Integrand size = 39, antiderivative size = 171 \[ \int \frac {(d+e x)^{9/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=-\frac {2 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}{3 c^3 d^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {4 e \left (c d^2-a e^2\right ) \sqrt {d+e x}}{c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {2 e^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c^3 d^3 \sqrt {d+e x}} \] Output:
-2/3*(-a*e^2+c*d^2)^2*(e*x+d)^(3/2)/c^3/d^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x ^2)^(3/2)-4*e*(-a*e^2+c*d^2)*(e*x+d)^(1/2)/c^3/d^3/(a*d*e+(a*e^2+c*d^2)*x+ c*d*e*x^2)^(1/2)+2*e^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^3/d^3/(e* x+d)^(1/2)
Time = 0.08 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.49 \[ \int \frac {(d+e x)^{9/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=-\frac {2 (d+e x)^{3/2} \left (-8 a^2 e^4+4 a c d e^2 (d-3 e x)+c^2 d^2 \left (d^2+6 d e x-3 e^2 x^2\right )\right )}{3 c^3 d^3 ((a e+c d x) (d+e x))^{3/2}} \] Input:
Integrate[(d + e*x)^(9/2)/(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)*(-8*a^2*e^4 + 4*a*c*d*e^2*(d - 3*e*x) + c^2*d^2*(d^2 + 6*d*e*x - 3*e^2*x^2)))/(3*c^3*d^3*((a*e + c*d*x)*(d + e*x))^(3/2))
Time = 0.52 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.01, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1128, 1128, 1122}
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.
\(\displaystyle \int \frac {(d+e x)^{9/2}}{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 1128 |
\(\displaystyle \frac {4 \left (d^2-\frac {a e^2}{c}\right ) \int \frac {(d+e x)^{7/2}}{\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{5/2}}dx}{d}+\frac {2 (d+e x)^{7/2}}{c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 1128 |
\(\displaystyle \frac {4 \left (d^2-\frac {a e^2}{c}\right ) \left (-\frac {2 \left (d^2-\frac {a e^2}{c}\right ) \int \frac {(d+e x)^{5/2}}{\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{5/2}}dx}{d}-\frac {2 (d+e x)^{5/2}}{c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\right )}{d}+\frac {2 (d+e x)^{7/2}}{c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 1122 |
\(\displaystyle \frac {2 (d+e x)^{7/2}}{c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac {4 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {4 (d+e x)^{3/2} \left (d^2-\frac {a e^2}{c}\right )}{3 c d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}-\frac {2 (d+e x)^{5/2}}{c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\right )}{d}\) |
Input:
Int[(d + e*x)^(9/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]
Output:
(2*(d + e*x)^(7/2))/(c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (4*(d^2 - (a*e^2)/c)*((4*(d^2 - (a*e^2)/c)*(d + e*x)^(3/2))/(3*c*d^2*(a*d* e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (2*(d + e*x)^(5/2))/(c*d*(a*d* e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))))/d
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(p + 1))), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[m + p, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Simp[Simplify[m + p]*((2*c*d - b*e)/(c*(m + 2*p + 1))) Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[Simplify[m + p], 0]
Time = 1.08 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.60
method | result | size |
default | \(\frac {2 \sqrt {\left (e x +d \right ) \left (c d x +a e \right )}\, \left (3 x^{2} c^{2} d^{2} e^{2}+12 x a c d \,e^{3}-6 x \,c^{2} d^{3} e +8 a^{2} e^{4}-4 a c \,d^{2} e^{2}-c^{2} d^{4}\right )}{3 \sqrt {e x +d}\, \left (c d x +a e \right )^{2} c^{3} d^{3}}\) | \(102\) |
gosper | \(\frac {2 \left (c d x +a e \right ) \left (3 x^{2} c^{2} d^{2} e^{2}+12 x a c d \,e^{3}-6 x \,c^{2} d^{3} e +8 a^{2} e^{4}-4 a c \,d^{2} e^{2}-c^{2} d^{4}\right ) \left (e x +d \right )^{\frac {5}{2}}}{3 d^{3} c^{3} \left (c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}\) | \(110\) |
orering | \(\frac {2 \left (3 x^{2} c^{2} d^{2} e^{2}+12 x a c d \,e^{3}-6 x \,c^{2} d^{3} e +8 a^{2} e^{4}-4 a c \,d^{2} e^{2}-c^{2} d^{4}\right ) \left (c d x +a e \right ) \left (e x +d \right )^{\frac {5}{2}}}{3 d^{3} c^{3} {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {5}{2}}}\) | \(111\) |
Input:
int((e*x+d)^(9/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(5/2),x,method=_RETURN VERBOSE)
Output:
2/3/(e*x+d)^(1/2)*((e*x+d)*(c*d*x+a*e))^(1/2)*(3*c^2*d^2*e^2*x^2+12*a*c*d* e^3*x-6*c^2*d^3*e*x+8*a^2*e^4-4*a*c*d^2*e^2-c^2*d^4)/(c*d*x+a*e)^2/c^3/d^3
Time = 0.10 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.02 \[ \int \frac {(d+e x)^{9/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (3 \, c^{2} d^{2} e^{2} x^{2} - c^{2} d^{4} - 4 \, a c d^{2} e^{2} + 8 \, a^{2} e^{4} - 6 \, {\left (c^{2} d^{3} e - 2 \, a c d e^{3}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{3 \, {\left (c^{5} d^{5} e x^{3} + a^{2} c^{3} d^{4} e^{2} + {\left (c^{5} d^{6} + 2 \, a c^{4} d^{4} e^{2}\right )} x^{2} + {\left (2 \, a c^{4} d^{5} e + a^{2} c^{3} d^{3} e^{3}\right )} x\right )}} \] Input:
integrate((e*x+d)^(9/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorit hm="fricas")
Output:
2/3*(3*c^2*d^2*e^2*x^2 - c^2*d^4 - 4*a*c*d^2*e^2 + 8*a^2*e^4 - 6*(c^2*d^3* e - 2*a*c*d*e^3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)/(c^5*d^5*e*x^3 + a^2*c^3*d^4*e^2 + (c^5*d^6 + 2*a*c^4*d^4*e^2)*x^2 + ( 2*a*c^4*d^5*e + a^2*c^3*d^3*e^3)*x)
Timed out. \[ \int \frac {(d+e x)^{9/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((e*x+d)**(9/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)
Output:
Timed out
Time = 0.06 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.54 \[ \int \frac {(d+e x)^{9/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (3 \, c^{2} d^{2} e^{2} x^{2} - c^{2} d^{4} - 4 \, a c d^{2} e^{2} + 8 \, a^{2} e^{4} - 6 \, {\left (c^{2} d^{3} e - 2 \, a c d e^{3}\right )} x\right )}}{3 \, {\left (c^{4} d^{4} x + a c^{3} d^{3} e\right )} \sqrt {c d x + a e}} \] Input:
integrate((e*x+d)^(9/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorit hm="maxima")
Output:
2/3*(3*c^2*d^2*e^2*x^2 - c^2*d^4 - 4*a*c*d^2*e^2 + 8*a^2*e^4 - 6*(c^2*d^3* e - 2*a*c*d*e^3)*x)/((c^4*d^4*x + a*c^3*d^3*e)*sqrt(c*d*x + a*e))
Time = 0.18 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^{9/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (\frac {3 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} e^{2}}{c d {\left | e \right |}} - \frac {c^{2} d^{4} e^{4} - 2 \, a c d^{2} e^{6} + a^{2} e^{8} + 6 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )} c d^{2} e^{3} - 6 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )} a e^{5}}{{\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c d {\left | e \right |}}\right )}}{3 \, c^{2} d^{2}} \] Input:
integrate((e*x+d)^(9/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorit hm="giac")
Output:
2/3*(3*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*e^2/(c*d*abs(e)) - (c^2*d^4 *e^4 - 2*a*c*d^2*e^6 + a^2*e^8 + 6*((e*x + d)*c*d*e - c*d^2*e + a*e^3)*c*d ^2*e^3 - 6*((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a*e^5)/(((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*c*d*abs(e)))/(c^2*d^2)
Time = 5.54 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.14 \[ \int \frac {(d+e x)^{9/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {4\,x\,\left (2\,a\,e^2-c\,d^2\right )\,\sqrt {d+e\,x}}{c^4\,d^4}+\frac {2\,e\,x^2\,\sqrt {d+e\,x}}{c^3\,d^3}-\frac {\sqrt {d+e\,x}\,\left (-16\,a^2\,e^4+8\,a\,c\,d^2\,e^2+2\,c^2\,d^4\right )}{3\,c^5\,d^5\,e}\right )}{x^3+\frac {a^2\,e}{c^2\,d}+\frac {a\,x\,\left (2\,c\,d^2+a\,e^2\right )}{c^2\,d^2}+\frac {x^2\,\left (3\,c^5\,d^6+6\,a\,c^4\,d^4\,e^2\right )}{3\,c^5\,d^5\,e}} \] Input:
int((d + e*x)^(9/2)/(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)*((4*x*(2*a*e^2 - c*d^2)*(d + e*x)^(1/2))/(c^4*d^4) + (2*e*x^2*(d + e*x)^(1/2))/(c^3*d^3) - ((d + e*x) ^(1/2)*(2*c^2*d^4 - 16*a^2*e^4 + 8*a*c*d^2*e^2))/(3*c^5*d^5*e)))/(x^3 + (a ^2*e)/(c^2*d) + (a*x*(a*e^2 + 2*c*d^2))/(c^2*d^2) + (x^2*(3*c^5*d^6 + 6*a* c^4*d^4*e^2))/(3*c^5*d^5*e))
Time = 0.19 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.52 \[ \int \frac {(d+e x)^{9/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {2 c^{2} d^{2} e^{2} x^{2}+8 a c d \,e^{3} x -4 c^{2} d^{3} e x +\frac {16}{3} a^{2} e^{4}-\frac {8}{3} a c \,d^{2} e^{2}-\frac {2}{3} c^{2} d^{4}}{\sqrt {c d x +a e}\, c^{3} d^{3} \left (c d x +a e \right )} \] Input:
int((e*x+d)^(9/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x)
Output:
(2*(8*a**2*e**4 - 4*a*c*d**2*e**2 + 12*a*c*d*e**3*x - c**2*d**4 - 6*c**2*d **3*e*x + 3*c**2*d**2*e**2*x**2))/(3*sqrt(a*e + c*d*x)*c**3*d**3*(a*e + c* d*x))