3.1931 \(\int (d+e x)^4 (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2} \, dx\)

Optimal. Leaf size=534 \[ \frac{143 \left (c d^2-a e^2\right )^8 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{131072 c^7 d^7 e^3}-\frac{143 \left (c d^2-a e^2\right )^6 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{49152 c^6 d^6 e^2}+\frac{143 \left (c d^2-a e^2\right )^4 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{15360 c^5 d^5 e}+\frac{143 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{4480 c^4 d^4}+\frac{143 (d+e x) \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{2880 c^3 d^3}+\frac{13 (d+e x)^2 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{180 c^2 d^2}-\frac{143 \left (c d^2-a e^2\right )^{10} \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{262144 c^{15/2} d^{15/2} e^{7/2}}+\frac{(d+e x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{10 c d} \]

[Out]

(143*(c*d^2 - a*e^2)^8*(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])/(131072*c^7*d^
7*e^3) - (143*(c*d^2 - a*e^2)^6*(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))/(49
152*c^6*d^6*e^2) + (143*(c*d^2 - a*e^2)^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)^
(5/2))/(15360*c^5*d^5*e) + (143*(c*d^2 - a*e^2)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(4480*c^4*d^4
) + (143*(c*d^2 - a*e^2)^2*(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(2880*c^3*d^3) + (13*(c*d^
2 - a*e^2)*(d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(180*c^2*d^2) + ((d + e*x)^3*(a*d*e + (c
*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(10*c*d) - (143*(c*d^2 - a*e^2)^10*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])])/(262144*c^(15/2)*d^(15/2)*e^(7/2))

________________________________________________________________________________________

Rubi [A]  time = 0.676372, antiderivative size = 534, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135, Rules used = {670, 640, 612, 621, 206} \[ \frac{143 \left (c d^2-a e^2\right )^8 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{131072 c^7 d^7 e^3}-\frac{143 \left (c d^2-a e^2\right )^6 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{49152 c^6 d^6 e^2}+\frac{143 \left (c d^2-a e^2\right )^4 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{15360 c^5 d^5 e}+\frac{143 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{4480 c^4 d^4}+\frac{143 (d+e x) \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{2880 c^3 d^3}+\frac{13 (d+e x)^2 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{180 c^2 d^2}-\frac{143 \left (c d^2-a e^2\right )^{10} \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{262144 c^{15/2} d^{15/2} e^{7/2}}+\frac{(d+e x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{10 c d} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]

[Out]

(143*(c*d^2 - a*e^2)^8*(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])/(131072*c^7*d^
7*e^3) - (143*(c*d^2 - a*e^2)^6*(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))/(49
152*c^6*d^6*e^2) + (143*(c*d^2 - a*e^2)^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)^
(5/2))/(15360*c^5*d^5*e) + (143*(c*d^2 - a*e^2)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(4480*c^4*d^4
) + (143*(c*d^2 - a*e^2)^2*(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(2880*c^3*d^3) + (13*(c*d^
2 - a*e^2)*(d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(180*c^2*d^2) + ((d + e*x)^3*(a*d*e + (c
*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(10*c*d) - (143*(c*d^2 - a*e^2)^10*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])])/(262144*c^(15/2)*d^(15/2)*e^(7/2))

