Integrand size = 48, antiderivative size = 310 \[ \int \frac {\sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\frac {(c d f-a e g)^2 \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 c d g^2 \sqrt {d+e x}}-\frac {(c d f-a e g) (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g^2 \sqrt {d+e x}}+\frac {(f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 g (d+e x)^{3/2}}+\frac {(c d f-a e g)^3 \sqrt {a e+c d x} \sqrt {d+e x} \text {arctanh}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {f+g x}}\right )}{8 c^{3/2} d^{3/2} g^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \]
1/3*(g*x+f)^(3/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/g/(e*x+d)^(3/2)+ 1/8*(-a*e*g+c*d*f)^3*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^(5/2)/(a*d*e +(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-1/4*(-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)^(1/2)/g^2/(e*x+d)^(1/2)+1/8*(-a*e*g+c*d*f)^2*(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^2/(e*x+d)^(1/2)
Time = 0.52 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.64 \[ \int \frac {\sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\frac {((a e+c d x) (d+e x))^{3/2} \left (\frac {\sqrt {c} \sqrt {d} \sqrt {g} \sqrt {f+g x} \left (3 a^2 e^2 g^2+2 a c d e g (4 f+7 g x)+c^2 d^2 \left (-3 f^2+2 f g x+8 g^2 x^2\right )\right )}{a e+c d x}+\frac {3 (c d f-a e g)^3 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {f+g x}}{\sqrt {g} \sqrt {a e+c d x}}\right )}{(a e+c d x)^{3/2}}\right )}{24 c^{3/2} d^{3/2} g^{5/2} (d+e x)^{3/2}} \]
(((a*e + c*d*x)*(d + e*x))^(3/2)*((Sqrt[c]*Sqrt[d]*Sqrt[g]*Sqrt[f + g*x]*( 3*a^2*e^2*g^2 + 2*a*c*d*e*g*(4*f + 7*g*x) + c^2*d^2*(-3*f^2 + 2*f*g*x + 8* g^2*x^2)))/(a*e + c*d*x) + (3*(c*d*f - a*e*g)^3*ArcTanh[(Sqrt[c]*Sqrt[d]*S qrt[f + g*x])/(Sqrt[g]*Sqrt[a*e + c*d*x])])/(a*e + c*d*x)^(3/2)))/(24*c^(3 /2)*d^(3/2)*g^(5/2)*(d + e*x)^(3/2))
Time = 0.61 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1250, 1250, 1253, 1268, 66, 221}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sqrt {f+g x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1250 |
| \(\displaystyle \frac {(f+g x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 g (d+e x)^{3/2}}-\frac {(c d f-a e g) \int \frac {\sqrt {f+g x} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\sqrt {d+e x}}dx}{2 g}\) |
| \(\Big \downarrow \) 1250 |
| \(\displaystyle \frac {(f+g x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 g (d+e x)^{3/2}}-\frac {(c d f-a e g) \left (\frac {(f+g x)^{3/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 g \sqrt {d+e x}}-\frac {(c d f-a e g) \int \frac {\sqrt {d+e x} \sqrt {f+g x}}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{4 g}\right )}{2 g}\) |
| \(\Big \downarrow \) 1253 |
| \(\displaystyle \frac {(f+g x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 g (d+e x)^{3/2}}-\frac {(c d f-a e g) \left (\frac {(f+g x)^{3/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 g \sqrt {d+e x}}-\frac {(c d f-a e g) \left (\frac {(c d f-a e g) \int \frac {\sqrt {d+e x}}{\sqrt {f+g x} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 c d}+\frac {\sqrt {f+g x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d \sqrt {d+e x}}\right )}{4 g}\right )}{2 g}\) |
| \(\Big \downarrow \) 1268 |
| \(\displaystyle \frac {(f+g x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 g (d+e x)^{3/2}}-\frac {(c d f-a e g) \left (\frac {(f+g x)^{3/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 g \sqrt {d+e x}}-\frac {(c d f-a e g) \left (\frac {\sqrt {d+e x} \sqrt {a e+c d x} (c d f-a e g) \int \frac {1}{\sqrt {a e+c d x} \sqrt {f+g x}}dx}{2 c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {\sqrt {f+g x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d \sqrt {d+e x}}\right )}{4 g}\right )}{2 g}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {(f+g x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 g (d+e x)^{3/2}}-\frac {(c d f-a e g) \left (\frac {(f+g x)^{3/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 g \sqrt {d+e x}}-\frac {(c d f-a e g) \left (\frac {\sqrt {d+e x} \sqrt {a e+c d x} (c d f-a e g) \int \frac {1}{c d-\frac {g (a e+c d x)}{f+g x}}d\frac {\sqrt {a e+c d x}}{\sqrt {f+g x}}}{c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {\sqrt {f+g x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d \sqrt {d+e x}}\right )}{4 g}\right )}{2 g}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {(f+g x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 g (d+e x)^{3/2}}-\frac {(c d f-a e g) \left (\frac {(f+g x)^{3/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 g \sqrt {d+e x}}-\frac {(c d f-a e g) \left (\frac {\sqrt {d+e x} \sqrt {a e+c d x} (c d f-a e g) \text {arctanh}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {f+g x}}\right )}{c^{3/2} d^{3/2} \sqrt {g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {\sqrt {f+g x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d \sqrt {d+e x}}\right )}{4 g}\right )}{2 g}\) |
((f + g*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(3*g*(d + e*x)^(3/2)) - ((c*d*f - a*e*g)*(((f + g*x)^(3/2)*Sqrt[a*d*e + (c*d^2 + a*e ^2)*x + c*d*e*x^2])/(2*g*Sqrt[d + e*x]) - ((c*d*f - a*e*g)*((Sqrt[f + g*x] *Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(c*d*Sqrt[d + e*x]) + ((c*d* f - a*e*g)*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])])/(c^(3/2)*d^(3/2)*Sqrt[g]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])))/(4*g)))/(2*g)
3.8.43.3.1 Defintions of rubi rules used
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 2 Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre eQ[{a, b, c, d}, x] && !GtQ[c - a*(d/b), 0]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
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] - Simp[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] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && 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]
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] - Simp[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] && EqQ[c*d^2 - b*d* e + a*e^2, 0] && EqQ[m + p, 0] && GtQ[n, 0] && NeQ[m - n - 1, 0] && (Intege rQ[2*p] || IntegerQ[n])
Int[((d_) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(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 ] && EqQ[c*d^2 - b*d*e + a*e^2, 0]
Time = 0.57 (sec) , antiderivative size = 504, normalized size of antiderivative = 1.63
| method | result | size |
| default | \(-\frac {\sqrt {g x +f}\, \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (3 \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) a^{3} e^{3} g^{3}-9 \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) a^{2} c d \,e^{2} f \,g^{2}+9 \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) a \,c^{2} d^{2} e \,f^{2} g -3 \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) c^{3} d^{3} f^{3}-16 c^{2} d^{2} g^{2} x^{2} \sqrt {c d g}\, \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}-28 \sqrt {c d g}\, \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, a c d e \,g^{2} x -4 \sqrt {c d g}\, \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, c^{2} d^{2} f g x -6 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}\, a^{2} e^{2} g^{2}-16 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}\, a c d e f g +6 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}\, c^{2} d^{2} f^{2}\right )}{48 \sqrt {e x +d}\, c d \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, g^{2} \sqrt {c d g}}\) | \(504\) |
int((g*x+f)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(3/2),x, method=_RETURNVERBOSE)
-1/48*(g*x+f)^(1/2)*((c*d*x+a*e)*(e*x+d))^(1/2)*(3*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*e^3 *g^3-9*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*d*e^2*f*g^2+9*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^2*d^2*e*f^2 *g-3*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^3*d^3*f^3-16*c^2*d^2*g^2*x^2*(c*d*g)^(1/2)*((g*x+f) *(c*d*x+a*e))^(1/2)-28*(c*d*g)^(1/2)*((g*x+f)*(c*d*x+a*e))^(1/2)*a*c*d*e*g ^2*x-4*(c*d*g)^(1/2)*((g*x+f)*(c*d*x+a*e))^(1/2)*c^2*d^2*f*g*x-6*((g*x+f)* (c*d*x+a*e))^(1/2)*(c*d*g)^(1/2)*a^2*e^2*g^2-16*((g*x+f)*(c*d*x+a*e))^(1/2 )*(c*d*g)^(1/2)*a*c*d*e*f*g+6*((g*x+f)*(c*d*x+a*e))^(1/2)*(c*d*g)^(1/2)*c^ 2*d^2*f^2)/(e*x+d)^(1/2)/c/d/((g*x+f)*(c*d*x+a*e))^(1/2)/g^2/(c*d*g)^(1/2)
Time = 1.01 (sec) , antiderivative size = 847, normalized size of antiderivative = 2.73 \[ \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=\left [\frac {4 \, {\left (8 \, c^{3} d^{3} g^{3} x^{2} - 3 \, c^{3} d^{3} f^{2} g + 8 \, a c^{2} d^{2} e f g^{2} + 3 \, a^{2} c d e^{2} g^{3} + 2 \, {\left (c^{3} d^{3} f g^{2} + 7 \, a c^{2} d^{2} e g^{3}\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} - 3 \, {\left (c^{3} d^{4} f^{3} - 3 \, a c^{2} d^{3} e f^{2} g + 3 \, a^{2} c d^{2} e^{2} f g^{2} - a^{3} d e^{3} g^{3} + {\left (c^{3} d^{3} e f^{3} - 3 \, a c^{2} d^{2} e^{2} f^{2} g + 3 \, a^{2} c d e^{3} f g^{2} - a^{3} e^{4} g^{3}\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 )}{96 \, {\left (c^{2} d^{2} e g^{3} x + c^{2} d^{3} g^{3}\right )}}, \frac {2 \, {\left (8 \, c^{3} d^{3} g^{3} x^{2} - 3 \, c^{3} d^{3} f^{2} g + 8 \, a c^{2} d^{2} e f g^{2} + 3 \, a^{2} c d e^{2} g^{3} + 2 \, {\left (c^{3} d^{3} f g^{2} + 7 \, a c^{2} d^{2} e g^{3}\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} - 3 \, {\left (c^{3} d^{4} f^{3} - 3 \, a c^{2} d^{3} e f^{2} g + 3 \, a^{2} c d^{2} e^{2} f g^{2} - a^{3} d e^{3} g^{3} + {\left (c^{3} d^{3} e f^{3} - 3 \, a c^{2} d^{2} e^{2} f^{2} g + 3 \, a^{2} c d e^{3} f g^{2} - a^{3} e^{4} g^{3}\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 )}{48 \, {\left (c^{2} d^{2} e g^{3} x + c^{2} d^{3} g^{3}\right )}}\right ] \]
integrate((g*x+f)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(3 /2),x, algorithm="fricas")
[1/96*(4*(8*c^3*d^3*g^3*x^2 - 3*c^3*d^3*f^2*g + 8*a*c^2*d^2*e*f*g^2 + 3*a^ 2*c*d*e^2*g^3 + 2*(c^3*d^3*f*g^2 + 7*a*c^2*d^2*e*g^3)*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) - 3*(c^3*d^4*f^3 - 3*a*c^2*d^3*e*f^2*g + 3*a^2*c*d^2*e^2*f*g^2 - a^3*d*e^3*g^3 + (c^3*d^3*e*f ^3 - 3*a*c^2*d^2*e^2*f^2*g + 3*a^2*c*d*e^3*f*g^2 - a^3*e^4*g^3)*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^3*x + c^2*d^3 *g^3), 1/48*(2*(8*c^3*d^3*g^3*x^2 - 3*c^3*d^3*f^2*g + 8*a*c^2*d^2*e*f*g^2 + 3*a^2*c*d*e^2*g^3 + 2*(c^3*d^3*f*g^2 + 7*a*c^2*d^2*e*g^3)*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) - 3*(c^3*d^4* f^3 - 3*a*c^2*d^3*e*f^2*g + 3*a^2*c*d^2*e^2*f*g^2 - a^3*d*e^3*g^3 + (c^3*d ^3*e*f^3 - 3*a*c^2*d^2*e^2*f^2*g + 3*a^2*c*d*e^3*f*g^2 - a^3*e^4*g^3)*x)*s qrt(-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^2*d^2*e*g^3*x + c^2*d^3*g^3)]
\[ \int \frac {\sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\int \frac {\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}} \sqrt {f + g x}}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {\sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}} \sqrt {g x + f}}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
integrate((g*x+f)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(3 /2),x, algorithm="maxima")
Leaf count of result is larger than twice the leaf count of optimal. 3057 vs. \(2 (262) = 524\).
Time = 1.21 (sec) , antiderivative size = 3057, normalized size of antiderivative = 9.86 \[ \int \frac {\sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\text {Too large to display} \]
integrate((g*x+f)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(3 /2),x, algorithm="giac")
1/24*(6*a*((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)*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*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) + s qrt(-c*d^2*e*g^2 + a*e^3*g^2))) + a^2*e^5*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))) + 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(-...
Timed out. \[ \int \frac {\sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\int \frac {\sqrt {f+g\,x}\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}}{{\left (d+e\,x\right )}^{3/2}} \,d x \]