3.1935 \(\int (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2} \, dx\)

Optimal. Leaf size=305 \[ \frac{5 \left (c d^2-a e^2\right )^4 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{512 c^3 d^3 e^3}-\frac{5 \left (c d^2-a e^2\right )^2 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{192 c^2 d^2 e^2}-\frac{5 \left (c d^2-a e^2\right )^6 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{1024 c^{7/2} d^{7/2} e^{7/2}}+\frac{\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{12 c d e} \]

[Out]

(5*(c*d^2 - a*e^2)^4*(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])/(512*c^3*d^3*e^3
) - (5*(c*d^2 - a*e^2)^2*(c*d^2 + a*e^2 + 2*c*d*e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(192*c^2*d
^2*e^2) + ((c*d^2 + a*e^2 + 2*c*d*e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(12*c*d*e) - (5*(c*d^2 -
a*e^2)^6*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])])/(1024*c^(7/2)*d^(7/2)*e^(7/2))

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Rubi [A]  time = 0.147549, antiderivative size = 305, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {612, 621, 206} \[ \frac{5 \left (c d^2-a e^2\right )^4 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{512 c^3 d^3 e^3}-\frac{5 \left (c d^2-a e^2\right )^2 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{192 c^2 d^2 e^2}-\frac{5 \left (c d^2-a e^2\right )^6 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{1024 c^{7/2} d^{7/2} e^{7/2}}+\frac{\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{12 c d e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*(c*d^2 - a*e^2)^4*(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])/(512*c^3*d^3*e^3
) - (5*(c*d^2 - a*e^2)^2*(c*d^2 + a*e^2 + 2*c*d*e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(192*c^2*d
^2*e^2) + ((c*d^2 + a*e^2 + 2*c*d*e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(12*c*d*e) - (5*(c*d^2 -
a*e^2)^6*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])])/(1024*c^(7/2)*d^(7/2)*e^(7/2))

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 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx &=\frac{\left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{12 c d e}-\frac{\left (5 \left (c d^2-a e^2\right )^2\right ) \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx}{24 c d e}\\ &=-\frac{5 \left (c d^2-a e^2\right )^2 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{192 c^2 d^2 e^2}+\frac{\left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{12 c d e}+\frac{\left (5 \left (c d^2-a e^2\right )^4\right ) \int \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{128 c^2 d^2 e^2}\\ &=\frac{5 \left (c d^2-a e^2\right )^4 \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}}{512 c^3 d^3 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 ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{192 c^2 d^2 e^2}+\frac{\left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{12 c d e}-\frac{\left (5 \left (c d^2-a e^2\right )^6\right ) \int \frac{1}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{1024 c^3 d^3 e^3}\\ &=\frac{5 \left (c d^2-a e^2\right )^4 \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}}{512 c^3 d^3 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 ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{192 c^2 d^2 e^2}+\frac{\left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{12 c d e}-\frac{\left (5 \left (c d^2-a e^2\right )^6\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 )}{512 c^3 d^3 e^3}\\ &=\frac{5 \left (c d^2-a e^2\right )^4 \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}}{512 c^3 d^3 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 ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{192 c^2 d^2 e^2}+\frac{\left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{12 c d e}-\frac{5 \left (c d^2-a e^2\right )^6 \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 )}{1024 c^{7/2} d^{7/2} e^{7/2}}\\ \end{align*}