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 (d+e x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx &=\frac{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{10 c d}+\frac{\left (13 \left (d^2-\frac{a e^2}{c}\right )\right ) \int (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx}{20 d}\\ &=\frac{13 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{180 c^2 d^2}+\frac{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{10 c d}+\frac{\left (143 \left (d^2-\frac{a e^2}{c}\right )^2\right ) \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx}{360 d^2}\\ &=\frac{143 \left (c d^2-a e^2\right )^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{2880 c^3 d^3}+\frac{13 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{180 c^2 d^2}+\frac{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{10 c d}+\frac{\left (143 \left (d^2-\frac{a e^2}{c}\right )^3\right ) \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx}{640 d^3}\\ &=\frac{143 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{4480 c^4 d^4}+\frac{143 \left (c d^2-a e^2\right )^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{2880 c^3 d^3}+\frac{13 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{180 c^2 d^2}+\frac{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{10 c d}+\frac{\left (143 \left (d^2-\frac{a e^2}{c}\right )^4\right ) \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx}{1280 d^4}\\ &=\frac{143 \left (c d^2-a e^2\right )^4 \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}}{15360 c^5 d^5 e}+\frac{143 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{4480 c^4 d^4}+\frac{143 \left (c d^2-a e^2\right )^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{2880 c^3 d^3}+\frac{13 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{180 c^2 d^2}+\frac{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{10 c d}-\frac{\left (143 \left (c d^2-a e^2\right )^6\right ) \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx}{6144 c^5 d^5 e}\\ &=-\frac{143 \left (c d^2-a e^2\right )^6 \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}}{49152 c^6 d^6 e^2}+\frac{143 \left (c d^2-a e^2\right )^4 \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}}{15360 c^5 d^5 e}+\frac{143 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{4480 c^4 d^4}+\frac{143 \left (c d^2-a e^2\right )^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{2880 c^3 d^3}+\frac{13 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{180 c^2 d^2}+\frac{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{10 c d}+\frac{\left (143 \left (c d^2-a e^2\right )^8\right ) \int \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{32768 c^6 d^6 e^2}\\ &=\frac{143 \left (c d^2-a e^2\right )^8 \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}}{131072 c^7 d^7 e^3}-\frac{143 \left (c d^2-a e^2\right )^6 \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}}{49152 c^6 d^6 e^2}+\frac{143 \left (c d^2-a e^2\right )^4 \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}}{15360 c^5 d^5 e}+\frac{143 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{4480 c^4 d^4}+\frac{143 \left (c d^2-a e^2\right )^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{2880 c^3 d^3}+\frac{13 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{180 c^2 d^2}+\frac{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{10 c d}-\frac{\left (143 \left (c d^2-a e^2\right )^{10}\right ) \int \frac{1}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{262144 c^7 d^7 e^3}\\ &=\frac{143 \left (c d^2-a e^2\right )^8 \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}}{131072 c^7 d^7 e^3}-\frac{143 \left (c d^2-a e^2\right )^6 \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}}{49152 c^6 d^6 e^2}+\frac{143 \left (c d^2-a e^2\right )^4 \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}}{15360 c^5 d^5 e}+\frac{143 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{4480 c^4 d^4}+\frac{143 \left (c d^2-a e^2\right )^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{2880 c^3 d^3}+\frac{13 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{180 c^2 d^2}+\frac{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{10 c d}-\frac{\left (143 \left (c d^2-a e^2\right )^{10}\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 )}{131072 c^7 d^7 e^3}\\ &=\frac{143 \left (c d^2-a e^2\right )^8 \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}}{131072 c^7 d^7 e^3}-\frac{143 \left (c d^2-a e^2\right )^6 \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}}{49152 c^6 d^6 e^2}+\frac{143 \left (c d^2-a e^2\right )^4 \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}}{15360 c^5 d^5 e}+\frac{143 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{4480 c^4 d^4}+\frac{143 \left (c d^2-a e^2\right )^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{2880 c^3 d^3}+\frac{13 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{180 c^2 d^2}+\frac{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{10 c d}-\frac{143 \left (c d^2-a e^2\right )^{10} \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 )}{262144 c^{15/2} d^{15/2} e^{7/2}}\\ \end{align*}

Mathematica [B]  time = 6.8119, size = 1439, normalized size = 2.69 \[ \frac{2 \left (c d^2-a e^2\right )^6 (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 )^{15/2} \left (\frac{1001 \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}{524288 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 )^7}+\frac{7}{20} \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{13}{18 \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{143}{288 \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{143}{448 \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 )^4}+\frac{143}{768 \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}+\frac{143}{1536 \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 )^6}+\frac{143}{4096 \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}\right )\right )}{7 c^7 d^7 \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 )^{13/2} (d+e x)^2 \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^4*(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)^6*(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))))^(15/2)*((7*(143/(4096*(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) + 143/(1536*(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))))^6) + 143/(768*(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) + 143/(448*(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))))^4) + 143
/(288*(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) + 13/(18*(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)))/20 + (1001*(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)))])))/(524288*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))))^7)))/(7*c^7*d^7*((c*d)/((c^2*d^3)/(
c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2)))^(13/2)*(d + e*x)^2*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])

________________________________________________________________________________________

Maple [B]  time = 0.065, size = 2973, normalized size = 5.6 \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)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x)

[Out]

-143/4480*e^6/d^4/c^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)*a^3-1001/131072/e*d^9*c*(a*d*e+(a*e^2+c*d^2)*x+c
*d*e*x^2)^(1/2)*a+1423/2880*e*d/c*x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)+1001/65536*e^5*d^3/c^2*(a*d*e+(a*e
^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^4+1/10*e^3*x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/d/c+1001/65536*e^7*d/c^3
*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^5-429/16384*e^8/d^2/c^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^5
+143/65536/e^2*d^10*c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x+143/131072*e^15/d^7/c^7*(a*d*e+(a*e^2+c*d^2)
*x+c*d*e*x^2)^(1/2)*a^9-1001/131072*e^13/d^5/c^6*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^8+715/32768*e^11/d^
3/c^5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^7-143/49152/e^2*d^8*c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+
143/131072/e^3*d^11*c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+715/49152*e^6/c^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e
*x^2)^(3/2)*a^4+67/180*e^2/c*x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)-10777/40320*e^2/c^2*(a*d*e+(a*e^2+c*d
^2)*x+c*d*e*x^2)^(7/2)*a+715/32768*e*d^7*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^2+143/15360/e*d^5*(a*d*e+(a
*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)+143/7680*d^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*x+715/49152*d^6*(a*d*e+(a*
e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a+1137/4480*d^2/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)-143/5120*e*d^3/c*(a*d*
e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*a+5863/40320*e^4/d^2/c^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)*a^2-143/26
2144/e^3*d^13*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)-143/24576/e*d^7*c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x+715/49152*e^4*d^2/c^2*(a*d*e+(a*e^2+c*d^
2)*x+c*d*e*x^2)^(3/2)*a^3-429/16384*e^2*d^4/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^2+143/15360*e^9/d^5/c^
5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*a^5-143/5120*e^7/d^3/c^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*a^4
-143/49152*e^12/d^6/c^6*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^7+715/49152*e^10/d^4/c^5*(a*d*e+(a*e^2+c*d^2
)*x+c*d*e*x^2)^(3/2)*a^6+143/7680*e^5/d/c^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*a^3+143/7680*e^3*d/c^2*(a*
d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*a^2+143/4096*e*d^5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*a+143/1280*e
^4/c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*x*a^2-1001/8192*e^8/c^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)
*x*a^5+1001/16384*e^2*d^6*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^2+2145/32768*e^3*d^7*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^3-1001/32768*e^3*d^5/c*(a
*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^3-1001/32768*e^9/d/c^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^6-143
/8192*d^8*c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a-39/160*e^3/d/c^2*x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(
7/2)*a-715/8192*e^3*d^3/c*a^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x-143/1920*e^2*d^2/c*(a*d*e+(a*e^2+c*d^2
)*x+c*d*e*x^2)^(5/2)*x*a+2145/32768*e^11/d/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^7-15015/131072*e^9*d/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+143/65536*e^14/d^6/c^6*(a*d*e+(a*e^2+c*d^2)*x+c*
d*e*x^2)^(1/2)*x*a^8-143/8192*e^12/d^4/c^5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^7+1001/16384*e^10/d^2/c
^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^6+715/131072/e*d^11*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-13/180*e^4/d^2/c^2*x^2*(a*d*e+(a*e^2+c*d^2)*x
+c*d*e*x^2)^(7/2)*a+9009/65536*e^7*d^3/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^5-15015/131072*e^5*d^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+715/6144*e^5*d/c^2*a^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*
x^2)^(3/2)*x+5005/32768*e^6*d^2/c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^4-1001/8192*e^4*d^4/c*(a*d*e+(
a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^3+143/2880*e^5/d^3/c^3*x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)*a^2-6435/
262144*e*d^9*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+143/4096*e^9/d^3/c^4*a^5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x-715/8192*e^7/d/c^3*a^4*(a*d*e+(a
*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x+143/7680*e^8/d^4/c^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*x*a^4-143/1920*e
^6/d^2/c^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*x*a^3-143/24576*e^11/d^5/c^5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x
^2)^(3/2)*x*a^6-143/262144*e^17/d^7/c^7*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^10+715/131072*e^15/d^5/c^6*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^9-6435/262144*e^13/d^3/c^5*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^8

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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)^4*(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 [B]  time = 4.02886, size = 5203, normalized size = 9.74 \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)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="fricas")

[Out]

[1/165150720*(45045*(c^10*d^20 - 10*a*c^9*d^18*e^2 + 45*a^2*c^8*d^16*e^4 - 120*a^3*c^7*d^14*e^6 + 210*a^4*c^6*
d^12*e^8 - 252*a^5*c^5*d^10*e^10 + 210*a^6*c^4*d^8*e^12 - 120*a^7*c^3*d^6*e^14 + 45*a^8*c^2*d^4*e^16 - 10*a^9*
c*d^2*e^18 + a^10*e^20)*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*(41
28768*c^10*d^10*e^10*x^9 + 45045*c^10*d^19*e - 435435*a*c^9*d^17*e^3 + 1885884*a^2*c^8*d^15*e^5 + 6983100*a^3*
c^7*d^13*e^7 - 9035650*a^4*c^6*d^11*e^9 + 8003710*a^5*c^5*d^9*e^11 - 4813380*a^6*c^4*d^7*e^13 + 1885884*a^7*c^
3*d^5*e^15 - 435435*a^8*c^2*d^3*e^17 + 45045*a^9*c*d*e^19 + 229376*(121*c^10*d^11*e^9 + 41*a*c^9*d^9*e^11)*x^8
+ 14336*(5503*c^10*d^12*e^8 + 4482*a*c^9*d^10*e^10 + 383*a^2*c^8*d^8*e^12)*x^7 + 1024*(119055*c^10*d^13*e^7 +
182129*a*c^9*d^11*e^9 + 37489*a^2*c^8*d^9*e^11 + 15*a^3*c^7*d^7*e^13)*x^6 + 256*(424895*c^10*d^14*e^6 + 11577
40*a*c^9*d^12*e^8 + 448938*a^2*c^8*d^10*e^10 + 620*a^3*c^7*d^8*e^12 - 65*a^4*c^6*d^6*e^14)*x^5 + 128*(419983*c
^10*d^15*e^5 + 2149035*a*c^9*d^13*e^7 + 1490630*a^2*c^8*d^11*e^9 + 5830*a^3*c^7*d^9*e^11 - 1365*a^4*c^6*d^7*e^
13 + 143*a^5*c^5*d^5*e^15)*x^4 + 16*(735993*c^10*d^16*e^4 + 9023498*a*c^9*d^14*e^6 + 11825815*a^2*c^8*d^12*e^8
+ 132300*a^3*c^7*d^10*e^10 - 52585*a^4*c^6*d^8*e^12 + 12298*a^5*c^5*d^6*e^14 - 1287*a^6*c^4*d^4*e^16)*x^3 + 8
*(3003*c^10*d^17*e^3 + 4394937*a*c^9*d^15*e^5 + 13885683*a^2*c^8*d^13*e^7 + 508825*a^3*c^7*d^11*e^9 - 310375*a
^4*c^6*d^9*e^11 + 123123*a^5*c^5*d^7*e^13 - 28743*a^6*c^4*d^5*e^15 + 3003*a^7*c^3*d^3*e^17)*x^2 - 2*(15015*c^1
0*d^18*e^2 - 144144*a*c^9*d^16*e^4 - 17075244*a^2*c^8*d^14*e^6 - 2878000*a^3*c^7*d^12*e^8 + 2579850*a^4*c^6*d^
10*e^10 - 1567280*a^5*c^5*d^8*e^12 + 619476*a^6*c^4*d^6*e^14 - 144144*a^7*c^3*d^4*e^16 + 15015*a^8*c^2*d^2*e^1
8)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^8*d^8*e^4), 1/82575360*(45045*(c^10*d^20 - 10*a*c^9*d^18
*e^2 + 45*a^2*c^8*d^16*e^4 - 120*a^3*c^7*d^14*e^6 + 210*a^4*c^6*d^12*e^8 - 252*a^5*c^5*d^10*e^10 + 210*a^6*c^4
*d^8*e^12 - 120*a^7*c^3*d^6*e^14 + 45*a^8*c^2*d^4*e^16 - 10*a^9*c*d^2*e^18 + a^10*e^20)*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*(4128768*c^10*d^10*e^10*x^9 + 45045*c^10*d^19*e - 435435*a*c^9*d^17*
e^3 + 1885884*a^2*c^8*d^15*e^5 + 6983100*a^3*c^7*d^13*e^7 - 9035650*a^4*c^6*d^11*e^9 + 8003710*a^5*c^5*d^9*e^1
1 - 4813380*a^6*c^4*d^7*e^13 + 1885884*a^7*c^3*d^5*e^15 - 435435*a^8*c^2*d^3*e^17 + 45045*a^9*c*d*e^19 + 22937
6*(121*c^10*d^11*e^9 + 41*a*c^9*d^9*e^11)*x^8 + 14336*(5503*c^10*d^12*e^8 + 4482*a*c^9*d^10*e^10 + 383*a^2*c^8
*d^8*e^12)*x^7 + 1024*(119055*c^10*d^13*e^7 + 182129*a*c^9*d^11*e^9 + 37489*a^2*c^8*d^9*e^11 + 15*a^3*c^7*d^7*
e^13)*x^6 + 256*(424895*c^10*d^14*e^6 + 1157740*a*c^9*d^12*e^8 + 448938*a^2*c^8*d^10*e^10 + 620*a^3*c^7*d^8*e^
12 - 65*a^4*c^6*d^6*e^14)*x^5 + 128*(419983*c^10*d^15*e^5 + 2149035*a*c^9*d^13*e^7 + 1490630*a^2*c^8*d^11*e^9
+ 5830*a^3*c^7*d^9*e^11 - 1365*a^4*c^6*d^7*e^13 + 143*a^5*c^5*d^5*e^15)*x^4 + 16*(735993*c^10*d^16*e^4 + 90234
98*a*c^9*d^14*e^6 + 11825815*a^2*c^8*d^12*e^8 + 132300*a^3*c^7*d^10*e^10 - 52585*a^4*c^6*d^8*e^12 + 12298*a^5*
c^5*d^6*e^14 - 1287*a^6*c^4*d^4*e^16)*x^3 + 8*(3003*c^10*d^17*e^3 + 4394937*a*c^9*d^15*e^5 + 13885683*a^2*c^8*
d^13*e^7 + 508825*a^3*c^7*d^11*e^9 - 310375*a^4*c^6*d^9*e^11 + 123123*a^5*c^5*d^7*e^13 - 28743*a^6*c^4*d^5*e^1
5 + 3003*a^7*c^3*d^3*e^17)*x^2 - 2*(15015*c^10*d^18*e^2 - 144144*a*c^9*d^16*e^4 - 17075244*a^2*c^8*d^14*e^6 -
2878000*a^3*c^7*d^12*e^8 + 2579850*a^4*c^6*d^10*e^10 - 1567280*a^5*c^5*d^8*e^12 + 619476*a^6*c^4*d^6*e^14 - 14
4144*a^7*c^3*d^4*e^16 + 15015*a^8*c^2*d^2*e^18)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^8*d^8*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((e*x+d)**4*(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 = 2.01313, size = 1397, normalized size = 2.62 \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)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="giac")

[Out]

1/41287680*sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*(2*(4*(2*(8*(2*(4*(14*(16*(18*c^2*d^2*x*e^6 + (121*c^11
*d^12*e^14 + 41*a*c^10*d^10*e^16)*e^(-9)/(c^9*d^9))*x + (5503*c^11*d^13*e^13 + 4482*a*c^10*d^11*e^15 + 383*a^2
*c^9*d^9*e^17)*e^(-9)/(c^9*d^9))*x + (119055*c^11*d^14*e^12 + 182129*a*c^10*d^12*e^14 + 37489*a^2*c^9*d^10*e^1
6 + 15*a^3*c^8*d^8*e^18)*e^(-9)/(c^9*d^9))*x + (424895*c^11*d^15*e^11 + 1157740*a*c^10*d^13*e^13 + 448938*a^2*
c^9*d^11*e^15 + 620*a^3*c^8*d^9*e^17 - 65*a^4*c^7*d^7*e^19)*e^(-9)/(c^9*d^9))*x + (419983*c^11*d^16*e^10 + 214
9035*a*c^10*d^14*e^12 + 1490630*a^2*c^9*d^12*e^14 + 5830*a^3*c^8*d^10*e^16 - 1365*a^4*c^7*d^8*e^18 + 143*a^5*c
^6*d^6*e^20)*e^(-9)/(c^9*d^9))*x + (735993*c^11*d^17*e^9 + 9023498*a*c^10*d^15*e^11 + 11825815*a^2*c^9*d^13*e^
13 + 132300*a^3*c^8*d^11*e^15 - 52585*a^4*c^7*d^9*e^17 + 12298*a^5*c^6*d^7*e^19 - 1287*a^6*c^5*d^5*e^21)*e^(-9
)/(c^9*d^9))*x + (3003*c^11*d^18*e^8 + 4394937*a*c^10*d^16*e^10 + 13885683*a^2*c^9*d^14*e^12 + 508825*a^3*c^8*
d^12*e^14 - 310375*a^4*c^7*d^10*e^16 + 123123*a^5*c^6*d^8*e^18 - 28743*a^6*c^5*d^6*e^20 + 3003*a^7*c^4*d^4*e^2
2)*e^(-9)/(c^9*d^9))*x - (15015*c^11*d^19*e^7 - 144144*a*c^10*d^17*e^9 - 17075244*a^2*c^9*d^15*e^11 - 2878000*
a^3*c^8*d^13*e^13 + 2579850*a^4*c^7*d^11*e^15 - 1567280*a^5*c^6*d^9*e^17 + 619476*a^6*c^5*d^7*e^19 - 144144*a^
7*c^4*d^5*e^21 + 15015*a^8*c^3*d^3*e^23)*e^(-9)/(c^9*d^9))*x + (45045*c^11*d^20*e^6 - 435435*a*c^10*d^18*e^8 +
1885884*a^2*c^9*d^16*e^10 + 6983100*a^3*c^8*d^14*e^12 - 9035650*a^4*c^7*d^12*e^14 + 8003710*a^5*c^6*d^10*e^16
- 4813380*a^6*c^5*d^8*e^18 + 1885884*a^7*c^4*d^6*e^20 - 435435*a^8*c^3*d^4*e^22 + 45045*a^9*c^2*d^2*e^24)*e^(
-9)/(c^9*d^9)) + 143/262144*(c^10*d^20 - 10*a*c^9*d^18*e^2 + 45*a^2*c^8*d^16*e^4 - 120*a^3*c^7*d^14*e^6 + 210*
a^4*c^6*d^12*e^8 - 252*a^5*c^5*d^10*e^10 + 210*a^6*c^4*d^8*e^12 - 120*a^7*c^3*d^6*e^14 + 45*a^8*c^2*d^4*e^16 -
10*a^9*c*d^2*e^18 + a^10*e^20)*sqrt(c*d)*e^(-7/2)*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 + c*d^2*x + a*x*e^2 + a*d*e))*c*d*e - sqrt(c*d)*a*e^(5/2)))/(c^8*d^8)