Integrand size = 35, antiderivative size = 557 \[ \int \frac {(A+B x) (d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {231 e^3 (b d-a e) (8 b B d+5 A b e-13 a B e) (a+b x) \sqrt {d+e x}}{64 b^7 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {77 e^3 (8 b B d+5 A b e-13 a B e) (a+b x) (d+e x)^{3/2}}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {231 e^3 (8 b B d+5 A b e-13 a B e) (a+b x) (d+e x)^{5/2}}{320 b^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {33 e^2 (8 b B d+5 A b e-13 a B e) (d+e x)^{7/2}}{64 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {11 e (8 b B d+5 A b e-13 a B e) (d+e x)^{9/2}}{96 b^3 (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(8 b B d+5 A b e-13 a B e) (d+e x)^{11/2}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{13/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 e^3 (b d-a e)^{3/2} (8 b B d+5 A b e-13 a B e) (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
77/64*e^3*(5*A*b*e-13*B*a*e+8*B*b*d)*(b*x+a)*(e*x+d)^(3/2)/b^6/((b*x+a)^2) ^(1/2)+231/320*e^3*(5*A*b*e-13*B*a*e+8*B*b*d)*(b*x+a)*(e*x+d)^(5/2)/b^5/(- a*e+b*d)/((b*x+a)^2)^(1/2)-33/64*e^2*(5*A*b*e-13*B*a*e+8*B*b*d)*(e*x+d)^(7 /2)/b^4/(-a*e+b*d)/((b*x+a)^2)^(1/2)-11/96*e*(5*A*b*e-13*B*a*e+8*B*b*d)*(e *x+d)^(9/2)/b^3/(-a*e+b*d)/(b*x+a)/((b*x+a)^2)^(1/2)-1/24*(5*A*b*e-13*B*a* e+8*B*b*d)*(e*x+d)^(11/2)/b^2/(-a*e+b*d)/(b*x+a)^2/((b*x+a)^2)^(1/2)-1/4*( A*b-B*a)*(e*x+d)^(13/2)/b/(-a*e+b*d)/(b*x+a)^3/((b*x+a)^2)^(1/2)-231/64*e^ 3*(-a*e+b*d)^(3/2)*(5*A*b*e-13*B*a*e+8*B*b*d)*(b*x+a)*arctanh(b^(1/2)*(e*x +d)^(1/2)/(-a*e+b*d)^(1/2))/b^(15/2)/((b*x+a)^2)^(1/2)+231/64*e^3*(-a*e+b* d)*(5*A*b*e-13*B*a*e+8*B*b*d)*(b*x+a)*(e*x+d)^(1/2)/b^7/((b*x+a)^2)^(1/2)
Time = 2.06 (sec) , antiderivative size = 583, normalized size of antiderivative = 1.05 \[ \int \frac {(A+B x) (d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {e^3 (a+b x) \left (-\frac {\sqrt {b} \sqrt {d+e x} \left (5 A b \left (3465 a^5 e^5+1155 a^4 b e^4 (-4 d+11 e x)+231 a^3 b^2 e^3 \left (3 d^2-74 d e x+73 e^2 x^2\right )+99 a^2 b^3 e^2 \left (2 d^3+27 d^2 e x-232 d e^2 x^2+93 e^3 x^3\right )+11 a b^4 e \left (8 d^4+68 d^3 e x+345 d^2 e^2 x^2-1162 d e^3 x^3+128 e^4 x^4\right )+b^5 \left (48 d^5+328 d^4 e x+1030 d^3 e^2 x^2+2295 d^2 e^3 x^3-2048 d e^4 x^4-128 e^5 x^5\right )\right )+B \left (-45045 a^6 e^5+1155 a^5 b e^4 (76 d-143 e x)-231 a^4 b^2 e^3 \left (199 d^2-1402 d e x+949 e^2 x^2\right )+33 a^3 b^3 e^2 \left (90 d^3-5197 d^2 e x+13136 d e^2 x^2-3627 e^3 x^3\right )+11 a^2 b^4 e \left (40 d^4+1060 d^3 e x-21189 d^2 e^2 x^2+21802 d e^3 x^3-1664 e^4 x^4\right )+8 b^6 x \left (40 d^5+310 d^4 e x+1335 d^3 e^2 x^2-2768 d^2 e^3 x^3-416 d e^4 x^4-48 e^5 x^5\right )+a b^5 \left (80 d^5+1720 d^4 e x+16970 d^3 e^2 x^2-132091 d^2 e^3 x^3+37888 d e^4 x^4+1664 e^5 x^5\right )\right )\right )}{e^3 (a+b x)^4}+3465 (-b d+a e)^{3/2} (8 b B d+5 A b e-13 a B e) \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )\right )}{960 b^{15/2} \sqrt {(a+b x)^2}} \]
