3.1937 \(\int \frac{(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^2} \, dx\)

Optimal. Leaf size=261 \[ -\frac{5 \left (c d^2-a e^2\right )^4 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{128 c^{3/2} d^{3/2} e^{7/2}}+\frac{5 \left (c d^2-a e^2\right )^2 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{64 c d e^3}+\frac{5}{24} \left (a-\frac{c d^2}{e^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}+\frac{(a e+c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 e} \]

[Out]

(5*(c*d^2 - a*e^2)^2*(c*d^2 + a*e^2 + 2*c*d*e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(64*c*d*e^3) + (
5*(a - (c*d^2)/e^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/24 + ((a*e + c*d*x)*(a*d*e + (c*d^2 + a*e^2
)*x + c*d*e*x^2)^(3/2))/(4*e) - (5*(c*d^2 - a*e^2)^4*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sq
rt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(128*c^(3/2)*d^(3/2)*e^(7/2))

________________________________________________________________________________________

Rubi [A]  time = 0.220456, antiderivative size = 261, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.162, Rules used = {654, 670, 640, 612, 621, 206} \[ -\frac{5 \left (c d^2-a e^2\right )^4 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{128 c^{3/2} d^{3/2} e^{7/2}}+\frac{5 \left (c d^2-a e^2\right )^2 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{64 c d e^3}+\frac{5}{24} \left (a-\frac{c d^2}{e^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}+\frac{(a e+c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*(c*d^2 - a*e^2)^2*(c*d^2 + a*e^2 + 2*c*d*e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(64*c*d*e^3) + (
5*(a - (c*d^2)/e^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/24 + ((a*e + c*d*x)*(a*d*e + (c*d^2 + a*e^2
)*x + c*d*e*x^2)^(3/2))/(4*e) - (5*(c*d^2 - a*e^2)^4*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sq
rt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(128*c^(3/2)*d^(3/2)*e^(7/2))

Rule 654

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Int[(a + b*x + c*x^2)^(m +
p)/(a/d + (c*x)/e)^m, 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]
&&  !IntegerQ[p] && IntegerQ[m] && RationalQ[p] && (LtQ[0, -m, p] || LtQ[p, -m, 0]) && NeQ[m, 2] && NeQ[m, -1
]

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 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{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^2} \, dx &=\int (a e+c d x)^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx\\ &=\frac{(a e+c d x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{4 e}+\frac{\left (5 \left (2 a c d e^2-c d \left (c d^2+a e^2\right )\right )\right ) \int (a e+c d x) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{8 c d e}\\ &=\frac{5}{24} \left (a-\frac{c d^2}{e^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}+\frac{(a e+c d x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{4 e}+\frac{\left (5 \left (c d^2-a e^2\right )^2\right ) \int \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{16 e^2}\\ &=\frac{5 \left (c d^2-a e^2\right )^2 \left (c d^2+a e^2+2 c d e x\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c d e^3}+\frac{5}{24} \left (a-\frac{c d^2}{e^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}+\frac{(a e+c d x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{4 e}-\frac{\left (5 \left (c d^2-a e^2\right )^4\right ) \int \frac{1}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{128 c d e^3}\\ &=\frac{5 \left (c d^2-a e^2\right )^2 \left (c d^2+a e^2+2 c d e x\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c d e^3}+\frac{5}{24} \left (a-\frac{c d^2}{e^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}+\frac{(a e+c d x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{4 e}-\frac{\left (5 \left (c d^2-a e^2\right )^4\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 )}{64 c d e^3}\\ &=\frac{5 \left (c d^2-a e^2\right )^2 \left (c d^2+a e^2+2 c d e x\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c d e^3}+\frac{5}{24} \left (a-\frac{c d^2}{e^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}+\frac{(a e+c d x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{4 e}-\frac{5 \left (c d^2-a e^2\right )^4 \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 )}{128 c^{3/2} d^{3/2} e^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.899307, size = 316, normalized size = 1.21 \[ \frac{\sqrt{c} \sqrt{d} \left (\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{c d} (d+e x) \left (a^2 c^2 d^2 e^3 \left (-55 d^2+109 d e x+254 e^2 x^2\right )+a^3 c d e^5 (73 d+133 e x)+15 a^4 e^7+a c^3 d^3 e \left (-65 d^2 e x+15 d^3+44 d e^2 x^2+184 e^3 x^3\right )+c^4 d^4 x \left (-10 d^2 e x+15 d^3+8 d e^2 x^2+48 e^3 x^3\right )\right )-15 \left (c d^2-a e^2\right )^{9/2} \sqrt{a e+c d x} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right )\right )}{192 e^{7/2} (c d)^{5/2} \sqrt{(d+e x) (a e+c d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c]*Sqrt[d]*(Sqrt[c]*Sqrt[d]*Sqrt[c*d]*Sqrt[e]*(d + e*x)*(15*a^4*e^7 + a^3*c*d*e^5*(73*d + 133*e*x) + a^2
*c^2*d^2*e^3*(-55*d^2 + 109*d*e*x + 254*e^2*x^2) + c^4*d^4*x*(15*d^3 - 10*d^2*e*x + 8*d*e^2*x^2 + 48*e^3*x^3)
+ a*c^3*d^3*e*(15*d^3 - 65*d^2*e*x + 44*d*e^2*x^2 + 184*e^3*x^3)) - 15*(c*d^2 - a*e^2)^(9/2)*Sqrt[a*e + c*d*x]
*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d
^2 - a*e^2])]))/(192*(c*d)^(5/2)*e^(7/2)*Sqrt[(a*e + c*d*x)*(d + e*x)])

________________________________________________________________________________________

Maple [B]  time = 0.049, size = 1455, normalized size = 5.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3/e^2/(a*e^2-c*d^2)/(d/e+x)^2*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(7/2)-2/3/e*d*c/(a*e^2-c*d^2)*(c*d*e*(
d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(5/2)-15/32*e^2*d^2*c/(a*e^2-c*d^2)*a^2*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x)
)^(1/2)*x+5/64*e^5/d/c/(a*e^2-c*d^2)*a^4*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)+5/32/e*d^5*c^2/(a*e^2-c
*d^2)*a*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)+25/128*e^5*d/(a*e^2-c*d^2)*a^4*ln((1/2*a*e^2-1/2*c*d^2+(
d/e+x)*c*d*e)/(d*e*c)^(1/2)+(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2))/(d*e*c)^(1/2)-25/64*e^3*d^3*c/(a*e^
2-c*d^2)*a^3*ln((1/2*a*e^2-1/2*c*d^2+(d/e+x)*c*d*e)/(d*e*c)^(1/2)+(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2
))/(d*e*c)^(1/2)-5/32/e^2*d^6*c^3/(a*e^2-c*d^2)*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)*x-5/24*e^2/(a*e^
2-c*d^2)*a^2*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(3/2)-5/12*e*d*c/(a*e^2-c*d^2)*a*(c*d*e*(d/e+x)^2+(a*e^2-
c*d^2)*(d/e+x))^(3/2)*x-5/32*e^3*d/(a*e^2-c*d^2)*a^3*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)+5/12/e*d^3*
c^2/(a*e^2-c*d^2)*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(3/2)*x+5/24/e^2*d^4*c^2/(a*e^2-c*d^2)*(c*d*e*(d/e+x
)^2+(a*e^2-c*d^2)*(d/e+x))^(3/2)-5/64/e^3*d^7*c^3/(a*e^2-c*d^2)*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)+
5/32*e^4/(a*e^2-c*d^2)*a^3*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)*x+15/32*d^4*c^2/(a*e^2-c*d^2)*a*(c*d*
e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)*x-5/128*e^7/d/c/(a*e^2-c*d^2)*a^5*ln((1/2*a*e^2-1/2*c*d^2+(d/e+x)*c*d
*e)/(d*e*c)^(1/2)+(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2))/(d*e*c)^(1/2)+25/64*e*d^5*c^2/(a*e^2-c*d^2)*a
^2*ln((1/2*a*e^2-1/2*c*d^2+(d/e+x)*c*d*e)/(d*e*c)^(1/2)+(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2))/(d*e*c)
^(1/2)-25/128/e*d^7*c^3/(a*e^2-c*d^2)*a*ln((1/2*a*e^2-1/2*c*d^2+(d/e+x)*c*d*e)/(d*e*c)^(1/2)+(c*d*e*(d/e+x)^2+
(a*e^2-c*d^2)*(d/e+x))^(1/2))/(d*e*c)^(1/2)+5/128/e^3*d^9*c^4/(a*e^2-c*d^2)*ln((1/2*a*e^2-1/2*c*d^2+(d/e+x)*c*
d*e)/(d*e*c)^(1/2)+(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2))/(d*e*c)^(1/2)

