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

Optimal. Leaf size=452 \[ \frac{\left (5 a^2 e^4+10 a c d^2 e^2+9 c^2 d^4\right ) \left (c d^2-a e^2\right )^3 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{1024 c^4 d^4 e^5}-\frac{\left (5 a^2 e^4+10 a c d^2 e^2+9 c^2 d^4\right ) \left (c d^2-a e^2\right ) \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{384 c^3 d^3 e^4}+\frac{\left (-35 a^2 e^4-10 c d e x \left (9 c d^2-5 a e^2\right )-20 a c d^2 e^2+63 c^2 d^4\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{840 c^2 d^2 e^3}-\frac{\left (5 a^2 e^4+10 a c d^2 e^2+9 c^2 d^4\right ) \left (c d^2-a e^2\right )^5 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2048 c^{9/2} d^{9/2} e^{11/2}}+\frac{x^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 e} \]

[Out]

((c*d^2 - a*e^2)^3*(9*c^2*d^4 + 10*a*c*d^2*e^2 + 5*a^2*e^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])/(1024*c^4*d^4*e^5) - ((c*d^2 - a*e^2)*(9*c^2*d^4 + 10*a*c*d^2*e^2 + 5*a^2*e^4)*(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))/(384*c^3*d^3*e^4) + (x^2*(a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2)^(5/2))/(7*e) + ((63*c^2*d^4 - 20*a*c*d^2*e^2 - 35*a^2*e^4 - 10*c*d*e*(9*c*d^2 - 5*a*e^2)
*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(840*c^2*d^2*e^3) - ((c*d^2 - a*e^2)^5*(9*c^2*d^4 + 10*a*c*
d^2*e^2 + 5*a^2*e^4)*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])])/(2048*c^(9/2)*d^(9/2)*e^(11/2))

________________________________________________________________________________________

Rubi [A]  time = 0.410182, antiderivative size = 452, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {851, 832, 779, 612, 621, 206} \[ \frac{\left (5 a^2 e^4+10 a c d^2 e^2+9 c^2 d^4\right ) \left (c d^2-a e^2\right )^3 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{1024 c^4 d^4 e^5}-\frac{\left (5 a^2 e^4+10 a c d^2 e^2+9 c^2 d^4\right ) \left (c d^2-a e^2\right ) \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{384 c^3 d^3 e^4}+\frac{\left (-35 a^2 e^4-10 c d e x \left (9 c d^2-5 a e^2\right )-20 a c d^2 e^2+63 c^2 d^4\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{840 c^2 d^2 e^3}-\frac{\left (5 a^2 e^4+10 a c d^2 e^2+9 c^2 d^4\right ) \left (c d^2-a e^2\right )^5 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2048 c^{9/2} d^{9/2} e^{11/2}}+\frac{x^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((c*d^2 - a*e^2)^3*(9*c^2*d^4 + 10*a*c*d^2*e^2 + 5*a^2*e^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])/(1024*c^4*d^4*e^5) - ((c*d^2 - a*e^2)*(9*c^2*d^4 + 10*a*c*d^2*e^2 + 5*a^2*e^4)*(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))/(384*c^3*d^3*e^4) + (x^2*(a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2)^(5/2))/(7*e) + ((63*c^2*d^4 - 20*a*c*d^2*e^2 - 35*a^2*e^4 - 10*c*d*e*(9*c*d^2 - 5*a*e^2)
*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(840*c^2*d^2*e^3) - ((c*d^2 - a*e^2)^5*(9*c^2*d^4 + 10*a*c*
d^2*e^2 + 5*a^2*e^4)*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])])/(2048*c^(9/2)*d^(9/2)*e^(11/2))

Rule 851

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :>
Int[((f + g*x)^n*(a + b*x + c*x^2)^(m + p))/(a/d + (c*x)/e)^m, x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] &&
NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[m, 0] && In
tegerQ[n] && (LtQ[n, 0] || GtQ[p, 0])

Rule 832

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m
- 1)*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m*(c*e*f + c*d*g - b*e*g) + e*(p
+ 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
&&  !(IGtQ[m, 0] && EqQ[f, 0])

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 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{x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{d+e x} \, dx &=\int x^2 (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} \, dx\\ &=\frac{x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 e}+\frac{\int x \left (-2 a c d^2 e-\frac{1}{2} c d \left (9 c d^2-5 a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx}{7 c d e}\\ &=\frac{x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 e}+\frac{\left (63 c^2 d^4-20 a c d^2 e^2-35 a^2 e^4-10 c d e \left (9 c d^2-5 a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{840 c^2 d^2 e^3}-\frac{\left (\left (c d^2-a e^2\right ) \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right )\right ) \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx}{48 c^2 d^2 e^3}\\ &=-\frac{\left (c d^2-a e^2\right ) \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right ) \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}}{384 c^3 d^3 e^4}+\frac{x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 e}+\frac{\left (63 c^2 d^4-20 a c d^2 e^2-35 a^2 e^4-10 c d e \left (9 c d^2-5 a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{840 c^2 d^2 e^3}+\frac{\left (\left (c d^2-a e^2\right )^3 \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right )\right ) \int \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{256 c^3 d^3 e^4}\\ &=\frac{\left (c d^2-a e^2\right )^3 \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right ) \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}}{1024 c^4 d^4 e^5}-\frac{\left (c d^2-a e^2\right ) \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right ) \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}}{384 c^3 d^3 e^4}+\frac{x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 e}+\frac{\left (63 c^2 d^4-20 a c d^2 e^2-35 a^2 e^4-10 c d e \left (9 c d^2-5 a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{840 c^2 d^2 e^3}-\frac{\left (\left (c d^2-a e^2\right )^5 \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right )\right ) \int \frac{1}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{2048 c^4 d^4 e^5}\\ &=\frac{\left (c d^2-a e^2\right )^3 \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right ) \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}}{1024 c^4 d^4 e^5}-\frac{\left (c d^2-a e^2\right ) \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right ) \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}}{384 c^3 d^3 e^4}+\frac{x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 e}+\frac{\left (63 c^2 d^4-20 a c d^2 e^2-35 a^2 e^4-10 c d e \left (9 c d^2-5 a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{840 c^2 d^2 e^3}-\frac{\left (\left (c d^2-a e^2\right )^5 \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\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 )}{1024 c^4 d^4 e^5}\\ &=\frac{\left (c d^2-a e^2\right )^3 \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right ) \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}}{1024 c^4 d^4 e^5}-\frac{\left (c d^2-a e^2\right ) \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right ) \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}}{384 c^3 d^3 e^4}+\frac{x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 e}+\frac{\left (63 c^2 d^4-20 a c d^2 e^2-35 a^2 e^4-10 c d e \left (9 c d^2-5 a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{840 c^2 d^2 e^3}-\frac{\left (c d^2-a e^2\right )^5 \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\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 )}{2048 c^{9/2} d^{9/2} e^{11/2}}\\ \end{align*}

Mathematica [B]  time = 6.14201, size = 1221, normalized size = 2.7 \[ \frac{x (d+e x) ((a e+c d x) (d+e x))^{3/2} (a e+c d x)^2}{7 c d e}+\frac{((a e+c d x) (d+e x))^{3/2} \left (\frac{\left (-9 c d^2-7 a e^2\right ) (d+e x)^{5/2} (a e+c d x)^{7/2}}{12 c d e}+\frac{\left (c d^2-a e^2\right ) \left (-6 a c d^2 e^2-\frac{1}{2} \left (-9 c d^2-7 a e^2\right ) \left (\frac{7 c d^2}{2}+\frac{5 a e^2}{2}\right )\right ) \sqrt{d+e x} \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 )^{5/2} \left (\frac{21 \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}{512 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 )^2}+\frac{7}{10} \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{3}{8 \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}\right )\right ) (a e+c d x)^{7/2}}{21 c^3 d^3 e \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 )^{3/2} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}}\right )}{7 c d e (d+e x)^{3/2} (a e+c d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(a*e + c*d*x)^2*(d + e*x)*((a*e + c*d*x)*(d + e*x))^(3/2))/(7*c*d*e) + (((a*e + c*d*x)*(d + e*x))^(3/2)*(((
-9*c*d^2 - 7*a*e^2)*(a*e + c*d*x)^(7/2)*(d + e*x)^(5/2))/(12*c*d*e) + ((c*d^2 - a*e^2)*(-6*a*c*d^2*e^2 - ((-9*
c*d^2 - 7*a*e^2)*((7*c*d^2)/2 + (5*a*e^2)/2))/2)*(a*e + c*d*x)^(7/2)*Sqrt[d + e*x]*(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))))^(5/2)*((7*(3/(8*(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)))/10 + (21*(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)))])))/(512*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))))^2)))/(21*c^3*d^3*e*((c*d)/((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*d*(d + e*x))/(c*d^2 - a*e^2)])))/(7*c*d*e*(a*e + c*d*x)^(3/2)*(d + e*x)^(3/2))

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Maple [B]  time = 0.072, size = 2731, normalized size = 6. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/384*e^4/d^3/c^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^4+5/128/e^3*d^6*c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2
)^(1/2)*a+5/192*e/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*a^2-1/24*e/d^2/c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*
x^2)^(5/2)*a^2-15/512/e^4*d^7*c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x+15/2048/e^5*d^10*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)+5/192*e^2/d/c^2*(a*d*
e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^3-1/12/d/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*x*a+5/64/e^3*d^4*c*(a*
d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x-5/128*e*d^2/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^3-5/1024*e^7/d^
4/c^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^6-25/192/e*d^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*a-15/
2048*e^5/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+3/64*d^7/e^4*c^2*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)*x-3/256*d^10/e^5*c^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)+1/8*d^2/e*a*(
c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(3/2)*x+3/64*d^2*e*a^3/c*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)-
1/8*d^4/e^3*c*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(3/2)*x+3/256*e^5*a^5/c^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)-225/2048*e*d^4*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^3-3/64*d^6/e^3*a*
c*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)+15/128*d^4*e*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)+1/5*d^2/e^3*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2
)*(d/e+x))^(5/2)-1/8/e^3*d^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)+1/16*d*a^2/c*(c*d*e*(d/e+x)^2+(a*e^2-c*d^
2)*(d/e+x))^(3/2)+9/64*d^3*a^2*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)*x-3/128*e^3*a^4/c^2*(c*d*e*(d/e+x
)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)-1/16*d^5/e^4*c*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(3/2)+3/128*d^8/e^5*c^
2*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)-15/1024/e^5*d^8*c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+5/
128/e^4*d^5*c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-1/6/e/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*a-1/4/e^
2*d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*x-15/1024/e*d^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^2+35/102
4*e^3/c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^4+1/7/e^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/d/c-5/19
2/e^2*d^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a-35/256*d^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^2-5
/96*d/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^2-9/64*d^5/e^2*a*c*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(
1/2)*x-15/128*d^6/e*a^2*c*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)+15/256*d^8/e^3*a*c^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)+125/2048*e^3*d^2/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+195/2048/e*d^6*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^2-15/2048*e^7/d^2/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+15/256*e^2*
d/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^3+5/2048*e^9/d^4/c^4*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^7-5/512*e^6/d^3/c^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*
e*x^2)^(1/2)*x*a^5+5/512*e^4/d/c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^4-85/2048/e^3*d^8*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+55/512/e^2*d^5*
c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a+5/192*e^3/d^2/c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*a^3-
15/256*d^2*e^3*a^4/c*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)-3/64*d*e^2*a^3/c*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)*x