(e^3*(a + b*x)*(-((Sqrt[b]*Sqrt[d + e*x]*(5*A*b*(3465*a^5*e^5 + 1155*a^4*b *e^4*(-4*d + 11*e*x) + 231*a^3*b^2*e^3*(3*d^2 - 74*d*e*x + 73*e^2*x^2) + 9 9*a^2*b^3*e^2*(2*d^3 + 27*d^2*e*x - 232*d*e^2*x^2 + 93*e^3*x^3) + 11*a*b^4 *e*(8*d^4 + 68*d^3*e*x + 345*d^2*e^2*x^2 - 1162*d*e^3*x^3 + 128*e^4*x^4) + b^5*(48*d^5 + 328*d^4*e*x + 1030*d^3*e^2*x^2 + 2295*d^2*e^3*x^3 - 2048*d* e^4*x^4 - 128*e^5*x^5)) + B*(-45045*a^6*e^5 + 1155*a^5*b*e^4*(76*d - 143*e *x) - 231*a^4*b^2*e^3*(199*d^2 - 1402*d*e*x + 949*e^2*x^2) + 33*a^3*b^3*e^ 2*(90*d^3 - 5197*d^2*e*x + 13136*d*e^2*x^2 - 3627*e^3*x^3) + 11*a^2*b^4*e* (40*d^4 + 1060*d^3*e*x - 21189*d^2*e^2*x^2 + 21802*d*e^3*x^3 - 1664*e^4*x^ 4) + 8*b^6*x*(40*d^5 + 310*d^4*e*x + 1335*d^3*e^2*x^2 - 2768*d^2*e^3*x^3 - 416*d*e^4*x^4 - 48*e^5*x^5) + a*b^5*(80*d^5 + 1720*d^4*e*x + 16970*d^3*e^ 2*x^2 - 132091*d^2*e^3*x^3 + 37888*d*e^4*x^4 + 1664*e^5*x^5))))/(e^3*(a + b*x)^4)) + 3465*(-(b*d) + a*e)^(3/2)*(8*b*B*d + 5*A*b*e - 13*a*B*e)*ArcTan [(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]]))/(960*b^(15/2)*Sqrt[(a + b*x )^2])
Time = 0.43 (sec) , antiderivative size = 315, normalized size of antiderivative = 0.57, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.314, Rules used = {1187, 27, 87, 51, 51, 51, 60, 60, 60, 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.
| \(\displaystyle \int \frac {(A+B x) (d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {b^5 (a+b x) \int \frac {(A+B x) (d+e x)^{11/2}}{b^5 (a+b x)^5}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(a+b x) \int \frac {(A+B x) (d+e x)^{11/2}}{(a+b x)^5}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {(a+b x) \left (\frac {(-13 a B e+5 A b e+8 b B d) \int \frac {(d+e x)^{11/2}}{(a+b x)^4}dx}{8 b (b d-a e)}-\frac {(d+e x)^{13/2} (A b-a B)}{4 b (a+b x)^4 (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {(a+b x) \left (\frac {(-13 a B e+5 A b e+8 b B d) \left (\frac {11 e \int \frac {(d+e x)^{9/2}}{(a+b x)^3}dx}{6 b}-\frac {(d+e x)^{11/2}}{3 b (a+b x)^3}\right )}{8 b (b d-a e)}-\frac {(d+e x)^{13/2} (A b-a B)}{4 b (a+b x)^4 (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {(a+b x) \left (\frac {(-13 a B e+5 A b e+8 b B d) \left (\frac {11 e \left (\frac {9 e \int \frac {(d+e x)^{7/2}}{(a+b x)^2}dx}{4 b}-\frac {(d+e x)^{9/2}}{2 b (a+b x)^2}\right )}{6 b}-\frac {(d+e x)^{11/2}}{3 b (a+b x)^3}\right )}{8 b (b d-a e)}-\frac {(d+e x)^{13/2} (A b-a B)}{4 b (a+b x)^4 (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {(a+b x) \left (\frac {(-13 a B e+5 A b e+8 b B d) \left (\frac {11 e \left (\frac {9 e \left (\frac {7 e \int \frac {(d+e x)^{5/2}}{a+b x}dx}{2 b}-\frac {(d+e x)^{7/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{9/2}}{2 b (a+b x)^2}\right )}{6 b}-\frac {(d+e x)^{11/2}}{3 b (a+b x)^3}\right )}{8 b (b d-a e)}-\frac {(d+e x)^{13/2} (A b-a B)}{4 b (a+b x)^4 (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(a+b x) \left (\frac {(-13 a B e+5 A b e+8 b B d) \left (\frac {11 e \left (\frac {9 e \left (\frac {7 e \left (\frac {(b d-a e) \int \frac {(d+e x)^{3/2}}{a+b x}dx}{b}+\frac {2 (d+e x)^{5/2}}{5 b}\right )}{2 b}-\frac {(d+e x)^{7/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{9/2}}{2 b (a+b x)^2}\right )}{6 b}-\frac {(d+e x)^{11/2}}{3 b (a+b x)^3}\right )}{8 b (b d-a e)}-\frac {(d+e x)^{13/2} (A b-a B)}{4 b (a+b x)^4 (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(a+b x) \left (\frac {(-13 a B e+5 A b e+8 b B d) \left (\frac {11 e \left (\frac {9 e \left (\frac {7 e \left (\frac {(b d-a e) \left (\frac {(b d-a e) \int \frac {\sqrt {d+e x}}{a+b x}dx}{b}+\frac {2 (d+e x)^{3/2}}{3 b}\right )}{b}+\frac {2 (d+e x)^{5/2}}{5 b}\right )}{2 b}-\frac {(d+e x)^{7/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{9/2}}{2 b (a+b x)^2}\right )}{6 b}-\frac {(d+e x)^{11/2}}{3 b (a+b x)^3}\right )}{8 b (b d-a e)}-\frac {(d+e x)^{13/2} (A b-a B)}{4 b (a+b x)^4 (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(a+b x) \left (\frac {(-13 a B e+5 A b e+8 b B d) \left (\frac {11 e \left (\frac {9 e \left (\frac {7 e \left (\frac {(b d-a e) \left (\frac {(b d-a e) \left (\frac {(b d-a e) \int \frac {1}{(a+b x) \sqrt {d+e x}}dx}{b}+\frac {2 \sqrt {d+e x}}{b}\right )}{b}+\frac {2 (d+e x)^{3/2}}{3 b}\right )}{b}+\frac {2 (d+e x)^{5/2}}{5 b}\right )}{2 b}-\frac {(d+e x)^{7/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{9/2}}{2 b (a+b x)^2}\right )}{6 b}-\frac {(d+e x)^{11/2}}{3 b (a+b x)^3}\right )}{8 b (b d-a e)}-\frac {(d+e x)^{13/2} (A b-a B)}{4 b (a+b x)^4 (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {(a+b x) \left (\frac {(-13 a B e+5 A b e+8 b B d) \left (\frac {11 e \left (\frac {9 e \left (\frac {7 e \left (\frac {(b d-a e) \left (\frac {(b d-a e) \left (\frac {2 (b d-a e) \int \frac {1}{a+\frac {b (d+e x)}{e}-\frac {b d}{e}}d\sqrt {d+e x}}{b e}+\frac {2 \sqrt {d+e x}}{b}\right )}{b}+\frac {2 (d+e x)^{3/2}}{3 b}\right )}{b}+\frac {2 (d+e x)^{5/2}}{5 b}\right )}{2 b}-\frac {(d+e x)^{7/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{9/2}}{2 b (a+b x)^2}\right )}{6 b}-\frac {(d+e x)^{11/2}}{3 b (a+b x)^3}\right )}{8 b (b d-a e)}-\frac {(d+e x)^{13/2} (A b-a B)}{4 b (a+b x)^4 (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {(a+b x) \left (\frac {(-13 a B e+5 A b e+8 b B d) \left (\frac {11 e \left (\frac {9 e \left (\frac {7 e \left (\frac {(b d-a e) \left (\frac {(b d-a e) \left (\frac {2 \sqrt {d+e x}}{b}-\frac {2 \sqrt {b d-a e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2}}\right )}{b}+\frac {2 (d+e x)^{3/2}}{3 b}\right )}{b}+\frac {2 (d+e x)^{5/2}}{5 b}\right )}{2 b}-\frac {(d+e x)^{7/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{9/2}}{2 b (a+b x)^2}\right )}{6 b}-\frac {(d+e x)^{11/2}}{3 b (a+b x)^3}\right )}{8 b (b d-a e)}-\frac {(d+e x)^{13/2} (A b-a B)}{4 b (a+b x)^4 (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