Mathematica [B]  time = 6.11832, size = 1119, normalized size = 3.67 \[ \frac{2 \left (c d^2-a e^2\right )^2 (a e+c d x) ((a e+c d x) (d+e x))^{5/2} \left (\frac{c d e (a e+c d x)}{\left (c d^2-a e^2\right ) \left (\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right )}+1\right )^{7/2} \left (\frac{35 \left (c d^2-a e^2\right )^4 \left (\frac{16 c^3 d^3 e^3 (a e+c d x)^3}{15 \left (c d^2-a e^2\right )^3 \left (\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right )^3}-\frac{4 c^2 d^2 e^2 (a e+c d x)^2}{3 \left (c d^2-a e^2\right )^2 \left (\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right )^2}+\frac{2 c d e (a e+c d x)}{\left (c d^2-a e^2\right ) \left (\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right )}-\frac{2 \sqrt{c} \sqrt{d} \sqrt{e} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d^2-a e^2} \sqrt{\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}}}\right ) \sqrt{a e+c d x}}{\sqrt{c d^2-a e^2} \sqrt{\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}} \sqrt{\frac{c d e (a e+c d x)}{\left (c d^2-a e^2\right ) \left (\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right )}+1}}\right ) \left (\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right )^4}{2048 c^4 d^4 e^4 (a e+c d x)^4 \left (\frac{c d e (a e+c d x)}{\left (c d^2-a e^2\right ) \left (\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right )}+1\right )^3}+\frac{7}{12} \left (\frac{1}{\frac{c d e (a e+c d x)}{\left (c d^2-a e^2\right ) \left (\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right )}+1}+\frac{1}{2 \left (\frac{c d e (a e+c d x)}{\left (c d^2-a e^2\right ) \left (\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right )}+1\right )^2}+\frac{3}{16 \left (\frac{c d e (a e+c d x)}{\left (c d^2-a e^2\right ) \left (\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}\right )}+1\right )^3}\right )\right )}{7 c^3 d^3 \left (\frac{c d}{\frac{c^2 d^3}{c d^2-a e^2}-\frac{a c d e^2}{c d^2-a e^2}}\right )^{5/2} (d+e x)^2 \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[(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*(a*e + c*d*x)*((a*e + c*d*x)*(d + e*x))^(5/2)*(1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)
*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^(7/2)*((7*(3/(16*(1 + (c*d*e*(a*e + c*d*x))/((c*d
^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^3) + 1/(2*(1 + (c*d*e*(a*e + c*d*x))/(
(c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^2) + (1 + (c*d*e*(a*e + c*d*x))/((
c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^(-1)))/12 + (35*(c*d^2 - a*e^2)^4*(
(c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))^4*((2*c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3
)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))) - (4*c^2*d^2*e^2*(a*e + c*d*x)^2)/(3*(c*d^2 - a*e^2)^2*((c^2
*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))^2) + (16*c^3*d^3*e^3*(a*e + c*d*x)^3)/(15*(c*d^2 - a*e^2)
^3*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))^3) - (2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x]
*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d^2 - a*e^2]*Sqrt[(c^2*d^3)/(c*d^2 - a*e^2) - (a*
c*d*e^2)/(c*d^2 - a*e^2)])])/(Sqrt[c*d^2 - a*e^2]*Sqrt[(c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2)
]*Sqrt[1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2)))])
))/(2048*c^4*d^4*e^4*(a*e + c*d*x)^4*(1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) -
(a*c*d*e^2)/(c*d^2 - a*e^2))))^3)))/(7*c^3*d^3*((c*d)/((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2)
))^(5/2)*(d + e*x)^2*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])

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Maple [B]  time = 0.045, size = 1247, normalized size = 4.1 \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),x)

[Out]

15/512*d^7/e*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)*a-5/1024/d^3*e^9/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^6+5/512*d^7/e^3*c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+5/256*d^3*e*(a*d*e+(a*e^2+
c*d^2)*x+c*d*e*x^2)^(1/2)*a^2+5/192*e^2/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^2-5/192*d^4/e^2*c*(a*d*e+(
a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+5/192*d^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a+15/512/d*e^7/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)*a^5-75/1024*d*e^5
/c*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^4-7
5/1024*d^5*e*c*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^2-5/96/d*e^3/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*a^2+5/256/d^2*e^6/c^2*(a*d*e+(a*e^2+c*d^2)*x
+c*d*e*x^2)^(1/2)*x*a^4+5/256*d^6/e^2*c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x+5/48*d*e*(a*d*e+(a*e^2+c*d
^2)*x+c*d*e*x^2)^(3/2)*x*a-5/192/d^2*e^4/c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^3-5/96*d^3/e*c*(a*d*e+(
a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x-15/512*d^5/e*c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a-5/64*d^4*c*(a*d*e+(
a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a-5/64*e^4/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^3+15/128*d^2*e^2*(a
*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^2+25/256*d^3*e^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^3+5/512/d^3*e^7/c^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/
2)*a^5-15/512/d*e^5/c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^4+5/256*d*e^3/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e
*x^2)^(1/2)*a^3-5/1024*d^9/e^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)+1/12*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c/d/e

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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((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 = 2.1914, size = 2199, normalized size = 7.21 \begin{align*} \text{result too large to display} \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),x, algorithm="fricas")

[Out]

[1/6144*(15*(c^6*d^12 - 6*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*e^4 - 20*a^3*c^3*d^6*e^6 + 15*a^4*c^2*d^4*e^8 - 6*a^
5*c*d^2*e^10 + a^6*e^12)*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*(2
56*c^6*d^6*e^6*x^5 + 15*c^6*d^11*e - 85*a*c^5*d^9*e^3 + 198*a^2*c^4*d^7*e^5 + 198*a^3*c^3*d^5*e^7 - 85*a^4*c^2
*d^3*e^9 + 15*a^5*c*d*e^11 + 640*(c^6*d^7*e^5 + a*c^5*d^5*e^7)*x^4 + 16*(27*c^6*d^8*e^4 + 106*a*c^5*d^6*e^6 +
27*a^2*c^4*d^4*e^8)*x^3 + 8*(c^6*d^9*e^3 + 159*a*c^5*d^7*e^5 + 159*a^2*c^4*d^5*e^7 + a^3*c^3*d^3*e^9)*x^2 - 2*
(5*c^6*d^10*e^2 - 28*a*c^5*d^8*e^4 - 594*a^2*c^4*d^6*e^6 - 28*a^3*c^3*d^4*e^8 + 5*a^4*c^2*d^2*e^10)*x)*sqrt(c*
d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^4*d^4*e^4), 1/3072*(15*(c^6*d^12 - 6*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*
e^4 - 20*a^3*c^3*d^6*e^6 + 15*a^4*c^2*d^4*e^8 - 6*a^5*c*d^2*e^10 + a^6*e^12)*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*(256*c^6*d^6*e^6*x^5 + 15*c^6*d^11*e - 85*a*c^5*d^9*e^3 + 198*a^2*c^4*d^7*e^5 +
198*a^3*c^3*d^5*e^7 - 85*a^4*c^2*d^3*e^9 + 15*a^5*c*d*e^11 + 640*(c^6*d^7*e^5 + a*c^5*d^5*e^7)*x^4 + 16*(27*c
^6*d^8*e^4 + 106*a*c^5*d^6*e^6 + 27*a^2*c^4*d^4*e^8)*x^3 + 8*(c^6*d^9*e^3 + 159*a*c^5*d^7*e^5 + 159*a^2*c^4*d^
5*e^7 + a^3*c^3*d^3*e^9)*x^2 - 2*(5*c^6*d^10*e^2 - 28*a*c^5*d^8*e^4 - 594*a^2*c^4*d^6*e^6 - 28*a^3*c^3*d^4*e^8
+ 5*a^4*c^2*d^2*e^10)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^4*d^4*e^4)]