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 3.17263, size = 2803, normalized size = 6.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/430080*(105*(9*c^7*d^14 - 35*a*c^6*d^12*e^2 + 45*a^2*c^5*d^10*e^4 - 15*a^3*c^4*d^8*e^6 - 5*a^4*c^3*d^6*e^8
- 9*a^5*c^2*d^4*e^10 + 15*a^6*c*d^2*e^12 - 5*a^7*e^14)*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*(15360*c^7*d^7*e^7*x^6 + 945*c^7*d^13*e - 3360*a*c^6*d^11*e^3 + 3689*a^2*c^5*d^9*
e^5 - 600*a^3*c^4*d^7*e^7 - 525*a^4*c^3*d^5*e^9 + 1400*a^5*c^2*d^3*e^11 - 525*a^6*c*d*e^13 + 1280*(15*c^7*d^8*
e^6 + 29*a*c^6*d^6*e^8)*x^5 + 128*(3*c^7*d^9*e^5 + 380*a*c^6*d^7*e^7 + 185*a^2*c^5*d^5*e^9)*x^4 - 16*(27*c^7*d
^10*e^4 - 93*a*c^6*d^8*e^6 - 2095*a^2*c^5*d^6*e^8 - 15*a^3*c^4*d^4*e^10)*x^3 + 8*(63*c^7*d^11*e^3 - 218*a*c^6*
d^9*e^5 + 228*a^2*c^5*d^7*e^7 + 90*a^3*c^4*d^5*e^9 - 35*a^4*c^3*d^3*e^11)*x^2 - 2*(315*c^7*d^12*e^2 - 1099*a*c
^6*d^10*e^4 + 1166*a^2*c^5*d^8*e^6 - 150*a^3*c^4*d^6*e^8 + 455*a^4*c^3*d^4*e^10 - 175*a^5*c^2*d^2*e^12)*x)*sqr
t(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^5*d^5*e^6), 1/215040*(105*(9*c^7*d^14 - 35*a*c^6*d^12*e^2 + 45*a^
2*c^5*d^10*e^4 - 15*a^3*c^4*d^8*e^6 - 5*a^4*c^3*d^6*e^8 - 9*a^5*c^2*d^4*e^10 + 15*a^6*c*d^2*e^12 - 5*a^7*e^14)
*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*(15360*c^7*d^7*e^7*x^6 + 945*c^7*d^13*e - 336
0*a*c^6*d^11*e^3 + 3689*a^2*c^5*d^9*e^5 - 600*a^3*c^4*d^7*e^7 - 525*a^4*c^3*d^5*e^9 + 1400*a^5*c^2*d^3*e^11 -
525*a^6*c*d*e^13 + 1280*(15*c^7*d^8*e^6 + 29*a*c^6*d^6*e^8)*x^5 + 128*(3*c^7*d^9*e^5 + 380*a*c^6*d^7*e^7 + 185
*a^2*c^5*d^5*e^9)*x^4 - 16*(27*c^7*d^10*e^4 - 93*a*c^6*d^8*e^6 - 2095*a^2*c^5*d^6*e^8 - 15*a^3*c^4*d^4*e^10)*x
^3 + 8*(63*c^7*d^11*e^3 - 218*a*c^6*d^9*e^5 + 228*a^2*c^5*d^7*e^7 + 90*a^3*c^4*d^5*e^9 - 35*a^4*c^3*d^3*e^11)*
x^2 - 2*(315*c^7*d^12*e^2 - 1099*a*c^6*d^10*e^4 + 1166*a^2*c^5*d^8*e^6 - 150*a^3*c^4*d^6*e^8 + 455*a^4*c^3*d^4
*e^10 - 175*a^5*c^2*d^2*e^12)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^5*d^5*e^6)]

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

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError