Integrand size = 30, antiderivative size = 435 \[ \int \frac {1}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {429 e^3}{64 (b d-a e)^4 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{4 (b d-a e) (a+b x)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {13 e}{24 (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {143 e^2}{96 (b d-a e)^3 (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3003 e^4 (a+b x)}{320 (b d-a e)^5 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1001 b e^4 (a+b x)}{64 (b d-a e)^6 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3003 b^2 e^4 (a+b x)}{64 (b d-a e)^7 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3003 b^{5/2} e^4 (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 (b d-a e)^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
429/64*e^3/(-a*e+b*d)^4/(e*x+d)^(5/2)/((b*x+a)^2)^(1/2)-1/4/(-a*e+b*d)/(b* x+a)^3/(e*x+d)^(5/2)/((b*x+a)^2)^(1/2)+13/24*e/(-a*e+b*d)^2/(b*x+a)^2/(e*x +d)^(5/2)/((b*x+a)^2)^(1/2)-143/96*e^2/(-a*e+b*d)^3/(b*x+a)/(e*x+d)^(5/2)/ ((b*x+a)^2)^(1/2)+3003/320*e^4*(b*x+a)/(-a*e+b*d)^5/(e*x+d)^(5/2)/((b*x+a) ^2)^(1/2)+1001/64*b*e^4*(b*x+a)/(-a*e+b*d)^6/(e*x+d)^(3/2)/((b*x+a)^2)^(1/ 2)-3003/64*b^(5/2)*e^4*(b*x+a)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1 /2))/(-a*e+b*d)^(15/2)/((b*x+a)^2)^(1/2)+3003/64*b^2*e^4*(b*x+a)/(-a*e+b*d )^7/(e*x+d)^(1/2)/((b*x+a)^2)^(1/2)
Time = 1.49 (sec) , antiderivative size = 378, normalized size of antiderivative = 0.87 \[ \int \frac {1}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {e^4 (a+b x)^5 \left (\frac {-384 a^6 e^6+128 a^5 b e^5 (31 d+13 e x)-128 a^4 b^2 e^4 \left (253 d^2+351 d e x+143 e^2 x^2\right )-a^3 b^3 e^3 \left (22155 d^3+196001 d^2 e x+285857 d e^2 x^2+119691 e^3 x^3\right )-a^2 b^4 e^2 \left (-7630 d^4+35945 d^3 e x+347919 d^2 e^2 x^2+517803 d e^3 x^3+219219 e^4 x^4\right )-a b^5 e \left (1960 d^5-5460 d^4 e x+25025 d^3 e^2 x^2+256971 d^2 e^3 x^3+387387 d e^4 x^4+165165 e^5 x^5\right )+b^6 \left (240 d^6-520 d^5 e x+1430 d^4 e^2 x^2-6435 d^3 e^3 x^3-69069 d^2 e^4 x^4-105105 d e^5 x^5-45045 e^6 x^6\right )}{e^4 (-b d+a e)^7 (a+b x)^4 (d+e x)^{5/2}}-\frac {45045 b^{5/2} \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{(-b d+a e)^{15/2}}\right )}{960 \left ((a+b x)^2\right )^{5/2}} \]
(e^4*(a + b*x)^5*((-384*a^6*e^6 + 128*a^5*b*e^5*(31*d + 13*e*x) - 128*a^4* b^2*e^4*(253*d^2 + 351*d*e*x + 143*e^2*x^2) - a^3*b^3*e^3*(22155*d^3 + 196 001*d^2*e*x + 285857*d*e^2*x^2 + 119691*e^3*x^3) - a^2*b^4*e^2*(-7630*d^4 + 35945*d^3*e*x + 347919*d^2*e^2*x^2 + 517803*d*e^3*x^3 + 219219*e^4*x^4) - a*b^5*e*(1960*d^5 - 5460*d^4*e*x + 25025*d^3*e^2*x^2 + 256971*d^2*e^3*x^ 3 + 387387*d*e^4*x^4 + 165165*e^5*x^5) + b^6*(240*d^6 - 520*d^5*e*x + 1430 *d^4*e^2*x^2 - 6435*d^3*e^3*x^3 - 69069*d^2*e^4*x^4 - 105105*d*e^5*x^5 - 4 5045*e^6*x^6))/(e^4*(-(b*d) + a*e)^7*(a + b*x)^4*(d + e*x)^(5/2)) - (45045 *b^(5/2)*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(-(b*d) + a*e )^(15/2)))/(960*((a + b*x)^2)^(5/2))
Time = 0.40 (sec) , antiderivative size = 349, normalized size of antiderivative = 0.80, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.367, Rules used = {1102, 27, 52, 52, 52, 52, 61, 61, 61, 73, 221}
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.
