Integrand size = 24, antiderivative size = 77 \[ \int \frac {(d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {2 (d+e x)}{5 e \left (d^2-e^2 x^2\right )^{5/2}}+\frac {x}{5 d^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 x}{5 d^4 \sqrt {d^2-e^2 x^2}} \] Output:
2/5*(e*x+d)/e/(-e^2*x^2+d^2)^(5/2)+1/5*x/d^2/(-e^2*x^2+d^2)^(3/2)+2/5*x/d^ 4/(-e^2*x^2+d^2)^(1/2)
Time = 0.34 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.91 \[ \int \frac {(d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (2 d^3+d^2 e x-4 d e^2 x^2+2 e^3 x^3\right )}{5 d^4 e (d-e x)^3 (d+e x)} \] Input:
Integrate[(d + e*x)^2/(d^2 - e^2*x^2)^(7/2),x]
Output:
(Sqrt[d^2 - e^2*x^2]*(2*d^3 + d^2*e*x - 4*d*e^2*x^2 + 2*e^3*x^3))/(5*d^4*e *(d - e*x)^3*(d + e*x))
Time = 0.30 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.06, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {457, 209, 208}
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.
\(\displaystyle \int \frac {(d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx\) |
\(\Big \downarrow \) 457 |
\(\displaystyle \frac {3}{5} \int \frac {1}{\left (d^2-e^2 x^2\right )^{5/2}}dx+\frac {2 (d+e x)}{5 e \left (d^2-e^2 x^2\right )^{5/2}}\) |
\(\Big \downarrow \) 209 |
\(\displaystyle \frac {3}{5} \left (\frac {2 \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}}dx}{3 d^2}+\frac {x}{3 d^2 \left (d^2-e^2 x^2\right )^{3/2}}\right )+\frac {2 (d+e x)}{5 e \left (d^2-e^2 x^2\right )^{5/2}}\) |
\(\Big \downarrow \) 208 |
\(\displaystyle \frac {2 (d+e x)}{5 e \left (d^2-e^2 x^2\right )^{5/2}}+\frac {3}{5} \left (\frac {x}{3 d^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 x}{3 d^4 \sqrt {d^2-e^2 x^2}}\right )\) |
Input:
Int[(d + e*x)^2/(d^2 - e^2*x^2)^(7/2),x]
Output:
(2*(d + e*x))/(5*e*(d^2 - e^2*x^2)^(5/2)) + (3*(x/(3*d^2*(d^2 - e^2*x^2)^( 3/2)) + (2*x)/(3*d^4*Sqrt[d^2 - e^2*x^2])))/5
Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b}, x]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && ILtQ[p + 3/2, 0]
Int[((c_) + (d_.)*(x_))^2*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[d*( c + d*x)*((a + b*x^2)^(p + 1)/(b*(p + 1))), x] - Simp[d^2*((p + 2)/(b*(p + 1))) Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, p}, x] && EqQ[ b*c^2 + a*d^2, 0] && LtQ[p, -1]
Time = 0.21 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.84
method | result | size |
gosper | \(\frac {\left (e x +d \right )^{3} \left (-e x +d \right ) \left (2 e^{3} x^{3}-4 d \,e^{2} x^{2}+d^{2} e x +2 d^{3}\right )}{5 d^{4} e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}\) | \(65\) |
orering | \(\frac {\left (e x +d \right )^{3} \left (-e x +d \right ) \left (2 e^{3} x^{3}-4 d \,e^{2} x^{2}+d^{2} e x +2 d^{3}\right )}{5 d^{4} e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}\) | \(65\) |
trager | \(\frac {\left (2 e^{3} x^{3}-4 d \,e^{2} x^{2}+d^{2} e x +2 d^{3}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{5 d^{4} \left (-e x +d \right )^{3} e \left (e x +d \right )}\) | \(67\) |
default | \(d^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )+e^{2} \left (\frac {x}{4 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {d^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )}{4 e^{2}}\right )+\frac {2 d}{5 e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\) | \(193\) |
Input:
int((e*x+d)^2/(-e^2*x^2+d^2)^(7/2),x,method=_RETURNVERBOSE)
Output:
1/5*(e*x+d)^3*(-e*x+d)*(2*e^3*x^3-4*d*e^2*x^2+d^2*e*x+2*d^3)/d^4/e/(-e^2*x ^2+d^2)^(7/2)
Time = 0.11 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.51 \[ \int \frac {(d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {2 \, e^{4} x^{4} - 4 \, d e^{3} x^{3} + 4 \, d^{3} e x - 2 \, d^{4} - {\left (2 \, e^{3} x^{3} - 4 \, d e^{2} x^{2} + d^{2} e x + 2 \, d^{3}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{5 \, {\left (d^{4} e^{5} x^{4} - 2 \, d^{5} e^{4} x^{3} + 2 \, d^{7} e^{2} x - d^{8} e\right )}} \] Input:
integrate((e*x+d)^2/(-e^2*x^2+d^2)^(7/2),x, algorithm="fricas")
Output:
1/5*(2*e^4*x^4 - 4*d*e^3*x^3 + 4*d^3*e*x - 2*d^4 - (2*e^3*x^3 - 4*d*e^2*x^ 2 + d^2*e*x + 2*d^3)*sqrt(-e^2*x^2 + d^2))/(d^4*e^5*x^4 - 2*d^5*e^4*x^3 + 2*d^7*e^2*x - d^8*e)
\[ \int \frac {(d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {\left (d + e x\right )^{2}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}\, dx \] Input:
integrate((e*x+d)**2/(-e**2*x**2+d**2)**(7/2),x)
Output:
Integral((d + e*x)**2/(-(-d + e*x)*(d + e*x))**(7/2), x)
Time = 0.04 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.01 \[ \int \frac {(d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {2 \, x}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {2 \, d}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e} + \frac {x}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2}} + \frac {2 \, x}{5 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{4}} \] Input:
integrate((e*x+d)^2/(-e^2*x^2+d^2)^(7/2),x, algorithm="maxima")
Output:
2/5*x/(-e^2*x^2 + d^2)^(5/2) + 2/5*d/((-e^2*x^2 + d^2)^(5/2)*e) + 1/5*x/(( -e^2*x^2 + d^2)^(3/2)*d^2) + 2/5*x/(sqrt(-e^2*x^2 + d^2)*d^4)
\[ \int \frac {(d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate((e*x+d)^2/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")
Output:
integrate((e*x + d)^2/(-e^2*x^2 + d^2)^(7/2), x)
Time = 6.58 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.86 \[ \int \frac {(d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\sqrt {d^2-e^2\,x^2}\,\left (2\,d^3+d^2\,e\,x-4\,d\,e^2\,x^2+2\,e^3\,x^3\right )}{5\,d^4\,e\,\left (d+e\,x\right )\,{\left (d-e\,x\right )}^3} \] Input:
int((d + e*x)^2/(d^2 - e^2*x^2)^(7/2),x)
Output:
((d^2 - e^2*x^2)^(1/2)*(2*d^3 + 2*e^3*x^3 - 4*d*e^2*x^2 + d^2*e*x))/(5*d^4 *e*(d + e*x)*(d - e*x)^3)
Time = 0.23 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.66 \[ \int \frac {(d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {-\sqrt {-e^{2} x^{2}+d^{2}}\, d^{2}+2 \sqrt {-e^{2} x^{2}+d^{2}}\, d e x -\sqrt {-e^{2} x^{2}+d^{2}}\, e^{2} x^{2}+4 d^{3}+2 d^{2} e x -8 d \,e^{2} x^{2}+4 e^{3} x^{3}}{10 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{4} e \left (e^{2} x^{2}-2 d e x +d^{2}\right )} \] Input:
int((e*x+d)^2/(-e^2*x^2+d^2)^(7/2),x)
Output:
( - sqrt(d**2 - e**2*x**2)*d**2 + 2*sqrt(d**2 - e**2*x**2)*d*e*x - sqrt(d* *2 - e**2*x**2)*e**2*x**2 + 4*d**3 + 2*d**2*e*x - 8*d*e**2*x**2 + 4*e**3*x **3)/(10*sqrt(d**2 - e**2*x**2)*d**4*e*(d**2 - 2*d*e*x + e**2*x**2))