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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((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 [A]  time = 1.29178, size = 678, normalized size = 2.22 \begin{align*} \frac{1}{1536} \, \sqrt{c d x^{2} e + c d^{2} x + a x e^{2} + a d e}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (2 \, c^{2} d^{2} x e^{2} + \frac{5 \,{\left (c^{7} d^{8} e^{6} + a c^{6} d^{6} e^{8}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x + \frac{{\left (27 \, c^{7} d^{9} e^{5} + 106 \, a c^{6} d^{7} e^{7} + 27 \, a^{2} c^{5} d^{5} e^{9}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x + \frac{{\left (c^{7} d^{10} e^{4} + 159 \, a c^{6} d^{8} e^{6} + 159 \, a^{2} c^{5} d^{6} e^{8} + a^{3} c^{4} d^{4} e^{10}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x - \frac{{\left (5 \, c^{7} d^{11} e^{3} - 28 \, a c^{6} d^{9} e^{5} - 594 \, a^{2} c^{5} d^{7} e^{7} - 28 \, a^{3} c^{4} d^{5} e^{9} + 5 \, a^{4} c^{3} d^{3} e^{11}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x + \frac{{\left (15 \, c^{7} d^{12} e^{2} - 85 \, a c^{6} d^{10} e^{4} + 198 \, a^{2} c^{5} d^{8} e^{6} + 198 \, a^{3} c^{4} d^{6} e^{8} - 85 \, a^{4} c^{3} d^{4} e^{10} + 15 \, a^{5} c^{2} d^{2} e^{12}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} + \frac{5 \,{\left (c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} + 15 \, a^{4} c^{2} d^{4} e^{8} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}\right )} \sqrt{c d} e^{\left (-\frac{7}{2}\right )} \log \left ({\left | -\sqrt{c d} c d^{2} e^{\frac{1}{2}} - 2 \,{\left (\sqrt{c d} x e^{\frac{1}{2}} - \sqrt{c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} c d e - \sqrt{c d} a e^{\frac{5}{2}} \right |}\right )}{1024 \, c^{4} d^{4}} \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),x, algorithm="giac")

[Out]

1/1536*sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*(2*(4*(2*(8*(2*c^2*d^2*x*e^2 + 5*(c^7*d^8*e^6 + a*c^6*d^6*e
^8)*e^(-5)/(c^5*d^5))*x + (27*c^7*d^9*e^5 + 106*a*c^6*d^7*e^7 + 27*a^2*c^5*d^5*e^9)*e^(-5)/(c^5*d^5))*x + (c^7
*d^10*e^4 + 159*a*c^6*d^8*e^6 + 159*a^2*c^5*d^6*e^8 + a^3*c^4*d^4*e^10)*e^(-5)/(c^5*d^5))*x - (5*c^7*d^11*e^3
- 28*a*c^6*d^9*e^5 - 594*a^2*c^5*d^7*e^7 - 28*a^3*c^4*d^5*e^9 + 5*a^4*c^3*d^3*e^11)*e^(-5)/(c^5*d^5))*x + (15*
c^7*d^12*e^2 - 85*a*c^6*d^10*e^4 + 198*a^2*c^5*d^8*e^6 + 198*a^3*c^4*d^6*e^8 - 85*a^4*c^3*d^4*e^10 + 15*a^5*c^
2*d^2*e^12)*e^(-5)/(c^5*d^5)) + 5/1024*(c^6*d^12 - 6*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*e^4 - 20*a^3*c^3*d^6*e^6
+ 15*a^4*c^2*d^4*e^8 - 6*a^5*c*d^2*e^10 + a^6*e^12)*sqrt(c*d)*e^(-7/2)*log(abs(-sqrt(c*d)*c*d^2*e^(1/2) - 2*(s
qrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))*c*d*e - sqrt(c*d)*a*e^(5/2)))/(c^4*d^4)