Integrand size = 31, antiderivative size = 145 \[ \int \frac {(d+e x)^3 (f+g x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {(e f+d g)^2 (d+e x)^3}{5 d e^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 (e f-4 d g) (e f+d g) (d+e x)^2}{15 d^2 e^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\left (2 e^2 f^2-6 d e f g+7 d^2 g^2\right ) (d+e x)}{15 d^3 e^3 \sqrt {d^2-e^2 x^2}} \]
1/5*(d*g+e*f)^2*(e*x+d)^3/d/e^3/(-e^2*x^2+d^2)^(5/2)+2/15*(-4*d*g+e*f)*(d* g+e*f)*(e*x+d)^2/d^2/e^3/(-e^2*x^2+d^2)^(3/2)+1/15*(7*d^2*g^2-6*d*e*f*g+2* e^2*f^2)*(e*x+d)/d^3/e^3/(-e^2*x^2+d^2)^(1/2)
Time = 0.56 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.72 \[ \int \frac {(d+e x)^3 (f+g x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (2 d^4 g^2+2 e^4 f^2 x^2-6 d^3 e g (f+g x)-6 d e^3 f x (f+g x)+d^2 e^2 \left (7 f^2+18 f g x+7 g^2 x^2\right )\right )}{15 d^3 e^3 (d-e x)^3} \]
(Sqrt[d^2 - e^2*x^2]*(2*d^4*g^2 + 2*e^4*f^2*x^2 - 6*d^3*e*g*(f + g*x) - 6* d*e^3*f*x*(f + g*x) + d^2*e^2*(7*f^2 + 18*f*g*x + 7*g^2*x^2)))/(15*d^3*e^3 *(d - e*x)^3)
Time = 0.33 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.08, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {691, 25, 27, 669, 453}
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.
| \(\displaystyle \int \frac {(d+e x)^3 (f+g x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx\) |
| \(\Big \downarrow \) 691 |
| \(\displaystyle \frac {(d+e x)^3 (d g+e f)^2}{5 d e^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int -\frac {(d+e x)^2 \left (e \left (2 f^2-\frac {6 d g f}{e}-\frac {3 d^2 g^2}{e^2}\right )-5 d g^2 x\right )}{e \left (d^2-e^2 x^2\right )^{5/2}}dx}{5 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {(d+e x)^2 \left (2 e f^2-6 d g f-\frac {3 d^2 g^2}{e}-5 d g^2 x\right )}{e \left (d^2-e^2 x^2\right )^{5/2}}dx}{5 d}+\frac {(d+e x)^3 (d g+e f)^2}{5 d e^3 \left (d^2-e^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(d+e x)^2 \left (2 e f^2-6 d g f-\frac {3 d^2 g^2}{e}-5 d g^2 x\right )}{\left (d^2-e^2 x^2\right )^{5/2}}dx}{5 d e}+\frac {(d+e x)^3 (d g+e f)^2}{5 d e^3 \left (d^2-e^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 669 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {2 e f^2}{d}+\frac {7 d g^2}{e}-6 f g\right ) \int \frac {d+e x}{\left (d^2-e^2 x^2\right )^{3/2}}dx+\frac {2 (d+e x)^2 (e f-4 d g) (d g+e f)}{3 d e^2 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d e}+\frac {(d+e x)^3 (d g+e f)^2}{5 d e^3 \left (d^2-e^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 453 |
| \(\displaystyle \frac {\frac {(d+e x) \left (\frac {2 e f^2}{d}+\frac {7 d g^2}{e}-6 f g\right )}{3 d e \sqrt {d^2-e^2 x^2}}+\frac {2 (d+e x)^2 (e f-4 d g) (d g+e f)}{3 d e^2 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d e}+\frac {(d+e x)^3 (d g+e f)^2}{5 d e^3 \left (d^2-e^2 x^2\right )^{5/2}}\) |
((e*f + d*g)^2*(d + e*x)^3)/(5*d*e^3*(d^2 - e^2*x^2)^(5/2)) + ((2*(e*f - 4 *d*g)*(e*f + d*g)*(d + e*x)^2)/(3*d*e^2*(d^2 - e^2*x^2)^(3/2)) + (((2*e*f^ 2)/d - 6*f*g + (7*d*g^2)/e)*(d + e*x))/(3*d*e*Sqrt[d^2 - e^2*x^2]))/(5*d*e )
3.6.82.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_) + (d_.)*(x_))/((a_) + (b_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[-(a* d - b*c*x)/(a*b*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b, c, d}, x]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^( p_), x_Symbol] :> Simp[(d*g + e*f)*(d + e*x)^m*((a + c*x^2)^(p + 1)/(2*c*d* (p + 1))), x] - Simp[e*((m*(d*g + e*f) + 2*e*f*(p + 1))/(2*c*d*(p + 1))) Int[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && EqQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_) ^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(f + g*x)^n, a*e + c*d* x, x], R = PolynomialRemainder[(f + g*x)^n, a*e + c*d*x, x]}, Simp[(-d)*R*( d + e*x)^m*((a + c*x^2)^(p + 1)/(2*a*e*(p + 1))), x] + Simp[d/(2*a*(p + 1)) Int[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1)*ExpandToSum[2*a*e*(p + 1)*Q + R*(m + 2*p + 2), x], x], x]] /; FreeQ[{a, c, d, e, f, g}, x] && IGtQ[n, 1] && IGtQ[m, 0] && LtQ[p, -1] && EqQ[c*d^2 + a*e^2, 0]
Time = 0.64 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.87
| method | result | size |
| trager | \(\frac {\left (7 x^{2} d^{2} e^{2} g^{2}-6 x^{2} d \,e^{3} f g +2 x^{2} e^{4} f^{2}-6 x \,d^{3} e \,g^{2}+18 x \,d^{2} e^{2} f g -6 x d \,e^{3} f^{2}+2 d^{4} g^{2}-6 f g e \,d^{3}+7 d^{2} e^{2} f^{2}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{15 d^{3} e^{3} \left (-e x +d \right )^{3}}\) | \(126\) |
| gosper | \(\frac {\left (-e x +d \right ) \left (e x +d \right )^{4} \left (7 x^{2} d^{2} e^{2} g^{2}-6 x^{2} d \,e^{3} f g +2 x^{2} e^{4} f^{2}-6 x \,d^{3} e \,g^{2}+18 x \,d^{2} e^{2} f g -6 x d \,e^{3} f^{2}+2 d^{4} g^{2}-6 f g e \,d^{3}+7 d^{2} e^{2} f^{2}\right )}{15 d^{3} e^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}\) | \(131\) |
| default | \(d^{3} f^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )+g^{2} e^{3} \left (\frac {x^{4}}{e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {4 d^{2} \left (\frac {x^{2}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {2 d^{2}}{15 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\right )}{e^{2}}\right )+\left (3 d \,g^{2} e^{2}+2 f g \,e^{3}\right ) \left (\frac {x^{3}}{2 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {3 d^{2} \left (\frac {x}{4 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {d^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )}{4 e^{2}}\right )}{2 e^{2}}\right )+\frac {2 d^{3} f g +3 d^{2} e \,f^{2}}{5 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\left (3 d^{2} g^{2} e +6 d f g \,e^{2}+f^{2} e^{3}\right ) \left (\frac {x^{2}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {2 d^{2}}{15 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\right )+\left (d^{3} g^{2}+6 d^{2} e f g +3 d \,e^{2} f^{2}\right ) \left (\frac {x}{4 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {d^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )}{4 e^{2}}\right )\) | \(532\) |
1/15*(7*d^2*e^2*g^2*x^2-6*d*e^3*f*g*x^2+2*e^4*f^2*x^2-6*d^3*e*g^2*x+18*d^2 *e^2*f*g*x-6*d*e^3*f^2*x+2*d^4*g^2-6*d^3*e*f*g+7*d^2*e^2*f^2)/d^3/e^3/(-e* x+d)^3*(-e^2*x^2+d^2)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 279 vs. \(2 (133) = 266\).
Time = 0.31 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.92 \[ \int \frac {(d+e x)^3 (f+g x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=-\frac {7 \, d^{3} e^{2} f^{2} - 6 \, d^{4} e f g + 2 \, d^{5} g^{2} - {\left (7 \, e^{5} f^{2} - 6 \, d e^{4} f g + 2 \, d^{2} e^{3} g^{2}\right )} x^{3} + 3 \, {\left (7 \, d e^{4} f^{2} - 6 \, d^{2} e^{3} f g + 2 \, d^{3} e^{2} g^{2}\right )} x^{2} - 3 \, {\left (7 \, d^{2} e^{3} f^{2} - 6 \, d^{3} e^{2} f g + 2 \, d^{4} e g^{2}\right )} x + {\left (7 \, d^{2} e^{2} f^{2} - 6 \, d^{3} e f g + 2 \, d^{4} g^{2} + {\left (2 \, e^{4} f^{2} - 6 \, d e^{3} f g + 7 \, d^{2} e^{2} g^{2}\right )} x^{2} - 6 \, {\left (d e^{3} f^{2} - 3 \, d^{2} e^{2} f g + d^{3} e g^{2}\right )} x\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d^{3} e^{6} x^{3} - 3 \, d^{4} e^{5} x^{2} + 3 \, d^{5} e^{4} x - d^{6} e^{3}\right )}} \]
-1/15*(7*d^3*e^2*f^2 - 6*d^4*e*f*g + 2*d^5*g^2 - (7*e^5*f^2 - 6*d*e^4*f*g + 2*d^2*e^3*g^2)*x^3 + 3*(7*d*e^4*f^2 - 6*d^2*e^3*f*g + 2*d^3*e^2*g^2)*x^2 - 3*(7*d^2*e^3*f^2 - 6*d^3*e^2*f*g + 2*d^4*e*g^2)*x + (7*d^2*e^2*f^2 - 6* d^3*e*f*g + 2*d^4*g^2 + (2*e^4*f^2 - 6*d*e^3*f*g + 7*d^2*e^2*g^2)*x^2 - 6* (d*e^3*f^2 - 3*d^2*e^2*f*g + d^3*e*g^2)*x)*sqrt(-e^2*x^2 + d^2))/(d^3*e^6* x^3 - 3*d^4*e^5*x^2 + 3*d^5*e^4*x - d^6*e^3)
\[ \int \frac {(d+e x)^3 (f+g x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {\left (d + e x\right )^{3} \left (f + g x\right )^{2}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 583 vs. \(2 (133) = 266\).
Time = 0.19 (sec) , antiderivative size = 583, normalized size of antiderivative = 4.02 \[ \int \frac {(d+e x)^3 (f+g x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {e g^{2} x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {4 \, d^{2} g^{2} x^{2}}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e} + \frac {d f^{2} x}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {3 \, d^{2} f^{2}}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e} + \frac {2 \, d^{3} f g}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} + \frac {8 \, d^{4} g^{2}}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{3}} + \frac {4 \, f^{2} x}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d} + \frac {8 \, f^{2} x}{15 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{3}} + \frac {{\left (2 \, e^{3} f g + 3 \, d e^{2} g^{2}\right )} x^{3}}{2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} + \frac {{\left (e^{3} f^{2} + 6 \, d e^{2} f g + 3 \, d^{2} e g^{2}\right )} x^{2}}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} - \frac {3 \, {\left (2 \, e^{3} f g + 3 \, d e^{2} g^{2}\right )} d^{2} x}{10 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}} + \frac {{\left (3 \, d e^{2} f^{2} + 6 \, d^{2} e f g + d^{3} g^{2}\right )} x}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} - \frac {2 \, {\left (e^{3} f^{2} + 6 \, d e^{2} f g + 3 \, d^{2} e g^{2}\right )} d^{2}}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}} + \frac {{\left (2 \, e^{3} f g + 3 \, d e^{2} g^{2}\right )} x}{10 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{4}} - \frac {{\left (3 \, d e^{2} f^{2} + 6 \, d^{2} e f g + d^{3} g^{2}\right )} x}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2} e^{2}} + \frac {{\left (2 \, e^{3} f g + 3 \, d e^{2} g^{2}\right )} x}{5 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2} e^{4}} - \frac {2 \, {\left (3 \, d e^{2} f^{2} + 6 \, d^{2} e f g + d^{3} g^{2}\right )} x}{15 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{4} e^{2}} \]
