\(\int \frac {\sqrt {f+g x} (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^{5/2}} \, dx\) [752]

Optimal result

Integrand size = 48, antiderivative size = 376 \[ \int \frac {\sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=-\frac {5 (c d f-a e g)^3 \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c d g^3 \sqrt {d+e x}}+\frac {5 (c d f-a e g)^2 (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{32 g^3 \sqrt {d+e x}}-\frac {5 (c d f-a e g) (f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{24 g^2 (d+e x)^{3/2}}+\frac {(f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{4 g (d+e x)^{5/2}}-\frac {5 (c d f-a e g)^4 \sqrt {a e+c d x} \sqrt {d+e x} \text {arctanh}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {f+g x}}\right )}{64 c^{3/2} d^{3/2} g^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \]

[Out]

-5/24*(-a*e*g+c*d*f)*(g*x+f)^(3/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/g^2/(e*x+d)^(3/2)+1/4*(g*x+f)^(3/2)
*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/g/(e*x+d)^(5/2)-5/64*(-a*e*g+c*d*f)^4*arctanh(g^(1/2)*(c*d*x+a*e)^(1/
2)/c^(1/2)/d^(1/2)/(g*x+f)^(1/2))*(c*d*x+a*e)^(1/2)*(e*x+d)^(1/2)/c^(3/2)/d^(3/2)/g^(7/2)/(a*d*e+(a*e^2+c*d^2)
*x+c*d*e*x^2)^(1/2)+5/32*(-a*e*g+c*d*f)^2*(g*x+f)^(3/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/g^3/(e*x+d)^(1
/2)-5/64*(-a*e*g+c*d*f)^3*(g*x+f)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c/d/g^3/(e*x+d)^(1/2)

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {878, 884, 905, 65, 223, 212} \[ \int \frac {\sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=-\frac {5 \sqrt {d+e x} \sqrt {a e+c d x} (c d f-a e g)^4 \text {arctanh}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {f+g x}}\right )}{64 c^{3/2} d^{3/2} g^{7/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {5 \sqrt {f+g x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^3}{64 c d g^3 \sqrt {d+e x}}+\frac {5 (f+g x)^{3/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2}{32 g^3 \sqrt {d+e x}}-\frac {5 (f+g x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)}{24 g^2 (d+e x)^{3/2}}+\frac {(f+g x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{4 g (d+e x)^{5/2}} \]

[In]

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

[Out]

(-5*(c*d*f - a*e*g)^3*Sqrt[f + g*x]*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]) +
(5*(c*d*f - a*e*g)^2*(f + g*x)^(3/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(32*g^3*Sqrt[d + e*x]) - (5*
(c*d*f - a*e*g)*(f + g*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(24*g^2*(d + e*x)^(3/2)) + ((f
+ g*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(4*g*(d + e*x)^(5/2)) - (5*(c*d*f - a*e*g)^4*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])])/(64*c^(3/2)*d
^(3/2)*g^(7/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])

Rule 65

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 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

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 878

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] &&  !Intege
rQ[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 884

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] &&  !Int
egerQ[p] && EqQ[m + p, 0] && GtQ[n, 0] && NeQ[m - n - 1, 0] && (IntegerQ[2*p] || IntegerQ[n])

Rule 905

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/e)*x)^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]

Rubi steps \begin{align*} \text {integral}& = \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}}{4 g (d+e x)^{5/2}}-\frac {(5 (c d f-a e g)) \int \frac {\sqrt {f+g x} \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}{8 g} \\ & = -\frac {5 (c d f-a e g) (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}}{24 g^2 (d+e x)^{3/2}}+\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}}{4 g (d+e x)^{5/2}}+\frac {\left (5 (c d f-a e g)^2\right ) \int \frac {\sqrt {f+g x} \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 {5 (c d f-a e g)^2 (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{32 g^3 \sqrt {d+e x}}-\frac {5 (c d f-a e g) (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}}{24 g^2 (d+e x)^{3/2}}+\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}}{4 g (d+e x)^{5/2}}-\frac {\left (5 (c d f-a e g)^3\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}{64 g^3} \\ & = -\frac {5 (c d f-a e g)^3 \sqrt {f+g x} \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 {5 (c d f-a e g)^2 (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{32 g^3 \sqrt {d+e x}}-\frac {5 (c d f-a e g) (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}}{24 g^2 (d+e x)^{3/2}}+\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}}{4 g (d+e x)^{5/2}}-\frac {\left (5 (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 {5 (c d f-a e g)^3 \sqrt {f+g x} \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 {5 (c d f-a e g)^2 (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{32 g^3 \sqrt {d+e x}}-\frac {5 (c d f-a e g) (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}}{24 g^2 (d+e x)^{3/2}}+\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}}{4 g (d+e x)^{5/2}}-\frac {\left (5 (c d f-a e g)^4 \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}{128 c d g^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \\ & = -\frac {5 (c d f-a e g)^3 \sqrt {f+g x} \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 {5 (c d f-a e g)^2 (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{32 g^3 \sqrt {d+e x}}-\frac {5 (c d f-a e g) (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}}{24 g^2 (d+e x)^{3/2}}+\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}}{4 g (d+e x)^{5/2}}-\frac {\left (5 (c d f-a e g)^4 \sqrt {a e+c d x} \sqrt {d+e x}\right ) \text {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 )}{64 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 {5 (c d f-a e g)^3 \sqrt {f+g x} \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 {5 (c d f-a e g)^2 (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{32 g^3 \sqrt {d+e x}}-\frac {5 (c d f-a e g) (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}}{24 g^2 (d+e x)^{3/2}}+\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}}{4 g (d+e x)^{5/2}}-\frac {\left (5 (c d f-a e g)^4 \sqrt {a e+c d x} \sqrt {d+e x}\right ) \text {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 )}{64 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 {5 (c d f-a e g)^3 \sqrt {f+g x} \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 {5 (c d f-a e g)^2 (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{32 g^3 \sqrt {d+e x}}-\frac {5 (c d f-a e g) (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}}{24 g^2 (d+e x)^{3/2}}+\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}}{4 g (d+e x)^{5/2}}-\frac {5 (c d f-a e g)^4 \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 )}{64 c^{3/2} d^{3/2} g^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.69 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.65 \[ \int \frac {\sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\frac {((a e+c d x) (d+e x))^{5/2} \left (\frac {\sqrt {c} \sqrt {d} \sqrt {g} \sqrt {f+g x} \left (15 a^3 e^3 g^3+a^2 c d e^2 g^2 (73 f+118 g x)+a c^2 d^2 e g \left (-55 f^2+36 f g x+136 g^2 x^2\right )+c^3 d^3 \left (15 f^3-10 f^2 g x+8 f g^2 x^2+48 g^3 x^3\right )\right )}{(a e+c d x)^2}-\frac {15 (c d f-a e g)^4 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {f+g x}}{\sqrt {g} \sqrt {a e+c d x}}\right )}{(a e+c d x)^{5/2}}\right )}{192 c^{3/2} d^{3/2} g^{7/2} (d+e x)^{5/2}} \]

[In]

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

[Out]

(((a*e + c*d*x)*(d + e*x))^(5/2)*((Sqrt[c]*Sqrt[d]*Sqrt[g]*Sqrt[f + g*x]*(15*a^3*e^3*g^3 + a^2*c*d*e^2*g^2*(73
*f + 118*g*x) + a*c^2*d^2*e*g*(-55*f^2 + 36*f*g*x + 136*g^2*x^2) + c^3*d^3*(15*f^3 - 10*f^2*g*x + 8*f*g^2*x^2
+ 48*g^3*x^3)))/(a*e + c*d*x)^2 - (15*(c*d*f - a*e*g)^4*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[f + g*x])/(Sqrt[g]*Sqrt[
a*e + c*d*x])])/(a*e + c*d*x)^(5/2)))/(192*c^(3/2)*d^(3/2)*g^(7/2)*(d + e*x)^(5/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(731\) vs. \(2(320)=640\).

Time = 0.59 (sec) , antiderivative size = 732, normalized size of antiderivative = 1.95

method	result	size
default	\(-\frac {\sqrt {g x +f}\, \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (-96 c^{3} d^{3} g^{3} x^{3} \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}+15 \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) a^{4} e^{4} g^{4}-60 \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) a^{3} c d \,e^{3} f \,g^{3}+90 \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) a^{2} c^{2} d^{2} e^{2} f^{2} g^{2}-60 \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) a \,c^{3} d^{3} e \,f^{3} g +15 \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) c^{4} d^{4} f^{4}-272 a \,c^{2} d^{2} e \,g^{3} x^{2} \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}-16 c^{3} d^{3} f \,g^{2} x^{2} \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}-236 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}\, a^{2} c d \,e^{2} g^{3} x -72 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}\, a \,c^{2} d^{2} e f \,g^{2} x +20 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}\, c^{3} d^{3} f^{2} g x -30 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}\, a^{3} e^{3} g^{3}-146 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}\, a^{2} c d \,e^{2} f \,g^{2}+110 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}\, a \,c^{2} d^{2} e \,f^{2} g -30 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}\, c^{3} d^{3} f^{3}\right )}{384 \sqrt {e x +d}\, c d \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, g^{3} \sqrt {c d g}}\)	\(732\)

[In]

int((g*x+f)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/384*(g*x+f)^(1/2)*((c*d*x+a*e)*(e*x+d))^(1/2)*(-96*c^3*d^3*g^3*x^3*((g*x+f)*(c*d*x+a*e))^(1/2)*(c*d*g)^(1/2
)+15*ln(1/2*(2*c*d*g*x+a*e*g+c*d*f+2*((g*x+f)*(c*d*x+a*e))^(1/2)*(c*d*g)^(1/2))/(c*d*g)^(1/2))*a^4*e^4*g^4-60*
ln(1/2*(2*c*d*g*x+a*e*g+c*d*f+2*((g*x+f)*(c*d*x+a*e))^(1/2)*(c*d*g)^(1/2))/(c*d*g)^(1/2))*a^3*c*d*e^3*f*g^3+90
*ln(1/2*(2*c*d*g*x+a*e*g+c*d*f+2*((g*x+f)*(c*d*x+a*e))^(1/2)*(c*d*g)^(1/2))/(c*d*g)^(1/2))*a^2*c^2*d^2*e^2*f^2
*g^2-60*ln(1/2*(2*c*d*g*x+a*e*g+c*d*f+2*((g*x+f)*(c*d*x+a*e))^(1/2)*(c*d*g)^(1/2))/(c*d*g)^(1/2))*a*c^3*d^3*e*
f^3*g+15*ln(1/2*(2*c*d*g*x+a*e*g+c*d*f+2*((g*x+f)*(c*d*x+a*e))^(1/2)*(c*d*g)^(1/2))/(c*d*g)^(1/2))*c^4*d^4*f^4
-272*a*c^2*d^2*e*g^3*x^2*((g*x+f)*(c*d*x+a*e))^(1/2)*(c*d*g)^(1/2)-16*c^3*d^3*f*g^2*x^2*((g*x+f)*(c*d*x+a*e))^
(1/2)*(c*d*g)^(1/2)-236*((g*x+f)*(c*d*x+a*e))^(1/2)*(c*d*g)^(1/2)*a^2*c*d*e^2*g^3*x-72*((g*x+f)*(c*d*x+a*e))^(
1/2)*(c*d*g)^(1/2)*a*c^2*d^2*e*f*g^2*x+20*((g*x+f)*(c*d*x+a*e))^(1/2)*(c*d*g)^(1/2)*c^3*d^3*f^2*g*x-30*((g*x+f
)*(c*d*x+a*e))^(1/2)*(c*d*g)^(1/2)*a^3*e^3*g^3-146*((g*x+f)*(c*d*x+a*e))^(1/2)*(c*d*g)^(1/2)*a^2*c*d*e^2*f*g^2
+110*((g*x+f)*(c*d*x+a*e))^(1/2)*(c*d*g)^(1/2)*a*c^2*d^2*e*f^2*g-30*((g*x+f)*(c*d*x+a*e))^(1/2)*(c*d*g)^(1/2)*
c^3*d^3*f^3)/(e*x+d)^(1/2)/c/d/((g*x+f)*(c*d*x+a*e))^(1/2)/g^3/(c*d*g)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 1.73 (sec) , antiderivative size = 1065, normalized size of antiderivative = 2.83 \[ \int \frac {\sqrt {f+g x} \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=\left [\frac {4 \, {\left (48 \, c^{4} d^{4} g^{4} x^{3} + 15 \, c^{4} d^{4} f^{3} g - 55 \, a c^{3} d^{3} e f^{2} g^{2} + 73 \, a^{2} c^{2} d^{2} e^{2} f g^{3} + 15 \, a^{3} c d e^{3} g^{4} + 8 \, {\left (c^{4} d^{4} f g^{3} + 17 \, a c^{3} d^{3} e g^{4}\right )} x^{2} - 2 \, {\left (5 \, c^{4} d^{4} f^{2} g^{2} - 18 \, a c^{3} d^{3} e f g^{3} - 59 \, a^{2} c^{2} d^{2} e^{2} g^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d} \sqrt {g x + f} + 15 \, {\left (c^{4} d^{5} f^{4} - 4 \, a c^{3} d^{4} e f^{3} g + 6 \, a^{2} c^{2} d^{3} e^{2} f^{2} g^{2} - 4 \, a^{3} c d^{2} e^{3} f g^{3} + a^{4} d e^{4} g^{4} + {\left (c^{4} d^{4} e f^{4} - 4 \, a c^{3} d^{3} e^{2} f^{3} g + 6 \, a^{2} c^{2} d^{2} e^{3} f^{2} g^{2} - 4 \, a^{3} c d e^{4} f g^{3} + a^{4} e^{5} g^{4}\right )} x\right )} \sqrt {c d g} \log \left (-\frac {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 + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d g x + c d f + a e g\right )} \sqrt {c d g} \sqrt {e x + d} \sqrt {g x + f} + 8 \, {\left (c^{2} d^{2} e f g + {\left (c^{2} d^{3} + a c d e^{2}\right )} g^{2}\right )} x^{2} + {\left (c^{2} d^{2} e f^{2} + 2 \, {\left (4 \, c^{2} d^{3} + 3 \, a c d e^{2}\right )} f g + {\left (8 \, a c d^{2} e + a^{2} e^{3}\right )} g^{2}\right )} x}{e x + d}\right )}{768 \, {\left (c^{2} d^{2} e g^{4} x + c^{2} d^{3} g^{4}\right )}}, \frac {2 \, {\left (48 \, c^{4} d^{4} g^{4} x^{3} + 15 \, c^{4} d^{4} f^{3} g - 55 \, a c^{3} d^{3} e f^{2} g^{2} + 73 \, a^{2} c^{2} d^{2} e^{2} f g^{3} + 15 \, a^{3} c d e^{3} g^{4} + 8 \, {\left (c^{4} d^{4} f g^{3} + 17 \, a c^{3} d^{3} e g^{4}\right )} x^{2} - 2 \, {\left (5 \, c^{4} d^{4} f^{2} g^{2} - 18 \, a c^{3} d^{3} e f g^{3} - 59 \, a^{2} c^{2} d^{2} e^{2} g^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d} \sqrt {g x + f} + 15 \, {\left (c^{4} d^{5} f^{4} - 4 \, a c^{3} d^{4} e f^{3} g + 6 \, a^{2} c^{2} d^{3} e^{2} f^{2} g^{2} - 4 \, a^{3} c d^{2} e^{3} f g^{3} + a^{4} d e^{4} g^{4} + {\left (c^{4} d^{4} e f^{4} - 4 \, a c^{3} d^{3} e^{2} f^{3} g + 6 \, a^{2} c^{2} d^{2} e^{3} f^{2} g^{2} - 4 \, a^{3} c d e^{4} f g^{3} + a^{4} e^{5} g^{4}\right )} x\right )} \sqrt {-c d g} \arctan \left (\frac {2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} 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 + {\left (c d e f + {\left (2 \, c d^{2} + a e^{2}\right )} g\right )} x}\right )}{384 \, {\left (c^{2} d^{2} e g^{4} x + c^{2} d^{3} g^{4}\right )}}\right ] \]

