Integrand size = 48, antiderivative size = 198 \[ \int \frac {\sqrt {d+e x}}{(f+g x)^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 (c d f-a e g) \sqrt {d+e x} (f+g x)^{5/2}}+\frac {8 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{15 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^{3/2}}+\frac {16 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{15 (c d f-a e g)^3 \sqrt {d+e x} \sqrt {f+g x}} \] Output:
2/5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e*g+c*d*f)/(e*x+d)^(1/2)/( g*x+f)^(5/2)+8/15*c*d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e*g+c*d* f)^2/(e*x+d)^(1/2)/(g*x+f)^(3/2)+16/15*c^2*d^2*(a*d*e+(a*e^2+c*d^2)*x+c*d* e*x^2)^(1/2)/(-a*e*g+c*d*f)^3/(e*x+d)^(1/2)/(g*x+f)^(1/2)
Time = 0.21 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.53 \[ \int \frac {\sqrt {d+e x}}{(f+g x)^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \sqrt {(a e+c d x) (d+e x)} \left (3 a^2 e^2 g^2-2 a c d e g (5 f+2 g x)+c^2 d^2 \left (15 f^2+20 f g x+8 g^2 x^2\right )\right )}{15 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)^{5/2}} \] Input:
Integrate[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]),x]
Output:
(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(3*a^2*e^2*g^2 - 2*a*c*d*e*g*(5*f + 2*g*x ) + c^2*d^2*(15*f^2 + 20*f*g*x + 8*g^2*x^2)))/(15*(c*d*f - a*e*g)^3*Sqrt[d + e*x]*(f + g*x)^(5/2))
Time = 0.74 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.07, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {1254, 1254, 1248}
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.
| \(\displaystyle \int \frac {\sqrt {d+e x}}{(f+g x)^{7/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \, dx\) |
| \(\Big \downarrow \) 1254 |
| \(\displaystyle \frac {4 c d \int \frac {\sqrt {d+e x}}{(f+g x)^{5/2} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{5 (c d f-a e g)}+\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 \sqrt {d+e x} (f+g x)^{5/2} (c d f-a e g)}\) |
| \(\Big \downarrow \) 1254 |
| \(\displaystyle \frac {4 c d \left (\frac {2 c d \int \frac {\sqrt {d+e x}}{(f+g x)^{3/2} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{3 (c d f-a e g)}+\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt {d+e x} (f+g x)^{3/2} (c d f-a e g)}\right )}{5 (c d f-a e g)}+\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 \sqrt {d+e x} (f+g x)^{5/2} (c d f-a e g)}\) |
| \(\Big \downarrow \) 1248 |
| \(\displaystyle \frac {4 c d \left (\frac {4 c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt {d+e x} \sqrt {f+g x} (c d f-a e g)^2}+\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt {d+e x} (f+g x)^{3/2} (c d f-a e g)}\right )}{5 (c d f-a e g)}+\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 \sqrt {d+e x} (f+g x)^{5/2} (c d f-a e g)}\) |
Input:
Int[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]),x]
Output:
(2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(5*(c*d*f - a*e*g)*Sqrt[d + e*x]*(f + g*x)^(5/2)) + (4*c*d*((2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d* e*x^2])/(3*(c*d*f - a*e*g)*Sqrt[d + e*x]*(f + g*x)^(3/2)) + (4*c*d*Sqrt[a* d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(3*(c*d*f - a*e*g)^2*Sqrt[d + e*x]*S qrt[f + g*x])))/(5*(c*d*f - a*e*g))
Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-e^2)*(d + e*x)^(m - 1)*(f + g*x)^ (n + 1)*((a + b*x + c*x^2)^(p + 1)/((n + 1)*(c*e*f + c*d*g - b*e*g))), x] / ; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[m + p, 0] && EqQ[m - n - 2, 0]
Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-e^2)*(d + e*x)^(m - 1)*(f + g*x)^ (n + 1)*((a + b*x + c*x^2)^(p + 1)/((n + 1)*(c*e*f + c*d*g - b*e*g))), x] - Simp[c*e*((m - n - 2)/((n + 1)*(c*e*f + c*d*g - b*e*g))) 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] && LtQ[n, -1 ] && IntegerQ[2*p]
Time = 2.82 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.56
| method | result | size |
| default | \(-\frac {2 \sqrt {\left (e x +d \right ) \left (c d x +a e \right )}\, \left (8 g^{2} x^{2} d^{2} c^{2}-4 a c d e \,g^{2} x +20 c^{2} d^{2} f g x +3 a^{2} e^{2} g^{2}-10 a c d e f g +15 f^{2} c^{2} d^{2}\right )}{15 \sqrt {e x +d}\, \left (g x +f \right )^{\frac {5}{2}} \left (a e g -d f c \right )^{3}}\) | \(111\) |
| gosper | \(-\frac {2 \left (c d x +a e \right ) \left (8 g^{2} x^{2} d^{2} c^{2}-4 a c d e \,g^{2} x +20 c^{2} d^{2} f g x +3 a^{2} e^{2} g^{2}-10 a c d e f g +15 f^{2} c^{2} d^{2}\right ) \sqrt {e x +d}}{15 \left (g x +f \right )^{\frac {5}{2}} \left (a^{3} e^{3} g^{3}-3 a^{2} c d \,e^{2} f \,g^{2}+3 a \,c^{2} d^{2} e \,f^{2} g -f^{3} d^{3} c^{3}\right ) \sqrt {c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e}}\) | \(169\) |
| orering | \(-\frac {2 \left (8 g^{2} x^{2} d^{2} c^{2}-4 a c d e \,g^{2} x +20 c^{2} d^{2} f g x +3 a^{2} e^{2} g^{2}-10 a c d e f g +15 f^{2} c^{2} d^{2}\right ) \left (c d x +a e \right ) \sqrt {e x +d}}{15 \left (a^{3} e^{3} g^{3}-3 a^{2} c d \,e^{2} f \,g^{2}+3 a \,c^{2} d^{2} e \,f^{2} g -f^{3} d^{3} c^{3}\right ) \left (g x +f \right )^{\frac {5}{2}} \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}\) | \(170\) |
Input:
int((e*x+d)^(1/2)/(g*x+f)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2),x, method=_RETURNVERBOSE)
Output:
-2/15/(e*x+d)^(1/2)/(g*x+f)^(5/2)*((e*x+d)*(c*d*x+a*e))^(1/2)*(8*c^2*d^2*g ^2*x^2-4*a*c*d*e*g^2*x+20*c^2*d^2*f*g*x+3*a^2*e^2*g^2-10*a*c*d*e*f*g+15*c^ 2*d^2*f^2)/(a*e*g-c*d*f)^3
Leaf count of result is larger than twice the leaf count of optimal. 572 vs. \(2 (174) = 348\).
Time = 0.44 (sec) , antiderivative size = 572, normalized size of antiderivative = 2.89 \[ \int \frac {\sqrt {d+e x}}{(f+g x)^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \, {\left (8 \, c^{2} d^{2} g^{2} x^{2} + 15 \, c^{2} d^{2} f^{2} - 10 \, a c d e f g + 3 \, a^{2} e^{2} g^{2} + 4 \, {\left (5 \, c^{2} d^{2} f g - a c d e g^{2}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d} \sqrt {g x + f}}{15 \, {\left (c^{3} d^{4} f^{6} - 3 \, a c^{2} d^{3} e f^{5} g + 3 \, a^{2} c d^{2} e^{2} f^{4} g^{2} - a^{3} d e^{3} f^{3} g^{3} + {\left (c^{3} d^{3} e f^{3} g^{3} - 3 \, a c^{2} d^{2} e^{2} f^{2} g^{4} + 3 \, a^{2} c d e^{3} f g^{5} - a^{3} e^{4} g^{6}\right )} x^{4} + {\left (3 \, c^{3} d^{3} e f^{4} g^{2} - a^{3} d e^{3} g^{6} + {\left (c^{3} d^{4} - 9 \, a c^{2} d^{2} e^{2}\right )} f^{3} g^{3} - 3 \, {\left (a c^{2} d^{3} e - 3 \, a^{2} c d e^{3}\right )} f^{2} g^{4} + 3 \, {\left (a^{2} c d^{2} e^{2} - a^{3} e^{4}\right )} f g^{5}\right )} x^{3} + 3 \, {\left (c^{3} d^{3} e f^{5} g - a^{3} d e^{3} f g^{5} + {\left (c^{3} d^{4} - 3 \, a c^{2} d^{2} e^{2}\right )} f^{4} g^{2} - 3 \, {\left (a c^{2} d^{3} e - a^{2} c d e^{3}\right )} f^{3} g^{3} + {\left (3 \, a^{2} c d^{2} e^{2} - a^{3} e^{4}\right )} f^{2} g^{4}\right )} x^{2} + {\left (c^{3} d^{3} e f^{6} - 3 \, a^{3} d e^{3} f^{2} g^{4} + 3 \, {\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} f^{5} g - 3 \, {\left (3 \, a c^{2} d^{3} e - a^{2} c d e^{3}\right )} f^{4} g^{2} + {\left (9 \, a^{2} c d^{2} e^{2} - a^{3} e^{4}\right )} f^{3} g^{3}\right )} x\right )}} \] Input:
integrate((e*x+d)^(1/2)/(g*x+f)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1 /2),x, algorithm="fricas")
Output:
2/15*(8*c^2*d^2*g^2*x^2 + 15*c^2*d^2*f^2 - 10*a*c*d*e*f*g + 3*a^2*e^2*g^2 + 4*(5*c^2*d^2*f*g - a*c*d*e*g^2)*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)/(c^3*d^4*f^6 - 3*a*c^2*d^3*e*f^5*g + 3* a^2*c*d^2*e^2*f^4*g^2 - a^3*d*e^3*f^3*g^3 + (c^3*d^3*e*f^3*g^3 - 3*a*c^2*d ^2*e^2*f^2*g^4 + 3*a^2*c*d*e^3*f*g^5 - a^3*e^4*g^6)*x^4 + (3*c^3*d^3*e*f^4 *g^2 - a^3*d*e^3*g^6 + (c^3*d^4 - 9*a*c^2*d^2*e^2)*f^3*g^3 - 3*(a*c^2*d^3* e - 3*a^2*c*d*e^3)*f^2*g^4 + 3*(a^2*c*d^2*e^2 - a^3*e^4)*f*g^5)*x^3 + 3*(c ^3*d^3*e*f^5*g - a^3*d*e^3*f*g^5 + (c^3*d^4 - 3*a*c^2*d^2*e^2)*f^4*g^2 - 3 *(a*c^2*d^3*e - a^2*c*d*e^3)*f^3*g^3 + (3*a^2*c*d^2*e^2 - a^3*e^4)*f^2*g^4 )*x^2 + (c^3*d^3*e*f^6 - 3*a^3*d*e^3*f^2*g^4 + 3*(c^3*d^4 - a*c^2*d^2*e^2) *f^5*g - 3*(3*a*c^2*d^3*e - a^2*c*d*e^3)*f^4*g^2 + (9*a^2*c*d^2*e^2 - a^3* e^4)*f^3*g^3)*x)
Timed out. \[ \int \frac {\sqrt {d+e x}}{(f+g x)^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\text {Timed out} \] Input:
integrate((e*x+d)**(1/2)/(g*x+f)**(7/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x** 2)**(1/2),x)
Output:
Timed out
\[ \int \frac {\sqrt {d+e x}}{(f+g x)^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int { \frac {\sqrt {e x + d}}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (g x + f\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate((e*x+d)^(1/2)/(g*x+f)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1 /2),x, algorithm="maxima")
Output:
integrate(sqrt(e*x + d)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(g*x + f)^(7/2)), x)
Leaf count of result is larger than twice the leaf count of optimal. 407 vs. \(2 (174) = 348\).
Time = 0.21 (sec) , antiderivative size = 407, normalized size of antiderivative = 2.06 \[ \int \frac {\sqrt {d+e x}}{(f+g x)^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {32 \, {\left (c^{2} d^{2} e^{4} f^{2} g^{2} - 2 \, a c d e^{5} f g^{3} + a^{2} e^{6} g^{4} + 5 \, {\left (\sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g} \sqrt {c d g} - \sqrt {-c d e^{2} f g + a e^{3} g^{2} + {\left (e^{2} f + {\left (e x + d\right )} e g - d e g\right )} c d g}\right )}^{2} c d e^{2} f g - 5 \, {\left (\sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g} \sqrt {c d g} - \sqrt {-c d e^{2} f g + a e^{3} g^{2} + {\left (e^{2} f + {\left (e x + d\right )} e g - d e g\right )} c d g}\right )}^{2} a e^{3} g^{2} + 10 \, {\left (\sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g} \sqrt {c d g} - \sqrt {-c d e^{2} f g + a e^{3} g^{2} + {\left (e^{2} f + {\left (e x + d\right )} e g - d e g\right )} c d g}\right )}^{4}\right )} \sqrt {c d g} c^{2} d^{2} e^{6} g^{3}}{15 \, {\left (c d e^{2} f g - a e^{3} g^{2} + {\left (\sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g} \sqrt {c d g} - \sqrt {-c d e^{2} f g + a e^{3} g^{2} + {\left (e^{2} f + {\left (e x + d\right )} e g - d e g\right )} c d g}\right )}^{2}\right )}^{5} {\left | g \right |}} \] Input:
integrate((e*x+d)^(1/2)/(g*x+f)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1 /2),x, algorithm="giac")
Output:
32/15*(c^2*d^2*e^4*f^2*g^2 - 2*a*c*d*e^5*f*g^3 + a^2*e^6*g^4 + 5*(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))^2*c*d*e^2*f*g - 5*(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))^2*a*e^3*g^2 + 10*(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))^4)*sqrt(c*d*g)*c^2*d^2*e^6*g^3/((c*d*e^2*f*g - a* e^3*g^2 + (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))^2)^5*abs(g))
Time = 7.65 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.22 \[ \int \frac {\sqrt {d+e x}}{(f+g x)^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=-\frac {\left (\frac {\sqrt {d+e\,x}\,\left (6\,a^2\,e^2\,g^2-20\,a\,c\,d\,e\,f\,g+30\,c^2\,d^2\,f^2\right )}{15\,e\,g^2\,{\left (a\,e\,g-c\,d\,f\right )}^3}+\frac {16\,c^2\,d^2\,x^2\,\sqrt {d+e\,x}}{15\,e\,{\left (a\,e\,g-c\,d\,f\right )}^3}-\frac {8\,c\,d\,x\,\left (a\,e\,g-5\,c\,d\,f\right )\,\sqrt {d+e\,x}}{15\,e\,g\,{\left (a\,e\,g-c\,d\,f\right )}^3}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{x^3\,\sqrt {f+g\,x}+\frac {d\,f^2\,\sqrt {f+g\,x}}{e\,g^2}+\frac {x^2\,\sqrt {f+g\,x}\,\left (d\,g+2\,e\,f\right )}{e\,g}+\frac {f\,x\,\sqrt {f+g\,x}\,\left (2\,d\,g+e\,f\right )}{e\,g^2}} \] Input:
int((d + e*x)^(1/2)/((f + g*x)^(7/2)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^ 2)^(1/2)),x)
Output:
-((((d + e*x)^(1/2)*(6*a^2*e^2*g^2 + 30*c^2*d^2*f^2 - 20*a*c*d*e*f*g))/(15 *e*g^2*(a*e*g - c*d*f)^3) + (16*c^2*d^2*x^2*(d + e*x)^(1/2))/(15*e*(a*e*g - c*d*f)^3) - (8*c*d*x*(a*e*g - 5*c*d*f)*(d + e*x)^(1/2))/(15*e*g*(a*e*g - c*d*f)^3))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(x^3*(f + g*x)^ (1/2) + (d*f^2*(f + g*x)^(1/2))/(e*g^2) + (x^2*(f + g*x)^(1/2)*(d*g + 2*e* f))/(e*g) + (f*x*(f + g*x)^(1/2)*(2*d*g + e*f))/(e*g^2))
Time = 0.28 (sec) , antiderivative size = 498, normalized size of antiderivative = 2.52 \[ \int \frac {\sqrt {d+e x}}{(f+g x)^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {-\frac {2 \sqrt {g x +f}\, \sqrt {c d x +a e}\, a^{2} e^{2} g^{3}}{5}+\frac {4 \sqrt {g x +f}\, \sqrt {c d x +a e}\, a c d e f \,g^{2}}{3}+\frac {8 \sqrt {g x +f}\, \sqrt {c d x +a e}\, a c d e \,g^{3} x}{15}-2 \sqrt {g x +f}\, \sqrt {c d x +a e}\, c^{2} d^{2} f^{2} g -\frac {8 \sqrt {g x +f}\, \sqrt {c d x +a e}\, c^{2} d^{2} f \,g^{2} x}{3}-\frac {16 \sqrt {g x +f}\, \sqrt {c d x +a e}\, c^{2} d^{2} g^{3} x^{2}}{15}+\frac {16 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, c^{2} d^{2} f^{3}}{15}+\frac {16 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, c^{2} d^{2} f^{2} g x}{5}+\frac {16 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, c^{2} d^{2} f \,g^{2} x^{2}}{5}+\frac {16 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, c^{2} d^{2} g^{3} x^{3}}{15}}{g \left (a^{3} e^{3} g^{6} x^{3}-3 a^{2} c d \,e^{2} f \,g^{5} x^{3}+3 a \,c^{2} d^{2} e \,f^{2} g^{4} x^{3}-c^{3} d^{3} f^{3} g^{3} x^{3}+3 a^{3} e^{3} f \,g^{5} x^{2}-9 a^{2} c d \,e^{2} f^{2} g^{4} x^{2}+9 a \,c^{2} d^{2} e \,f^{3} g^{3} x^{2}-3 c^{3} d^{3} f^{4} g^{2} x^{2}+3 a^{3} e^{3} f^{2} g^{4} x -9 a^{2} c d \,e^{2} f^{3} g^{3} x +9 a \,c^{2} d^{2} e \,f^{4} g^{2} x -3 c^{3} d^{3} f^{5} g x +a^{3} e^{3} f^{3} g^{3}-3 a^{2} c d \,e^{2} f^{4} g^{2}+3 a \,c^{2} d^{2} e \,f^{5} g -c^{3} d^{3} f^{6}\right )} \] Input:
int((e*x+d)^(1/2)/(g*x+f)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x)
Output:
(2*( - 3*sqrt(f + g*x)*sqrt(a*e + c*d*x)*a**2*e**2*g**3 + 10*sqrt(f + g*x) *sqrt(a*e + c*d*x)*a*c*d*e*f*g**2 + 4*sqrt(f + g*x)*sqrt(a*e + c*d*x)*a*c* d*e*g**3*x - 15*sqrt(f + g*x)*sqrt(a*e + c*d*x)*c**2*d**2*f**2*g - 20*sqrt (f + g*x)*sqrt(a*e + c*d*x)*c**2*d**2*f*g**2*x - 8*sqrt(f + g*x)*sqrt(a*e + c*d*x)*c**2*d**2*g**3*x**2 + 8*sqrt(g)*sqrt(d)*sqrt(c)*c**2*d**2*f**3 + 24*sqrt(g)*sqrt(d)*sqrt(c)*c**2*d**2*f**2*g*x + 24*sqrt(g)*sqrt(d)*sqrt(c) *c**2*d**2*f*g**2*x**2 + 8*sqrt(g)*sqrt(d)*sqrt(c)*c**2*d**2*g**3*x**3))/( 15*g*(a**3*e**3*f**3*g**3 + 3*a**3*e**3*f**2*g**4*x + 3*a**3*e**3*f*g**5*x **2 + a**3*e**3*g**6*x**3 - 3*a**2*c*d*e**2*f**4*g**2 - 9*a**2*c*d*e**2*f* *3*g**3*x - 9*a**2*c*d*e**2*f**2*g**4*x**2 - 3*a**2*c*d*e**2*f*g**5*x**3 + 3*a*c**2*d**2*e*f**5*g + 9*a*c**2*d**2*e*f**4*g**2*x + 9*a*c**2*d**2*e*f* *3*g**3*x**2 + 3*a*c**2*d**2*e*f**2*g**4*x**3 - c**3*d**3*f**6 - 3*c**3*d* *3*f**5*g*x - 3*c**3*d**3*f**4*g**2*x**2 - c**3*d**3*f**3*g**3*x**3))