Integrand size = 48, antiderivative size = 227 \[ \int \frac {(d+e x)^{3/2} (f+g x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 \sqrt {d+e x} (f+g x)^{3/2}}{c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {3 g \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c^2 d^2 \sqrt {d+e x}}+\frac {3 \sqrt {g} (c d f-a e g) \sqrt {a e+c d x} \sqrt {d+e x} \text {arctanh}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {f+g x}}\right )}{c^{5/2} d^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \]
-2*(g*x+f)^(3/2)*(e*x+d)^(1/2)/c/d/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2) +3*(-a*e*g+c*d*f)*arctanh(g^(1/2)*(c*d*x+a*e)^(1/2)/c^(1/2)/d^(1/2)/(g*x+f )^(1/2))*g^(1/2)*(c*d*x+a*e)^(1/2)*(e*x+d)^(1/2)/c^(5/2)/d^(5/2)/(a*d*e+(a *e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+3*g*(g*x+f)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c* d*e*x^2)^(1/2)/c^2/d^2/(e*x+d)^(1/2)
Time = 0.30 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.63 \[ \int \frac {(d+e x)^{3/2} (f+g x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {\sqrt {d+e x} \left (\sqrt {c} \sqrt {d} \sqrt {f+g x} (-2 c d f+3 a e g+c d g x)+3 \sqrt {g} (c d f-a e g) \sqrt {a e+c d x} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {f+g x}}{\sqrt {g} \sqrt {a e+c d x}}\right )\right )}{c^{5/2} d^{5/2} \sqrt {(a e+c d x) (d+e x)}} \]
(Sqrt[d + e*x]*(Sqrt[c]*Sqrt[d]*Sqrt[f + g*x]*(-2*c*d*f + 3*a*e*g + c*d*g* x) + 3*Sqrt[g]*(c*d*f - a*e*g)*Sqrt[a*e + c*d*x]*ArcTanh[(Sqrt[c]*Sqrt[d]* Sqrt[f + g*x])/(Sqrt[g]*Sqrt[a*e + c*d*x])]))/(c^(5/2)*d^(5/2)*Sqrt[(a*e + c*d*x)*(d + e*x)])
Time = 0.49 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.104, Rules used = {1251, 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.
| \(\displaystyle \int \frac {(d+e x)^{3/2} (f+g x)^{3/2}}{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1251 |
| \(\displaystyle \frac {3 g \int \frac {\sqrt {d+e x} \sqrt {f+g x}}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{c d}-\frac {2 \sqrt {d+e x} (f+g x)^{3/2}}{c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 1253 |
| \(\displaystyle \frac {3 g \left (\frac {(c d f-a e g) \int \frac {\sqrt {d+e x}}{\sqrt {f+g x} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 c d}+\frac {\sqrt {f+g x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d \sqrt {d+e x}}\right )}{c d}-\frac {2 \sqrt {d+e x} (f+g x)^{3/2}}{c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 1268 |
| \(\displaystyle \frac {3 g \left (\frac {\sqrt {d+e x} \sqrt {a e+c d x} (c d f-a e g) \int \frac {1}{\sqrt {a e+c d x} \sqrt {f+g x}}dx}{2 c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {\sqrt {f+g x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d \sqrt {d+e x}}\right )}{c d}-\frac {2 \sqrt {d+e x} (f+g x)^{3/2}}{c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {3 g \left (\frac {\sqrt {d+e x} \sqrt {a e+c d x} (c d f-a e g) \int \frac {1}{c d-\frac {g (a e+c d x)}{f+g x}}d\frac {\sqrt {a e+c d x}}{\sqrt {f+g x}}}{c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {\sqrt {f+g x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d \sqrt {d+e x}}\right )}{c d}-\frac {2 \sqrt {d+e x} (f+g x)^{3/2}}{c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {3 g \left (\frac {\sqrt {d+e x} \sqrt {a e+c d x} (c d f-a e g) \text {arctanh}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {f+g x}}\right )}{c^{3/2} d^{3/2} \sqrt {g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {\sqrt {f+g x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d \sqrt {d+e x}}\right )}{c d}-\frac {2 \sqrt {d+e x} (f+g x)^{3/2}}{c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
(-2*Sqrt[d + e*x]*(f + g*x)^(3/2))/(c*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c *d*e*x^2]) + (3*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])))/(c*d )
3.8.21.3.1 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[e*(d + e*x)^(m - 1)*(f + g*x)^n*((a + b*x + c*x^2)^(p + 1)/(c*(p + 1))), x] - Simp[e*g*(n/(c*(p + 1))) Int[( d + e*x)^(m - 1)*(f + g*x)^(n - 1)*(a + b*x + c*x^2)^(p + 1), x], x] /; Fre eQ[{a, b, c, d, e, f, g}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[m + p, 0] && LtQ[p, -1] && GtQ[n, 0]
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 = 0.55 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.70
| method | result | size |
| default | \(-\frac {\left (3 \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) a c d e \,g^{2} x -3 \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) c^{2} d^{2} f g x +3 \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) a^{2} e^{2} g^{2}-3 \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) a c d e f g -2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}\, c d g x -6 \sqrt {c d g}\, \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, a e g +4 \sqrt {c d g}\, \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, c d f \right ) \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \sqrt {g x +f}}{2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \left (c d x +a e \right ) \sqrt {c d g}\, c^{2} d^{2} \sqrt {e x +d}}\) | \(386\) |
int((e*x+d)^(3/2)*(g*x+f)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, method=_RETURNVERBOSE)
-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*c*d*e*g^2*x-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^2*d^2*f*g*x+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^2*e^2*g^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*c*d*e*f*g-2*((g*x+f)*(c*d*x+ a*e))^(1/2)*(c*d*g)^(1/2)*c*d*g*x-6*(c*d*g)^(1/2)*((g*x+f)*(c*d*x+a*e))^(1 /2)*a*e*g+4*(c*d*g)^(1/2)*((g*x+f)*(c*d*x+a*e))^(1/2)*c*d*f)*((c*d*x+a*e)* (e*x+d))^(1/2)*(g*x+f)^(1/2)/((g*x+f)*(c*d*x+a*e))^(1/2)/(c*d*x+a*e)/(c*d* g)^(1/2)/c^2/d^2/(e*x+d)^(1/2)
Time = 0.75 (sec) , antiderivative size = 725, normalized size of antiderivative = 3.19 \[ \int \frac {(d+e x)^{3/2} (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}} \, dx=\left [\frac {4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (c d g x - 2 \, c d f + 3 \, a e g\right )} \sqrt {e x + d} \sqrt {g x + f} - 3 \, {\left (a c d^{2} e f - a^{2} d e^{2} g + {\left (c^{2} d^{2} e f - a c d e^{2} g\right )} x^{2} + {\left ({\left (c^{2} d^{3} + a c d e^{2}\right )} f - {\left (a c d^{2} e + a^{2} e^{3}\right )} g\right )} x\right )} \sqrt {\frac {g}{c d}} \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} + 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} - 4 \, {\left (2 \, c^{2} d^{2} g x + c^{2} d^{2} f + a c d e g\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} \sqrt {\frac {g}{c d}} + {\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 )}{4 \, {\left (c^{3} d^{3} e x^{2} + a c^{2} d^{3} e + {\left (c^{3} d^{4} + a c^{2} d^{2} e^{2}\right )} x\right )}}, \frac {2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (c d g x - 2 \, c d f + 3 \, a e g\right )} \sqrt {e x + d} \sqrt {g x + f} - 3 \, {\left (a c d^{2} e f - a^{2} d e^{2} g + {\left (c^{2} d^{2} e f - a c d e^{2} g\right )} x^{2} + {\left ({\left (c^{2} d^{3} + a c d e^{2}\right )} f - {\left (a c d^{2} e + a^{2} e^{3}\right )} g\right )} x\right )} \sqrt {-\frac {g}{c d}} \arctan \left (\frac {2 \, \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} c d \sqrt {-\frac {g}{c d}}}{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 )}{2 \, {\left (c^{3} d^{3} e x^{2} + a c^{2} d^{3} e + {\left (c^{3} d^{4} + a c^{2} d^{2} e^{2}\right )} x\right )}}\right ] \]
integrate((e*x+d)^(3/2)*(g*x+f)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3 /2),x, algorithm="fricas")
[1/4*(4*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(c*d*g*x - 2*c*d*f + 3 *a*e*g)*sqrt(e*x + d)*sqrt(g*x + f) - 3*(a*c*d^2*e*f - a^2*d*e^2*g + (c^2* d^2*e*f - a*c*d*e^2*g)*x^2 + ((c^2*d^3 + a*c*d*e^2)*f - (a*c*d^2*e + a^2*e ^3)*g)*x)*sqrt(g/(c*d))*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 + 8*(c^2*d^2*e*f*g + (c^2*d^3 + a*c*d*e^2)*g^2)*x^ 2 - 4*(2*c^2*d^2*g*x + c^2*d^2*f + a*c*d*e*g)*sqrt(c*d*e*x^2 + a*d*e + (c* d^2 + a*e^2)*x)*sqrt(e*x + d)*sqrt(g*x + f)*sqrt(g/(c*d)) + (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*x^2 + a*c^2*d^3*e + (c^3*d^4 + a*c^2*d^2*e^2)*x), 1/2*(2 *sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(c*d*g*x - 2*c*d*f + 3*a*e*g) *sqrt(e*x + d)*sqrt(g*x + f) - 3*(a*c*d^2*e*f - a^2*d*e^2*g + (c^2*d^2*e*f - a*c*d*e^2*g)*x^2 + ((c^2*d^3 + a*c*d*e^2)*f - (a*c*d^2*e + a^2*e^3)*g)* x)*sqrt(-g/(c*d))*arctan(2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqr t(e*x + d)*sqrt(g*x + f)*c*d*sqrt(-g/(c*d))/(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*x^2 + a*c^2*d^3*e + (c^3*d^4 + a*c^2*d^2*e^2)*x)]
Timed out. \[ \int \frac {(d+e x)^{3/2} (f+g x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(d+e x)^{3/2} (f+g x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{\frac {3}{2}}}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}}} \,d x } \]
integrate((e*x+d)^(3/2)*(g*x+f)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3 /2),x, algorithm="maxima")
Leaf count of result is larger than twice the leaf count of optimal. 527 vs. \(2 (193) = 386\).
Time = 0.52 (sec) , antiderivative size = 527, normalized size of antiderivative = 2.32 \[ \int \frac {(d+e x)^{3/2} (f+g x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {\sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g} {\left (\frac {{\left (e^{2} f + {\left (e x + d\right )} e g - d e g\right )} g^{2}}{c d e^{2} {\left | g \right |}} - \frac {3 \, {\left (c^{2} d^{2} e^{2} f g^{2} - a c d e^{3} g^{3}\right )}}{c^{3} d^{3} e^{2} {\left | g \right |}}\right )}}{\sqrt {-c d e^{2} f g + a e^{3} g^{2} + {\left (e^{2} f + {\left (e x + d\right )} e g - d e g\right )} c d g}} - \frac {3 \, {\left (c d f g^{2} - a e g^{3}\right )} \log \left ({\left | -\sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g} \sqrt {c d g} + \sqrt {-c d e^{2} f g + a e^{3} g^{2} + {\left (e^{2} f + {\left (e x + d\right )} e g - d e g\right )} c d g} \right |}\right )}{\sqrt {c d g} c^{2} d^{2} {\left | g \right |}} + \frac {3 \, \sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} c d e f g^{2} \log \left ({\left | -\sqrt {e^{2} f - d e g} \sqrt {c d g} + \sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} \right |}\right ) - 3 \, \sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} a e^{2} g^{3} \log \left ({\left | -\sqrt {e^{2} f - d e g} \sqrt {c d g} + \sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} \right |}\right ) + 2 \, \sqrt {e^{2} f - d e g} \sqrt {c d g} c d e f g^{2} + \sqrt {e^{2} f - d e g} \sqrt {c d g} c d^{2} g^{3} - 3 \, \sqrt {e^{2} f - d e g} \sqrt {c d g} a e^{2} g^{3}}{\sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} \sqrt {c d g} c^{2} d^{2} e {\left | g \right |}} \]
integrate((e*x+d)^(3/2)*(g*x+f)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3 /2),x, algorithm="giac")
sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*((e^2*f + (e*x + d)*e*g - d*e*g)*g^2/( c*d*e^2*abs(g)) - 3*(c^2*d^2*e^2*f*g^2 - a*c*d*e^3*g^3)/(c^3*d^3*e^2*abs(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) - 3*(c*d*f*g^2 - a*e*g^3)*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)*c^2*d^2*abs(g)) + (3*sqrt(-c*d^2*e*g^2 + a*e^3*g^2 )*c*d*e*f*g^2*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)*a*e^2*g^3*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*sqrt(e^ 2*f - d*e*g)*sqrt(c*d*g)*c*d*e*f*g^2 + sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*c*d ^2*g^3 - 3*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a*e^2*g^3)/(sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(c*d*g)*c^2*d^2*e*abs(g))
Timed out. \[ \int \frac {(d+e x)^{3/2} (f+g x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int \frac {{\left (f+g\,x\right )}^{3/2}\,{\left (d+e\,x\right )}^{3/2}}{{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}} \,d x \]