| \(\displaystyle \int \frac {1}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2} (d+e x)^{7/2}} \, dx\) |
| \(\Big \downarrow \) 1102 |
| \(\displaystyle \frac {b^5 (a+b x) \int \frac {1}{b^5 (a+b x)^5 (d+e x)^{7/2}}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(a+b x) \int \frac {1}{(a+b x)^5 (d+e x)^{7/2}}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {(a+b x) \left (-\frac {13 e \int \frac {1}{(a+b x)^4 (d+e x)^{7/2}}dx}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 (d+e x)^{5/2} (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {(a+b x) \left (-\frac {13 e \left (-\frac {11 e \int \frac {1}{(a+b x)^3 (d+e x)^{7/2}}dx}{6 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{5/2} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 (d+e x)^{5/2} (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {(a+b x) \left (-\frac {13 e \left (-\frac {11 e \left (-\frac {9 e \int \frac {1}{(a+b x)^2 (d+e x)^{7/2}}dx}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{5/2} (b d-a e)}\right )}{6 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{5/2} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 (d+e x)^{5/2} (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {(a+b x) \left (-\frac {13 e \left (-\frac {11 e \left (-\frac {9 e \left (-\frac {7 e \int \frac {1}{(a+b x) (d+e x)^{7/2}}dx}{2 (b d-a e)}-\frac {1}{(a+b x) (d+e x)^{5/2} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{5/2} (b d-a e)}\right )}{6 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{5/2} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 (d+e x)^{5/2} (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {(a+b x) \left (-\frac {13 e \left (-\frac {11 e \left (-\frac {9 e \left (-\frac {7 e \left (\frac {b \int \frac {1}{(a+b x) (d+e x)^{5/2}}dx}{b d-a e}+\frac {2}{5 (d+e x)^{5/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) (d+e x)^{5/2} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{5/2} (b d-a e)}\right )}{6 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{5/2} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 (d+e x)^{5/2} (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {(a+b x) \left (-\frac {13 e \left (-\frac {11 e \left (-\frac {9 e \left (-\frac {7 e \left (\frac {b \left (\frac {b \int \frac {1}{(a+b x) (d+e x)^{3/2}}dx}{b d-a e}+\frac {2}{3 (d+e x)^{3/2} (b d-a e)}\right )}{b d-a e}+\frac {2}{5 (d+e x)^{5/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) (d+e x)^{5/2} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{5/2} (b d-a e)}\right )}{6 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{5/2} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 (d+e x)^{5/2} (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {(a+b x) \left (-\frac {13 e \left (-\frac {11 e \left (-\frac {9 e \left (-\frac {7 e \left (\frac {b \left (\frac {b \left (\frac {b \int \frac {1}{(a+b x) \sqrt {d+e x}}dx}{b d-a e}+\frac {2}{\sqrt {d+e x} (b d-a e)}\right )}{b d-a e}+\frac {2}{3 (d+e x)^{3/2} (b d-a e)}\right )}{b d-a e}+\frac {2}{5 (d+e x)^{5/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) (d+e x)^{5/2} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{5/2} (b d-a e)}\right )}{6 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{5/2} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 (d+e x)^{5/2} (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {(a+b x) \left (-\frac {13 e \left (-\frac {11 e \left (-\frac {9 e \left (-\frac {7 e \left (\frac {b \left (\frac {b \left (\frac {2 b \int \frac {1}{a+\frac {b (d+e x)}{e}-\frac {b d}{e}}d\sqrt {d+e x}}{e (b d-a e)}+\frac {2}{\sqrt {d+e x} (b d-a e)}\right )}{b d-a e}+\frac {2}{3 (d+e x)^{3/2} (b d-a e)}\right )}{b d-a e}+\frac {2}{5 (d+e x)^{5/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) (d+e x)^{5/2} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{5/2} (b d-a e)}\right )}{6 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{5/2} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 (d+e x)^{5/2} (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {(a+b x) \left (-\frac {13 e \left (-\frac {11 e \left (-\frac {9 e \left (-\frac {7 e \left (\frac {b \left (\frac {b \left (\frac {2}{\sqrt {d+e x} (b d-a e)}-\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{3/2}}\right )}{b d-a e}+\frac {2}{3 (d+e x)^{3/2} (b d-a e)}\right )}{b d-a e}+\frac {2}{5 (d+e x)^{5/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) (d+e x)^{5/2} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{5/2} (b d-a e)}\right )}{6 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{5/2} (b d-a e)}\right )}{8 (b d-a e)}-\frac {1}{4 (a+b x)^4 (d+e x)^{5/2} (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
((a + b*x)*(-1/4*1/((b*d - a*e)*(a + b*x)^4*(d + e*x)^(5/2)) - (13*e*(-1/3 *1/((b*d - a*e)*(a + b*x)^3*(d + e*x)^(5/2)) - (11*e*(-1/2*1/((b*d - a*e)* (a + b*x)^2*(d + e*x)^(5/2)) - (9*e*(-(1/((b*d - a*e)*(a + b*x)*(d + e*x)^ (5/2))) - (7*e*(2/(5*(b*d - a*e)*(d + e*x)^(5/2)) + (b*(2/(3*(b*d - a*e)*( d + e*x)^(3/2)) + (b*(2/((b*d - a*e)*Sqrt[d + e*x]) - (2*Sqrt[b]*ArcTanh[( Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(b*d - a*e)^(3/2)))/(b*d - a*e))) /(b*d - a*e)))/(2*(b*d - a*e))))/(4*(b*d - a*e))))/(6*(b*d - a*e))))/(8*(b *d - a*e))))/Sqrt[a^2 + 2*a*b*x + b^2*x^2]
3.18.30.3.1 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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*F racPart[p])) Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(950\) vs. \(2(307)=614\).
Time = 2.57 (sec) , antiderivative size = 951, normalized size of antiderivative = 2.19
| method | result | size |
| default | \(-\frac {\left (256971 \sqrt {\left (a e -b d \right ) b}\, a \,b^{5} d^{2} e^{4} x^{3}+347919 \sqrt {\left (a e -b d \right ) b}\, a^{2} b^{4} d^{2} e^{4} x^{2}+180180 \left (e x +d \right )^{\frac {5}{2}} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a \,b^{6} e^{4} x^{3}+270270 \left (e x +d \right )^{\frac {5}{2}} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{2} b^{5} e^{4} x^{2}+25025 \sqrt {\left (a e -b d \right ) b}\, a \,b^{5} d^{3} e^{3} x^{2}+196001 \sqrt {\left (a e -b d \right ) b}\, a^{3} b^{3} d^{2} e^{4} x +35945 \sqrt {\left (a e -b d \right ) b}\, a^{2} b^{4} d^{3} e^{3} x -5460 \sqrt {\left (a e -b d \right ) b}\, a \,b^{5} d^{4} e^{2} x +180180 \left (e x +d \right )^{\frac {5}{2}} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{3} b^{4} e^{4} x +165165 \sqrt {\left (a e -b d \right ) b}\, a \,b^{5} e^{6} x^{5}+105105 \sqrt {\left (a e -b d \right ) b}\, b^{6} d \,e^{5} x^{5}+219219 \sqrt {\left (a e -b d \right ) b}\, a^{2} b^{4} e^{6} x^{4}+119691 \sqrt {\left (a e -b d \right ) b}\, a^{3} b^{3} e^{6} x^{3}+18304 \sqrt {\left (a e -b d \right ) b}\, a^{4} b^{2} e^{6} x^{2}-1664 \sqrt {\left (a e -b d \right ) b}\, a^{5} b \,e^{6} x -3968 \sqrt {\left (a e -b d \right ) b}\, a^{5} b d \,e^{5}+384 \sqrt {\left (a e -b d \right ) b}\, a^{6} e^{6}-240 \sqrt {\left (a e -b d \right ) b}\, b^{6} d^{6}+45045 \sqrt {\left (a e -b d \right ) b}\, b^{6} e^{6} x^{6}+45045 \left (e x +d \right )^{\frac {5}{2}} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) b^{7} e^{4} x^{4}+45045 \left (e x +d \right )^{\frac {5}{2}} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{4} b^{3} e^{4}+387387 \sqrt {\left (a e -b d \right ) b}\, a \,b^{5} d \,e^{5} x^{4}+517803 \sqrt {\left (a e -b d \right ) b}\, a^{2} b^{4} d \,e^{5} x^{3}+285857 \sqrt {\left (a e -b d \right ) b}\, a^{3} b^{3} d \,e^{5} x^{2}+44928 \sqrt {\left (a e -b d \right ) b}\, a^{4} b^{2} d \,e^{5} x +69069 \sqrt {\left (a e -b d \right ) b}\, b^{6} d^{2} e^{4} x^{4}+6435 \sqrt {\left (a e -b d \right ) b}\, b^{6} d^{3} e^{3} x^{3}-1430 \sqrt {\left (a e -b d \right ) b}\, b^{6} d^{4} e^{2} x^{2}+520 \sqrt {\left (a e -b d \right ) b}\, b^{6} d^{5} e x +32384 \sqrt {\left (a e -b d \right ) b}\, a^{4} b^{2} d^{2} e^{4}+22155 \sqrt {\left (a e -b d \right ) b}\, a^{3} b^{3} d^{3} e^{3}-7630 \sqrt {\left (a e -b d \right ) b}\, a^{2} b^{4} d^{4} e^{2}+1960 \sqrt {\left (a e -b d \right ) b}\, a \,b^{5} d^{5} e \right ) \left (b x +a \right )}{960 \sqrt {\left (a e -b d \right ) b}\, \left (e x +d \right )^{\frac {5}{2}} \left (a e -b d \right )^{7} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}\) | \(951\) |
