3.1933 \(\int (d+e x)^2 (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2} \, dx\)

Optimal. Leaf size=414 \[ \frac{45 \left (c d^2-a e^2\right )^6 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{16384 c^5 d^5 e^3}-\frac{15 \left (c d^2-a e^2\right )^4 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{2048 c^4 d^4 e^2}+\frac{3 \left (c d^2-a e^2\right )^2 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{128 c^3 d^3 e}+\frac{9 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{112 c^2 d^2}-\frac{45 \left (c d^2-a e^2\right )^8 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{32768 c^{11/2} d^{11/2} e^{7/2}}+\frac{(d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{8 c d} \]

[Out]

(45*(c*d^2 - a*e^2)^6*(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])/(16384*c^5*d^5*
e^3) - (15*(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)^(3/2))/(2048*
c^4*d^4*e^2) + (3*(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)^(5/2))
/(128*c^3*d^3*e) + (9*(c*d^2 - a*e^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(112*c^2*d^2) + ((d + e*x
)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(8*c*d) - (45*(c*d^2 - a*e^2)^8*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])])/(32768*c^(11/2)*d^(11/2)*e^(7
/2))

________________________________________________________________________________________

Rubi [A]  time = 0.35434, antiderivative size = 414, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{45 \left (c d^2-a e^2\right )^6 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{16384 c^5 d^5 e^3}-\frac{15 \left (c d^2-a e^2\right )^4 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{2048 c^4 d^4 e^2}+\frac{3 \left (c d^2-a e^2\right )^2 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{128 c^3 d^3 e}+\frac{9 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{112 c^2 d^2}-\frac{45 \left (c d^2-a e^2\right )^8 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{32768 c^{11/2} d^{11/2} e^{7/2}}+\frac{(d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{8 c d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(45*(c*d^2 - a*e^2)^6*(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])/(16384*c^5*d^5*
e^3) - (15*(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)^(3/2))/(2048*
c^4*d^4*e^2) + (3*(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)^(5/2))
/(128*c^3*d^3*e) + (9*(c*d^2 - a*e^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(112*c^2*d^2) + ((d + e*x
)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(8*c*d) - (45*(c*d^2 - a*e^2)^8*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])])/(32768*c^(11/2)*d^(11/2)*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)^2 \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) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{8 c d}+\frac{\left (9 \left (d^2-\frac{a e^2}{c}\right )\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}{16 d}\\ &=\frac{9 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{112 c^2 d^2}+\frac{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{8 c d}+\frac{\left (9 \left (d^2-\frac{a e^2}{c}\right )^2\right ) \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx}{32 d^2}\\ &=\frac{3 \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 )^{5/2}}{128 c^3 d^3 e}+\frac{9 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{112 c^2 d^2}+\frac{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{8 c d}-\frac{\left (15 \left (c d^2-a e^2\right )^4\right ) \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx}{256 c^3 d^3 e}\\ &=-\frac{15 \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 )^{3/2}}{2048 c^4 d^4 e^2}+\frac{3 \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 )^{5/2}}{128 c^3 d^3 e}+\frac{9 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{112 c^2 d^2}+\frac{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{8 c d}+\frac{\left (45 \left (c d^2-a e^2\right )^6\right ) \int \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{4096 c^4 d^4 e^2}\\ &=\frac{45 \left (c d^2-a e^2\right )^6 \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}}{16384 c^5 d^5 e^3}-\frac{15 \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 )^{3/2}}{2048 c^4 d^4 e^2}+\frac{3 \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 )^{5/2}}{128 c^3 d^3 e}+\frac{9 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{112 c^2 d^2}+\frac{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{8 c d}-\frac{\left (45 \left (c d^2-a e^2\right )^8\right ) \int \frac{1}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{32768 c^5 d^5 e^3}\\ &=\frac{45 \left (c d^2-a e^2\right )^6 \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}}{16384 c^5 d^5 e^3}-\frac{15 \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 )^{3/2}}{2048 c^4 d^4 e^2}+\frac{3 \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 )^{5/2}}{128 c^3 d^3 e}+\frac{9 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{112 c^2 d^2}+\frac{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{8 c d}-\frac{\left (45 \left (c d^2-a e^2\right )^8\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 )}{16384 c^5 d^5 e^3}\\ &=\frac{45 \left (c d^2-a e^2\right )^6 \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}}{16384 c^5 d^5 e^3}-\frac{15 \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 )^{3/2}}{2048 c^4 d^4 e^2}+\frac{3 \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 )^{5/2}}{128 c^3 d^3 e}+\frac{9 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{112 c^2 d^2}+\frac{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{8 c d}-\frac{45 \left (c d^2-a e^2\right )^8 \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 )}{32768 c^{11/2} d^{11/2} e^{7/2}}\\ \end{align*}

Mathematica [B]  time = 6.31834, size = 1279, normalized size = 3.09 \[ \frac{2 \left (c d^2-a e^2\right )^4 (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 )^{11/2} \left (\frac{315 \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}{65536 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 )^5}+\frac{7}{16} \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{9}{14 \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}{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 )^3}+\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 )^4}+\frac{9}{128 \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}\right )\right )}{7 c^5 d^5 \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 )^{9/2} (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)^2*(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)^4*(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))))^(11/2)*((7*(9/(128*(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) + 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))))^4) + 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))))^3) + 9/(14*(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)))/16 + (315*(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)))])))/(65536*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))))^5)))/(7*c^5*d^5*((c*d)/((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(
c*d^2 - a*e^2)))^(9/2)*(d + e*x)^2*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])

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Maple [B]  time = 0.054, size = 2044, normalized size = 4.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9/112*e^2/d^2/c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)*a-225/16384/e*d^7*c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^
2)^(1/2)*a+3/128*e^5/d^3/c^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*a^3-135/4096*d^6*c*(a*d*e+(a*e^2+c*d^2)*x
+c*d*e*x^2)^(1/2)*x*a-3/128*e^3/d/c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*a^2-15/1024*e^2*d^2/c*(a*d*e+(a*
e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^2+1/8*e*x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/d/c+315/4096*e^3*d^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^3+675/8192*e^
6/c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^4+15/256*e*a*d^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x-1
5/1024/e*d^5*c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x+675/8192*e^2*d^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1
/2)*x*a^2-3/32*e^2/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*x*a-225/16384*e^5*d/c^2*(a*d*e+(a*e^2+c*d^2)*x+c*
d*e*x^2)^(1/2)*a^4+45/8192/e^2*d^8*c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x+45/16384*e^11/d^5/c^5*(a*d*e+
(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^7-225/16384*e^9/d^3/c^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^6-225/163
84*e^3*d^3/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^3-45/32768/e^3*d^11*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)-15/2048*e^8/d^4/c^4*(a*d*e+(a*e^2+c*d^2
)*x+c*d*e*x^2)^(3/2)*a^5+45/2048*e^6/d^2/c^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^4+3/64*d^2*(a*d*e+(a*e^
2+c*d^2)*x+c*d*e*x^2)^(5/2)*x+45/2048*d^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a+3/128/e*d^3*(a*d*e+(a*e^2+
c*d^2)*x+c*d*e*x^2)^(5/2)-3/128*e*d/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*a+405/16384*e^7/d/c^3*(a*d*e+(a*
e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^5+23/112/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)+405/16384*e*d^5*(a*d*e+(a*e
^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^2-15/1024*e^4/c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^3-15/2048/e^2*d^6*c
*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+45/16384/e^3*d^9*c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-315/8192
*e*d^7*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+45/4096/e*d^9*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-45/32768*e^13/d^5/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+45/4096*e^11/d^3/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^7-315/8192*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+315/4096*e^7*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)*a^5+45/8192*e^10/d^4/c^4*(a*d*e+
(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^6-135/4096*e^8/d^2/c^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^5+15/2
56*e^5*a^3/c^2/d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x-225/2048*e^4*d^2/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2
)^(1/2)*x*a^3-15/1024*e^7/d^3/c^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*a^4+3/64*e^4/d^2/c^2*(a*d*e+(a*e^2
+c*d^2)*x+c*d*e*x^2)^(5/2)*x*a^2-45/512*e^3*d/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*a^2-1575/16384*e^5*d
^3/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

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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)^2*(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 = 3.04319, size = 3398, normalized size = 8.21 \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)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="fricas")

[Out]

[1/458752*(315*(c^8*d^16 - 8*a*c^7*d^14*e^2 + 28*a^2*c^6*d^12*e^4 - 56*a^3*c^5*d^10*e^6 + 70*a^4*c^4*d^8*e^8 -
56*a^5*c^3*d^6*e^10 + 28*a^6*c^2*d^4*e^12 - 8*a^7*c*d^2*e^14 + a^8*e^16)*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*(14336*c^8*d^8*e^8*x^7 + 315*c^8*d^15*e - 2415*a*c^7*d^13*e^3 +
8043*a^2*c^6*d^11*e^5 + 17609*a^3*c^5*d^9*e^7 - 15159*a^4*c^4*d^7*e^9 + 8043*a^5*c^3*d^5*e^11 - 2415*a^6*c^2*
d^3*e^13 + 315*a^7*c*d*e^15 + 1024*(65*c^8*d^9*e^7 + 33*a*c^7*d^7*e^9)*x^6 + 768*(155*c^8*d^10*e^6 + 210*a*c^7
*d^8*e^8 + 27*a^2*c^6*d^6*e^10)*x^5 + 128*(769*c^8*d^11*e^5 + 2343*a*c^7*d^9*e^7 + 807*a^2*c^6*d^7*e^9 + a^3*c
^5*d^5*e^11)*x^4 + 16*(2039*c^8*d^12*e^4 + 16452*a*c^7*d^10*e^6 + 12810*a^2*c^6*d^8*e^8 + 68*a^3*c^5*d^6*e^10
- 9*a^4*c^4*d^4*e^12)*x^3 + 24*(7*c^8*d^13*e^3 + 4043*a*c^7*d^11*e^5 + 8366*a^2*c^6*d^9*e^7 + 174*a^3*c^5*d^7*
e^9 - 53*a^4*c^4*d^5*e^11 + 7*a^5*c^3*d^3*e^13)*x^2 - 2*(105*c^8*d^14*e^2 - 798*a*c^7*d^12*e^4 - 46521*a^2*c^6
*d^10*e^6 - 4900*a^3*c^5*d^8*e^8 + 2631*a^4*c^4*d^6*e^10 - 798*a^5*c^3*d^4*e^12 + 105*a^6*c^2*d^2*e^14)*x)*sqr
t(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^6*d^6*e^4), 1/229376*(315*(c^8*d^16 - 8*a*c^7*d^14*e^2 + 28*a^2*c
^6*d^12*e^4 - 56*a^3*c^5*d^10*e^6 + 70*a^4*c^4*d^8*e^8 - 56*a^5*c^3*d^6*e^10 + 28*a^6*c^2*d^4*e^12 - 8*a^7*c*d
^2*e^14 + a^8*e^16)*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*(14336*c^8*d^8*e^8*x^7 + 3
15*c^8*d^15*e - 2415*a*c^7*d^13*e^3 + 8043*a^2*c^6*d^11*e^5 + 17609*a^3*c^5*d^9*e^7 - 15159*a^4*c^4*d^7*e^9 +
8043*a^5*c^3*d^5*e^11 - 2415*a^6*c^2*d^3*e^13 + 315*a^7*c*d*e^15 + 1024*(65*c^8*d^9*e^7 + 33*a*c^7*d^7*e^9)*x^
6 + 768*(155*c^8*d^10*e^6 + 210*a*c^7*d^8*e^8 + 27*a^2*c^6*d^6*e^10)*x^5 + 128*(769*c^8*d^11*e^5 + 2343*a*c^7*
d^9*e^7 + 807*a^2*c^6*d^7*e^9 + a^3*c^5*d^5*e^11)*x^4 + 16*(2039*c^8*d^12*e^4 + 16452*a*c^7*d^10*e^6 + 12810*a
^2*c^6*d^8*e^8 + 68*a^3*c^5*d^6*e^10 - 9*a^4*c^4*d^4*e^12)*x^3 + 24*(7*c^8*d^13*e^3 + 4043*a*c^7*d^11*e^5 + 83
66*a^2*c^6*d^9*e^7 + 174*a^3*c^5*d^7*e^9 - 53*a^4*c^4*d^5*e^11 + 7*a^5*c^3*d^3*e^13)*x^2 - 2*(105*c^8*d^14*e^2
- 798*a*c^7*d^12*e^4 - 46521*a^2*c^6*d^10*e^6 - 4900*a^3*c^5*d^8*e^8 + 2631*a^4*c^4*d^6*e^10 - 798*a^5*c^3*d^
4*e^12 + 105*a^6*c^2*d^2*e^14)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^6*d^6*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)**2*(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 = 1.84902, size = 1006, normalized size = 2.43 \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)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="giac")

[Out]

1/114688*sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*(2*(4*(2*(8*(2*(4*(14*c^2*d^2*x*e^4 + (65*c^9*d^10*e^10 +
33*a*c^8*d^8*e^12)*e^(-7)/(c^7*d^7))*x + 3*(155*c^9*d^11*e^9 + 210*a*c^8*d^9*e^11 + 27*a^2*c^7*d^7*e^13)*e^(-
7)/(c^7*d^7))*x + (769*c^9*d^12*e^8 + 2343*a*c^8*d^10*e^10 + 807*a^2*c^7*d^8*e^12 + a^3*c^6*d^6*e^14)*e^(-7)/(
c^7*d^7))*x + (2039*c^9*d^13*e^7 + 16452*a*c^8*d^11*e^9 + 12810*a^2*c^7*d^9*e^11 + 68*a^3*c^6*d^7*e^13 - 9*a^4
*c^5*d^5*e^15)*e^(-7)/(c^7*d^7))*x + 3*(7*c^9*d^14*e^6 + 4043*a*c^8*d^12*e^8 + 8366*a^2*c^7*d^10*e^10 + 174*a^
3*c^6*d^8*e^12 - 53*a^4*c^5*d^6*e^14 + 7*a^5*c^4*d^4*e^16)*e^(-7)/(c^7*d^7))*x - (105*c^9*d^15*e^5 - 798*a*c^8
*d^13*e^7 - 46521*a^2*c^7*d^11*e^9 - 4900*a^3*c^6*d^9*e^11 + 2631*a^4*c^5*d^7*e^13 - 798*a^5*c^4*d^5*e^15 + 10
5*a^6*c^3*d^3*e^17)*e^(-7)/(c^7*d^7))*x + (315*c^9*d^16*e^4 - 2415*a*c^8*d^14*e^6 + 8043*a^2*c^7*d^12*e^8 + 17
609*a^3*c^6*d^10*e^10 - 15159*a^4*c^5*d^8*e^12 + 8043*a^5*c^4*d^6*e^14 - 2415*a^6*c^3*d^4*e^16 + 315*a^7*c^2*d
^2*e^18)*e^(-7)/(c^7*d^7)) + 45/32768*(c^8*d^16 - 8*a*c^7*d^14*e^2 + 28*a^2*c^6*d^12*e^4 - 56*a^3*c^5*d^10*e^6
+ 70*a^4*c^4*d^8*e^8 - 56*a^5*c^3*d^6*e^10 + 28*a^6*c^2*d^4*e^12 - 8*a^7*c*d^2*e^14 + a^8*e^16)*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^6*d^6)