((a + b*x)*(-1/4*((A*b - a*B)*(d + e*x)^(13/2))/(b*(b*d - a*e)*(a + b*x)^4 ) + ((8*b*B*d + 5*A*b*e - 13*a*B*e)*(-1/3*(d + e*x)^(11/2)/(b*(a + b*x)^3) + (11*e*(-1/2*(d + e*x)^(9/2)/(b*(a + b*x)^2) + (9*e*(-((d + e*x)^(7/2)/( b*(a + b*x))) + (7*e*((2*(d + e*x)^(5/2))/(5*b) + ((b*d - a*e)*((2*(d + e* x)^(3/2))/(3*b) + ((b*d - a*e)*((2*Sqrt[d + e*x])/b - (2*Sqrt[b*d - a*e]*A rcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/b^(3/2)))/b))/b))/(2*b))) /(4*b)))/(6*b)))/(8*b*(b*d - a*e))))/Sqrt[a^2 + 2*a*b*x + b^2*x^2]
3.19.74.3.1 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/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[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_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
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_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x + c*x^2)^FracPart[p]/(c^ IntPart[p]*(b/2 + c*x)^(2*FracPart[p])) Int[(d + e*x)^m*(f + g*x)^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p]
Time = 0.34 (sec) , antiderivative size = 513, normalized size of antiderivative = 0.92
| method | result | size |
| risch | \(-\frac {2 e^{3} \left (-3 B \,b^{2} e^{2} x^{2}-5 A \,b^{2} e^{2} x +25 B a b \,e^{2} x -26 B \,b^{2} d e x +75 A a b \,e^{2}-80 A \,b^{2} d e -225 a^{2} B \,e^{2}+400 B a b d e -173 B \,b^{2} d^{2}\right ) \sqrt {e x +d}\, \sqrt {\left (b x +a \right )^{2}}}{15 b^{7} \left (b x +a \right )}+\frac {\left (2 e^{2} a^{2}-4 a b d e +2 b^{2} d^{2}\right ) e^{3} \left (\frac {\left (-\frac {765}{128} A \,b^{4} e +\frac {1477}{128} B e \,b^{3} a -\frac {89}{16} B \,b^{4} d \right ) \left (e x +d \right )^{\frac {7}{2}}-\frac {b^{2} \left (5855 A a b \,e^{2}-5855 A \,b^{2} d e -11767 a^{2} B \,e^{2}+17679 B a b d e -5912 B \,b^{2} d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{384}+\left (-\frac {5153}{384} A \,a^{2} b^{2} e^{3}+\frac {5153}{192} A a \,b^{3} d \,e^{2}-\frac {5153}{384} A \,b^{4} d^{2} e +\frac {10633}{384} B \,e^{3} b \,a^{3}-\frac {13373}{192} B \,a^{2} b^{2} d \,e^{2}+\frac {21593}{384} B a \,b^{3} d^{2} e -\frac {685}{48} B \,b^{4} d^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {515}{128} A \,a^{3} b \,e^{4}+\frac {1545}{128} A \,a^{2} b^{2} d \,e^{3}-\frac {1545}{128} A a \,b^{3} d^{2} e^{2}+\frac {515}{128} A \,b^{4} d^{3} e +\frac {1083}{128} B \,a^{4} e^{4}-\frac {3817}{128} B \,a^{3} b d \,e^{3}+\frac {4953}{128} B \,a^{2} b^{2} d^{2} e^{2}-\frac {2787}{128} B a \,b^{3} d^{3} e +\frac {71}{16} b^{4} B \,d^{4}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{4}}+\frac {231 \left (5 A b e -13 B a e +8 B b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{128 \sqrt {\left (a e -b d \right ) b}}\right ) \sqrt {\left (b x +a \right )^{2}}}{b^{7} \left (b x +a \right )}\) | \(513\) |
| default | \(\text {Expression too large to display}\) | \(3768\) |
-2/15*e^3*(-3*B*b^2*e^2*x^2-5*A*b^2*e^2*x+25*B*a*b*e^2*x-26*B*b^2*d*e*x+75 *A*a*b*e^2-80*A*b^2*d*e-225*B*a^2*e^2+400*B*a*b*d*e-173*B*b^2*d^2)*(e*x+d) ^(1/2)/b^7*((b*x+a)^2)^(1/2)/(b*x+a)+1/b^7*(2*a^2*e^2-4*a*b*d*e+2*b^2*d^2) *e^3*(((-765/128*A*b^4*e+1477/128*B*e*b^3*a-89/16*B*b^4*d)*(e*x+d)^(7/2)-1 /384*b^2*(5855*A*a*b*e^2-5855*A*b^2*d*e-11767*B*a^2*e^2+17679*B*a*b*d*e-59 12*B*b^2*d^2)*(e*x+d)^(5/2)+(-5153/384*A*a^2*b^2*e^3+5153/192*A*a*b^3*d*e^ 2-5153/384*A*b^4*d^2*e+10633/384*B*e^3*b*a^3-13373/192*B*a^2*b^2*d*e^2+215 93/384*B*a*b^3*d^2*e-685/48*B*b^4*d^3)*(e*x+d)^(3/2)+(-515/128*A*a^3*b*e^4 +1545/128*A*a^2*b^2*d*e^3-1545/128*A*a*b^3*d^2*e^2+515/128*A*b^4*d^3*e+108 3/128*B*a^4*e^4-3817/128*B*a^3*b*d*e^3+4953/128*B*a^2*b^2*d^2*e^2-2787/128 *B*a*b^3*d^3*e+71/16*b^4*B*d^4)*(e*x+d)^(1/2))/(b*(e*x+d)+a*e-b*d)^4+231/1 28*(5*A*b*e-13*B*a*e+8*B*b*d)/((a*e-b*d)*b)^(1/2)*arctan(b*(e*x+d)^(1/2)/( (a*e-b*d)*b)^(1/2)))*((b*x+a)^2)^(1/2)/(b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 998 vs. \(2 (429) = 858\).
Time = 0.52 (sec) , antiderivative size = 2006, normalized size of antiderivative = 3.60 \[ \int \frac {(A+B x) (d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\text {Too large to display} \]
[1/1920*(3465*(8*B*a^4*b^2*d^2*e^3 - (21*B*a^5*b - 5*A*a^4*b^2)*d*e^4 + (1 3*B*a^6 - 5*A*a^5*b)*e^5 + (8*B*b^6*d^2*e^3 - (21*B*a*b^5 - 5*A*b^6)*d*e^4 + (13*B*a^2*b^4 - 5*A*a*b^5)*e^5)*x^4 + 4*(8*B*a*b^5*d^2*e^3 - (21*B*a^2* b^4 - 5*A*a*b^5)*d*e^4 + (13*B*a^3*b^3 - 5*A*a^2*b^4)*e^5)*x^3 + 6*(8*B*a^ 2*b^4*d^2*e^3 - (21*B*a^3*b^3 - 5*A*a^2*b^4)*d*e^4 + (13*B*a^4*b^2 - 5*A*a ^3*b^3)*e^5)*x^2 + 4*(8*B*a^3*b^3*d^2*e^3 - (21*B*a^4*b^2 - 5*A*a^3*b^3)*d *e^4 + (13*B*a^5*b - 5*A*a^4*b^2)*e^5)*x)*sqrt((b*d - a*e)/b)*log((b*e*x + 2*b*d - a*e - 2*sqrt(e*x + d)*b*sqrt((b*d - a*e)/b))/(b*x + a)) + 2*(384* B*b^6*e^5*x^6 - 80*(B*a*b^5 + 3*A*b^6)*d^5 - 440*(B*a^2*b^4 + A*a*b^5)*d^4 *e - 990*(3*B*a^3*b^3 + A*a^2*b^4)*d^3*e^2 + 231*(199*B*a^4*b^2 - 15*A*a^3 *b^3)*d^2*e^3 - 4620*(19*B*a^5*b - 5*A*a^4*b^2)*d*e^4 + 3465*(13*B*a^6 - 5 *A*a^5*b)*e^5 + 128*(26*B*b^6*d*e^4 - (13*B*a*b^5 - 5*A*b^6)*e^5)*x^5 + 12 8*(173*B*b^6*d^2*e^3 - 8*(37*B*a*b^5 - 10*A*b^6)*d*e^4 + 11*(13*B*a^2*b^4 - 5*A*a*b^5)*e^5)*x^4 - (10680*B*b^6*d^3*e^2 - (132091*B*a*b^5 - 11475*A*b ^6)*d^2*e^3 + 22*(10901*B*a^2*b^4 - 2905*A*a*b^5)*d*e^4 - 9207*(13*B*a^3*b ^3 - 5*A*a^2*b^4)*e^5)*x^3 - (2480*B*b^6*d^4*e + 10*(1697*B*a*b^5 + 515*A* b^6)*d^3*e^2 - 33*(7063*B*a^2*b^4 - 575*A*a*b^5)*d^2*e^3 + 264*(1642*B*a^3 *b^3 - 435*A*a^2*b^4)*d*e^4 - 16863*(13*B*a^4*b^2 - 5*A*a^3*b^3)*e^5)*x^2 - (320*B*b^6*d^5 + 40*(43*B*a*b^5 + 41*A*b^6)*d^4*e + 220*(53*B*a^2*b^4 + 17*A*a*b^5)*d^3*e^2 - 33*(5197*B*a^3*b^3 - 405*A*a^2*b^4)*d^2*e^3 + 462...
