 6*a^2*c^5*d^7*e^6 - 4*a^3*c^4*d^5*e^8 +
a^4*c^3*d^3*e^10))/(c*d*x^2*e + a*d*e + (c*d^2 + a*e^2)*x)^(3/2) - 5/2*(c*d^2*e^2 - a*e^4)*sqrt(c*d)*e^(-1/2)*
log(abs(-sqrt(c*d)*c*d^2*e^(1/2) - 2*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + a*d*e + (c*d^2 + a*e^2)*x))*c*d*e
- sqrt(c*d)*a*e^(5/2)))/(c^4*d^4)