3.751 \(\int \frac{(f+g x)^{3/2} (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^{5/2}} \, dx\)

Optimal. Leaf size=448 \[ -\frac{3 \sqrt{f+g x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^4}{128 c^2 d^2 g^3 \sqrt{d+e x}}-\frac{3 \sqrt{d+e x} \sqrt{a e+c d x} (c d f-a e g)^5 \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right )}{128 c^{5/2} d^{5/2} g^{7/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{(f+g x)^{3/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^3}{64 c d g^3 \sqrt{d+e x}}+\frac{(f+g x)^{5/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2}{16 g^3 \sqrt{d+e x}}-\frac{(f+g x)^{5/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)}{8 g^2 (d+e x)^{3/2}}+\frac{(f+g x)^{5/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2}} \]

[Out]

(-3*(c*d*f - a*e*g)^4*Sqrt[f + g*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(128*c^2*d^2*g^3*Sqrt[d + e*x
]) - ((c*d*f - a*e*g)^3*(f + g*x)^(3/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(64*c*d*g^3*Sqrt[d + e*x]
) + ((c*d*f - a*e*g)^2*(f + g*x)^(5/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(16*g^3*Sqrt[d + e*x]) - (
(c*d*f - a*e*g)*(f + g*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(8*g^2*(d + e*x)^(3/2)) + ((f +
g*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(5*g*(d + e*x)^(5/2)) - (3*(c*d*f - a*e*g)^5*Sqrt[a
*e + c*d*x]*Sqrt[d + e*x]*ArcTanh[(Sqrt[g]*Sqrt[a*e + c*d*x])/(Sqrt[c]*Sqrt[d]*Sqrt[f + g*x])])/(128*c^(5/2)*d
^(5/2)*g^(7/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.888996, antiderivative size = 448, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 48, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {864, 870, 891, 63, 217, 206} \[ -\frac{3 \sqrt{f+g x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^4}{128 c^2 d^2 g^3 \sqrt{d+e x}}-\frac{3 \sqrt{d+e x} \sqrt{a e+c d x} (c d f-a e g)^5 \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right )}{128 c^{5/2} d^{5/2} g^{7/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{(f+g x)^{3/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^3}{64 c d g^3 \sqrt{d+e x}}+\frac{(f+g x)^{5/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2}{16 g^3 \sqrt{d+e x}}-\frac{(f+g x)^{5/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)}{8 g^2 (d+e x)^{3/2}}+\frac{(f+g x)^{5/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[((f + g*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(d + e*x)^(5/2),x]

[Out]

(-3*(c*d*f - a*e*g)^4*Sqrt[f + g*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(128*c^2*d^2*g^3*Sqrt[d + e*x
]) - ((c*d*f - a*e*g)^3*(f + g*x)^(3/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(64*c*d*g^3*Sqrt[d + e*x]
) + ((c*d*f - a*e*g)^2*(f + g*x)^(5/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(16*g^3*Sqrt[d + e*x]) - (
(c*d*f - a*e*g)*(f + g*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(8*g^2*(d + e*x)^(3/2)) + ((f +
g*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(5*g*(d + e*x)^(5/2)) - (3*(c*d*f - a*e*g)^5*Sqrt[a
*e + c*d*x]*Sqrt[d + e*x]*ArcTanh[(Sqrt[g]*Sqrt[a*e + c*d*x])/(Sqrt[c]*Sqrt[d]*Sqrt[f + g*x])])/(128*c^(5/2)*d
^(5/2)*g^(7/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])

Rule 864

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

Rule 870

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

Rule 891

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :>
Dist[(a + b*x + c*x^2)^FracPart[p]/((d + e*x)^FracPart[p]*(a/d + (c*x)/e)^FracPart[p]), Int[(d + e*x)^(m + p)*
(f + g*x)^n*(a/d + (c*x)/e)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, 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] &&  !IGtQ[m, 0] &&  !IGtQ[n, 0]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 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{(f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx &=\frac{(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2}}-\frac{(c d f-a e g) \int \frac{(f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx}{2 g}\\ &=-\frac{(c d f-a e g) (f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 g^2 (d+e x)^{3/2}}+\frac{(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2}}+\frac{\left (3 (c d f-a e g)^2\right ) \int \frac{(f+g x)^{3/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{d+e x}} \, dx}{16 g^2}\\ &=\frac{(c d f-a e g)^2 (f+g x)^{5/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 g^3 \sqrt{d+e x}}-\frac{(c d f-a e g) (f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 g^2 (d+e x)^{3/2}}+\frac{(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2}}-\frac{(c d f-a e g)^3 \int \frac{\sqrt{d+e x} (f+g x)^{3/2}}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{32 g^3}\\ &=-\frac{(c d f-a e g)^3 (f+g x)^{3/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c d g^3 \sqrt{d+e x}}+\frac{(c d f-a e g)^2 (f+g x)^{5/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 g^3 \sqrt{d+e x}}-\frac{(c d f-a e g) (f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 g^2 (d+e x)^{3/2}}+\frac{(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2}}-\frac{\left (3 (c d f-a e g)^4\right ) \int \frac{\sqrt{d+e x} \sqrt{f+g x}}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{128 c d g^3}\\ &=-\frac{3 (c d f-a e g)^4 \sqrt{f+g x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 c^2 d^2 g^3 \sqrt{d+e x}}-\frac{(c d f-a e g)^3 (f+g x)^{3/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c d g^3 \sqrt{d+e x}}+\frac{(c d f-a e g)^2 (f+g x)^{5/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 g^3 \sqrt{d+e x}}-\frac{(c d f-a e g) (f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 g^2 (d+e x)^{3/2}}+\frac{(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2}}-\frac{\left (3 (c d f-a e g)^5\right ) \int \frac{\sqrt{d+e x}}{\sqrt{f+g x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{256 c^2 d^2 g^3}\\ &=-\frac{3 (c d f-a e g)^4 \sqrt{f+g x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 c^2 d^2 g^3 \sqrt{d+e x}}-\frac{(c d f-a e g)^3 (f+g x)^{3/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c d g^3 \sqrt{d+e x}}+\frac{(c d f-a e g)^2 (f+g x)^{5/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 g^3 \sqrt{d+e x}}-\frac{(c d f-a e g) (f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 g^2 (d+e x)^{3/2}}+\frac{(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2}}-\frac{\left (3 (c d f-a e g)^5 \sqrt{a e+c d x} \sqrt{d+e x}\right ) \int \frac{1}{\sqrt{a e+c d x} \sqrt{f+g x}} \, dx}{256 c^2 d^2 g^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ &=-\frac{3 (c d f-a e g)^4 \sqrt{f+g x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 c^2 d^2 g^3 \sqrt{d+e x}}-\frac{(c d f-a e g)^3 (f+g x)^{3/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c d g^3 \sqrt{d+e x}}+\frac{(c d f-a e g)^2 (f+g x)^{5/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 g^3 \sqrt{d+e x}}-\frac{(c d f-a e g) (f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 g^2 (d+e x)^{3/2}}+\frac{(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2}}-\frac{\left (3 (c d f-a e g)^5 \sqrt{a e+c d x} \sqrt{d+e x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{f-\frac{a e g}{c d}+\frac{g x^2}{c d}}} \, dx,x,\sqrt{a e+c d x}\right )}{128 c^3 d^3 g^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ &=-\frac{3 (c d f-a e g)^4 \sqrt{f+g x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 c^2 d^2 g^3 \sqrt{d+e x}}-\frac{(c d f-a e g)^3 (f+g x)^{3/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c d g^3 \sqrt{d+e x}}+\frac{(c d f-a e g)^2 (f+g x)^{5/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 g^3 \sqrt{d+e x}}-\frac{(c d f-a e g) (f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 g^2 (d+e x)^{3/2}}+\frac{(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2}}-\frac{\left (3 (c d f-a e g)^5 \sqrt{a e+c d x} \sqrt{d+e x}\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{g x^2}{c d}} \, dx,x,\frac{\sqrt{a e+c d x}}{\sqrt{f+g x}}\right )}{128 c^3 d^3 g^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ &=-\frac{3 (c d f-a e g)^4 \sqrt{f+g x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 c^2 d^2 g^3 \sqrt{d+e x}}-\frac{(c d f-a e g)^3 (f+g x)^{3/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c d g^3 \sqrt{d+e x}}+\frac{(c d f-a e g)^2 (f+g x)^{5/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 g^3 \sqrt{d+e x}}-\frac{(c d f-a e g) (f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 g^2 (d+e x)^{3/2}}+\frac{(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2}}-\frac{3 (c d f-a e g)^5 \sqrt{a e+c d x} \sqrt{d+e x} \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right )}{128 c^{5/2} d^{5/2} g^{7/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ \end{align*}

Mathematica [B]  time = 6.129, size = 974, normalized size = 2.17 \[ \frac{2 (c d f-a e g) (a e+c d x) ((a e+c d x) (d+e x))^{5/2} \sqrt{f+g x} \left (\frac{c d g (a e+c d x)}{(c d f-a e g) \left (\frac{c^2 d^2 f}{c d f-a e g}-\frac{a c d e g}{c d f-a e g}\right )}+1\right )^{5/2} \left (\frac{21 (c d f-a e g)^4 \left (\frac{16 c^3 d^3 g^3 (a e+c d x)^3}{15 (c d f-a e g)^3 \left (\frac{c^2 d^2 f}{c d f-a e g}-\frac{a c d e g}{c d f-a e g}\right )^3}-\frac{4 c^2 d^2 g^2 (a e+c d x)^2}{3 (c d f-a e g)^2 \left (\frac{c^2 d^2 f}{c d f-a e g}-\frac{a c d e g}{c d f-a e g}\right )^2}+\frac{2 c d g (a e+c d x)}{(c d f-a e g) \left (\frac{c^2 d^2 f}{c d f-a e g}-\frac{a c d e g}{c d f-a e g}\right )}-\frac{2 \sqrt{c} \sqrt{d} \sqrt{g} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{g} \sqrt{a e+c d x}}{\sqrt{c d f-a e g} \sqrt{\frac{c^2 d^2 f}{c d f-a e g}-\frac{a c d e g}{c d f-a e g}}}\right ) \sqrt{a e+c d x}}{\sqrt{c d f-a e g} \sqrt{\frac{c^2 d^2 f}{c d f-a e g}-\frac{a c d e g}{c d f-a e g}} \sqrt{\frac{c d g (a e+c d x)}{(c d f-a e g) \left (\frac{c^2 d^2 f}{c d f-a e g}-\frac{a c d e g}{c d f-a e g}\right )}+1}}\right ) \left (\frac{c^2 d^2 f}{c d f-a e g}-\frac{a c d e g}{c d f-a e g}\right )^4}{512 c^4 d^4 g^4 (a e+c d x)^4 \left (\frac{c d g (a e+c d x)}{(c d f-a e g) \left (\frac{c^2 d^2 f}{c d f-a e g}-\frac{a c d e g}{c d f-a e g}\right )}+1\right )^2}+\frac{7}{10} \left (\frac{1}{\frac{c d g (a e+c d x)}{(c d f-a e g) \left (\frac{c^2 d^2 f}{c d f-a e g}-\frac{a c d e g}{c d f-a e g}\right )}+1}+\frac{3}{8 \left (\frac{c d g (a e+c d x)}{(c d f-a e g) \left (\frac{c^2 d^2 f}{c d f-a e g}-\frac{a c d e g}{c d f-a e g}\right )}+1\right )^2}\right )\right )}{7 c^2 d^2 \left (\frac{c d}{\frac{c^2 d^2 f}{c d f-a e g}-\frac{a c d e g}{c d f-a e g}}\right )^{3/2} (d+e x)^{5/2} \sqrt{\frac{c d (f+g x)}{c d f-a e g}}} \]

Antiderivative was successfully verified.

[In]

Integrate[((f + g*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(d + e*x)^(5/2),x]

[Out]

(2*(c*d*f - a*e*g)*(a*e + c*d*x)*((a*e + c*d*x)*(d + e*x))^(5/2)*Sqrt[f + g*x]*(1 + (c*d*g*(a*e + c*d*x))/((c*
d*f - a*e*g)*((c^2*d^2*f)/(c*d*f - a*e*g) - (a*c*d*e*g)/(c*d*f - a*e*g))))^(5/2)*((7*(3/(8*(1 + (c*d*g*(a*e +
c*d*x))/((c*d*f - a*e*g)*((c^2*d^2*f)/(c*d*f - a*e*g) - (a*c*d*e*g)/(c*d*f - a*e*g))))^2) + (1 + (c*d*g*(a*e +
c*d*x))/((c*d*f - a*e*g)*((c^2*d^2*f)/(c*d*f - a*e*g) - (a*c*d*e*g)/(c*d*f - a*e*g))))^(-1)))/10 + (21*(c*d*f
- a*e*g)^4*((c^2*d^2*f)/(c*d*f - a*e*g) - (a*c*d*e*g)/(c*d*f - a*e*g))^4*((2*c*d*g*(a*e + c*d*x))/((c*d*f - a
*e*g)*((c^2*d^2*f)/(c*d*f - a*e*g) - (a*c*d*e*g)/(c*d*f - a*e*g))) - (4*c^2*d^2*g^2*(a*e + c*d*x)^2)/(3*(c*d*f
- a*e*g)^2*((c^2*d^2*f)/(c*d*f - a*e*g) - (a*c*d*e*g)/(c*d*f - a*e*g))^2) + (16*c^3*d^3*g^3*(a*e + c*d*x)^3)/
(15*(c*d*f - a*e*g)^3*((c^2*d^2*f)/(c*d*f - a*e*g) - (a*c*d*e*g)/(c*d*f - a*e*g))^3) - (2*Sqrt[c]*Sqrt[d]*Sqrt
[g]*Sqrt[a*e + c*d*x]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[g]*Sqrt[a*e + c*d*x])/(Sqrt[c*d*f - a*e*g]*Sqrt[(c^2*d^2*f
)/(c*d*f - a*e*g) - (a*c*d*e*g)/(c*d*f - a*e*g)])])/(Sqrt[c*d*f - a*e*g]*Sqrt[(c^2*d^2*f)/(c*d*f - a*e*g) - (a
*c*d*e*g)/(c*d*f - a*e*g)]*Sqrt[1 + (c*d*g*(a*e + c*d*x))/((c*d*f - a*e*g)*((c^2*d^2*f)/(c*d*f - a*e*g) - (a*c
*d*e*g)/(c*d*f - a*e*g)))])))/(512*c^4*d^4*g^4*(a*e + c*d*x)^4*(1 + (c*d*g*(a*e + c*d*x))/((c*d*f - a*e*g)*((c
^2*d^2*f)/(c*d*f - a*e*g) - (a*c*d*e*g)/(c*d*f - a*e*g))))^2)))/(7*c^2*d^2*((c*d)/((c^2*d^2*f)/(c*d*f - a*e*g)
- (a*c*d*e*g)/(c*d*f - a*e*g)))^(3/2)*(d + e*x)^(5/2)*Sqrt[(c*d*(f + g*x))/(c*d*f - a*e*g)])

________________________________________________________________________________________

Maple [B]  time = 0.41, size = 1191, normalized size = 2.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)^(3/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2),x)

[Out]