-1/960*(256971*((a*e-b*d)*b)^(1/2)*a*b^5*d^2*e^4*x^3+347919*((a*e-b*d)*b)^ (1/2)*a^2*b^4*d^2*e^4*x^2+180180*(e*x+d)^(5/2)*arctan(b*(e*x+d)^(1/2)/((a* e-b*d)*b)^(1/2))*a*b^6*e^4*x^3+270270*(e*x+d)^(5/2)*arctan(b*(e*x+d)^(1/2) /((a*e-b*d)*b)^(1/2))*a^2*b^5*e^4*x^2+25025*((a*e-b*d)*b)^(1/2)*a*b^5*d^3* e^3*x^2+196001*((a*e-b*d)*b)^(1/2)*a^3*b^3*d^2*e^4*x+35945*((a*e-b*d)*b)^( 1/2)*a^2*b^4*d^3*e^3*x-5460*((a*e-b*d)*b)^(1/2)*a*b^5*d^4*e^2*x+180180*(e* x+d)^(5/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*a^3*b^4*e^4*x+16516 5*((a*e-b*d)*b)^(1/2)*a*b^5*e^6*x^5+105105*((a*e-b*d)*b)^(1/2)*b^6*d*e^5*x ^5+219219*((a*e-b*d)*b)^(1/2)*a^2*b^4*e^6*x^4+119691*((a*e-b*d)*b)^(1/2)*a ^3*b^3*e^6*x^3+18304*((a*e-b*d)*b)^(1/2)*a^4*b^2*e^6*x^2-1664*((a*e-b*d)*b )^(1/2)*a^5*b*e^6*x-3968*((a*e-b*d)*b)^(1/2)*a^5*b*d*e^5+384*((a*e-b*d)*b) ^(1/2)*a^6*e^6-240*((a*e-b*d)*b)^(1/2)*b^6*d^6+45045*((a*e-b*d)*b)^(1/2)*b ^6*e^6*x^6+45045*(e*x+d)^(5/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)) *b^7*e^4*x^4+45045*(e*x+d)^(5/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2 ))*a^4*b^3*e^4+387387*((a*e-b*d)*b)^(1/2)*a*b^5*d*e^5*x^4+517803*((a*e-b*d )*b)^(1/2)*a^2*b^4*d*e^5*x^3+285857*((a*e-b*d)*b)^(1/2)*a^3*b^3*d*e^5*x^2+ 44928*((a*e-b*d)*b)^(1/2)*a^4*b^2*d*e^5*x+69069*((a*e-b*d)*b)^(1/2)*b^6*d^ 2*e^4*x^4+6435*((a*e-b*d)*b)^(1/2)*b^6*d^3*e^3*x^3-1430*((a*e-b*d)*b)^(1/2 )*b^6*d^4*e^2*x^2+520*((a*e-b*d)*b)^(1/2)*b^6*d^5*e*x+32384*((a*e-b*d)*b)^ (1/2)*a^4*b^2*d^2*e^4+22155*((a*e-b*d)*b)^(1/2)*a^3*b^3*d^3*e^3-7630*((...
Leaf count of result is larger than twice the leaf count of optimal. 1679 vs. \(2 (307) = 614\).
Time = 1.43 (sec) , antiderivative size = 3368, normalized size of antiderivative = 7.74 \[ \int \frac {1}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\text {Too large to display} \]
[-1/1920*(45045*(b^6*e^7*x^7 + a^4*b^2*d^3*e^4 + (3*b^6*d*e^6 + 4*a*b^5*e^ 7)*x^6 + 3*(b^6*d^2*e^5 + 4*a*b^5*d*e^6 + 2*a^2*b^4*e^7)*x^5 + (b^6*d^3*e^ 4 + 12*a*b^5*d^2*e^5 + 18*a^2*b^4*d*e^6 + 4*a^3*b^3*e^7)*x^4 + (4*a*b^5*d^ 3*e^4 + 18*a^2*b^4*d^2*e^5 + 12*a^3*b^3*d*e^6 + a^4*b^2*e^7)*x^3 + 3*(2*a^ 2*b^4*d^3*e^4 + 4*a^3*b^3*d^2*e^5 + a^4*b^2*d*e^6)*x^2 + (4*a^3*b^3*d^3*e^ 4 + 3*a^4*b^2*d^2*e^5)*x)*sqrt(b/(b*d - a*e))*log((b*e*x + 2*b*d - a*e + 2 *(b*d - a*e)*sqrt(e*x + d)*sqrt(b/(b*d - a*e)))/(b*x + a)) - 2*(45045*b^6* e^6*x^6 - 240*b^6*d^6 + 1960*a*b^5*d^5*e - 7630*a^2*b^4*d^4*e^2 + 22155*a^ 3*b^3*d^3*e^3 + 32384*a^4*b^2*d^2*e^4 - 3968*a^5*b*d*e^5 + 384*a^6*e^6 + 1 5015*(7*b^6*d*e^5 + 11*a*b^5*e^6)*x^5 + 3003*(23*b^6*d^2*e^4 + 129*a*b^5*d *e^5 + 73*a^2*b^4*e^6)*x^4 + 429*(15*b^6*d^3*e^3 + 599*a*b^5*d^2*e^4 + 120 7*a^2*b^4*d*e^5 + 279*a^3*b^3*e^6)*x^3 - 143*(10*b^6*d^4*e^2 - 175*a*b^5*d ^3*e^3 - 2433*a^2*b^4*d^2*e^4 - 1999*a^3*b^3*d*e^5 - 128*a^4*b^2*e^6)*x^2 + 13*(40*b^6*d^5*e - 420*a*b^5*d^4*e^2 + 2765*a^2*b^4*d^3*e^3 + 15077*a^3* b^3*d^2*e^4 + 3456*a^4*b^2*d*e^5 - 128*a^5*b*e^6)*x)*sqrt(e*x + d))/(a^4*b ^7*d^10 - 7*a^5*b^6*d^9*e + 21*a^6*b^5*d^8*e^2 - 35*a^7*b^4*d^7*e^3 + 35*a ^8*b^3*d^6*e^4 - 21*a^9*b^2*d^5*e^5 + 7*a^10*b*d^4*e^6 - a^11*d^3*e^7 + (b ^11*d^7*e^3 - 7*a*b^10*d^6*e^4 + 21*a^2*b^9*d^5*e^5 - 35*a^3*b^8*d^4*e^6 + 35*a^4*b^7*d^3*e^7 - 21*a^5*b^6*d^2*e^8 + 7*a^6*b^5*d*e^9 - a^7*b^4*e^10) *x^7 + (3*b^11*d^8*e^2 - 17*a*b^10*d^7*e^3 + 35*a^2*b^9*d^6*e^4 - 21*a^...
\[ \int \frac {1}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int \frac {1}{\left (d + e x\right )^{\frac {7}{2}} \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {1}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 743 vs. \(2 (307) = 614\).
Time = 0.32 (sec) , antiderivative size = 743, normalized size of antiderivative = 1.71 \[ \int \frac {1}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {3003 \, b^{3} e^{4} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{64 \, {\left (b^{7} d^{7} \mathrm {sgn}\left (b x + a\right ) - 7 \, a b^{6} d^{6} e \mathrm {sgn}\left (b x + a\right ) + 21 \, a^{2} b^{5} d^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) - 35 \, a^{3} b^{4} d^{4} e^{3} \mathrm {sgn}\left (b x + a\right ) + 35 \, a^{4} b^{3} d^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) - 21 \, a^{5} b^{2} d^{2} e^{5} \mathrm {sgn}\left (b x + a\right ) + 7 \, a^{6} b d e^{6} \mathrm {sgn}\left (b x + a\right ) - a^{7} e^{7} \mathrm {sgn}\left (b x + a\right )\right )} \sqrt {-b^{2} d + a b e}} + \frac {2 \, {\left (225 \, {\left (e x + d\right )}^{2} b^{2} e^{4} + 25 \, {\left (e x + d\right )} b^{2} d e^{4} + 3 \, b^{2} d^{2} e^{4} - 25 \, {\left (e x + d\right )} a b e^{5} - 6 \, a b d e^{5} + 3 \, a^{2} e^{6}\right )}}{15 \, {\left (b^{7} d^{7} \mathrm {sgn}\left (b x + a\right ) - 7 \, a b^{6} d^{6} e \mathrm {sgn}\left (b x + a\right ) + 21 \, a^{2} b^{5} d^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) - 35 \, a^{3} b^{4} d^{4} e^{3} \mathrm {sgn}\left (b x + a\right ) + 35 \, a^{4} b^{3} d^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) - 21 \, a^{5} b^{2} d^{2} e^{5} \mathrm {sgn}\left (b x + a\right ) + 7 \, a^{6} b d e^{6} \mathrm {sgn}\left (b x + a\right ) - a^{7} e^{7} \mathrm {sgn}\left (b x + a\right )\right )} {\left (e x + d\right )}^{\frac {5}{2}}} + \frac {3249 