[In]

integrate((g*x+f)^(1/2)*(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]

[1/768*(4*(48*c^4*d^4*g^4*x^3 + 15*c^4*d^4*f^3*g - 55*a*c^3*d^3*e*f^2*g^2 + 73*a^2*c^2*d^2*e^2*f*g^3 + 15*a^3*
c*d*e^3*g^4 + 8*(c^4*d^4*f*g^3 + 17*a*c^3*d^3*e*g^4)*x^2 - 2*(5*c^4*d^4*f^2*g^2 - 18*a*c^3*d^3*e*f*g^3 - 59*a^
2*c^2*d^2*e^2*g^4)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)*sqrt(g*x + f) + 15*(c^4*d^5*f^
4 - 4*a*c^3*d^4*e*f^3*g + 6*a^2*c^2*d^3*e^2*f^2*g^2 - 4*a^3*c*d^2*e^3*f*g^3 + a^4*d*e^4*g^4 + (c^4*d^4*e*f^4 -
4*a*c^3*d^3*e^2*f^3*g + 6*a^2*c^2*d^2*e^3*f^2*g^2 - 4*a^3*c*d*e^4*f*g^3 + a^4*e^5*g^4)*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^2*d^2*e*g^4*x + c^2*d^3*g^4), 1/384*(2*(48*c^4*d^4*g^4*x^3 + 15*c^4*d^4*f^3*g - 55*a*c^3*d^3*e*f^2*g^2 +
73*a^2*c^2*d^2*e^2*f*g^3 + 15*a^3*c*d*e^3*g^4 + 8*(c^4*d^4*f*g^3 + 17*a*c^3*d^3*e*g^4)*x^2 - 2*(5*c^4*d^4*f^2*
g^2 - 18*a*c^3*d^3*e*f*g^3 - 59*a^2*c^2*d^2*e^2*g^4)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x +
d)*sqrt(g*x + f) + 15*(c^4*d^5*f^4 - 4*a*c^3*d^4*e*f^3*g + 6*a^2*c^2*d^3*e^2*f^2*g^2 - 4*a^3*c*d^2*e^3*f*g^3
+ a^4*d*e^4*g^4 + (c^4*d^4*e*f^4 - 4*a*c^3*d^3*e^2*f^3*g + 6*a^2*c^2*d^2*e^3*f^2*g^2 - 4*a^3*c*d*e^4*f*g^3 + a
^4*e^5*g^4)*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)*sq
rt(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^2*d^2*e*g^4*x + c^2*d
^3*g^4)]

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate((g*x+f)**(1/2)*(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

Maxima [F]

\[ \int \frac {\sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}} \sqrt {g x + f}}{{\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate((g*x+f)^(1/2)*(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)*sqrt(g*x + f)/(e*x + d)^(5/2), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6756 vs. \(2 (320) = 640\).

Time = 2.20 (sec) , antiderivative size = 6756, normalized size of antiderivative = 17.97 \[ \int \frac {\sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\text {Too large to display} \]

[In]

integrate((g*x+f)^(1/2)*(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]

1/192*(48*a^2*((4*((c*d*e^2*f*g - a*e^3*g^2)*log(abs(-sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*sqrt(c*d*g) + sqrt(-
c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)))/sqrt(c*d*g) + sqrt(-c*d*e^2*f*g + a*e^3*g^2
+ (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)*sqrt(e^2*f + (e*x + d)*e*g - d*e*g))*e*f*abs(g)/g^2 - 4*((c*d*e^2*f*
g - a*e^3*g^2)*log(abs(-sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*sqrt(c*d*g) + sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2
*f + (e*x + d)*e*g - d*e*g)*c*d*g)))/sqrt(c*d*g) + sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*
e*g)*c*d*g)*sqrt(e^2*f + (e*x + d)*e*g - d*e*g))*d*abs(g)/g + (sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x +
d)*e*g - d*e*g)*c*d*g)*(2*e^2*f + 2*(e*x + d)*e*g - 2*d*e*g - (5*c^2*d^2*e^2*f - 4*c^2*d^3*e*g - a*c*d*e^3*g)
/(c^2*d^2))*sqrt(e^2*f + (e*x + d)*e*g - d*e*g) - (3*c^2*d^2*e^4*f^2*g - 4*c^2*d^3*e^3*f*g^2 - 2*a*c*d*e^5*f*g
^2 + 4*a*c*d^2*e^4*g^3 - a^2*e^6*g^3)*log(abs(-sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*sqrt(c*d*g) + sqrt(-c*d*e^2
*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)))/(sqrt(c*d*g)*c*d))*abs(g)/(e*g^2))/g - (c^2*d^2*e^
3*f^2*g*abs(g)*log(abs(-sqrt(e^2*f - d*e*g)*sqrt(c*d*g) + sqrt(-c*d^2*e*g^2 + a*e^3*g^2))) - 2*a*c*d*e^4*f*g^2
*abs(g)*log(abs(-sqrt(e^2*f - d*e*g)*sqrt(c*d*g) + sqrt(-c*d^2*e*g^2 + a*e^3*g^2))) + a^2*e^5*g^3*abs(g)*log(a
bs(-sqrt(e^2*f - d*e*g)*sqrt(c*d*g) + sqrt(-c*d^2*e*g^2 + a*e^3*g^2))) + sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e
^2*f - d*e*g)*sqrt(c*d*g)*c*d*e*f*abs(g) - 2*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*c*
d^2*g*abs(g) + sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a*e^2*g*abs(g))/(sqrt(c*d*g)*c*d
*g^3))*abs(e)^2/e^2 + c^2*d^2*((192*((c*d*e^2*f*g - a*e^3*g^2)*log(abs(-sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*sq
rt(c*d*g) + sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)))/sqrt(c*d*g) + sqrt(-c*d*e
^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)*sqrt(e^2*f + (e*x + d)*e*g - d*e*g))*d^2*e*f*abs(g
)/g^2 - 192*((c*d*e^2*f*g - a*e^3*g^2)*log(abs(-sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*sqrt(c*d*g) + sqrt(-c*d*e^
2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)))/sqrt(c*d*g) + sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^
2*f + (e*x + d)*e*g - d*e*g)*c*d*g)*sqrt(e^2*f + (e*x + d)*e*g - d*e*g))*d^3*abs(g)/g + 8*(sqrt(-c*d*e^2*f*g +
a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)*sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*(2*(e^2*f + (e*x + d)*
e*g - d*e*g)*(4*(e^2*f + (e*x + d)*e*g - d*e*g)/(e^2*g^2) - (13*c^4*d^4*e^3*f*g^5 - 12*c^4*d^5*e^2*g^6 - a*c^3
*d^3*e^4*g^6)/(c^4*d^4*e^3*g^7)) + 3*(11*c^4*d^4*e^5*f^2*g^5 - 20*c^4*d^5*e^4*f*g^6 - 2*a*c^3*d^3*e^6*f*g^6 +
8*c^4*d^6*e^3*g^7 + 4*a*c^3*d^4*e^5*g^7 - a^2*c^2*d^2*e^7*g^7)/(c^4*d^4*e^3*g^7)) + 3*(5*c^3*d^3*e^4*f^3 - 12*
c^3*d^4*e^3*f^2*g - 3*a*c^2*d^2*e^5*f^2*g + 8*c^3*d^5*e^2*f*g^2 + 8*a*c^2*d^3*e^4*f*g^2 - a^2*c*d*e^6*f*g^2 -
8*a*c^2*d^4*e^3*g^3 + 4*a^2*c*d^2*e^5*g^3 - a^3*e^7*g^3)*log(abs(-sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*sqrt(c*d
*g) + sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)))/(sqrt(c*d*g)*c^2*d^2*g))*e*f*ab
s(g)/g^2 - 24*(sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)*sqrt(e^2*f + (e*x + d)*e
*g - d*e*g)*(2*(e^2*f + (e*x + d)*e*g - d*e*g)*(4*(e^2*f + (e*x + d)*e*g - d*e*g)/(e^2*g^2) - (13*c^4*d^4*e^3*
f*g^5 - 12*c^4*d^5*e^2*g^6 - a*c^3*d^3*e^4*g^6)/(c^4*d^4*e^3*g^7)) + 3*(11*c^4*d^4*e^5*f^2*g^5 - 20*c^4*d^5*e^
4*f*g^6 - 2*a*c^3*d^3*e^6*f*g^6 + 8*c^4*d^6*e^3*g^7 + 4*a*c^3*d^4*e^5*g^7 - a^2*c^2*d^2*e^7*g^7)/(c^4*d^4*e^3*
g^7)) + 3*(5*c^3*d^3*e^4*f^3 - 12*c^3*d^4*e^3*f^2*g - 3*a*c^2*d^2*e^5*f^2*g + 8*c^3*d^5*e^2*f*g^2 + 8*a*c^2*d^
3*e^4*f*g^2 - a^2*c*d*e^6*f*g^2 - 8*a*c^2*d^4*e^3*g^3 + 4*a^2*c*d^2*e^5*g^3 - a^3*e^7*g^3)*log(abs(-sqrt(e^2*f
+ (e*x + d)*e*g - d*e*g)*sqrt(c*d*g) + sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)
))/(sqrt(c*d*g)*c^2*d^2*g))*d*abs(g)/g - 96*(sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c
*d*g)*(2*e^2*f + 2*(e*x + d)*e*g - 2*d*e*g - (5*c^2*d^2*e^2*f - 4*c^2*d^3*e*g - a*c*d*e^3*g)/(c^2*d^2))*sqrt(e
^2*f + (e*x + d)*e*g - d*e*g) - (3*c^2*d^2*e^4*f^2*g - 4*c^2*d^3*e^3*f*g^2 - 2*a*c*d*e^5*f*g^2 + 4*a*c*d^2*e^4
*g^3 - a^2*e^6*g^3)*log(abs(-sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*sqrt(c*d*g) + sqrt(-c*d*e^2*f*g + a*e^3*g^2 +
(e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)))/(sqrt(c*d*g)*c*d))*d*f*abs(g)/g^3 + 144*(sqrt(-c*d*e^2*f*g + a*e^3*g
^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)*(2*e^2*f + 2*(e*x + d)*e*g - 2*d*e*g - (5*c^2*d^2*e^2*f - 4*c^2*d^
3*e*g - a*c*d*e^3*g)/(c^2*d^2))*sqrt(e^2*f + (e*x + d)*e*g - d*e*g) - (3*c^2*d^2*e^4*f^2*g - 4*c^2*d^3*e^3*f*g
^2 - 2*a*c*d*e^5*f*g^2 + 4*a*c*d^2*e^4*g^3 - a^2*e^6*g^3)*log(abs(-sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*sqrt(c*
d*g) + sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)))/(sqrt(c*d*g)*c*d))*d^2*abs(g)/
(e*g^2) + (sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)*sqrt(e^2*f + (e*x + d)*e*g -
d*e*g)*(2*(e^2*f + (e*x + d)*e*g - d*e*g)*(4*(e^2*f + (e*x + d)*e*g - d*e*g)*(6*(e^2*f + (e*x + d)*e*g - d*e*
g)/(e^3*g^3) - (25*c^6*d^6*e^5*f*g^11 - 24*c^6*d^7*e^4*g^12 - a*c^5*d^5*e^6*g^12)/(c^6*d^6*e^6*g^14)) + (163*c
^6*d^6*e^7*f^2*g^11 - 312*c^6*d^7*e^6*f*g^12 - 14*a*c^5*d^5*e^8*f*g^12 + 144*c^6*d^8*e^5*g^13 + 24*a*c^5*d^6*e
^7*g^13 - 5*a^2*c^4*d^4*e^9*g^13)/(c^6*d^6*e^6*g^14)) - 3*(93*c^6*d^6*e^9*f^3*g^11 - 264*c^6*d^7*e^8*f^2*g^12
- 15*a*c^5*d^5*e^10*f^2*g^12 + 240*c^6*d^8*e^7*f*g^13 + 48*a*c^5*d^6*e^9*f*g^13 - 9*a^2*c^4*d^4*e^11*f*g^13 -
64*c^6*d^9*e^6*g^14 - 48*a*c^5*d^7*e^8*g^14 + 24*a^2*c^4*d^5*e^10*g^14 - 5*a^3*c^3*d^3*e^12*g^14)/(c^6*d^6*e^6
*g^14)) - 3*(35*c^4*d^4*e^5*f^4 - 120*c^4*d^5*e^4*f^3*g - 20*a*c^3*d^3*e^6*f^3*g + 144*c^4*d^6*e^3*f^2*g^2 + 7
2*a*c^3*d^4*e^5*f^2*g^2 - 6*a^2*c^2*d^2*e^7*f^2*g^2 - 64*c^4*d^7*e^2*f*g^3 - 96*a*c^3*d^5*e^4*f*g^3 + 24*a^2*c
^2*d^3*e^6*f*g^3 - 4*a^3*c*d*e^8*f*g^3 + 64*a*c^3*d^6*e^3*g^4 - 48*a^2*c^2*d^4*e^5*g^4 + 24*a^3*c*d^2*e^7*g^4
- 5*a^4*e^9*g^4)*log(abs(-sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*sqrt(c*d*g) + sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e
^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)))/(sqrt(c*d*g)*c^3*d^3*g^2))*abs(g)/g)/(e^2*g) - (15*c^4*d^4*e^5*f^4*g*ab
s(g)*log(abs(-sqrt(e^2*f - d*e*g)*sqrt(c*d*g) + sqrt(-c*d^2*e*g^2 + a*e^3*g^2))) - 12*a*c^3*d^3*e^6*f^3*g^2*ab
s(g)*log(abs(-sqrt(e^2*f - d*e*g)*sqrt(c*d*g) + sqrt(-c*d^2*e*g^2 + a*e^3*g^2))) - 6*a^2*c^2*d^2*e^7*f^2*g^3*a
bs(g)*log(abs(-sqrt(e^2*f - d*e*g)*sqrt(c*d*g) + sqrt(-c*d^2*e*g^2 + a*e^3*g^2))) - 12*a^3*c*d*e^8*f*g^4*abs(g
)*log(abs(-sqrt(e^2*f - d*e*g)*sqrt(c*d*g) + sqrt(-c*d^2*e*g^2 + a*e^3*g^2))) + 15*a^4*e^9*g^5*abs(g)*log(abs(
-sqrt(e^2*f - d*e*g)*sqrt(c*d*g) + sqrt(-c*d^2*e*g^2 + a*e^3*g^2))) + 15*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e
^2*f - d*e*g)*sqrt(c*d*g)*c^3*d^3*e^3*f^3*abs(g) + 10*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(
c*d*g)*c^3*d^4*e^2*f^2*g*abs(g) - 7*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a*c^2*d^2*e
^4*f^2*g*abs(g) + 8*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*c^3*d^5*e*f*g^2*abs(g) - 4*
sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a*c^2*d^3*e^3*f*g^2*abs(g) - 7*sqrt(-c*d^2*e*g^
2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^2*c*d*e^5*f*g^2*abs(g) - 48*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sq
rt(e^2*f - d*e*g)*sqrt(c*d*g)*c^3*d^6*g^3*abs(g) + 8*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c
*d*g)*a*c^2*d^4*e^2*g^3*abs(g) + 10*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^2*c*d^2*e
^4*g^3*abs(g) + 15*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^3*e^6*g^3*abs(g))/(sqrt(c*
d*g)*c^3*d^3*e^2*g^5))*abs(e)^2/e^4 - 16*a*c*d*((24*((c*d*e^2*f*g - a*e^3*g^2)*log(abs(-sqrt(e^2*f + (e*x + d)
*e*g - d*e*g)*sqrt(c*d*g) + sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)))/sqrt(c*d*
g) + sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)*sqrt(e^2*f + (e*x + d)*e*g - d*e*g
))*d*e*f*abs(g)/g^2 - 24*((c*d*e^2*f*g - a*e^3*g^2)*log(abs(-sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*sqrt(c*d*g) +
sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)))/sqrt(c*d*g) + sqrt(-c*d*e^2*f*g + a*
e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)*sqrt(e^2*f + (e*x + d)*e*g - d*e*g))*d^2*abs(g)/g - (sqrt(-c*
d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)*sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*(2*(e^2*f +
(e*x + d)*e*g - d*e*g)*(4*(e^2*f + (e*x + d)*e*g - d*e*g)/(e^2*g^2) - (13*c^4*d^4*e^3*f*g^5 - 12*c^4*d^5*e^2*
g^6 - a*c^3*d^3*e^4*g^6)/(c^4*d^4*e^3*g^7)) + 3*(11*c^4*d^4*e^5*f^2*g^5 - 20*c^4*d^5*e^4*f*g^6 - 2*a*c^3*d^3*e
^6*f*g^6 + 8*c^4*d^6*e^3*g^7 + 4*a*c^3*d^4*e^5*g^7 - a^2*c^2*d^2*e^7*g^7)/(c^4*d^4*e^3*g^7)) + 3*(5*c^3*d^3*e^
4*f^3 - 12*c^3*d^4*e^3*f^2*g - 3*a*c^2*d^2*e^5*f^2*g + 8*c^3*d^5*e^2*f*g^2 + 8*a*c^2*d^3*e^4*f*g^2 - a^2*c*d*e
^6*f*g^2 - 8*a*c^2*d^4*e^3*g^3 + 4*a^2*c*d^2*e^5*g^3 - a^3*e^7*g^3)*log(abs(-sqrt(e^2*f + (e*x + d)*e*g - d*e*
g)*sqrt(c*d*g) + sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)))/(sqrt(c*d*g)*c^2*d^2
*g))*abs(g)/g - 6*(sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)*(2*e^2*f + 2*(e*x +
d)*e*g - 2*d*e*g - (5*c^2*d^2*e^2*f - 4*c^2*d^3*e*g - a*c*d*e^3*g)/(c^2*d^2))*sqrt(e^2*f + (e*x + d)*e*g - d*e
*g) - (3*c^2*d^2*e^4*f^2*g - 4*c^2*d^3*e^3*f*g^2 - 2*a*c*d*e^5*f*g^2 + 4*a*c*d^2*e^4*g^3 - a^2*e^6*g^3)*log(ab
s(-sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*sqrt(c*d*g) + sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g -
d*e*g)*c*d*g)))/(sqrt(c*d*g)*c*d))*f*abs(g)/g^3 + 12*(sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g -
d*e*g)*c*d*g)*(2*e^2*f + 2*(e*x + d)*e*g - 2*d*e*g - (5*c^2*d^2*e^2*f - 4*c^2*d^3*e*g - a*c*d*e^3*g)/(c^2*d^2
))*sqrt(e^2*f + (e*x + d)*e*g - d*e*g) - (3*c^2*d^2*e^4*f^2*g - 4*c^2*d^3*e^3*f*g^2 - 2*a*c*d*e^5*f*g^2 + 4*a*
c*d^2*e^4*g^3 - a^2*e^6*g^3)*log(abs(-sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*sqrt(c*d*g) + sqrt(-c*d*e^2*f*g + a*
e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)))/(sqrt(c*d*g)*c*d))*d*abs(g)/(e*g^2))/g - (3*c^3*d^3*e^4*f^3
*g*abs(g)*log(abs(-sqrt(e^2*f - d*e*g)*sqrt(c*d*g) + sqrt(-c*d^2*e*g^2 + a*e^3*g^2))) - 3*a*c^2*d^2*e^5*f^2*g^
2*abs(g)*log(abs(-sqrt(e^2*f - d*e*g)*sqrt(c*d*g) + sqrt(-c*d^2*e*g^2 + a*e^3*g^2))) - 3*a^2*c*d*e^6*f*g^3*abs
(g)*log(abs(-sqrt(e^2*f - d*e*g)*sqrt(c*d*g) + sqrt(-c*d^2*e*g^2 + a*e^3*g^2))) + 3*a^3*e^7*g^4*abs(g)*log(abs
(-sqrt(e^2*f - d*e*g)*sqrt(c*d*g) + sqrt(-c*d^2*e*g^2 + a*e^3*g^2))) + 3*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e
^2*f - d*e*g)*sqrt(c*d*g)*c^2*d^2*e^2*f^2*abs(g) + 2*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c
*d*g)*c^2*d^3*e*f*g*abs(g) - 2*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a*c*d*e^3*f*g*ab
s(g) - 8*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*c^2*d^4*g^2*abs(g) + 2*sqrt(-c*d^2*e*g
^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a*c*d^2*e^2*g^2*abs(g) + 3*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt
(e^2*f - d*e*g)*sqrt(c*d*g)*a^2*e^4*g^2*abs(g))/(sqrt(c*d*g)*c^2*d^2*g^4))*abs(e)^2/e^4)/e

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\int \frac {\sqrt {f+g\,x}\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}}{{\left (d+e\,x\right )}^{5/2}} \,d x \]

[In]

int(((f + g*x)^(1/2)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2))/(d + e*x)^(5/2),x)

[Out]

int(((f + g*x)^(1/2)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2))/(d + e*x)^(5/2), x)