1/1280*(g*x+f)^(1/2)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)*(256*x^4*c^4*d^4*g^4*(c*d*g*x^2+a*e*g*x+c*d*f*x+a
*e*f)^(1/2)*(c*d*g)^(1/2)+672*x^3*a*c^3*d^3*e*g^4*(c*d*g*x^2+a*e*g*x+c*d*f*x+a*e*f)^(1/2)*(c*d*g)^(1/2)+352*x^
3*c^4*d^4*f*g^3*(c*d*g*x^2+a*e*g*x+c*d*f*x+a*e*f)^(1/2)*(c*d*g)^(1/2)+15*ln(1/2*(2*x*c*d*g+a*e*g+c*d*f+2*(c*d*
g*x^2+a*e*g*x+c*d*f*x+a*e*f)^(1/2)*(c*d*g)^(1/2))/(c*d*g)^(1/2))*a^5*e^5*g^5-75*ln(1/2*(2*x*c*d*g+a*e*g+c*d*f+
2*(c*d*g*x^2+a*e*g*x+c*d*f*x+a*e*f)^(1/2)*(c*d*g)^(1/2))/(c*d*g)^(1/2))*a^4*c*d*e^4*f*g^4+150*ln(1/2*(2*x*c*d*
g+a*e*g+c*d*f+2*(c*d*g*x^2+a*e*g*x+c*d*f*x+a*e*f)^(1/2)*(c*d*g)^(1/2))/(c*d*g)^(1/2))*a^3*c^2*d^2*e^3*f^2*g^3-
150*ln(1/2*(2*x*c*d*g+a*e*g+c*d*f+2*(c*d*g*x^2+a*e*g*x+c*d*f*x+a*e*f)^(1/2)*(c*d*g)^(1/2))/(c*d*g)^(1/2))*a^2*
c^3*d^3*e^2*f^3*g^2+75*ln(1/2*(2*x*c*d*g+a*e*g+c*d*f+2*(c*d*g*x^2+a*e*g*x+c*d*f*x+a*e*f)^(1/2)*(c*d*g)^(1/2))/
(c*d*g)^(1/2))*a*c^4*d^4*e*f^4*g-15*ln(1/2*(2*x*c*d*g+a*e*g+c*d*f+2*(c*d*g*x^2+a*e*g*x+c*d*f*x+a*e*f)^(1/2)*(c
*d*g)^(1/2))/(c*d*g)^(1/2))*c^5*d^5*f^5+496*x^2*a^2*c^2*d^2*e^2*g^4*(c*d*g*x^2+a*e*g*x+c*d*f*x+a*e*f)^(1/2)*(c
*d*g)^(1/2)+1024*x^2*a*c^3*d^3*e*f*g^3*(c*d*g*x^2+a*e*g*x+c*d*f*x+a*e*f)^(1/2)*(c*d*g)^(1/2)+16*x^2*c^4*d^4*f^
2*g^2*(c*d*g*x^2+a*e*g*x+c*d*f*x+a*e*f)^(1/2)*(c*d*g)^(1/2)+20*(c*d*g)^(1/2)*(c*d*g*x^2+a*e*g*x+c*d*f*x+a*e*f)
^(1/2)*x*a^3*c*d*e^3*g^4+932*(c*d*g)^(1/2)*(c*d*g*x^2+a*e*g*x+c*d*f*x+a*e*f)^(1/2)*x*a^2*c^2*d^2*e^2*f*g^3+92*
(c*d*g)^(1/2)*(c*d*g*x^2+a*e*g*x+c*d*f*x+a*e*f)^(1/2)*x*a*c^3*d^3*e*f^2*g^2-20*(c*d*g)^(1/2)*(c*d*g*x^2+a*e*g*
x+c*d*f*x+a*e*f)^(1/2)*x*c^4*d^4*f^3*g-30*(c*d*g)^(1/2)*(c*d*g*x^2+a*e*g*x+c*d*f*x+a*e*f)^(1/2)*a^4*e^4*g^4+14
0*(c*d*g)^(1/2)*(c*d*g*x^2+a*e*g*x+c*d*f*x+a*e*f)^(1/2)*a^3*c*d*e^3*f*g^3+256*a^2*c^2*d^2*e^2*f^2*g^2*(c*d*g*x
^2+a*e*g*x+c*d*f*x+a*e*f)^(1/2)*(c*d*g)^(1/2)-140*(c*d*g)^(1/2)*(c*d*g*x^2+a*e*g*x+c*d*f*x+a*e*f)^(1/2)*a*c^3*
d^3*e*f^3*g+30*(c*d*g)^(1/2)*(c*d*g*x^2+a*e*g*x+c*d*f*x+a*e*f)^(1/2)*c^4*d^4*f^4)/(e*x+d)^(1/2)/c^2/d^2/g^3/(c
*d*g*x^2+a*e*g*x+c*d*f*x+a*e*f)^(1/2)/(c*d*g)^(1/2)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac{5}{2}}{\left (g x + f\right )}^{\frac{3}{2}}}{{\left (e x + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^(3/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2),x, algorithm="maxima")

[Out]

integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)*(g*x + f)^(3/2)/(e*x + d)^(5/2), x)

________________________________________________________________________________________

Fricas [A]  time = 28.5776, size = 2781, normalized size = 6.21 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^(3/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2),x, algorithm="fricas")

[Out]

[1/2560*(4*(128*c^5*d^5*g^5*x^4 + 15*c^5*d^5*f^4*g - 70*a*c^4*d^4*e*f^3*g^2 + 128*a^2*c^3*d^3*e^2*f^2*g^3 + 70
*a^3*c^2*d^2*e^3*f*g^4 - 15*a^4*c*d*e^4*g^5 + 16*(11*c^5*d^5*f*g^4 + 21*a*c^4*d^4*e*g^5)*x^3 + 8*(c^5*d^5*f^2*
g^3 + 64*a*c^4*d^4*e*f*g^4 + 31*a^2*c^3*d^3*e^2*g^5)*x^2 - 2*(5*c^5*d^5*f^3*g^2 - 23*a*c^4*d^4*e*f^2*g^3 - 233
*a^2*c^3*d^3*e^2*f*g^4 - 5*a^3*c^2*d^2*e^3*g^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)*s
qrt(g*x + f) - 15*(c^5*d^6*f^5 - 5*a*c^4*d^5*e*f^4*g + 10*a^2*c^3*d^4*e^2*f^3*g^2 - 10*a^3*c^2*d^3*e^3*f^2*g^3
+ 5*a^4*c*d^2*e^4*f*g^4 - a^5*d*e^5*g^5 + (c^5*d^5*e*f^5 - 5*a*c^4*d^4*e^2*f^4*g + 10*a^2*c^3*d^3*e^3*f^3*g^2
- 10*a^3*c^2*d^2*e^4*f^2*g^3 + 5*a^4*c*d*e^5*f*g^4 - a^5*e^6*g^5)*x)*sqrt(c*d*g)*log(-(8*c^2*d^2*e*g^2*x^3 +
c^2*d^3*f^2 + 6*a*c*d^2*e*f*g + a^2*d*e^2*g^2 + 4*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*c*d*g*x + c*d
*f + a*e*g)*sqrt(c*d*g)*sqrt(e*x + d)*sqrt(g*x + f) + 8*(c^2*d^2*e*f*g + (c^2*d^3 + a*c*d*e^2)*g^2)*x^2 + (c^2
*d^2*e*f^2 + 2*(4*c^2*d^3 + 3*a*c*d*e^2)*f*g + (8*a*c*d^2*e + a^2*e^3)*g^2)*x)/(e*x + d)))/(c^3*d^3*e*g^4*x +
c^3*d^4*g^4), 1/1280*(2*(128*c^5*d^5*g^5*x^4 + 15*c^5*d^5*f^4*g - 70*a*c^4*d^4*e*f^3*g^2 + 128*a^2*c^3*d^3*e^2
*f^2*g^3 + 70*a^3*c^2*d^2*e^3*f*g^4 - 15*a^4*c*d*e^4*g^5 + 16*(11*c^5*d^5*f*g^4 + 21*a*c^4*d^4*e*g^5)*x^3 + 8*
(c^5*d^5*f^2*g^3 + 64*a*c^4*d^4*e*f*g^4 + 31*a^2*c^3*d^3*e^2*g^5)*x^2 - 2*(5*c^5*d^5*f^3*g^2 - 23*a*c^4*d^4*e*
f^2*g^3 - 233*a^2*c^3*d^3*e^2*f*g^4 - 5*a^3*c^2*d^2*e^3*g^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sq
rt(e*x + d)*sqrt(g*x + f) + 15*(c^5*d^6*f^5 - 5*a*c^4*d^5*e*f^4*g + 10*a^2*c^3*d^4*e^2*f^3*g^2 - 10*a^3*c^2*d^
3*e^3*f^2*g^3 + 5*a^4*c*d^2*e^4*f*g^4 - a^5*d*e^5*g^5 + (c^5*d^5*e*f^5 - 5*a*c^4*d^4*e^2*f^4*g + 10*a^2*c^3*d^
3*e^3*f^3*g^2 - 10*a^3*c^2*d^2*e^4*f^2*g^3 + 5*a^4*c*d*e^5*f*g^4 - a^5*e^6*g^5)*x)*sqrt(-c*d*g)*arctan(2*sqrt(
c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-c*d*g)*sqrt(e*x + d)*sqrt(g*x + f)/(2*c*d*e*g*x^2 + c*d^2*f + a*d
*e*g + (c*d*e*f + (2*c*d^2 + a*e^2)*g)*x)))/(c^3*d^3*e*g^4*x + c^3*d^4*g^4)]

________________________________________________________________________________________

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((g*x+f)**(3/2)*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/(e*x+d)**(5/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac{5}{2}}{\left (g x + f\right )}^{\frac{3}{2}}}{{\left (e x + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^(3/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2),x, algorithm="giac")

[Out]

integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)*(g*x + f)^(3/2)/(e*x + d)^(5/2), x)