Timed out. \[ \int \frac {(A+B x) (d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(A+B x) (d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int { \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {11}{2}}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 1121 vs. \(2 (429) = 858\).
Time = 0.33 (sec) , antiderivative size = 1121, normalized size of antiderivative = 2.01 \[ \int \frac {(A+B x) (d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\text {Too large to display} \]
231/64*(8*B*b^3*d^3*e^3 - 29*B*a*b^2*d^2*e^4 + 5*A*b^3*d^2*e^4 + 34*B*a^2* b*d*e^5 - 10*A*a*b^2*d*e^5 - 13*B*a^3*e^6 + 5*A*a^2*b*e^6)*arctan(sqrt(e*x + d)*b/sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d + a*b*e)*b^7*sgn(b*x + a)) - 1/ 192*(2136*(e*x + d)^(7/2)*B*b^6*d^3*e^3 - 5912*(e*x + d)^(5/2)*B*b^6*d^4*e ^3 + 5480*(e*x + d)^(3/2)*B*b^6*d^5*e^3 - 1704*sqrt(e*x + d)*B*b^6*d^6*e^3 - 8703*(e*x + d)^(7/2)*B*a*b^5*d^2*e^4 + 2295*(e*x + d)^(7/2)*A*b^6*d^2*e ^4 + 29503*(e*x + d)^(5/2)*B*a*b^5*d^3*e^4 - 5855*(e*x + d)^(5/2)*A*b^6*d^ 3*e^4 - 32553*(e*x + d)^(3/2)*B*a*b^5*d^4*e^4 + 5153*(e*x + d)^(3/2)*A*b^6 *d^4*e^4 + 11769*sqrt(e*x + d)*B*a*b^5*d^5*e^4 - 1545*sqrt(e*x + d)*A*b^6* d^5*e^4 + 10998*(e*x + d)^(7/2)*B*a^2*b^4*d*e^5 - 4590*(e*x + d)^(7/2)*A*a *b^5*d*e^5 - 53037*(e*x + d)^(5/2)*B*a^2*b^4*d^2*e^5 + 17565*(e*x + d)^(5/ 2)*A*a*b^5*d^2*e^5 + 75412*(e*x + d)^(3/2)*B*a^2*b^4*d^3*e^5 - 20612*(e*x + d)^(3/2)*A*a*b^5*d^3*e^5 - 33285*sqrt(e*x + d)*B*a^2*b^4*d^4*e^5 + 7725* sqrt(e*x + d)*A*a*b^5*d^4*e^5 - 4431*(e*x + d)^(7/2)*B*a^3*b^3*e^6 + 2295* (e*x + d)^(7/2)*A*a^2*b^4*e^6 + 41213*(e*x + d)^(5/2)*B*a^3*b^3*d*e^6 - 17 565*(e*x + d)^(5/2)*A*a^2*b^4*d*e^6 - 85718*(e*x + d)^(3/2)*B*a^3*b^3*d^2* e^6 + 30918*(e*x + d)^(3/2)*A*a^2*b^4*d^2*e^6 + 49530*sqrt(e*x + d)*B*a^3* b^3*d^3*e^6 - 15450*sqrt(e*x + d)*A*a^2*b^4*d^3*e^6 - 11767*(e*x + d)^(5/2 )*B*a^4*b^2*e^7 + 5855*(e*x + d)^(5/2)*A*a^3*b^3*e^7 + 48012*(e*x + d)^(3/ 2)*B*a^4*b^2*d*e^7 - 20612*(e*x + d)^(3/2)*A*a^3*b^3*d*e^7 - 41010*sqrt...
Timed out. \[ \int \frac {(A+B x) (d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^{11/2}}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \]