Integrand size = 39, antiderivative size = 171 \[ \int \frac {(d+e x)^{7/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 \left (c d^2-a e^2\right )^2 \sqrt {d+e x}}{c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {4 e \left (c d^2-a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c^3 d^3 \sqrt {d+e x}}+\frac {2 e^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 c^3 d^3 (d+e x)^{3/2}} \] Output:
-2*(-a*e^2+c*d^2)^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)+4*e*(-a*e^2+c*d^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)+2/3*e^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c^3/d^3/(e* x+d)^(3/2)
Time = 0.07 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.51 \[ \int \frac {(d+e x)^{7/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 \sqrt {d+e x} \left (8 a^2 e^4+4 a c d e^2 (-3 d+e x)+c^2 d^2 \left (3 d^2-6 d e x-e^2 x^2\right )\right )}{3 c^3 d^3 \sqrt {(a e+c d x) (d+e x)}} \] Input:
Integrate[(d + e*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]*(8*a^2*e^4 + 4*a*c*d*e^2*(-3*d + e*x) + c^2*d^2*(3*d^2 - 6*d*e*x - e^2*x^2)))/(3*c^3*d^3*Sqrt[(a*e + c*d*x)*(d + e*x)])
Time = 0.51 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.02, 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.
\(\displaystyle \int \frac {(d+e x)^{7/2}}{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 1128 |
\(\displaystyle \frac {4 \left (d^2-\frac {a e^2}{c}\right ) \int \frac {(d+e x)^{5/2}}{\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{3 d}+\frac {2 (d+e x)^{5/2}}{3 c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
\(\Big \downarrow \) 1128 |
\(\displaystyle \frac {4 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {2 \left (d^2-\frac {a e^2}{c}\right ) \int \frac {(d+e x)^{3/2}}{\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{d}+\frac {2 (d+e x)^{3/2}}{c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{3 d}+\frac {2 (d+e x)^{5/2}}{3 c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
\(\Big \downarrow \) 1122 |
\(\displaystyle \frac {2 (d+e x)^{5/2}}{3 c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {4 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {2 (d+e x)^{3/2}}{c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {4 \sqrt {d+e x} \left (d^2-\frac {a e^2}{c}\right )}{c d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{3 d}\) |
Input:
Int[(d + e*x)^(7/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]
Output:
(2*(d + e*x)^(5/2))/(3*c*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (4*(d^2 - (a*e^2)/c)*((-4*(d^2 - (a*e^2)/c)*Sqrt[d + e*x])/(c*d^2*Sqrt[a*d *e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (2*(d + e*x)^(3/2))/(c*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])))/(3*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 = 1.04 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.60
method | result | size |
default | \(-\frac {2 \sqrt {\left (e x +d \right ) \left (c d x +a e \right )}\, \left (-x^{2} c^{2} d^{2} e^{2}+4 x a c d \,e^{3}-6 x \,c^{2} d^{3} e +8 a^{2} e^{4}-12 a c \,d^{2} e^{2}+3 c^{2} d^{4}\right )}{3 \sqrt {e x +d}\, \left (c d x +a e \right ) c^{3} d^{3}}\) | \(102\) |
gosper | \(-\frac {2 \left (c d x +a e \right ) \left (-x^{2} c^{2} d^{2} e^{2}+4 x a c d \,e^{3}-6 x \,c^{2} d^{3} e +8 a^{2} e^{4}-12 a c \,d^{2} e^{2}+3 c^{2} d^{4}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3 d^{3} c^{3} \left (c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}\) | \(110\) |
orering | \(-\frac {2 \left (-x^{2} c^{2} d^{2} e^{2}+4 x a c d \,e^{3}-6 x \,c^{2} d^{3} e +8 a^{2} e^{4}-12 a c \,d^{2} e^{2}+3 c^{2} d^{4}\right ) \left (c d x +a e \right ) \left (e x +d \right )^{\frac {3}{2}}}{3 d^{3} c^{3} {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {3}{2}}}\) | \(111\) |
Input:
int((e*x+d)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2),x,method=_RETURN VERBOSE)
Output:
-2/3/(e*x+d)^(1/2)*((e*x+d)*(c*d*x+a*e))^(1/2)*(-c^2*d^2*e^2*x^2+4*a*c*d*e ^3*x-6*c^2*d^3*e*x+8*a^2*e^4-12*a*c*d^2*e^2+3*c^2*d^4)/(c*d*x+a*e)/c^3/d^3
Time = 0.09 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.82 \[ \int \frac {(d+e x)^{7/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {2 \, {\left (c^{2} d^{2} e^{2} x^{2} - 3 \, c^{2} d^{4} + 12 \, a c d^{2} e^{2} - 8 \, a^{2} e^{4} + 2 \, {\left (3 \, c^{2} d^{3} e - 2 \, a c d e^{3}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{3 \, {\left (c^{4} d^{4} e x^{2} + a c^{3} d^{4} e + {\left (c^{4} d^{5} + a c^{3} d^{3} e^{2}\right )} x\right )}} \] Input:
integrate((e*x+d)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorit hm="fricas")
Output:
2/3*(c^2*d^2*e^2*x^2 - 3*c^2*d^4 + 12*a*c*d^2*e^2 - 8*a^2*e^4 + 2*(3*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^4*d^4*e*x^2 + a*c^3*d^4*e + (c^4*d^5 + a*c^3*d^3*e^2)*x)
Timed out. \[ \int \frac {(d+e x)^{7/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((e*x+d)**(7/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)
Output:
Timed out
Time = 0.05 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.46 \[ \int \frac {(d+e x)^{7/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {2 \, {\left (c^{2} d^{2} e^{2} x^{2} - 3 \, c^{2} d^{4} + 12 \, a c d^{2} e^{2} - 8 \, a^{2} e^{4} + 2 \, {\left (3 \, c^{2} d^{3} e - 2 \, a c d e^{3}\right )} x\right )}}{3 \, \sqrt {c d x + a e} c^{3} d^{3}} \] Input:
integrate((e*x+d)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorit hm="maxima")
Output:
2/3*(c^2*d^2*e^2*x^2 - 3*c^2*d^4 + 12*a*c*d^2*e^2 - 8*a^2*e^4 + 2*(3*c^2*d ^3*e - 2*a*c*d*e^3)*x)/(sqrt(c*d*x + a*e)*c^3*d^3)
Time = 0.15 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.67 \[ \int \frac {(d+e x)^{7/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 \, {\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}}{\sqrt {c d x + a e} c^{3} d^{3}} + \frac {2 \, {\left (6 \, \sqrt {c d x + a e} c^{7} d^{8} e - 6 \, \sqrt {c d x + a e} a c^{6} d^{6} e^{3} + {\left (c d x + a e\right )}^{\frac {3}{2}} c^{6} d^{6} e^{2}\right )}}{3 \, c^{9} d^{9}} \] Input:
integrate((e*x+d)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorit hm="giac")
Output:
-2*(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)/(sqrt(c*d*x + a*e)*c^3*d^3) + 2/3*( 6*sqrt(c*d*x + a*e)*c^7*d^8*e - 6*sqrt(c*d*x + a*e)*a*c^6*d^6*e^3 + (c*d*x + a*e)^(3/2)*c^6*d^6*e^2)/(c^9*d^9)
Time = 5.85 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.02 \[ \int \frac {(d+e x)^{7/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {\left (\frac {2\,e\,x^2\,\sqrt {d+e\,x}}{3\,c^2\,d^2}-\frac {\sqrt {d+e\,x}\,\left (16\,a^2\,e^4-24\,a\,c\,d^2\,e^2+6\,c^2\,d^4\right )}{3\,c^4\,d^4\,e}+\frac {x\,\left (12\,c^2\,d^3\,e-8\,a\,c\,d\,e^3\right )\,\sqrt {d+e\,x}}{3\,c^4\,d^4\,e}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{\frac {a}{c}+x^2+\frac {x\,\left (3\,c^4\,d^5+3\,a\,c^3\,d^3\,e^2\right )}{3\,c^4\,d^4\,e}} \] Input:
int((d + e*x)^(7/2)/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2),x)
Output:
(((2*e*x^2*(d + e*x)^(1/2))/(3*c^2*d^2) - ((d + e*x)^(1/2)*(16*a^2*e^4 + 6 *c^2*d^4 - 24*a*c*d^2*e^2))/(3*c^4*d^4*e) + (x*(12*c^2*d^3*e - 8*a*c*d*e^3 )*(d + e*x)^(1/2))/(3*c^4*d^4*e))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^ (1/2))/(a/c + x^2 + (x*(3*c^4*d^5 + 3*a*c^3*d^3*e^2))/(3*c^4*d^4*e))
Time = 0.22 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.46 \[ \int \frac {(d+e x)^{7/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {\frac {2}{3} c^{2} d^{2} e^{2} x^{2}-\frac {8}{3} a c d \,e^{3} x +4 c^{2} d^{3} e x -\frac {16}{3} a^{2} e^{4}+8 a c \,d^{2} e^{2}-2 c^{2} d^{4}}{\sqrt {c d x +a e}\, c^{3} d^{3}} \] Input:
int((e*x+d)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x)
Output:
(2*( - 8*a**2*e**4 + 12*a*c*d**2*e**2 - 4*a*c*d*e**3*x - 3*c**2*d**4 + 6*c **2*d**3*e*x + c**2*d**2*e**2*x**2))/(3*sqrt(a*e + c*d*x)*c**3*d**3)