e*g^2*x^4/(-e^2*x^2 + d^2)^(5/2) - 4/3*d^2*g^2*x^2/((-e^2*x^2 + d^2)^(5/2) *e) + 1/5*d*f^2*x/(-e^2*x^2 + d^2)^(5/2) + 3/5*d^2*f^2/((-e^2*x^2 + d^2)^( 5/2)*e) + 2/5*d^3*f*g/((-e^2*x^2 + d^2)^(5/2)*e^2) + 8/15*d^4*g^2/((-e^2*x ^2 + d^2)^(5/2)*e^3) + 4/15*f^2*x/((-e^2*x^2 + d^2)^(3/2)*d) + 8/15*f^2*x/ (sqrt(-e^2*x^2 + d^2)*d^3) + 1/2*(2*e^3*f*g + 3*d*e^2*g^2)*x^3/((-e^2*x^2 + d^2)^(5/2)*e^2) + 1/3*(e^3*f^2 + 6*d*e^2*f*g + 3*d^2*e*g^2)*x^2/((-e^2*x ^2 + d^2)^(5/2)*e^2) - 3/10*(2*e^3*f*g + 3*d*e^2*g^2)*d^2*x/((-e^2*x^2 + d ^2)^(5/2)*e^4) + 1/5*(3*d*e^2*f^2 + 6*d^2*e*f*g + d^3*g^2)*x/((-e^2*x^2 + d^2)^(5/2)*e^2) - 2/15*(e^3*f^2 + 6*d*e^2*f*g + 3*d^2*e*g^2)*d^2/((-e^2*x^ 2 + d^2)^(5/2)*e^4) + 1/10*(2*e^3*f*g + 3*d*e^2*g^2)*x/((-e^2*x^2 + d^2)^( 3/2)*e^4) - 1/15*(3*d*e^2*f^2 + 6*d^2*e*f*g + d^3*g^2)*x/((-e^2*x^2 + d^2) ^(3/2)*d^2*e^2) + 1/5*(2*e^3*f*g + 3*d*e^2*g^2)*x/(sqrt(-e^2*x^2 + d^2)*d^ 2*e^4) - 2/15*(3*d*e^2*f^2 + 6*d^2*e*f*g + d^3*g^2)*x/(sqrt(-e^2*x^2 + d^2 )*d^4*e^2)
Leaf count of result is larger than twice the leaf count of optimal. 370 vs. \(2 (133) = 266\).
Time = 0.30 (sec) , antiderivative size = 370, normalized size of antiderivative = 2.55 \[ \int \frac {(d+e x)^3 (f+g x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {2 \, {\left (7 \, e^{2} f^{2} - 6 \, d e f g + 2 \, d^{2} g^{2} - \frac {20 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} f^{2}}{x} + \frac {30 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d f g}{e x} - \frac {10 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{2} g^{2}}{e^{2} x} + \frac {40 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} f^{2}}{e^{2} x^{2}} - \frac {30 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d f g}{e^{3} x^{2}} + \frac {20 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{2} g^{2}}{e^{4} x^{2}} - \frac {30 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} f^{2}}{e^{4} x^{3}} + \frac {30 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d f g}{e^{5} x^{3}} + \frac {15 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} f^{2}}{e^{6} x^{4}}\right )}}{15 \, d^{3} e^{2} {\left (\frac {d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}}{e^{2} x} - 1\right )}^{5} {\left | e \right |}} \]
2/15*(7*e^2*f^2 - 6*d*e*f*g + 2*d^2*g^2 - 20*(d*e + sqrt(-e^2*x^2 + d^2)*a bs(e))*f^2/x + 30*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*d*f*g/(e*x) - 10*(d* e + sqrt(-e^2*x^2 + d^2)*abs(e))*d^2*g^2/(e^2*x) + 40*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*f^2/(e^2*x^2) - 30*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2 *d*f*g/(e^3*x^2) + 20*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*d^2*g^2/(e^4*x ^2) - 30*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3*f^2/(e^4*x^3) + 30*(d*e + s qrt(-e^2*x^2 + d^2)*abs(e))^3*d*f*g/(e^5*x^3) + 15*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4*f^2/(e^6*x^4))/(d^3*e^2*((d*e + sqrt(-e^2*x^2 + d^2)*abs(e) )/(e^2*x) - 1)^5*abs(e))
Time = 12.12 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.86 \[ \int \frac {(d+e x)^3 (f+g x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\sqrt {d^2-e^2\,x^2}\,\left (2\,d^4\,g^2-6\,d^3\,e\,f\,g-6\,d^3\,e\,g^2\,x+7\,d^2\,e^2\,f^2+18\,d^2\,e^2\,f\,g\,x+7\,d^2\,e^2\,g^2\,x^2-6\,d\,e^3\,f^2\,x-6\,d\,e^3\,f\,g\,x^2+2\,e^4\,f^2\,x^2\right )}{15\,d^3\,e^3\,{\left (d-e\,x\right )}^3} \]