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{6} e^{4} - 10633 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{6} d e^{4} + 11767 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{6} d^{2} e^{4} - 4431 \, \sqrt {e x + d} b^{6} d^{3} e^{4} + 10633 \, {\left (e x + d\right )}^{\frac {5}{2}} a b^{5} e^{5} - 23534 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{5} d e^{5} + 13293 \, \sqrt {e x + d} a b^{5} d^{2} e^{5} + 11767 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} b^{4} e^{6} - 13293 \, \sqrt {e x + d} a^{2} b^{4} d e^{6} + 4431 \, \sqrt {e x + d} a^{3} b^{3} e^{7}}{192 \, {\left (b^{7} d^{7} \mathrm {sgn}\left (b x + a\right ) - 7 \, a b^{6} d^{6} e \mathrm {sgn}\left (b x + a\right ) + 21 \, a^{2} b^{5} d^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) - 35 \, a^{3} b^{4} d^{4} e^{3} \mathrm {sgn}\left (b x + a\right ) + 35 \, a^{4} b^{3} d^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) - 21 \, a^{5} b^{2} d^{2} e^{5} \mathrm {sgn}\left (b x + a\right ) + 7 \, a^{6} b d e^{6} \mathrm {sgn}\left (b x + a\right ) - a^{7} e^{7} \mathrm {sgn}\left (b x + a\right )\right )} {\left ({\left (e x + d\right )} b - b d + a e\right )}^{4}} \]
3003/64*b^3*e^4*arctan(sqrt(e*x + d)*b/sqrt(-b^2*d + a*b*e))/((b^7*d^7*sgn (b*x + a) - 7*a*b^6*d^6*e*sgn(b*x + a) + 21*a^2*b^5*d^5*e^2*sgn(b*x + a) - 35*a^3*b^4*d^4*e^3*sgn(b*x + a) + 35*a^4*b^3*d^3*e^4*sgn(b*x + a) - 21*a^ 5*b^2*d^2*e^5*sgn(b*x + a) + 7*a^6*b*d*e^6*sgn(b*x + a) - a^7*e^7*sgn(b*x + a))*sqrt(-b^2*d + a*b*e)) + 2/15*(225*(e*x + d)^2*b^2*e^4 + 25*(e*x + d) *b^2*d*e^4 + 3*b^2*d^2*e^4 - 25*(e*x + d)*a*b*e^5 - 6*a*b*d*e^5 + 3*a^2*e^ 6)/((b^7*d^7*sgn(b*x + a) - 7*a*b^6*d^6*e*sgn(b*x + a) + 21*a^2*b^5*d^5*e^ 2*sgn(b*x + a) - 35*a^3*b^4*d^4*e^3*sgn(b*x + a) + 35*a^4*b^3*d^3*e^4*sgn( b*x + a) - 21*a^5*b^2*d^2*e^5*sgn(b*x + a) + 7*a^6*b*d*e^6*sgn(b*x + a) - a^7*e^7*sgn(b*x + a))*(e*x + d)^(5/2)) + 1/192*(3249*(e*x + d)^(7/2)*b^6*e ^4 - 10633*(e*x + d)^(5/2)*b^6*d*e^4 + 11767*(e*x + d)^(3/2)*b^6*d^2*e^4 - 4431*sqrt(e*x + d)*b^6*d^3*e^4 + 10633*(e*x + d)^(5/2)*a*b^5*e^5 - 23534* (e*x + d)^(3/2)*a*b^5*d*e^5 + 13293*sqrt(e*x + d)*a*b^5*d^2*e^5 + 11767*(e *x + d)^(3/2)*a^2*b^4*e^6 - 13293*sqrt(e*x + d)*a^2*b^4*d*e^6 + 4431*sqrt( e*x + d)*a^3*b^3*e^7)/((b^7*d^7*sgn(b*x + a) - 7*a*b^6*d^6*e*sgn(b*x + a) + 21*a^2*b^5*d^5*e^2*sgn(b*x + a) - 35*a^3*b^4*d^4*e^3*sgn(b*x + a) + 35*a ^4*b^3*d^3*e^4*sgn(b*x + a) - 21*a^5*b^2*d^2*e^5*sgn(b*x + a) + 7*a^6*b*d* e^6*sgn(b*x + a) - a^7*e^7*sgn(b*x + a))*((e*x + d)*b - b*d + a*e)^4)
Timed out. \[ \int \frac {1}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int \frac {1}{{\left (d+e\,x\right )}^{7/2}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \]