\(\int \frac {(f+g x)^{5/2} \sqrt {a d e+(c d^2+a e^2) x+c d e x^2}}{\sqrt {d+e x}} \, dx\) [733]

Optimal result

Integrand size = 48, antiderivative size = 385 \[ \int \frac {(f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx=-\frac {5 (c d f-a e g)^3 \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c^3 d^3 g \sqrt {d+e x}}-\frac {5 (c d f-a e g)^2 (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{96 c^2 d^2 g \sqrt {d+e x}}+\frac {\left (\frac {a e}{c d}-\frac {f}{g}\right ) (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 \sqrt {d+e x}}+\frac {(f+g x)^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g \sqrt {d+e x}}-\frac {5 (c d f-a e g)^4 \sqrt {a e+c d x} \sqrt {d+e x} \text {arctanh}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {f+g x}}\right )}{64 c^{7/2} d^{7/2} g^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \]

[Out]

-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^(7/2)/d^(7/2)/g^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-5/96*(-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)/c^2/d^2/g/(e*x+d)^(1/2)+1/24*(a*e/c/d-f/g)*(g*x+f)^(5/2)*(a*d*e+(a*e^2
+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^(1/2)+1/4*(g*x+f)^(7/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/g/(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^3/d^3/g/(e*x+d)^(1/2)

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 385, 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} \[ \int \frac {(f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx=-\frac {5 \sqrt {d+e x} \sqrt {a e+c d x} (c d f-a e g)^4 \text {arctanh}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {f+g x}}\right )}{64 c^{7/2} d^{7/2} g^{3/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {5 \sqrt {f+g x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^3}{64 c^3 d^3 g \sqrt {d+e x}}-\frac {5 (f+g x)^{3/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2}{96 c^2 d^2 g \sqrt {d+e x}}+\frac {(f+g x)^{7/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 g \sqrt {d+e x}}+\frac {(f+g x)^{5/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (\frac {a e}{c d}-\frac {f}{g}\right )}{24 \sqrt {d+e x}} \]

[In]

Int[((f + g*x)^(5/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/Sqrt[d + e*x],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^3*d^3*g*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])/(96*c^2*d^2*g*Sqrt[d + e*x
]) + (((a*e)/(c*d) - f/g)*(f + g*x)^(5/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(24*Sqrt[d + e*x]) + ((
f + g*x)^(7/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(4*g*Sqrt[d + e*x]) - (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^(7/2)*d^(
7/2)*g^(3/2)*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)^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g \sqrt {d+e x}}-\frac {(c d f-a e g) \int \frac {\sqrt {d+e x} (f+g x)^{5/2}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{8 g} \\ & = \frac {\left (\frac {a e}{c d}-\frac {f}{g}\right ) (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 \sqrt {d+e x}}+\frac {(f+g x)^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g \sqrt {d+e x}}-\frac {\left (5 (c d f-a e g)^2\right ) \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}{48 c d g} \\ & = -\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}}{96 c^2 d^2 g \sqrt {d+e x}}+\frac {\left (\frac {a e}{c d}-\frac {f}{g}\right ) (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 \sqrt {d+e x}}+\frac {(f+g x)^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g \sqrt {d+e x}}-\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 c^2 d^2 g} \\ & = -\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^3 d^3 g \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}}{96 c^2 d^2 g \sqrt {d+e x}}+\frac {\left (\frac {a e}{c d}-\frac {f}{g}\right ) (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 \sqrt {d+e x}}+\frac {(f+g x)^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g \sqrt {d+e x}}-\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^3 d^3 g} \\ & = -\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^3 d^3 g \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}}{96 c^2 d^2 g \sqrt {d+e x}}+\frac {\left (\frac {a e}{c d}-\frac {f}{g}\right ) (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 \sqrt {d+e x}}+\frac {(f+g x)^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g \sqrt {d+e x}}-\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^3 d^3 g \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^3 d^3 g \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}}{96 c^2 d^2 g \sqrt {d+e x}}+\frac {\left (\frac {a e}{c d}-\frac {f}{g}\right ) (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 \sqrt {d+e x}}+\frac {(f+g x)^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g \sqrt {d+e x}}-\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^4 d^4 g \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^3 d^3 g \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}}{96 c^2 d^2 g \sqrt {d+e x}}+\frac {\left (\frac {a e}{c d}-\frac {f}{g}\right ) (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 \sqrt {d+e x}}+\frac {(f+g x)^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g \sqrt {d+e x}}-\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^4 d^4 g \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^3 d^3 g \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}}{96 c^2 d^2 g \sqrt {d+e x}}+\frac {\left (\frac {a e}{c d}-\frac {f}{g}\right ) (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 \sqrt {d+e x}}+\frac {(f+g x)^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g \sqrt {d+e x}}-\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^{7/2} d^{7/2} g^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.52 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.61 \[ \int \frac {(f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (\sqrt {c} \sqrt {d} \sqrt {g} \sqrt {f+g x} \left (15 a^3 e^3 g^3-5 a^2 c d e^2 g^2 (11 f+2 g x)+a c^2 d^2 e g \left (73 f^2+36 f g x+8 g^2 x^2\right )+c^3 d^3 \left (15 f^3+118 f^2 g x+136 f g^2 x^2+48 g^3 x^3\right )\right )-\frac {15 (c d f-a e g)^4 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {f+g x}}{\sqrt {g} \sqrt {a e+c d x}}\right )}{\sqrt {a e+c d x}}\right )}{192 c^{7/2} d^{7/2} g^{3/2} \sqrt {d+e x}} \]

[In]

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

[Out]

(Sqrt[(a*e + c*d*x)*(d + e*x)]*(Sqrt[c]*Sqrt[d]*Sqrt[g]*Sqrt[f + g*x]*(15*a^3*e^3*g^3 - 5*a^2*c*d*e^2*g^2*(11*
f + 2*g*x) + a*c^2*d^2*e*g*(73*f^2 + 36*f*g*x + 8*g^2*x^2) + c^3*d^3*(15*f^3 + 118*f^2*g*x + 136*f*g^2*x^2 + 4
8*g^3*x^3)) - (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])])/Sqrt
[a*e + c*d*x]))/(192*c^(7/2)*d^(7/2)*g^(3/2)*Sqrt[d + e*x])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(731\) vs. \(2(329)=658\).

Time = 0.57 (sec) , antiderivative size = 732, normalized size of antiderivative = 1.90

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}-16 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}-272 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}+20 \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 -236 \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}+110 \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}-146 \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}\, g \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, c^{3} d^{3} \sqrt {c d g}}\)	\(732\)

[In]

int((g*x+f)^(5/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^(1/2),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
-16*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)-272*c^3*d^3*f*g^2*x^2*((g*x+f)*(c*d*x+a*e))^
(1/2)*(c*d*g)^(1/2)+20*((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-236*((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+110*((g*x+f)*(c*d*x+a*e))^(1/2)*(c*d*g)^(1/2)*a^2*c*d*e^2*f*g^2
-146*((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)/g/((g*x+f)*(c*d*x+a*e))^(1/2)/c^3/d^3/(c*d*g)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 2.54 (sec) , antiderivative size = 1065, normalized size of antiderivative = 2.77 \[ \int \frac {(f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx=\left [\frac {4 \, {\left (48 \, c^{4} d^{4} g^{4} x^{3} + 15 \, c^{4} d^{4} f^{3} g + 73 \, a c^{3} d^{3} e f^{2} g^{2} - 55 \, a^{2} c^{2} d^{2} e^{2} f g^{3} + 15 \, a^{3} c d e^{3} g^{4} + 8 \, {\left (17 \, c^{4} d^{4} f g^{3} + a c^{3} d^{3} e g^{4}\right )} x^{2} + 2 \, {\left (59 \, c^{4} d^{4} f^{2} g^{2} + 18 \, a c^{3} d^{3} e f g^{3} - 5 \, 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^{4} d^{4} e g^{2} x + c^{4} d^{5} g^{2}\right )}}, \frac {2 \, {\left (48 \, c^{4} d^{4} g^{4} x^{3} + 15 \, c^{4} d^{4} f^{3} g + 73 \, a c^{3} d^{3} e f^{2} g^{2} - 55 \, a^{2} c^{2} d^{2} e^{2} f g^{3} + 15 \, a^{3} c d e^{3} g^{4} + 8 \, {\left (17 \, c^{4} d^{4} f g^{3} + a c^{3} d^{3} e g^{4}\right )} x^{2} + 2 \, {\left (59 \, c^{4} d^{4} f^{2} g^{2} + 18 \, a c^{3} d^{3} e f g^{3} - 5 \, 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^{4} d^{4} e g^{2} x + c^{4} d^{5} g^{2}\right )}}\right ] \]

[In]

integrate((g*x+f)^(5/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

integrate((g*x+f)^(5/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6752 vs. \(2 (329) = 658\).

Time = 2.15 (sec) , antiderivative size = 6752, normalized size of antiderivative = 17.54 \[ \int \frac {(f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx=\text {Too large to display} \]

[In]

integrate((g*x+f)^(5/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

1/192*(48*f^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^4 + 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^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*abs(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^1
4)) - 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 + 72*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*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))) - 12*a*c^3*d^3*e^6*f^3*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))) - 6*a^2*c^2*d^2*e^7*f^2*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))) - 12*a^3*c*d*e^8*f*g^4*abs(g)*lo
g(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(-sqr
t(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)*sqrt(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*f*g*((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) + s
qrt(-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)*sqr
t(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))*a
bs(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(abs(-sqr
t(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))*sqr
t(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)*lo
g(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*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^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^5)/e

Mupad [F(-1)]

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

[In]

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

[Out]

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