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.9126, size = 1424, normalized size = 5.46 \begin{align*} \left [\frac{15 \,{\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} \sqrt{c d e} \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} - 4 \, \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{c d e} + 8 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right ) + 4 \,{\left (48 \, c^{4} d^{4} e^{4} x^{3} + 15 \, c^{4} d^{7} e - 55 \, a c^{3} d^{5} e^{3} + 73 \, a^{2} c^{2} d^{3} e^{5} + 15 \, a^{3} c d e^{7} + 8 \,{\left (c^{4} d^{5} e^{3} + 17 \, a c^{3} d^{3} e^{5}\right )} x^{2} - 2 \,{\left (5 \, c^{4} d^{6} e^{2} - 18 \, a c^{3} d^{4} e^{4} - 59 \, a^{2} c^{2} d^{2} e^{6}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{768 \, c^{2} d^{2} e^{4}}, \frac{15 \,{\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} \sqrt{-c d e} \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{-c d e}}{2 \,{\left (c^{2} d^{2} e^{2} x^{2} + a c d^{2} e^{2} +{\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )}}\right ) + 2 \,{\left (48 \, c^{4} d^{4} e^{4} x^{3} + 15 \, c^{4} d^{7} e - 55 \, a c^{3} d^{5} e^{3} + 73 \, a^{2} c^{2} d^{3} e^{5} + 15 \, a^{3} c d e^{7} + 8 \,{\left (c^{4} d^{5} e^{3} + 17 \, a c^{3} d^{3} e^{5}\right )} x^{2} - 2 \,{\left (5 \, c^{4} d^{6} e^{2} - 18 \, a c^{3} d^{4} e^{4} - 59 \, a^{2} c^{2} d^{2} e^{6}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{384 \, c^{2} d^{2} e^{4}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/768*(15*(c^4*d^8 - 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 - 4*a^3*c*d^2*e^6 + a^4*e^8)*sqrt(c*d*e)*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 - 4*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(c*d*e) + 8*(c^2*d^3*e + a*c*d*e^3)*x) + 4*(48*c^4*d^4*e^4*x^3 + 15*c^4*d^7*e - 55*a*c^3*d^5*
e^3 + 73*a^2*c^2*d^3*e^5 + 15*a^3*c*d*e^7 + 8*(c^4*d^5*e^3 + 17*a*c^3*d^3*e^5)*x^2 - 2*(5*c^4*d^6*e^2 - 18*a*c
^3*d^4*e^4 - 59*a^2*c^2*d^2*e^6)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^2*d^2*e^4), 1/384*(15*(c^4
*d^8 - 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 - 4*a^3*c*d^2*e^6 + a^4*e^8)*sqrt(-c*d*e)*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(-c*d*e)/(c^2*d^2*e^2*x^2 + a*c*d^2*e^2 + (c^2*d
^3*e + a*c*d*e^3)*x)) + 2*(48*c^4*d^4*e^4*x^3 + 15*c^4*d^7*e - 55*a*c^3*d^5*e^3 + 73*a^2*c^2*d^3*e^5 + 15*a^3*
c*d*e^7 + 8*(c^4*d^5*e^3 + 17*a*c^3*d^3*e^5)*x^2 - 2*(5*c^4*d^6*e^2 - 18*a*c^3*d^4*e^4 - 59*a^2*c^2*d^2*e^6)*x
)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^2*d^2*e^4)]

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

[Out]

Timed out