Integrand size = 48, antiderivative size = 260 \[ \int \frac {(d+e x)^{5/2}}{(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=-\frac {2 (d+e x)^{3/2}}{3 (c d f-a e g) (f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {4 g \sqrt {d+e x}}{(c d f-a e g)^2 (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {16 g^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)^{3/2}}+\frac {32 c d g^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 (c d f-a e g)^4 \sqrt {d+e x} \sqrt {f+g x}} \] Output:
-2/3*(e*x+d)^(3/2)/(-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)+4*g*(e*x+d)^(1/2)/(-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)+16/3*g^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)^(3/2)+32/3*c*d*g^2*(a*d*e+(a*e^ 2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e*g+c*d*f)^4/(e*x+d)^(1/2)/(g*x+f)^(1/2)
Time = 0.31 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.58 \[ \int \frac {(d+e x)^{5/2}}{(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {2 (d+e x)^{3/2} \left (-a^3 e^3 g^3+3 a^2 c d e^2 g^2 (3 f+2 g x)+3 a c^2 d^2 e g \left (3 f^2+12 f g x+8 g^2 x^2\right )+c^3 d^3 \left (-f^3+6 f^2 g x+24 f g^2 x^2+16 g^3 x^3\right )\right )}{3 (c d f-a e g)^4 ((a e+c d x) (d+e x))^{3/2} (f+g x)^{3/2}} \] Input:
Integrate[(d + e*x)^(5/2)/((f + g*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c* d*e*x^2)^(5/2)),x]
Output:
(2*(d + e*x)^(3/2)*(-(a^3*e^3*g^3) + 3*a^2*c*d*e^2*g^2*(3*f + 2*g*x) + 3*a *c^2*d^2*e*g*(3*f^2 + 12*f*g*x + 8*g^2*x^2) + c^3*d^3*(-f^3 + 6*f^2*g*x + 24*f*g^2*x^2 + 16*g^3*x^3)))/(3*(c*d*f - a*e*g)^4*((a*e + c*d*x)*(d + e*x) )^(3/2)*(f + g*x)^(3/2))
Time = 0.96 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.10, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1252, 1252, 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.
| \(\displaystyle \int \frac {(d+e x)^{5/2}}{(f+g x)^{5/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1252 |
| \(\displaystyle -\frac {2 g \int \frac {(d+e x)^{3/2}}{(f+g x)^{5/2} \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{c d f-a e g}-\frac {2 (d+e x)^{3/2}}{3 (f+g x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)}\) |
| \(\Big \downarrow \) 1252 |
| \(\displaystyle -\frac {2 g \left (-\frac {4 g \int \frac {\sqrt {d+e x}}{(f+g x)^{5/2} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{c d f-a e g}-\frac {2 \sqrt {d+e x}}{(f+g x)^{3/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}\right )}{c d f-a e g}-\frac {2 (d+e x)^{3/2}}{3 (f+g x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)}\) |
| \(\Big \downarrow \) 1254 |
| \(\displaystyle -\frac {2 g \left (-\frac {4 g \left (\frac {2 c d \int \frac {\sqrt {d+e x}}{(f+g x)^{3/2} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{3 (c d f-a e g)}+\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt {d+e x} (f+g x)^{3/2} (c d f-a e g)}\right )}{c d f-a e g}-\frac {2 \sqrt {d+e x}}{(f+g x)^{3/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}\right )}{c d f-a e g}-\frac {2 (d+e x)^{3/2}}{3 (f+g x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)}\) |
| \(\Big \downarrow \) 1248 |
| \(\displaystyle -\frac {2 (d+e x)^{3/2}}{3 (f+g x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)}-\frac {2 g \left (-\frac {4 g \left (\frac {4 c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt {d+e x} \sqrt {f+g x} (c d f-a e g)^2}+\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt {d+e x} (f+g x)^{3/2} (c d f-a e g)}\right )}{c d f-a e g}-\frac {2 \sqrt {d+e x}}{(f+g x)^{3/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}\right )}{c d f-a e g}\) |
Input:
Int[(d + e*x)^(5/2)/((f + g*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^ 2)^(5/2)),x]
Output:
(-2*(d + e*x)^(3/2))/(3*(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)) - (2*g*((-2*Sqrt[d + e*x])/((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]) - (4*g*((2*Sqr t[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]*Sqrt[f + g*x])))/(c*d*f - a*e*g)))/(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)/((p + 1)*(c*e*f + c*d*g - b*e*g))), x] + Si mp[e^2*g*((m - n - 2)/((p + 1)*(c*e*f + c*d*g - b*e*g))) 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] && LtQ[p, -1] && RationalQ[n]
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 = 2.92 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.73
| method | result | size |
| default | \(-\frac {2 \sqrt {\left (e x +d \right ) \left (c d x +a e \right )}\, \left (-16 x^{3} g^{3} d^{3} c^{3}-24 a \,c^{2} d^{2} e \,g^{3} x^{2}-24 c^{3} d^{3} f \,g^{2} x^{2}-6 a^{2} c d \,e^{2} g^{3} x -36 a \,c^{2} d^{2} e f \,g^{2} x -6 c^{3} d^{3} f^{2} g x +a^{3} e^{3} g^{3}-9 a^{2} c d \,e^{2} f \,g^{2}-9 a \,c^{2} d^{2} e \,f^{2} g +f^{3} d^{3} c^{3}\right )}{3 \sqrt {e x +d}\, \left (g x +f \right )^{\frac {3}{2}} \left (c d x +a e \right )^{2} \left (a e g -d f c \right )^{4}}\) | \(191\) |
| gosper | \(-\frac {2 \left (c d x +a e \right ) \left (-16 x^{3} g^{3} d^{3} c^{3}-24 a \,c^{2} d^{2} e \,g^{3} x^{2}-24 c^{3} d^{3} f \,g^{2} x^{2}-6 a^{2} c d \,e^{2} g^{3} x -36 a \,c^{2} d^{2} e f \,g^{2} x -6 c^{3} d^{3} f^{2} g x +a^{3} e^{3} g^{3}-9 a^{2} c d \,e^{2} f \,g^{2}-9 a \,c^{2} d^{2} e \,f^{2} g +f^{3} d^{3} c^{3}\right ) \left (e x +d \right )^{\frac {5}{2}}}{3 \left (g x +f \right )^{\frac {3}{2}} \left (a^{4} e^{4} g^{4}-4 a^{3} c d \,e^{3} f \,g^{3}+6 a^{2} c^{2} d^{2} e^{2} f^{2} g^{2}-4 a \,c^{3} d^{3} e \,f^{3} g +f^{4} d^{4} c^{4}\right ) \left (c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}\) | \(258\) |
| orering | \(-\frac {2 \left (-16 x^{3} g^{3} d^{3} c^{3}-24 a \,c^{2} d^{2} e \,g^{3} x^{2}-24 c^{3} d^{3} f \,g^{2} x^{2}-6 a^{2} c d \,e^{2} g^{3} x -36 a \,c^{2} d^{2} e f \,g^{2} x -6 c^{3} d^{3} f^{2} g x +a^{3} e^{3} g^{3}-9 a^{2} c d \,e^{2} f \,g^{2}-9 a \,c^{2} d^{2} e \,f^{2} g +f^{3} d^{3} c^{3}\right ) \left (c d x +a e \right ) \left (e x +d \right )^{\frac {5}{2}}}{3 \left (a^{4} e^{4} g^{4}-4 a^{3} c d \,e^{3} f \,g^{3}+6 a^{2} c^{2} d^{2} e^{2} f^{2} g^{2}-4 a \,c^{3} d^{3} e \,f^{3} g +f^{4} d^{4} c^{4}\right ) \left (g x +f \right )^{\frac {3}{2}} {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {5}{2}}}\) | \(259\) |
Input:
int((e*x+d)^(5/2)/(g*x+f)^(5/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(5/2),x, method=_RETURNVERBOSE)
Output:
-2/3/(e*x+d)^(1/2)/(g*x+f)^(3/2)*((e*x+d)*(c*d*x+a*e))^(1/2)*(-16*c^3*d^3* g^3*x^3-24*a*c^2*d^2*e*g^3*x^2-24*c^3*d^3*f*g^2*x^2-6*a^2*c*d*e^2*g^3*x-36 *a*c^2*d^2*e*f*g^2*x-6*c^3*d^3*f^2*g*x+a^3*e^3*g^3-9*a^2*c*d*e^2*f*g^2-9*a *c^2*d^2*e*f^2*g+c^3*d^3*f^3)/(c*d*x+a*e)^2/(a*e*g-c*d*f)^4
Leaf count of result is larger than twice the leaf count of optimal. 1065 vs. \(2 (230) = 460\).
Time = 0.83 (sec) , antiderivative size = 1065, normalized size of antiderivative = 4.10 \[ \int \frac {(d+e x)^{5/2}}{(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:
integrate((e*x+d)^(5/2)/(g*x+f)^(5/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5 /2),x, algorithm="fricas")
Output:
2/3*(16*c^3*d^3*g^3*x^3 - c^3*d^3*f^3 + 9*a*c^2*d^2*e*f^2*g + 9*a^2*c*d*e^ 2*f*g^2 - a^3*e^3*g^3 + 24*(c^3*d^3*f*g^2 + a*c^2*d^2*e*g^3)*x^2 + 6*(c^3* d^3*f^2*g + 6*a*c^2*d^2*e*f*g^2 + a^2*c*d*e^2*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)/(a^2*c^4*d^5*e^2*f^6 - 4*a^3*c^3*d^4*e^3*f^5*g + 6*a^4*c^2*d^3*e^4*f^4*g^2 - 4*a^5*c*d^2*e^5*f^3 *g^3 + a^6*d*e^6*f^2*g^4 + (c^6*d^6*e*f^4*g^2 - 4*a*c^5*d^5*e^2*f^3*g^3 + 6*a^2*c^4*d^4*e^3*f^2*g^4 - 4*a^3*c^3*d^3*e^4*f*g^5 + a^4*c^2*d^2*e^5*g^6) *x^5 + (2*c^6*d^6*e*f^5*g + (c^6*d^7 - 6*a*c^5*d^5*e^2)*f^4*g^2 - 4*(a*c^5 *d^6*e - a^2*c^4*d^4*e^3)*f^3*g^3 + 2*(3*a^2*c^4*d^5*e^2 + 2*a^3*c^3*d^3*e ^4)*f^2*g^4 - 2*(2*a^3*c^3*d^4*e^3 + 3*a^4*c^2*d^2*e^5)*f*g^5 + (a^4*c^2*d ^3*e^4 + 2*a^5*c*d*e^6)*g^6)*x^4 + (c^6*d^6*e*f^6 + 2*c^6*d^7*f^5*g - 6*a^ 4*c^2*d^3*e^4*f*g^5 - 3*(2*a*c^5*d^6*e + 3*a^2*c^4*d^4*e^3)*f^4*g^2 + 4*(a ^2*c^4*d^5*e^2 + 4*a^3*c^3*d^3*e^4)*f^3*g^3 + (4*a^3*c^3*d^4*e^3 - 9*a^4*c ^2*d^2*e^5)*f^2*g^4 + (2*a^5*c*d^2*e^5 + a^6*e^7)*g^6)*x^3 - (6*a^2*c^4*d^ 4*e^3*f^5*g - 2*a^6*e^7*f*g^5 - a^6*d*e^6*g^6 - (c^6*d^7 + 2*a*c^5*d^5*e^2 )*f^6 + (9*a^2*c^4*d^5*e^2 - 4*a^3*c^3*d^3*e^4)*f^4*g^2 - 4*(4*a^3*c^3*d^4 *e^3 + a^4*c^2*d^2*e^5)*f^3*g^3 + 3*(3*a^4*c^2*d^3*e^4 + 2*a^5*c*d*e^6)*f^ 2*g^4)*x^2 + (2*a^6*d*e^6*f*g^5 + (2*a*c^5*d^6*e + a^2*c^4*d^4*e^3)*f^6 - 2*(3*a^2*c^4*d^5*e^2 + 2*a^3*c^3*d^3*e^4)*f^5*g + 2*(2*a^3*c^3*d^4*e^3 + 3 *a^4*c^2*d^2*e^5)*f^4*g^2 + 4*(a^4*c^2*d^3*e^4 - a^5*c*d*e^6)*f^3*g^3 -...
Timed out. \[ \int \frac {(d+e x)^{5/2}}{(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((e*x+d)**(5/2)/(g*x+f)**(5/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x** 2)**(5/2),x)
Output:
Timed out
\[ \int \frac {(d+e x)^{5/2}}{(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {5}{2}}}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}} {\left (g x + f\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((e*x+d)^(5/2)/(g*x+f)^(5/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5 /2),x, algorithm="maxima")
Output:
integrate((e*x + d)^(5/2)/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)*( g*x + f)^(5/2)), x)
Leaf count of result is larger than twice the leaf count of optimal. 1119 vs. \(2 (230) = 460\).
Time = 0.52 (sec) , antiderivative size = 1119, normalized size of antiderivative = 4.30 \[ \int \frac {(d+e x)^{5/2}}{(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:
integrate((e*x+d)^(5/2)/(g*x+f)^(5/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5 /2),x, algorithm="giac")
Output:
2/3*e^4*(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)*(8*(c^7*d^7*e^2*f^3*g^4*abs(g) - 3*a*c^6*d^6*e^3*f^2*g^5*abs(g) + 3*a^2*c^5*d^5*e^4*f*g^6*abs(g) - a^3*c^4 *d^4*e^5*g^7*abs(g))*(e^2*f + (e*x + d)*e*g - d*e*g)/(c^8*d^8*e^6*f^7*g^2 - 7*a*c^7*d^7*e^7*f^6*g^3 + 21*a^2*c^6*d^6*e^8*f^5*g^4 - 35*a^3*c^5*d^5*e^ 9*f^4*g^5 + 35*a^4*c^4*d^4*e^10*f^3*g^6 - 21*a^5*c^3*d^3*e^11*f^2*g^7 + 7* a^6*c^2*d^2*e^12*f*g^8 - a^7*c*d*e^13*g^9) - 9*(c^7*d^7*e^4*f^4*g^4*abs(g) - 4*a*c^6*d^6*e^5*f^3*g^5*abs(g) + 6*a^2*c^5*d^5*e^6*f^2*g^6*abs(g) - 4*a ^3*c^4*d^4*e^7*f*g^7*abs(g) + a^4*c^3*d^3*e^8*g^8*abs(g))/(c^8*d^8*e^6*f^7 *g^2 - 7*a*c^7*d^7*e^7*f^6*g^3 + 21*a^2*c^6*d^6*e^8*f^5*g^4 - 35*a^3*c^5*d ^5*e^9*f^4*g^5 + 35*a^4*c^4*d^4*e^10*f^3*g^6 - 21*a^5*c^3*d^3*e^11*f^2*g^7 + 7*a^6*c^2*d^2*e^12*f*g^8 - a^7*c*d*e^13*g^9))/(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 + 4*(4*sqrt(c*d*g)*c^3*d^3*e^4* f^2*g^5 - 8*sqrt(c*d*g)*a*c^2*d^2*e^5*f*g^6 + 4*sqrt(c*d*g)*a^2*c*d*e^6*g^ 7 + 9*sqrt(c*d*g)*(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^2*d ^2*e^2*f*g^4 - 9*sqrt(c*d*g)*(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*c*d*e^3*g^5 + 3*sqrt(c*d*g)*(sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*s qrt(c*d*g) - sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d...
Time = 7.79 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.60 \[ \int \frac {(d+e x)^{5/2}}{(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {16\,g\,x^2\,\left (a\,e\,g+c\,d\,f\right )\,\sqrt {d+e\,x}}{e\,{\left (a\,e\,g-c\,d\,f\right )}^4}-\frac {\sqrt {d+e\,x}\,\left (2\,a^3\,e^3\,g^3-18\,a^2\,c\,d\,e^2\,f\,g^2-18\,a\,c^2\,d^2\,e\,f^2\,g+2\,c^3\,d^3\,f^3\right )}{3\,c^2\,d^2\,e\,g\,{\left (a\,e\,g-c\,d\,f\right )}^4}+\frac {32\,c\,d\,g^2\,x^3\,\sqrt {d+e\,x}}{3\,e\,{\left (a\,e\,g-c\,d\,f\right )}^4}+\frac {4\,x\,\sqrt {d+e\,x}\,\left (a^2\,e^2\,g^2+6\,a\,c\,d\,e\,f\,g+c^2\,d^2\,f^2\right )}{c\,d\,e\,{\left (a\,e\,g-c\,d\,f\right )}^4}\right )}{x^4\,\sqrt {f+g\,x}+\frac {x^2\,\sqrt {f+g\,x}\,\left (g\,a^2\,e^3+2\,g\,a\,c\,d^2\,e+2\,f\,a\,c\,d\,e^2+f\,c^2\,d^3\right )}{c^2\,d^2\,e\,g}+\frac {a\,x\,\sqrt {f+g\,x}\,\left (2\,c\,f\,d^2+a\,g\,d\,e+a\,f\,e^2\right )}{c^2\,d^2\,g}+\frac {a^2\,e\,f\,\sqrt {f+g\,x}}{c^2\,d\,g}+\frac {x^3\,\sqrt {f+g\,x}\,\left (c\,g\,d^2+c\,f\,d\,e+2\,a\,g\,e^2\right )}{c\,d\,e\,g}} \] Input:
int((d + e*x)^(5/2)/((f + g*x)^(5/2)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^ 2)^(5/2)),x)
Output:
((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*((16*g*x^2*(a*e*g + c*d*f)* (d + e*x)^(1/2))/(e*(a*e*g - c*d*f)^4) - ((d + e*x)^(1/2)*(2*a^3*e^3*g^3 + 2*c^3*d^3*f^3 - 18*a*c^2*d^2*e*f^2*g - 18*a^2*c*d*e^2*f*g^2))/(3*c^2*d^2* e*g*(a*e*g - c*d*f)^4) + (32*c*d*g^2*x^3*(d + e*x)^(1/2))/(3*e*(a*e*g - c* d*f)^4) + (4*x*(d + e*x)^(1/2)*(a^2*e^2*g^2 + c^2*d^2*f^2 + 6*a*c*d*e*f*g) )/(c*d*e*(a*e*g - c*d*f)^4)))/(x^4*(f + g*x)^(1/2) + (x^2*(f + g*x)^(1/2)* (a^2*e^3*g + c^2*d^3*f + 2*a*c*d*e^2*f + 2*a*c*d^2*e*g))/(c^2*d^2*e*g) + ( a*x*(f + g*x)^(1/2)*(a*e^2*f + 2*c*d^2*f + a*d*e*g))/(c^2*d^2*g) + (a^2*e* f*(f + g*x)^(1/2))/(c^2*d*g) + (x^3*(f + g*x)^(1/2)*(2*a*e^2*g + c*d^2*g + c*d*e*f))/(c*d*e*g))
Time = 0.28 (sec) , antiderivative size = 760, normalized size of antiderivative = 2.92 \[ \int \frac {(d+e x)^{5/2}}{(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {-\frac {32 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, a c d e \,f^{2} g}{3}-\frac {64 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, a c d e f \,g^{2} x}{3}-\frac {32 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, a c d e \,g^{3} x^{2}}{3}-\frac {32 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, c^{2} d^{2} f^{2} g x}{3}-\frac {64 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, c^{2} d^{2} f \,g^{2} x^{2}}{3}-\frac {32 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, c^{2} d^{2} g^{3} x^{3}}{3}-\frac {2 \sqrt {g x +f}\, a^{3} e^{3} g^{3}}{3}+6 \sqrt {g x +f}\, a^{2} c d \,e^{2} f \,g^{2}+4 \sqrt {g x +f}\, a^{2} c d \,e^{2} g^{3} x +6 \sqrt {g x +f}\, a \,c^{2} d^{2} e \,f^{2} g +24 \sqrt {g x +f}\, a \,c^{2} d^{2} e f \,g^{2} x +16 \sqrt {g x +f}\, a \,c^{2} d^{2} e \,g^{3} x^{2}-\frac {2 \sqrt {g x +f}\, c^{3} d^{3} f^{3}}{3}+4 \sqrt {g x +f}\, c^{3} d^{3} f^{2} g x +16 \sqrt {g x +f}\, c^{3} d^{3} f \,g^{2} x^{2}+\frac {32 \sqrt {g x +f}\, c^{3} d^{3} g^{3} x^{3}}{3}}{\sqrt {c d x +a e}\, \left (a^{4} c d \,e^{4} g^{6} x^{3}-4 a^{3} c^{2} d^{2} e^{3} f \,g^{5} x^{3}+6 a^{2} c^{3} d^{3} e^{2} f^{2} g^{4} x^{3}-4 a \,c^{4} d^{4} e \,f^{3} g^{3} x^{3}+c^{5} d^{5} f^{4} g^{2} x^{3}+a^{5} e^{5} g^{6} x^{2}-2 a^{4} c d \,e^{4} f \,g^{5} x^{2}-2 a^{3} c^{2} d^{2} e^{3} f^{2} g^{4} x^{2}+8 a^{2} c^{3} d^{3} e^{2} f^{3} g^{3} x^{2}-7 a \,c^{4} d^{4} e \,f^{4} g^{2} x^{2}+2 c^{5} d^{5} f^{5} g \,x^{2}+2 a^{5} e^{5} f \,g^{5} x -7 a^{4} c d \,e^{4} f^{2} g^{4} x +8 a^{3} c^{2} d^{2} e^{3} f^{3} g^{3} x -2 a^{2} c^{3} d^{3} e^{2} f^{4} g^{2} x -2 a \,c^{4} d^{4} e \,f^{5} g x +c^{5} d^{5} f^{6} x +a^{5} e^{5} f^{2} g^{4}-4 a^{4} c d \,e^{4} f^{3} g^{3}+6 a^{3} c^{2} d^{2} e^{3} f^{4} g^{2}-4 a^{2} c^{3} d^{3} e^{2} f^{5} g +a \,c^{4} d^{4} e \,f^{6}\right )} \] Input:
int((e*x+d)^(5/2)/(g*x+f)^(5/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x)
Output:
(2*( - 16*sqrt(g)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a*c*d*e*f**2*g - 32*sq rt(g)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a*c*d*e*f*g**2*x - 16*sqrt(g)*sqrt (d)*sqrt(c)*sqrt(a*e + c*d*x)*a*c*d*e*g**3*x**2 - 16*sqrt(g)*sqrt(d)*sqrt( c)*sqrt(a*e + c*d*x)*c**2*d**2*f**2*g*x - 32*sqrt(g)*sqrt(d)*sqrt(c)*sqrt( a*e + c*d*x)*c**2*d**2*f*g**2*x**2 - 16*sqrt(g)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*c**2*d**2*g**3*x**3 - sqrt(f + g*x)*a**3*e**3*g**3 + 9*sqrt(f + g* x)*a**2*c*d*e**2*f*g**2 + 6*sqrt(f + g*x)*a**2*c*d*e**2*g**3*x + 9*sqrt(f + g*x)*a*c**2*d**2*e*f**2*g + 36*sqrt(f + g*x)*a*c**2*d**2*e*f*g**2*x + 24 *sqrt(f + g*x)*a*c**2*d**2*e*g**3*x**2 - sqrt(f + g*x)*c**3*d**3*f**3 + 6* sqrt(f + g*x)*c**3*d**3*f**2*g*x + 24*sqrt(f + g*x)*c**3*d**3*f*g**2*x**2 + 16*sqrt(f + g*x)*c**3*d**3*g**3*x**3))/(3*sqrt(a*e + c*d*x)*(a**5*e**5*f **2*g**4 + 2*a**5*e**5*f*g**5*x + a**5*e**5*g**6*x**2 - 4*a**4*c*d*e**4*f* *3*g**3 - 7*a**4*c*d*e**4*f**2*g**4*x - 2*a**4*c*d*e**4*f*g**5*x**2 + a**4 *c*d*e**4*g**6*x**3 + 6*a**3*c**2*d**2*e**3*f**4*g**2 + 8*a**3*c**2*d**2*e **3*f**3*g**3*x - 2*a**3*c**2*d**2*e**3*f**2*g**4*x**2 - 4*a**3*c**2*d**2* e**3*f*g**5*x**3 - 4*a**2*c**3*d**3*e**2*f**5*g - 2*a**2*c**3*d**3*e**2*f* *4*g**2*x + 8*a**2*c**3*d**3*e**2*f**3*g**3*x**2 + 6*a**2*c**3*d**3*e**2*f **2*g**4*x**3 + a*c**4*d**4*e*f**6 - 2*a*c**4*d**4*e*f**5*g*x - 7*a*c**4*d **4*e*f**4*g**2*x**2 - 4*a*c**4*d**4*e*f**3*g**3*x**3 + c**5*d**5*f**6*x + 2*c**5*d**5*f**5*g*x**2 + c**5*d**5*f**4*g**2*x**3))