Integrand size = 48, antiderivative size = 129 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^{9/2}} \, dx=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 (c d f-a e g) (d+e x)^{5/2} (f+g x)^{7/2}}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{35 (c d f-a e g)^2 (d+e x)^{5/2} (f+g x)^{5/2}} \]
2/7*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(-a*e*g+c*d*f)/(e*x+d)^(5/2)/( g*x+f)^(7/2)+4/35*c*d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(-a*e*g+c*d* f)^2/(e*x+d)^(5/2)/(g*x+f)^(5/2)
Time = 0.12 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.53 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^{9/2}} \, dx=\frac {2 ((a e+c d x) (d+e x))^{5/2} (-5 a e g+c d (7 f+2 g x))}{35 (c d f-a e g)^2 (d+e x)^{5/2} (f+g x)^{7/2}} \]
(2*((a*e + c*d*x)*(d + e*x))^(5/2)*(-5*a*e*g + c*d*(7*f + 2*g*x)))/(35*(c* d*f - a*e*g)^2*(d + e*x)^(5/2)*(f + g*x)^(7/2))
Time = 0.33 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {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.
| \(\displaystyle \int \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^{9/2}} \, dx\) |
| \(\Big \downarrow \) 1254 |
| \(\displaystyle \frac {2 c d \int \frac {\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^{7/2}}dx}{7 (c d f-a e g)}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 (d+e x)^{5/2} (f+g x)^{7/2} (c d f-a e g)}\) |
| \(\Big \downarrow \) 1248 |
| \(\displaystyle \frac {4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{35 (d+e x)^{5/2} (f+g x)^{5/2} (c d f-a e g)^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 (d+e x)^{5/2} (f+g x)^{7/2} (c d f-a e g)}\) |
(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(7*(c*d*f - a*e*g)*(d + e*x)^(5/2)*(f + g*x)^(7/2)) + (4*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^ 2)^(5/2))/(35*(c*d*f - a*e*g)^2*(d + e*x)^(5/2)*(f + g*x)^(5/2))
3.8.48.3.1 Defintions of rubi rules used
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 = 0.54 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.77
| method | result | size |
| gosper | \(-\frac {2 \left (c d x +a e \right ) \left (-2 c d g x +5 a e g -7 c d f \right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}{35 \left (g x +f \right )^{\frac {7}{2}} \left (a^{2} e^{2} g^{2}-2 a c d e f g +c^{2} d^{2} f^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}\) | \(99\) |
| default | \(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (-2 g \,x^{2} c^{2} d^{2}+3 a c d e g x -7 c^{2} d^{2} f x +5 a^{2} e^{2} g -7 a c d e f \right ) \left (c d x +a e \right )}{35 \sqrt {e x +d}\, \left (g x +f \right )^{\frac {7}{2}} \left (a e g -c d f \right )^{2}}\) | \(100\) |
int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(3/2)/(g*x+f)^(9/2),x, method=_RETURNVERBOSE)
-2/35*(c*d*x+a*e)*(-2*c*d*g*x+5*a*e*g-7*c*d*f)*(c*d*e*x^2+a*e^2*x+c*d^2*x+ a*d*e)^(3/2)/(g*x+f)^(7/2)/(a^2*e^2*g^2-2*a*c*d*e*f*g+c^2*d^2*f^2)/(e*x+d) ^(3/2)
Leaf count of result is larger than twice the leaf count of optimal. 526 vs. \(2 (113) = 226\).
Time = 0.50 (sec) , antiderivative size = 526, normalized size of antiderivative = 4.08 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^{9/2}} \, dx=\frac {2 \, {\left (2 \, c^{3} d^{3} g x^{3} + 7 \, a^{2} c d e^{2} f - 5 \, a^{3} e^{3} g + {\left (7 \, c^{3} d^{3} f - a c^{2} d^{2} e g\right )} x^{2} + 2 \, {\left (7 \, a c^{2} d^{2} e f - 4 \, a^{2} c d e^{2} g\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d} \sqrt {g x + f}}{35 \, {\left (c^{2} d^{3} f^{6} - 2 \, a c d^{2} e f^{5} g + a^{2} d e^{2} f^{4} g^{2} + {\left (c^{2} d^{2} e f^{2} g^{4} - 2 \, a c d e^{2} f g^{5} + a^{2} e^{3} g^{6}\right )} x^{5} + {\left (4 \, c^{2} d^{2} e f^{3} g^{3} + a^{2} d e^{2} g^{6} + {\left (c^{2} d^{3} - 8 \, a c d e^{2}\right )} f^{2} g^{4} - 2 \, {\left (a c d^{2} e - 2 \, a^{2} e^{3}\right )} f g^{5}\right )} x^{4} + 2 \, {\left (3 \, c^{2} d^{2} e f^{4} g^{2} + 2 \, a^{2} d e^{2} f g^{5} + 2 \, {\left (c^{2} d^{3} - 3 \, a c d e^{2}\right )} f^{3} g^{3} - {\left (4 \, a c d^{2} e - 3 \, a^{2} e^{3}\right )} f^{2} g^{4}\right )} x^{3} + 2 \, {\left (2 \, c^{2} d^{2} e f^{5} g + 3 \, a^{2} d e^{2} f^{2} g^{4} + {\left (3 \, c^{2} d^{3} - 4 \, a c d e^{2}\right )} f^{4} g^{2} - 2 \, {\left (3 \, a c d^{2} e - a^{2} e^{3}\right )} f^{3} g^{3}\right )} x^{2} + {\left (c^{2} d^{2} e f^{6} + 4 \, a^{2} d e^{2} f^{3} g^{3} + 2 \, {\left (2 \, c^{2} d^{3} - a c d e^{2}\right )} f^{5} g - {\left (8 \, a c d^{2} e - a^{2} e^{3}\right )} f^{4} g^{2}\right )} x\right )}} \]
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(3/2)/(g*x+f)^(9 /2),x, algorithm="fricas")
2/35*(2*c^3*d^3*g*x^3 + 7*a^2*c*d*e^2*f - 5*a^3*e^3*g + (7*c^3*d^3*f - a*c ^2*d^2*e*g)*x^2 + 2*(7*a*c^2*d^2*e*f - 4*a^2*c*d*e^2*g)*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^2*d^3*f^6 - 2* a*c*d^2*e*f^5*g + a^2*d*e^2*f^4*g^2 + (c^2*d^2*e*f^2*g^4 - 2*a*c*d*e^2*f*g ^5 + a^2*e^3*g^6)*x^5 + (4*c^2*d^2*e*f^3*g^3 + a^2*d*e^2*g^6 + (c^2*d^3 - 8*a*c*d*e^2)*f^2*g^4 - 2*(a*c*d^2*e - 2*a^2*e^3)*f*g^5)*x^4 + 2*(3*c^2*d^2 *e*f^4*g^2 + 2*a^2*d*e^2*f*g^5 + 2*(c^2*d^3 - 3*a*c*d*e^2)*f^3*g^3 - (4*a* c*d^2*e - 3*a^2*e^3)*f^2*g^4)*x^3 + 2*(2*c^2*d^2*e*f^5*g + 3*a^2*d*e^2*f^2 *g^4 + (3*c^2*d^3 - 4*a*c*d*e^2)*f^4*g^2 - 2*(3*a*c*d^2*e - a^2*e^3)*f^3*g ^3)*x^2 + (c^2*d^2*e*f^6 + 4*a^2*d*e^2*f^3*g^3 + 2*(2*c^2*d^3 - a*c*d*e^2) *f^5*g - (8*a*c*d^2*e - a^2*e^3)*f^4*g^2)*x)
Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^{9/2}} \, dx=\text {Timed out} \]
\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^{9/2}} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{\frac {9}{2}}} \,d x } \]
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(3/2)/(g*x+f)^(9 /2),x, algorithm="maxima")
integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)/((e*x + d)^(3/2)*( g*x + f)^(9/2)), x)
Leaf count of result is larger than twice the leaf count of optimal. 1001 vs. \(2 (113) = 226\).
Time = 0.74 (sec) , antiderivative size = 1001, normalized size of antiderivative = 7.76 \[ \int \frac {\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} (f+g x)^{9/2}} \, dx=-\frac {2 \, {\left (7 \, \sqrt {-c d^{2} e + a e^{3}} c^{3} d^{5} e f {\left | c \right |} {\left | d \right |} - 14 \, \sqrt {-c d^{2} e + a e^{3}} a c^{2} d^{3} e^{3} f {\left | c \right |} {\left | d \right |} + 7 \, \sqrt {-c d^{2} e + a e^{3}} a^{2} c d e^{5} f {\left | c \right |} {\left | d \right |} - 2 \, \sqrt {-c d^{2} e + a e^{3}} c^{3} d^{6} g {\left | c \right |} {\left | d \right |} - \sqrt {-c d^{2} e + a e^{3}} a c^{2} d^{4} e^{2} g {\left | c \right |} {\left | d \right |} + 8 \, \sqrt {-c d^{2} e + a e^{3}} a^{2} c d^{2} e^{4} g {\left | c \right |} {\left | d \right |} - 5 \, \sqrt {-c d^{2} e + a e^{3}} a^{3} e^{6} g {\left | c \right |} {\left | d \right |}\right )}}{35 \, {\left (\sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} c^{2} d^{2} e^{3} f^{5} - 3 \, \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} c^{2} d^{3} e^{2} f^{4} g - 2 \, \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} a c d e^{4} f^{4} g + 3 \, \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} c^{2} d^{4} e f^{3} g^{2} + 6 \, \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} a c d^{2} e^{3} f^{3} g^{2} + \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} a^{2} e^{5} f^{3} g^{2} - \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} c^{2} d^{5} f^{2} g^{3} - 6 \, \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} a c d^{3} e^{2} f^{2} g^{3} - 3 \, \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} a^{2} d e^{4} f^{2} g^{3} + 2 \, \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} a c d^{4} e f g^{4} + 3 \, \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} a^{2} d^{2} e^{3} f g^{4} - \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} a^{2} d^{3} e^{2} g^{5}\right )}} + \frac {2 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} {\left (\frac {2 \, {\left (c^{7} d^{7} e^{6} f g^{4} {\left | c \right |} {\left | d \right |} - a c^{6} d^{6} e^{7} g^{5} {\left | c \right |} {\left | d \right |}\right )} {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}}{c^{3} d^{3} e^{6} f^{3} g^{3} - 3 \, a c^{2} d^{2} e^{7} f^{2} g^{4} + 3 \, a^{2} c d e^{8} f g^{5} - a^{3} e^{9} g^{6}} + \frac {7 \, {\left (c^{8} d^{8} e^{8} f^{2} g^{3} {\left | c \right |} {\left | d \right |} - 2 \, a c^{7} d^{7} e^{9} f g^{4} {\left | c \right |} {\left | d \right |} + a^{2} c^{6} d^{6} e^{10} g^{5} {\left | c \right |} {\left | d \right |}\right )}}{c^{3} d^{3} e^{6} f^{3} g^{3} - 3 \, a c^{2} d^{2} e^{7} f^{2} g^{4} + 3 \, a^{2} c d e^{8} f g^{5} - a^{3} e^{9} g^{6}}\right )}}{35 \, {\left (c^{2} d^{2} e^{2} f - a c d e^{3} g + {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )} c d g\right )}^{\frac {7}{2}}} \]
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(3/2)/(g*x+f)^(9 /2),x, algorithm="giac")
-2/35*(7*sqrt(-c*d^2*e + a*e^3)*c^3*d^5*e*f*abs(c)*abs(d) - 14*sqrt(-c*d^2 *e + a*e^3)*a*c^2*d^3*e^3*f*abs(c)*abs(d) + 7*sqrt(-c*d^2*e + a*e^3)*a^2*c *d*e^5*f*abs(c)*abs(d) - 2*sqrt(-c*d^2*e + a*e^3)*c^3*d^6*g*abs(c)*abs(d) - sqrt(-c*d^2*e + a*e^3)*a*c^2*d^4*e^2*g*abs(c)*abs(d) + 8*sqrt(-c*d^2*e + a*e^3)*a^2*c*d^2*e^4*g*abs(c)*abs(d) - 5*sqrt(-c*d^2*e + a*e^3)*a^3*e^6*g *abs(c)*abs(d))/(sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*c^2*d^2*e^3*f^5 - 3*sqr t(c^2*d^2*e^2*f - c^2*d^3*e*g)*c^2*d^3*e^2*f^4*g - 2*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a*c*d*e^4*f^4*g + 3*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*c^2*d^4 *e*f^3*g^2 + 6*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a*c*d^2*e^3*f^3*g^2 + sqr t(c^2*d^2*e^2*f - c^2*d^3*e*g)*a^2*e^5*f^3*g^2 - sqrt(c^2*d^2*e^2*f - c^2* d^3*e*g)*c^2*d^5*f^2*g^3 - 6*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a*c*d^3*e^2 *f^2*g^3 - 3*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a^2*d*e^4*f^2*g^3 + 2*sqrt( c^2*d^2*e^2*f - c^2*d^3*e*g)*a*c*d^4*e*f*g^4 + 3*sqrt(c^2*d^2*e^2*f - c^2* d^3*e*g)*a^2*d^2*e^3*f*g^4 - sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a^2*d^3*e^2 *g^5) + 2/35*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*(2*(c^7*d^7*e^6*f*g ^4*abs(c)*abs(d) - a*c^6*d^6*e^7*g^5*abs(c)*abs(d))*((e*x + d)*c*d*e - c*d ^2*e + a*e^3)/(c^3*d^3*e^6*f^3*g^3 - 3*a*c^2*d^2*e^7*f^2*g^4 + 3*a^2*c*d*e ^8*f*g^5 - a^3*e^9*g^6) + 7*(c^8*d^8*e^8*f^2*g^3*abs(c)*abs(d) - 2*a*c^7*d ^7*e^9*f*g^4*abs(c)*abs(d) + a^2*c^6*d^6*e^10*g^5*abs(c)*abs(d))/(c^3*d^3* e^6*f^3*g^3 - 3*a*c^2*d^2*e^7*f^2*g^4 + 3*a^2*c*d*e^8*f*g^5 - a^3*e^9*g...
Time = 12.66 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.91 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^{9/2}} \, dx=-\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {2\,a^2\,e^2\,\left (5\,a\,e\,g-7\,c\,d\,f\right )}{35\,g^3\,{\left (a\,e\,g-c\,d\,f\right )}^2}-\frac {4\,c^3\,d^3\,x^3}{35\,g^2\,{\left (a\,e\,g-c\,d\,f\right )}^2}+\frac {2\,c^2\,d^2\,x^2\,\left (a\,e\,g-7\,c\,d\,f\right )}{35\,g^3\,{\left (a\,e\,g-c\,d\,f\right )}^2}+\frac {4\,a\,c\,d\,e\,x\,\left (4\,a\,e\,g-7\,c\,d\,f\right )}{35\,g^3\,{\left (a\,e\,g-c\,d\,f\right )}^2}\right )}{x^3\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}+\frac {f^3\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^3}+\frac {3\,f\,x^2\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g}+\frac {3\,f^2\,x\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^2}} \]
-((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*((2*a^2*e^2*(5*a*e*g - 7*c *d*f))/(35*g^3*(a*e*g - c*d*f)^2) - (4*c^3*d^3*x^3)/(35*g^2*(a*e*g - c*d*f )^2) + (2*c^2*d^2*x^2*(a*e*g - 7*c*d*f))/(35*g^3*(a*e*g - c*d*f)^2) + (4*a *c*d*e*x*(4*a*e*g - 7*c*d*f))/(35*g^3*(a*e*g - c*d*f)^2)))/(x^3*(f + g*x)^ (1/2)*(d + e*x)^(1/2) + (f^3*(f + g*x)^(1/2)*(d + e*x)^(1/2))/g^3 + (3*f*x ^2*(f + g*x)^(1/2)*(d + e*x)^(1/2))/g + (3*f^2*x*(f + g*x)^(1/2)*(d + e*x) ^(1/2))/g^2)