Integrand size = 48, antiderivative size = 262 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 \sqrt {d+e x}}{(c d f-a e g) (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {12 g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^{5/2}}-\frac {16 c d g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)^{3/2}}-\frac {32 c^2 d^2 g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 (c d f-a e g)^4 \sqrt {d+e x} \sqrt {f+g x}} \] Output:
-2*(e*x+d)^(1/2)/(-a*e*g+c*d*f)/(g*x+f)^(5/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e *x^2)^(1/2)-12/5*g*(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)^(5/2)-16/5*c*d*g*(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)-32/5*c^2*d^2*g*(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 = 0.28 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.57 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 \sqrt {d+e x} \left (a^3 e^3 g^3-a^2 c d e^2 g^2 (5 f+2 g x)+a c^2 d^2 e g \left (15 f^2+20 f g x+8 g^2 x^2\right )+c^3 d^3 \left (5 f^3+30 f^2 g x+40 f g^2 x^2+16 g^3 x^3\right )\right )}{5 (c d f-a e g)^4 \sqrt {(a e+c d x) (d+e x)} (f+g x)^{5/2}} \] Input:
Integrate[(d + e*x)^(3/2)/((f + g*x)^(7/2)*(a*d*e + (c*d^2 + a*e^2)*x + c* d*e*x^2)^(3/2)),x]
Output:
(-2*Sqrt[d + e*x]*(a^3*e^3*g^3 - a^2*c*d*e^2*g^2*(5*f + 2*g*x) + a*c^2*d^2 *e*g*(15*f^2 + 20*f*g*x + 8*g^2*x^2) + c^3*d^3*(5*f^3 + 30*f^2*g*x + 40*f* g^2*x^2 + 16*g^3*x^3)))/(5*(c*d*f - a*e*g)^4*Sqrt[(a*e + c*d*x)*(d + e*x)] *(f + g*x)^(5/2))
Time = 0.95 (sec) , antiderivative size = 288, 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, 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.
| \(\displaystyle \int \frac {(d+e x)^{3/2}}{(f+g x)^{7/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1252 |
| \(\displaystyle -\frac {6 g \int \frac {\sqrt {d+e x}}{(f+g x)^{7/2} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{c d f-a e g}-\frac {2 \sqrt {d+e x}}{(f+g x)^{5/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}\) |
| \(\Big \downarrow \) 1254 |
| \(\displaystyle -\frac {6 g \left (\frac {4 c d \int \frac {\sqrt {d+e x}}{(f+g x)^{5/2} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{5 (c d f-a e g)}+\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 \sqrt {d+e x} (f+g x)^{5/2} (c d f-a e g)}\right )}{c d f-a e g}-\frac {2 \sqrt {d+e x}}{(f+g x)^{5/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}\) |
| \(\Big \downarrow \) 1254 |
| \(\displaystyle -\frac {6 g \left (\frac {4 c d \left (\frac {2 c d \int \frac {\sqrt {d+e x}}{(f+g x)^{3/2} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{3 (c d f-a e g)}+\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt {d+e x} (f+g x)^{3/2} (c d f-a e g)}\right )}{5 (c d f-a e g)}+\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 \sqrt {d+e x} (f+g x)^{5/2} (c d f-a e g)}\right )}{c d f-a e g}-\frac {2 \sqrt {d+e x}}{(f+g x)^{5/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}\) |
| \(\Big \downarrow \) 1248 |
| \(\displaystyle -\frac {6 g \left (\frac {4 c d \left (\frac {4 c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt {d+e x} \sqrt {f+g x} (c d f-a e g)^2}+\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt {d+e x} (f+g x)^{3/2} (c d f-a e g)}\right )}{5 (c d f-a e g)}+\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 \sqrt {d+e x} (f+g x)^{5/2} (c d f-a e g)}\right )}{c d f-a e g}-\frac {2 \sqrt {d+e x}}{(f+g x)^{5/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}\) |
Input:
Int[(d + e*x)^(3/2)/((f + g*x)^(7/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^ 2)^(3/2)),x]
Output:
(-2*Sqrt[d + e*x])/((c*d*f - a*e*g)*(f + g*x)^(5/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - (6*g*((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]*Sqrt[f + g*x])))/(5*(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 = 2.89 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.73
| method | result | size |
| default | \(-\frac {2 \sqrt {\left (e x +d \right ) \left (c d x +a e \right )}\, \left (16 x^{3} g^{3} d^{3} c^{3}+8 a \,c^{2} d^{2} e \,g^{3} x^{2}+40 c^{3} d^{3} f \,g^{2} x^{2}-2 a^{2} c d \,e^{2} g^{3} x +20 a \,c^{2} d^{2} e f \,g^{2} x +30 c^{3} d^{3} f^{2} g x +a^{3} e^{3} g^{3}-5 a^{2} c d \,e^{2} f \,g^{2}+15 a \,c^{2} d^{2} e \,f^{2} g +5 f^{3} d^{3} c^{3}\right )}{5 \sqrt {e x +d}\, \left (g x +f \right )^{\frac {5}{2}} \left (c d x +a e \right ) \left (a e g -d f c \right )^{4}}\) | \(192\) |
| gosper | \(-\frac {2 \left (c d x +a e \right ) \left (16 x^{3} g^{3} d^{3} c^{3}+8 a \,c^{2} d^{2} e \,g^{3} x^{2}+40 c^{3} d^{3} f \,g^{2} x^{2}-2 a^{2} c d \,e^{2} g^{3} x +20 a \,c^{2} d^{2} e f \,g^{2} x +30 c^{3} d^{3} f^{2} g x +a^{3} e^{3} g^{3}-5 a^{2} c d \,e^{2} f \,g^{2}+15 a \,c^{2} d^{2} e \,f^{2} g +5 f^{3} d^{3} c^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}}{5 \left (g x +f \right )^{\frac {5}{2}} \left (a^{4} e^{4} g^{4}-4 a^{3} c d \,e^{3} f \,g^{3}+6 a^{2} c^{2} d^{2} e^{2} f^{2} g^{2}-4 a \,c^{3} d^{3} e \,f^{3} g +f^{4} d^{4} c^{4}\right ) \left (c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}\) | \(259\) |
| orering | \(-\frac {2 \left (16 x^{3} g^{3} d^{3} c^{3}+8 a \,c^{2} d^{2} e \,g^{3} x^{2}+40 c^{3} d^{3} f \,g^{2} x^{2}-2 a^{2} c d \,e^{2} g^{3} x +20 a \,c^{2} d^{2} e f \,g^{2} x +30 c^{3} d^{3} f^{2} g x +a^{3} e^{3} g^{3}-5 a^{2} c d \,e^{2} f \,g^{2}+15 a \,c^{2} d^{2} e \,f^{2} g +5 f^{3} d^{3} c^{3}\right ) \left (c d x +a e \right ) \left (e x +d \right )^{\frac {3}{2}}}{5 \left (a^{4} e^{4} g^{4}-4 a^{3} c d \,e^{3} f \,g^{3}+6 a^{2} c^{2} d^{2} e^{2} f^{2} g^{2}-4 a \,c^{3} d^{3} e \,f^{3} g +f^{4} d^{4} c^{4}\right ) \left (g x +f \right )^{\frac {5}{2}} {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {3}{2}}}\) | \(260\) |
Input:
int((e*x+d)^(3/2)/(g*x+f)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2),x, method=_RETURNVERBOSE)
Output:
-2/5/(e*x+d)^(1/2)/(g*x+f)^(5/2)*((e*x+d)*(c*d*x+a*e))^(1/2)*(16*c^3*d^3*g ^3*x^3+8*a*c^2*d^2*e*g^3*x^2+40*c^3*d^3*f*g^2*x^2-2*a^2*c*d*e^2*g^3*x+20*a *c^2*d^2*e*f*g^2*x+30*c^3*d^3*f^2*g*x+a^3*e^3*g^3-5*a^2*c*d*e^2*f*g^2+15*a *c^2*d^2*e*f^2*g+5*c^3*d^3*f^3)/(c*d*x+a*e)/(a*e*g-c*d*f)^4
Leaf count of result is larger than twice the leaf count of optimal. 1062 vs. \(2 (232) = 464\).
Time = 1.27 (sec) , antiderivative size = 1062, normalized size of antiderivative = 4.05 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
integrate((e*x+d)^(3/2)/(g*x+f)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3 /2),x, algorithm="fricas")
Output:
-2/5*(16*c^3*d^3*g^3*x^3 + 5*c^3*d^3*f^3 + 15*a*c^2*d^2*e*f^2*g - 5*a^2*c* d*e^2*f*g^2 + a^3*e^3*g^3 + 8*(5*c^3*d^3*f*g^2 + a*c^2*d^2*e*g^3)*x^2 + 2* (15*c^3*d^3*f^2*g + 10*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*c^4*d^5*e* f^7 - 4*a^2*c^3*d^4*e^2*f^6*g + 6*a^3*c^2*d^3*e^3*f^5*g^2 - 4*a^4*c*d^2*e^ 4*f^4*g^3 + a^5*d*e^5*f^3*g^4 + (c^5*d^5*e*f^4*g^3 - 4*a*c^4*d^4*e^2*f^3*g ^4 + 6*a^2*c^3*d^3*e^3*f^2*g^5 - 4*a^3*c^2*d^2*e^4*f*g^6 + a^4*c*d*e^5*g^7 )*x^5 + (3*c^5*d^5*e*f^5*g^2 + (c^5*d^6 - 11*a*c^4*d^4*e^2)*f^4*g^3 - 2*(2 *a*c^4*d^5*e - 7*a^2*c^3*d^3*e^3)*f^3*g^4 + 6*(a^2*c^3*d^4*e^2 - a^3*c^2*d ^2*e^4)*f^2*g^5 - (4*a^3*c^2*d^3*e^3 + a^4*c*d*e^5)*f*g^6 + (a^4*c*d^2*e^4 + a^5*e^6)*g^7)*x^4 + (3*c^5*d^5*e*f^6*g + a^5*d*e^5*g^7 + 3*(c^5*d^6 - 3 *a*c^4*d^4*e^2)*f^5*g^2 - (11*a*c^4*d^5*e - 6*a^2*c^3*d^3*e^3)*f^4*g^3 + 2 *(7*a^2*c^3*d^4*e^2 + 3*a^3*c^2*d^2*e^4)*f^3*g^4 - 3*(2*a^3*c^2*d^3*e^3 + 3*a^4*c*d*e^5)*f^2*g^5 - (a^4*c*d^2*e^4 - 3*a^5*e^6)*f*g^6)*x^3 + (c^5*d^5 *e*f^7 + 3*a^5*d*e^5*f*g^6 + (3*c^5*d^6 - a*c^4*d^4*e^2)*f^6*g - 3*(3*a*c^ 4*d^5*e + 2*a^2*c^3*d^3*e^3)*f^5*g^2 + 2*(3*a^2*c^3*d^4*e^2 + 7*a^3*c^2*d^ 2*e^4)*f^4*g^3 + (6*a^3*c^2*d^3*e^3 - 11*a^4*c*d*e^5)*f^3*g^4 - 3*(3*a^4*c *d^2*e^4 - a^5*e^6)*f^2*g^5)*x^2 + (3*a^5*d*e^5*f^2*g^5 + (c^5*d^6 + a*c^4 *d^4*e^2)*f^7 - (a*c^4*d^5*e + 4*a^2*c^3*d^3*e^3)*f^6*g - 6*(a^2*c^3*d^4*e ^2 - a^3*c^2*d^2*e^4)*f^5*g^2 + 2*(7*a^3*c^2*d^3*e^3 - 2*a^4*c*d*e^5)*f...
Timed out. \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((e*x+d)**(3/2)/(g*x+f)**(7/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x** 2)**(3/2),x)
Output:
Timed out
\[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {3}{2}}}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate((e*x+d)^(3/2)/(g*x+f)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3 /2),x, algorithm="maxima")
Output:
integrate((e*x + d)^(3/2)/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*( g*x + f)^(7/2)), x)
Leaf count of result is larger than twice the leaf count of optimal. 1476 vs. \(2 (232) = 464\).
Time = 0.62 (sec) , antiderivative size = 1476, normalized size of antiderivative = 5.63 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:
integrate((e*x+d)^(3/2)/(g*x+f)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3 /2),x, algorithm="giac")
Output:
2/5*(5*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)*c^3*d^3*g^2/((c^4*d^4*e^4*f^4*abs( g) - 4*a*c^3*d^3*e^5*f^3*g*abs(g) + 6*a^2*c^2*d^2*e^6*f^2*g^2*abs(g) - 4*a ^3*c*d*e^7*f*g^3*abs(g) + a^4*e^8*g^4*abs(g))*(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*(11*sqrt(c*d*g)*c^6*d^6*e^8*f^4 *g^6 - 44*sqrt(c*d*g)*a*c^5*d^5*e^9*f^3*g^7 + 66*sqrt(c*d*g)*a^2*c^4*d^4*e ^10*f^2*g^8 - 44*sqrt(c*d*g)*a^3*c^3*d^3*e^11*f*g^9 + 11*sqrt(c*d*g)*a^4*c ^2*d^2*e^12*g^10 + 50*sqrt(c*d*g)*(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))^2*c^5*d^5*e^6*f^3*g^5 - 150*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^4*d^4*e^7*f^2*g^6 + 150*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^2*c^3*d^3*e^8*f*g^7 - 50*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^3*c^2*d^2*e^9 *g^8 + 80*sqrt(c*d*g)*(sqrt(e^2*f + (e*x + d)*e*g - d*e*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))^4*c ^4*d^4*e^4*f^2*g^4 - 160*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 - ...
Time = 7.97 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.58 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {4\,x\,\sqrt {d+e\,x}\,\left (-a^2\,e^2\,g^2+10\,a\,c\,d\,e\,f\,g+15\,c^2\,d^2\,f^2\right )}{5\,e\,g\,{\left (a\,e\,g-c\,d\,f\right )}^4}+\frac {\sqrt {d+e\,x}\,\left (\frac {2\,a^3\,e^3\,g^3}{5}-2\,a^2\,c\,d\,e^2\,f\,g^2+6\,a\,c^2\,d^2\,e\,f^2\,g+2\,c^3\,d^3\,f^3\right )}{c\,d\,e\,g^2\,{\left (a\,e\,g-c\,d\,f\right )}^4}+\frac {32\,c^2\,d^2\,g\,x^3\,\sqrt {d+e\,x}}{5\,e\,{\left (a\,e\,g-c\,d\,f\right )}^4}+\frac {16\,c\,d\,x^2\,\left (a\,e\,g+5\,c\,d\,f\right )\,\sqrt {d+e\,x}}{5\,e\,{\left (a\,e\,g-c\,d\,f\right )}^4}\right )}{x^4\,\sqrt {f+g\,x}+\frac {a\,f^2\,\sqrt {f+g\,x}}{c\,g^2}+\frac {x^2\,\sqrt {f+g\,x}\,\left (2\,c\,d^2\,f\,g+c\,d\,e\,f^2+a\,d\,e\,g^2+2\,a\,e^2\,f\,g\right )}{c\,d\,e\,g^2}+\frac {x^3\,\sqrt {f+g\,x}\,\left (c\,g\,d^2+2\,c\,f\,d\,e+a\,g\,e^2\right )}{c\,d\,e\,g}+\frac {f\,x\,\sqrt {f+g\,x}\,\left (c\,f\,d^2+2\,a\,g\,d\,e+a\,f\,e^2\right )}{c\,d\,e\,g^2}} \] Input:
int((d + e*x)^(3/2)/((f + g*x)^(7/2)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^ 2)^(3/2)),x)
Output:
-((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*((4*x*(d + e*x)^(1/2)*(15* c^2*d^2*f^2 - a^2*e^2*g^2 + 10*a*c*d*e*f*g))/(5*e*g*(a*e*g - c*d*f)^4) + ( (d + e*x)^(1/2)*((2*a^3*e^3*g^3)/5 + 2*c^3*d^3*f^3 + 6*a*c^2*d^2*e*f^2*g - 2*a^2*c*d*e^2*f*g^2))/(c*d*e*g^2*(a*e*g - c*d*f)^4) + (32*c^2*d^2*g*x^3*( d + e*x)^(1/2))/(5*e*(a*e*g - c*d*f)^4) + (16*c*d*x^2*(a*e*g + 5*c*d*f)*(d + e*x)^(1/2))/(5*e*(a*e*g - c*d*f)^4)))/(x^4*(f + g*x)^(1/2) + (a*f^2*(f + g*x)^(1/2))/(c*g^2) + (x^2*(f + g*x)^(1/2)*(a*d*e*g^2 + c*d*e*f^2 + 2*a* e^2*f*g + 2*c*d^2*f*g))/(c*d*e*g^2) + (x^3*(f + g*x)^(1/2)*(a*e^2*g + c*d^ 2*g + 2*c*d*e*f))/(c*d*e*g) + (f*x*(f + g*x)^(1/2)*(a*e^2*f + c*d^2*f + 2* a*d*e*g))/(c*d*e*g^2))
Time = 0.33 (sec) , antiderivative size = 664, normalized size of antiderivative = 2.53 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {\frac {32 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, c^{2} d^{2} f^{3}}{5}+\frac {96 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, c^{2} d^{2} f^{2} g x}{5}+\frac {96 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, c^{2} d^{2} f \,g^{2} x^{2}}{5}+\frac {32 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, c^{2} d^{2} g^{3} x^{3}}{5}-\frac {2 \sqrt {g x +f}\, a^{3} e^{3} g^{3}}{5}+2 \sqrt {g x +f}\, a^{2} c d \,e^{2} f \,g^{2}+\frac {4 \sqrt {g x +f}\, a^{2} c d \,e^{2} g^{3} x}{5}-6 \sqrt {g x +f}\, a \,c^{2} d^{2} e \,f^{2} g -8 \sqrt {g x +f}\, a \,c^{2} d^{2} e f \,g^{2} x -\frac {16 \sqrt {g x +f}\, a \,c^{2} d^{2} e \,g^{3} x^{2}}{5}-2 \sqrt {g x +f}\, c^{3} d^{3} f^{3}-12 \sqrt {g x +f}\, c^{3} d^{3} f^{2} g x -16 \sqrt {g x +f}\, c^{3} d^{3} f \,g^{2} x^{2}-\frac {32 \sqrt {g x +f}\, c^{3} d^{3} g^{3} x^{3}}{5}}{\sqrt {c d x +a e}\, \left (a^{4} e^{4} g^{7} x^{3}-4 a^{3} c d \,e^{3} f \,g^{6} x^{3}+6 a^{2} c^{2} d^{2} e^{2} f^{2} g^{5} x^{3}-4 a \,c^{3} d^{3} e \,f^{3} g^{4} x^{3}+c^{4} d^{4} f^{4} g^{3} x^{3}+3 a^{4} e^{4} f \,g^{6} x^{2}-12 a^{3} c d \,e^{3} f^{2} g^{5} x^{2}+18 a^{2} c^{2} d^{2} e^{2} f^{3} g^{4} x^{2}-12 a \,c^{3} d^{3} e \,f^{4} g^{3} x^{2}+3 c^{4} d^{4} f^{5} g^{2} x^{2}+3 a^{4} e^{4} f^{2} g^{5} x -12 a^{3} c d \,e^{3} f^{3} g^{4} x +18 a^{2} c^{2} d^{2} e^{2} f^{4} g^{3} x -12 a \,c^{3} d^{3} e \,f^{5} g^{2} x +3 c^{4} d^{4} f^{6} g x +a^{4} e^{4} f^{3} g^{4}-4 a^{3} c d \,e^{3} f^{4} g^{3}+6 a^{2} c^{2} d^{2} e^{2} f^{5} g^{2}-4 a \,c^{3} d^{3} e \,f^{6} g +c^{4} d^{4} f^{7}\right )} \] Input:
int((e*x+d)^(3/2)/(g*x+f)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x)
Output:
(2*(16*sqrt(g)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*c**2*d**2*f**3 + 48*sqrt( g)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*c**2*d**2*f**2*g*x + 48*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)*sq rt(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 + 5*sqrt(f + g*x)*a**2*c*d*e**2*f*g**2 + 2*sqrt(f + g*x)*a**2*c*d*e**2*g* *3*x - 15*sqrt(f + g*x)*a*c**2*d**2*e*f**2*g - 20*sqrt(f + g*x)*a*c**2*d** 2*e*f*g**2*x - 8*sqrt(f + g*x)*a*c**2*d**2*e*g**3*x**2 - 5*sqrt(f + g*x)*c **3*d**3*f**3 - 30*sqrt(f + g*x)*c**3*d**3*f**2*g*x - 40*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))/(5*sqrt(a*e + c*d*x)*(a**4*e**4*f**3*g**4 + 3*a**4*e**4*f**2*g**5*x + 3*a**4*e**4*f*g**6 *x**2 + a**4*e**4*g**7*x**3 - 4*a**3*c*d*e**3*f**4*g**3 - 12*a**3*c*d*e**3 *f**3*g**4*x - 12*a**3*c*d*e**3*f**2*g**5*x**2 - 4*a**3*c*d*e**3*f*g**6*x* *3 + 6*a**2*c**2*d**2*e**2*f**5*g**2 + 18*a**2*c**2*d**2*e**2*f**4*g**3*x + 18*a**2*c**2*d**2*e**2*f**3*g**4*x**2 + 6*a**2*c**2*d**2*e**2*f**2*g**5* x**3 - 4*a*c**3*d**3*e*f**6*g - 12*a*c**3*d**3*e*f**5*g**2*x - 12*a*c**3*d **3*e*f**4*g**3*x**2 - 4*a*c**3*d**3*e*f**3*g**4*x**3 + c**4*d**4*f**7 + 3 *c**4*d**4*f**6*g*x + 3*c**4*d**4*f**5*g**2*x**2 + c**4*d**4*f**4*g**3*x** 3))