Integrand size = 48, antiderivative size = 63 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^{7/2}} \, dx=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 (c d f-a e g) (d+e x)^{5/2} (f+g x)^{5/2}} \] Output:
2/5*(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)^(5/2)
Time = 0.09 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.83 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^{7/2}} \, dx=\frac {2 ((a e+c d x) (d+e x))^{5/2}}{5 (c d f-a e g) (d+e x)^{5/2} (f+g x)^{5/2}} \] Input:
Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/((d + e*x)^(3/2)*( f + g*x)^(7/2)),x]
Output:
(2*((a*e + c*d*x)*(d + e*x))^(5/2))/(5*(c*d*f - a*e*g)*(d + e*x)^(5/2)*(f + g*x)^(5/2))
Time = 0.37 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.021, Rules used = {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.
\(\displaystyle \int \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^{7/2}} \, dx\) |
\(\Big \downarrow \) 1248 |
\(\displaystyle \frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 (d+e x)^{5/2} (f+g x)^{5/2} (c d f-a e g)}\) |
Input:
Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/((d + e*x)^(3/2)*(f + g* x)^(7/2)),x]
Output:
(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(5*(c*d*f - a*e*g)*(d + e*x)^(5/2)*(f + g*x)^(5/2))
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]
Time = 2.62 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.87
method | result | size |
default | \(-\frac {2 \sqrt {\left (e x +d \right ) \left (c d x +a e \right )}\, \left (c d x +a e \right )^{2}}{5 \sqrt {e x +d}\, \left (g x +f \right )^{\frac {5}{2}} \left (a e g -d f c \right )}\) | \(55\) |
gosper | \(-\frac {2 \left (c d x +a e \right ) \left (c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}{5 \left (g x +f \right )^{\frac {5}{2}} \left (a e g -d f c \right ) \left (e x +d \right )^{\frac {3}{2}}}\) | \(63\) |
orering | \(-\frac {2 \left (c d x +a e \right ) {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {3}{2}}}{5 \left (g x +f \right )^{\frac {5}{2}} \left (a e g -d f c \right ) \left (e x +d \right )^{\frac {3}{2}}}\) | \(64\) |
Input:
int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)/(e*x+d)^(3/2)/(g*x+f)^(7/2),x, method=_RETURNVERBOSE)
Output:
-2/5*((e*x+d)*(c*d*x+a*e))^(1/2)/(e*x+d)^(1/2)/(g*x+f)^(5/2)*(c*d*x+a*e)^2 /(a*e*g-c*d*f)
Leaf count of result is larger than twice the leaf count of optimal. 232 vs. \(2 (55) = 110\).
Time = 0.10 (sec) , antiderivative size = 232, normalized size of antiderivative = 3.68 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^{7/2}} \, dx=\frac {2 \, {\left (c^{2} d^{2} x^{2} + 2 \, a c d e x + a^{2} e^{2}\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d} \sqrt {g x + f}}{5 \, {\left (c d^{2} f^{4} - a d e f^{3} g + {\left (c d e f g^{3} - a e^{2} g^{4}\right )} x^{4} + {\left (3 \, c d e f^{2} g^{2} - a d e g^{4} + {\left (c d^{2} - 3 \, a e^{2}\right )} f g^{3}\right )} x^{3} + 3 \, {\left (c d e f^{3} g - a d e f g^{3} + {\left (c d^{2} - a e^{2}\right )} f^{2} g^{2}\right )} x^{2} + {\left (c d e f^{4} - 3 \, a d e f^{2} g^{2} + {\left (3 \, c d^{2} - a e^{2}\right )} f^{3} g\right )} x\right )}} \] Input:
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)^(7 /2),x, algorithm="fricas")
Output:
2/5*(c^2*d^2*x^2 + 2*a*c*d*e*x + a^2*e^2)*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*d^2*f^4 - a*d*e*f^3*g + (c*d*e* f*g^3 - a*e^2*g^4)*x^4 + (3*c*d*e*f^2*g^2 - a*d*e*g^4 + (c*d^2 - 3*a*e^2)* f*g^3)*x^3 + 3*(c*d*e*f^3*g - a*d*e*f*g^3 + (c*d^2 - a*e^2)*f^2*g^2)*x^2 + (c*d*e*f^4 - 3*a*d*e*f^2*g^2 + (3*c*d^2 - a*e^2)*f^3*g)*x)
Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^{7/2}} \, dx=\text {Timed out} \] Input:
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)**(7/2),x)
Output:
Timed out
\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^{7/2}} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{\frac {7}{2}}} \,d x } \] Input:
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)^(7 /2),x, algorithm="maxima")
Output:
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)^(7/2)), x)
Leaf count of result is larger than twice the leaf count of optimal. 111 vs. \(2 (55) = 110\).
Time = 0.25 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.76 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^{7/2}} \, dx=\frac {2 \, {\left (c^{5} d^{5} f g^{2} {\left | c \right |} {\left | d \right |} - a c^{4} d^{4} e g^{3} {\left | c \right |} {\left | d \right |}\right )} {\left (c d x + a e\right )}^{\frac {5}{2}}}{5 \, {\left (c^{2} d^{2} f^{2} g^{2} - 2 \, a c d e f g^{3} + a^{2} e^{2} g^{4}\right )} {\left (c^{2} d^{2} f - a c d e g + {\left (c d x + a e\right )} c d g\right )}^{\frac {5}{2}}} \] Input:
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)^(7 /2),x, algorithm="giac")
Output:
2/5*(c^5*d^5*f*g^2*abs(c)*abs(d) - a*c^4*d^4*e*g^3*abs(c)*abs(d))*(c*d*x + a*e)^(5/2)/((c^2*d^2*f^2*g^2 - 2*a*c*d*e*f*g^3 + a^2*e^2*g^4)*(c^2*d^2*f - a*c*d*e*g + (c*d*x + a*e)*c*d*g)^(5/2))
Time = 6.82 (sec) , antiderivative size = 232, normalized size of antiderivative = 3.68 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^{7/2}} \, dx=-\frac {\left (\frac {2\,a^2\,e^2}{5\,a\,e\,g^3-5\,c\,d\,f\,g^2}+\frac {2\,c^2\,d^2\,x^2}{5\,a\,e\,g^3-5\,c\,d\,f\,g^2}+\frac {4\,a\,c\,d\,e\,x}{5\,a\,e\,g^3-5\,c\,d\,f\,g^2}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{x^2\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}-\frac {\sqrt {f+g\,x}\,\left (5\,c\,d\,f^3-5\,a\,e\,f^2\,g\right )\,\sqrt {d+e\,x}}{5\,a\,e\,g^3-5\,c\,d\,f\,g^2}+\frac {x\,\sqrt {f+g\,x}\,\left (10\,a\,e\,f\,g^2-10\,c\,d\,f^2\,g\right )\,\sqrt {d+e\,x}}{5\,a\,e\,g^3-5\,c\,d\,f\,g^2}} \] Input:
int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)/((f + g*x)^(7/2)*(d + e* x)^(3/2)),x)
Output:
-(((2*a^2*e^2)/(5*a*e*g^3 - 5*c*d*f*g^2) + (2*c^2*d^2*x^2)/(5*a*e*g^3 - 5* c*d*f*g^2) + (4*a*c*d*e*x)/(5*a*e*g^3 - 5*c*d*f*g^2))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(x^2*(f + g*x)^(1/2)*(d + e*x)^(1/2) - ((f + g* x)^(1/2)*(5*c*d*f^3 - 5*a*e*f^2*g)*(d + e*x)^(1/2))/(5*a*e*g^3 - 5*c*d*f*g ^2) + (x*(f + g*x)^(1/2)*(10*a*e*f*g^2 - 10*c*d*f^2*g)*(d + e*x)^(1/2))/(5 *a*e*g^3 - 5*c*d*f*g^2))
Time = 0.65 (sec) , antiderivative size = 244, normalized size of antiderivative = 3.87 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^{7/2}} \, dx=\frac {-\frac {2 \sqrt {g x +f}\, \sqrt {c d x +a e}\, a^{2} e^{2} g^{3}}{5}-\frac {4 \sqrt {g x +f}\, \sqrt {c d x +a e}\, a c d e \,g^{3} x}{5}-\frac {2 \sqrt {g x +f}\, \sqrt {c d x +a e}\, c^{2} d^{2} g^{3} x^{2}}{5}-\frac {2 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, c^{2} d^{2} f^{3}}{5}-\frac {6 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, c^{2} d^{2} f^{2} g x}{5}-\frac {6 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, c^{2} d^{2} f \,g^{2} x^{2}}{5}-\frac {2 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, c^{2} d^{2} g^{3} x^{3}}{5}}{g^{3} \left (a e \,g^{4} x^{3}-c d f \,g^{3} x^{3}+3 a e f \,g^{3} x^{2}-3 c d \,f^{2} g^{2} x^{2}+3 a e \,f^{2} g^{2} x -3 c d \,f^{3} g x +a e \,f^{3} g -c d \,f^{4}\right )} \] Input:
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)^(7/2),x)
Output:
(2*( - sqrt(f + g*x)*sqrt(a*e + c*d*x)*a**2*e**2*g**3 - 2*sqrt(f + g*x)*sq rt(a*e + c*d*x)*a*c*d*e*g**3*x - sqrt(f + g*x)*sqrt(a*e + c*d*x)*c**2*d**2 *g**3*x**2 - sqrt(g)*sqrt(d)*sqrt(c)*c**2*d**2*f**3 - 3*sqrt(g)*sqrt(d)*sq rt(c)*c**2*d**2*f**2*g*x - 3*sqrt(g)*sqrt(d)*sqrt(c)*c**2*d**2*f*g**2*x**2 - sqrt(g)*sqrt(d)*sqrt(c)*c**2*d**2*g**3*x**3))/(5*g**3*(a*e*f**3*g + 3*a *e*f**2*g**2*x + 3*a*e*f*g**3*x**2 + a*e*g**4*x**3 - c*d*f**4 - 3*c*d*f**3 *g*x - 3*c*d*f**2*g**2*x**2 - c*d*f*g**3*x**3))