### 3.1919 $$\int (d+e x)^3 (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2} \, dx$$

Optimal. Leaf size=401 $-\frac{9 \left (c d^2-a e^2\right )^5 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{1024 c^5 d^5 e^2}+\frac{3 \left (c d^2-a e^2\right )^3 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{128 c^4 d^4 e}+\frac{3 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{40 c^3 d^3}+\frac{3 (d+e x) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{28 c^2 d^2}+\frac{9 \left (c d^2-a e^2\right )^7 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2048 c^{11/2} d^{11/2} e^{5/2}}+\frac{(d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 c d}$

[Out]

(-9*(c*d^2 - a*e^2)^5*(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^5*d^5*e
^2) + (3*(c*d^2 - a*e^2)^3*(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))/(128*c^4
*d^4*e) + (3*(c*d^2 - a*e^2)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(40*c^3*d^3) + (3*(c*d^2 - a*e^2
)*(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(28*c^2*d^2) + ((d + e*x)^2*(a*d*e + (c*d^2 + a*e^2
)*x + c*d*e*x^2)^(5/2))/(7*c*d) + (9*(c*d^2 - a*e^2)^7*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^(11/2)*d^(11/2)*e^(5/2))

________________________________________________________________________________________

Rubi [A]  time = 0.372587, antiderivative size = 401, 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} $-\frac{9 \left (c d^2-a e^2\right )^5 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{1024 c^5 d^5 e^2}+\frac{3 \left (c d^2-a e^2\right )^3 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{128 c^4 d^4 e}+\frac{3 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{40 c^3 d^3}+\frac{3 (d+e x) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{28 c^2 d^2}+\frac{9 \left (c d^2-a e^2\right )^7 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2048 c^{11/2} d^{11/2} e^{5/2}}+\frac{(d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 c d}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]

[Out]

(-9*(c*d^2 - a*e^2)^5*(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^5*d^5*e
^2) + (3*(c*d^2 - a*e^2)^3*(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))/(128*c^4
*d^4*e) + (3*(c*d^2 - a*e^2)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(40*c^3*d^3) + (3*(c*d^2 - a*e^2
)*(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(28*c^2*d^2) + ((d + e*x)^2*(a*d*e + (c*d^2 + a*e^2
)*x + c*d*e*x^2)^(5/2))/(7*c*d) + (9*(c*d^2 - a*e^2)^7*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^(11/2)*d^(11/2)*e^(5/2))

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)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx &=\frac{(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}}{7 c d}+\frac{\left (9 \left (d^2-\frac{a e^2}{c}\right )\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 )^{3/2} \, dx}{14 d}\\ &=\frac{3 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{28 c^2 d^2}+\frac{(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}}{7 c d}+\frac{\left (3 \left (d^2-\frac{a e^2}{c}\right )^2\right ) \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx}{8 d^2}\\ &=\frac{3 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{40 c^3 d^3}+\frac{3 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{28 c^2 d^2}+\frac{(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}}{7 c d}+\frac{\left (3 \left (d^2-\frac{a e^2}{c}\right )^3\right ) \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx}{16 d^3}\\ &=\frac{3 \left (c d^2-a e^2\right )^3 \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}}{128 c^4 d^4 e}+\frac{3 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{40 c^3 d^3}+\frac{3 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{28 c^2 d^2}+\frac{(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}}{7 c d}-\frac{\left (9 \left (c d^2-a e^2\right )^5\right ) \int \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{256 c^4 d^4 e}\\ &=-\frac{9 \left (c d^2-a e^2\right )^5 \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^5 d^5 e^2}+\frac{3 \left (c d^2-a e^2\right )^3 \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}}{128 c^4 d^4 e}+\frac{3 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{40 c^3 d^3}+\frac{3 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{28 c^2 d^2}+\frac{(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}}{7 c d}+\frac{\left (9 \left (c d^2-a e^2\right )^7\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^5 d^5 e^2}\\ &=-\frac{9 \left (c d^2-a e^2\right )^5 \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^5 d^5 e^2}+\frac{3 \left (c d^2-a e^2\right )^3 \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}}{128 c^4 d^4 e}+\frac{3 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{40 c^3 d^3}+\frac{3 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{28 c^2 d^2}+\frac{(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}}{7 c d}+\frac{\left (9 \left (c d^2-a e^2\right )^7\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^5 d^5 e^2}\\ &=-\frac{9 \left (c d^2-a e^2\right )^5 \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^5 d^5 e^2}+\frac{3 \left (c d^2-a e^2\right )^3 \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}}{128 c^4 d^4 e}+\frac{3 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{40 c^3 d^3}+\frac{3 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{28 c^2 d^2}+\frac{(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}}{7 c d}+\frac{9 \left (c d^2-a e^2\right )^7 \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^{11/2} d^{11/2} e^{5/2}}\\ \end{align*}

Mathematica [B]  time = 6.25364, size = 1196, normalized size = 2.98 $\frac{2 \left (c d^2-a e^2\right )^4 (a e+c d x) ((a e+c d x) (d+e x))^{3/2} \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{5}{14} \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}{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{21}{40 \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{21}{64 \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{21}{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 )-\frac{45 \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 \left (-\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 )}{4096 c^3 d^3 e^3 (a e+c d x)^3 \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 )}{5 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) \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]

[Out]

(2*(c*d^2 - a*e^2)^4*(a*e + c*d*x)*((a*e + c*d*x)*(d + e*x))^(3/2)*(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)*((5*(21/(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) + 21/(64*(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) + 21/(40*(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) + 3/(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) + (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)))/14 - (45*(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*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) - (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)))])))/(4096*c^3*d^3*e^3*(a*e + c*d*x)^3*(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)))/(5*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)*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])

________________________________________________________________________________________

Maple [B]  time = 0.057, size = 1586, normalized size = 4. \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)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x)

[Out]

3/64*d^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x+3/128/e*d^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+13/40*d
/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)+9/256*d^5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a+9/2048/e^2*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)-9/35
*e^2/d/c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*a-3/64*e*d^2/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a+45
/256*e^5/c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^3+189/2048*e^2*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^2+45/512*e*d^4*(a*d*e+(a*e^2+c*d^2)*x+c*
d*e*x^2)^(1/2)*x*a-63/2048*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-9/1024/e^2*d^7*c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+11/28*e/c*x*(a*d*e+(a*e^2+
c*d^2)*x+c*d*e*x^2)^(5/2)-3/128*e^7/d^4/c^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^4-9/256*e^8/d^3/c^4*(a*d
*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^5+9/1024*e^10/d^5/c^5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^6+1/7*e^
2*x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/d/c-45/1024*e^2*d^3/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^
2+45/1024*e^6/d/c^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^4+3/40*e^4/d^3/c^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*
x^2)^(5/2)*a^2-9/512/e*d^6*c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x+3/64*e^5/d^2/c^3*(a*d*e+(a*e^2+c*d^2)*x
+c*d*e*x^2)^(3/2)*a^3-3/64*e^6/d^3/c^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*a^3+9/64*e^4/d/c^2*(a*d*e+(a*
e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*a^2-189/2048*e^8/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^5+315/2048*e^6*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^4-3/28*e^3/d^2/c^2*x*(a*d*e+(a*e^2+c*d^2)*x+c*
d*e*x^2)^(5/2)*a+9/512*e^9/d^4/c^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^5-45/512*e^7/d^2/c^3*(a*d*e+(a*
e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^4-9/64*e^2*d/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*a-315/2048*e^4*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^3-9/
2048*e^12/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^7+63/2048*e^10/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^6-45/256*e^3*d^2/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^2

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 2.4543, size = 2804, normalized size = 6.99 \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)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="fricas")

[Out]

[1/143360*(315*(c^7*d^14 - 7*a*c^6*d^12*e^2 + 21*a^2*c^5*d^10*e^4 - 35*a^3*c^4*d^8*e^6 + 35*a^4*c^3*d^6*e^8 -
21*a^5*c^2*d^4*e^10 + 7*a^6*c*d^2*e^12 - 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*(5120*c^7*d^7*e^7*x^6 - 315*c^7*d^13*e + 2100*a*c^6*d^11*e^3 + 8393*a^2*c^5*d^9*e^5 -
9216*a^3*c^4*d^7*e^7 + 5943*a^4*c^3*d^5*e^9 - 2100*a^5*c^2*d^3*e^11 + 315*a^6*c*d*e^13 + 1280*(19*c^7*d^8*e^6
+ 5*a*c^6*d^6*e^8)*x^5 + 128*(351*c^7*d^9*e^5 + 248*a*c^6*d^7*e^7 + a^2*c^5*d^5*e^9)*x^4 + 16*(2441*c^7*d^10*e
^4 + 3909*a*c^6*d^8*e^6 + 59*a^2*c^5*d^6*e^8 - 9*a^3*c^4*d^4*e^10)*x^3 + 8*(1771*c^7*d^11*e^3 + 7562*a*c^6*d^9
*e^5 + 384*a^2*c^5*d^7*e^7 - 138*a^3*c^4*d^5*e^9 + 21*a^4*c^3*d^3*e^11)*x^2 + 2*(105*c^7*d^12*e^2 + 13643*a*c^
6*d^10*e^4 + 2962*a^2*c^5*d^8*e^6 - 1938*a^3*c^4*d^6*e^8 + 693*a^4*c^3*d^4*e^10 - 105*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^6*d^6*e^3), -1/71680*(315*(c^7*d^14 - 7*a*c^6*d^12*e^2 + 21*a^2*c
^5*d^10*e^4 - 35*a^3*c^4*d^8*e^6 + 35*a^4*c^3*d^6*e^8 - 21*a^5*c^2*d^4*e^10 + 7*a^6*c*d^2*e^12 - a^7*e^14)*sqr
t(-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*(5120*c^7*d^7*e^7*x^6 - 315*c^7*d^13*e + 2100*a*c
^6*d^11*e^3 + 8393*a^2*c^5*d^9*e^5 - 9216*a^3*c^4*d^7*e^7 + 5943*a^4*c^3*d^5*e^9 - 2100*a^5*c^2*d^3*e^11 + 315
*a^6*c*d*e^13 + 1280*(19*c^7*d^8*e^6 + 5*a*c^6*d^6*e^8)*x^5 + 128*(351*c^7*d^9*e^5 + 248*a*c^6*d^7*e^7 + a^2*c
^5*d^5*e^9)*x^4 + 16*(2441*c^7*d^10*e^4 + 3909*a*c^6*d^8*e^6 + 59*a^2*c^5*d^6*e^8 - 9*a^3*c^4*d^4*e^10)*x^3 +
8*(1771*c^7*d^11*e^3 + 7562*a*c^6*d^9*e^5 + 384*a^2*c^5*d^7*e^7 - 138*a^3*c^4*d^5*e^9 + 21*a^4*c^3*d^3*e^11)*x
^2 + 2*(105*c^7*d^12*e^2 + 13643*a*c^6*d^10*e^4 + 2962*a^2*c^5*d^8*e^6 - 1938*a^3*c^4*d^6*e^8 + 693*a^4*c^3*d^
4*e^10 - 105*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^6*d^6*e^3)]

________________________________________________________________________________________

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)**3*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.31157, size = 824, normalized size = 2.05 \begin{align*} \frac{1}{35840} \, \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 (10 \,{\left (4 \, c d x e^{4} + \frac{{\left (19 \, c^{7} d^{8} e^{9} + 5 \, a c^{6} d^{6} e^{11}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} x + \frac{{\left (351 \, c^{7} d^{9} e^{8} + 248 \, a c^{6} d^{7} e^{10} + a^{2} c^{5} d^{5} e^{12}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} x + \frac{{\left (2441 \, c^{7} d^{10} e^{7} + 3909 \, a c^{6} d^{8} e^{9} + 59 \, a^{2} c^{5} d^{6} e^{11} - 9 \, a^{3} c^{4} d^{4} e^{13}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} x + \frac{{\left (1771 \, c^{7} d^{11} e^{6} + 7562 \, a c^{6} d^{9} e^{8} + 384 \, a^{2} c^{5} d^{7} e^{10} - 138 \, a^{3} c^{4} d^{5} e^{12} + 21 \, a^{4} c^{3} d^{3} e^{14}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} x + \frac{{\left (105 \, c^{7} d^{12} e^{5} + 13643 \, a c^{6} d^{10} e^{7} + 2962 \, a^{2} c^{5} d^{8} e^{9} - 1938 \, a^{3} c^{4} d^{6} e^{11} + 693 \, a^{4} c^{3} d^{4} e^{13} - 105 \, a^{5} c^{2} d^{2} e^{15}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} x - \frac{{\left (315 \, c^{7} d^{13} e^{4} - 2100 \, a c^{6} d^{11} e^{6} - 8393 \, a^{2} c^{5} d^{9} e^{8} + 9216 \, a^{3} c^{4} d^{7} e^{10} - 5943 \, a^{4} c^{3} d^{5} e^{12} + 2100 \, a^{5} c^{2} d^{3} e^{14} - 315 \, a^{6} c d e^{16}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} - \frac{9 \,{\left (c^{7} d^{14} - 7 \, a c^{6} d^{12} e^{2} + 21 \, a^{2} c^{5} d^{10} e^{4} - 35 \, a^{3} c^{4} d^{8} e^{6} + 35 \, a^{4} c^{3} d^{6} e^{8} - 21 \, a^{5} c^{2} d^{4} e^{10} + 7 \, a^{6} c d^{2} e^{12} - a^{7} e^{14}\right )} \sqrt{c d} e^{\left (-\frac{5}{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 )}{2048 \, c^{6} d^{6}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="giac")

[Out]

1/35840*sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*(2*(4*(2*(8*(10*(4*c*d*x*e^4 + (19*c^7*d^8*e^9 + 5*a*c^6*d
^6*e^11)*e^(-6)/(c^6*d^6))*x + (351*c^7*d^9*e^8 + 248*a*c^6*d^7*e^10 + a^2*c^5*d^5*e^12)*e^(-6)/(c^6*d^6))*x +
(2441*c^7*d^10*e^7 + 3909*a*c^6*d^8*e^9 + 59*a^2*c^5*d^6*e^11 - 9*a^3*c^4*d^4*e^13)*e^(-6)/(c^6*d^6))*x + (17
71*c^7*d^11*e^6 + 7562*a*c^6*d^9*e^8 + 384*a^2*c^5*d^7*e^10 - 138*a^3*c^4*d^5*e^12 + 21*a^4*c^3*d^3*e^14)*e^(-
6)/(c^6*d^6))*x + (105*c^7*d^12*e^5 + 13643*a*c^6*d^10*e^7 + 2962*a^2*c^5*d^8*e^9 - 1938*a^3*c^4*d^6*e^11 + 69
3*a^4*c^3*d^4*e^13 - 105*a^5*c^2*d^2*e^15)*e^(-6)/(c^6*d^6))*x - (315*c^7*d^13*e^4 - 2100*a*c^6*d^11*e^6 - 839
3*a^2*c^5*d^9*e^8 + 9216*a^3*c^4*d^7*e^10 - 5943*a^4*c^3*d^5*e^12 + 2100*a^5*c^2*d^3*e^14 - 315*a^6*c*d*e^16)*
e^(-6)/(c^6*d^6)) - 9/2048*(c^7*d^14 - 7*a*c^6*d^12*e^2 + 21*a^2*c^5*d^10*e^4 - 35*a^3*c^4*d^8*e^6 + 35*a^4*c^
3*d^6*e^8 - 21*a^5*c^2*d^4*e^10 + 7*a^6*c*d^2*e^12 - a^7*e^14)*sqrt(c*d)*e^(-5/2)*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)