Integrand size = 39, antiderivative size = 110 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\frac {2 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{7 c^2 d^2 (d+e x)^{7/2}}+\frac {2 e \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{9/2}}{9 c^2 d^2 (d+e x)^{9/2}} \] Output:
2/7*(-a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c^2/d^2/(e*x+d) ^(7/2)+2/9*e*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(9/2)/c^2/d^2/(e*x+d)^(9/2)
Time = 0.06 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.59 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\frac {2 (a e+c d x)^3 \sqrt {(a e+c d x) (d+e x)} \left (-2 a e^2+c d (9 d+7 e x)\right )}{63 c^2 d^2 \sqrt {d+e x}} \] Input:
Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(3/2),x]
Output:
(2*(a*e + c*d*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-2*a*e^2 + c*d*(9*d + 7* e*x)))/(63*c^2*d^2*Sqrt[d + e*x])
Time = 0.39 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {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.
\(\displaystyle \int \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 1128 |
\(\displaystyle \frac {2 \left (d^2-\frac {a e^2}{c}\right ) \int \frac {\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{5/2}}{(d+e x)^{5/2}}dx}{9 d}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{9 c d (d+e x)^{5/2}}\) |
\(\Big \downarrow \) 1122 |
\(\displaystyle \frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{9 c d (d+e x)^{5/2}}+\frac {4 \left (d^2-\frac {a e^2}{c}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{63 c d^2 (d+e x)^{7/2}}\) |
Input:
Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(3/2),x]
Output:
(4*(d^2 - (a*e^2)/c)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(63*c* d^2*(d + e*x)^(7/2)) + (2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/( 9*c*d*(d + e*x)^(5/2))
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 = 1.18 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.55
method | result | size |
default | \(-\frac {2 \sqrt {\left (e x +d \right ) \left (c d x +a e \right )}\, \left (c d x +a e \right )^{3} \left (-7 c d x e +2 a \,e^{2}-9 c \,d^{2}\right )}{63 \sqrt {e x +d}\, c^{2} d^{2}}\) | \(61\) |
gosper | \(-\frac {2 \left (c d x +a e \right ) \left (-7 c d x e +2 a \,e^{2}-9 c \,d^{2}\right ) \left (c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}{63 c^{2} d^{2} \left (e x +d \right )^{\frac {5}{2}}}\) | \(69\) |
orering | \(-\frac {2 \left (-7 c d x e +2 a \,e^{2}-9 c \,d^{2}\right ) \left (c d x +a e \right ) {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {5}{2}}}{63 c^{2} d^{2} \left (e x +d \right )^{\frac {5}{2}}}\) | \(70\) |
Input:
int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(5/2)/(e*x+d)^(3/2),x,method=_RETURN VERBOSE)
Output:
-2/63*((e*x+d)*(c*d*x+a*e))^(1/2)/(e*x+d)^(1/2)*(c*d*x+a*e)^3*(-7*c*d*e*x+ 2*a*e^2-9*c*d^2)/c^2/d^2
Time = 0.09 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.54 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (7 \, c^{4} d^{4} e x^{4} + 9 \, a^{3} c d^{2} e^{3} - 2 \, a^{4} e^{5} + {\left (9 \, c^{4} d^{5} + 19 \, a c^{3} d^{3} e^{2}\right )} x^{3} + 3 \, {\left (9 \, a c^{3} d^{4} e + 5 \, a^{2} c^{2} d^{2} e^{3}\right )} x^{2} + {\left (27 \, a^{2} c^{2} d^{3} e^{2} + a^{3} c d e^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{63 \, {\left (c^{2} d^{2} e x + c^{2} d^{3}\right )}} \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(3/2),x, algorit hm="fricas")
Output:
2/63*(7*c^4*d^4*e*x^4 + 9*a^3*c*d^2*e^3 - 2*a^4*e^5 + (9*c^4*d^5 + 19*a*c^ 3*d^3*e^2)*x^3 + 3*(9*a*c^3*d^4*e + 5*a^2*c^2*d^2*e^3)*x^2 + (27*a^2*c^2*d ^3*e^2 + a^3*c*d*e^4)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt( e*x + d)/(c^2*d^2*e*x + c^2*d^3)
\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\int \frac {\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {5}{2}}}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/(e*x+d)**(3/2),x)
Output:
Integral(((d + e*x)*(a*e + c*d*x))**(5/2)/(d + e*x)**(3/2), x)
Time = 0.06 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.20 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (7 \, c^{4} d^{4} e x^{4} + 9 \, a^{3} c d^{2} e^{3} - 2 \, a^{4} e^{5} + {\left (9 \, c^{4} d^{5} + 19 \, a c^{3} d^{3} e^{2}\right )} x^{3} + 3 \, {\left (9 \, a c^{3} d^{4} e + 5 \, a^{2} c^{2} d^{2} e^{3}\right )} x^{2} + {\left (27 \, a^{2} c^{2} d^{3} e^{2} + a^{3} c d e^{4}\right )} x\right )} \sqrt {c d x + a e}}{63 \, c^{2} d^{2}} \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(3/2),x, algorit hm="maxima")
Output:
2/63*(7*c^4*d^4*e*x^4 + 9*a^3*c*d^2*e^3 - 2*a^4*e^5 + (9*c^4*d^5 + 19*a*c^ 3*d^3*e^2)*x^3 + 3*(9*a*c^3*d^4*e + 5*a^2*c^2*d^2*e^3)*x^2 + (27*a^2*c^2*d ^3*e^2 + a^3*c*d*e^4)*x)*sqrt(c*d*x + a*e)/(c^2*d^2)
Leaf count of result is larger than twice the leaf count of optimal. 520 vs. \(2 (98) = 196\).
Time = 0.22 (sec) , antiderivative size = 520, normalized size of antiderivative = 4.73 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (\frac {105 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{2} {\left | e \right |}}{c e} - \frac {42 \, {\left (5 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a e^{3} - 3 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}}\right )} a {\left | e \right |}}{c e^{4}} - \frac {21 \, {\left (5 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a e^{3} - 3 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}}\right )} a^{2} {\left | e \right |}}{c^{2} d^{2} e^{2}} + \frac {3 \, {\left (35 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{2} e^{6} - 42 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a e^{3} + 15 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}}\right )} {\left | e \right |}}{c e^{7}} + \frac {6 \, {\left (35 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{2} e^{6} - 42 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a e^{3} + 15 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}}\right )} a {\left | e \right |}}{c^{2} d^{2} e^{5}} - \frac {{\left (105 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{3} e^{9} - 189 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a^{2} e^{6} + 135 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}} a e^{3} - 35 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {9}{2}}\right )} {\left | e \right |}}{c^{2} d^{2} e^{8}}\right )}}{315 \, e} \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(3/2),x, algorit hm="giac")
Output:
2/315*(105*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^2*abs(e)/(c*e) - 42 *(5*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a*e^3 - 3*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2))*a*abs(e)/(c*e^4) - 21*(5*((e*x + d)*c*d*e - c*d^2 *e + a*e^3)^(3/2)*a*e^3 - 3*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2))*a^2 *abs(e)/(c^2*d^2*e^2) + 3*(35*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^ 2*e^6 - 42*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a*e^3 + 15*((e*x + d) *c*d*e - c*d^2*e + a*e^3)^(7/2))*abs(e)/(c*e^7) + 6*(35*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^2*e^6 - 42*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^( 5/2)*a*e^3 + 15*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(7/2))*a*abs(e)/(c^2*d ^2*e^5) - (105*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^3*e^9 - 189*((e *x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a^2*e^6 + 135*((e*x + d)*c*d*e - c* d^2*e + a*e^3)^(7/2)*a*e^3 - 35*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(9/2)) *abs(e)/(c^2*d^2*e^8))/e
Time = 5.41 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.42 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {x^3\,\left (18\,c^4\,d^5+38\,a\,c^3\,d^3\,e^2\right )}{63\,c^2\,d^2}-\frac {4\,a^4\,e^5-18\,a^3\,c\,d^2\,e^3}{63\,c^2\,d^2}+\frac {2\,c^2\,d^2\,e\,x^4}{9}+\frac {2\,a\,e\,x^2\,\left (9\,c\,d^2+5\,a\,e^2\right )}{21}+\frac {2\,a^2\,e^2\,x\,\left (27\,c\,d^2+a\,e^2\right )}{63\,c\,d}\right )}{\sqrt {d+e\,x}} \] Input:
int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/(d + e*x)^(3/2),x)
Output:
((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*((x^3*(18*c^4*d^5 + 38*a*c^ 3*d^3*e^2))/(63*c^2*d^2) - (4*a^4*e^5 - 18*a^3*c*d^2*e^3)/(63*c^2*d^2) + ( 2*c^2*d^2*e*x^4)/9 + (2*a*e*x^2*(5*a*e^2 + 9*c*d^2))/21 + (2*a^2*e^2*x*(a* e^2 + 27*c*d^2))/(63*c*d)))/(d + e*x)^(1/2)
Time = 0.27 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.19 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\frac {2 \sqrt {c d x +a e}\, \left (7 c^{4} d^{4} e \,x^{4}+19 a \,c^{3} d^{3} e^{2} x^{3}+9 c^{4} d^{5} x^{3}+15 a^{2} c^{2} d^{2} e^{3} x^{2}+27 a \,c^{3} d^{4} e \,x^{2}+a^{3} c d \,e^{4} x +27 a^{2} c^{2} d^{3} e^{2} x -2 a^{4} e^{5}+9 a^{3} c \,d^{2} e^{3}\right )}{63 c^{2} d^{2}} \] Input:
int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(3/2),x)
Output:
(2*sqrt(a*e + c*d*x)*( - 2*a**4*e**5 + 9*a**3*c*d**2*e**3 + a**3*c*d*e**4* x + 27*a**2*c**2*d**3*e**2*x + 15*a**2*c**2*d**2*e**3*x**2 + 27*a*c**3*d** 4*e*x**2 + 19*a*c**3*d**3*e**2*x**3 + 9*c**4*d**5*x**3 + 7*c**4*d**4*e*x** 4))/(63*c**2*d**2)