Integrand size = 33, antiderivative size = 313 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {7 e^3 (10 b B d-A b e-9 a B e) \sqrt {d+e x}}{128 b^5 (b d-a e) (a+b x)}-\frac {7 e^2 (10 b B d-A b e-9 a B e) (d+e x)^{3/2}}{192 b^4 (b d-a e) (a+b x)^2}-\frac {7 e (10 b B d-A b e-9 a B e) (d+e x)^{5/2}}{240 b^3 (b d-a e) (a+b x)^3}-\frac {(10 b B d-A b e-9 a B e) (d+e x)^{7/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{9/2}}{5 b (b d-a e) (a+b x)^5}-\frac {7 e^4 (10 b B d-A b e-9 a B e) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{11/2} (b d-a e)^{3/2}} \]
-7/192*e^2*(-A*b*e-9*B*a*e+10*B*b*d)*(e*x+d)^(3/2)/b^4/(-a*e+b*d)/(b*x+a)^ 2-7/240*e*(-A*b*e-9*B*a*e+10*B*b*d)*(e*x+d)^(5/2)/b^3/(-a*e+b*d)/(b*x+a)^3 -1/40*(-A*b*e-9*B*a*e+10*B*b*d)*(e*x+d)^(7/2)/b^2/(-a*e+b*d)/(b*x+a)^4-1/5 *(A*b-B*a)*(e*x+d)^(9/2)/b/(-a*e+b*d)/(b*x+a)^5-7/128*e^4*(-A*b*e-9*B*a*e+ 10*B*b*d)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))/b^(11/2)/(-a*e+b *d)^(3/2)-7/128*e^3*(-A*b*e-9*B*a*e+10*B*b*d)*(e*x+d)^(1/2)/b^5/(-a*e+b*d) /(b*x+a)
Time = 3.35 (sec) , antiderivative size = 421, normalized size of antiderivative = 1.35 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {\sqrt {d+e x} \left (B \left (-945 a^5 e^4+210 a^4 b e^3 (2 d-21 e x)+14 a^3 b^2 e^2 \left (14 d^2+143 d e x-576 e^2 x^2\right )+2 a^2 b^3 e \left (64 d^3+458 d^2 e x+1879 d e^2 x^2-3555 e^3 x^3\right )+10 b^5 d x \left (48 d^3+200 d^2 e x+326 d e^2 x^2+279 e^3 x^3\right )+a b^4 \left (96 d^4+592 d^3 e x+1676 d^2 e^2 x^2+3430 d e^3 x^3-2895 e^4 x^4\right )\right )+A b \left (-105 a^4 e^4-70 a^3 b e^3 (d+7 e x)-14 a^2 b^2 e^2 \left (4 d^2+23 d e x+64 e^2 x^2\right )-2 a b^3 e \left (24 d^3+128 d^2 e x+289 d e^2 x^2+395 e^3 x^3\right )+b^4 \left (384 d^4+1488 d^3 e x+2104 d^2 e^2 x^2+1210 d e^3 x^3+105 e^4 x^4\right )\right )\right )}{1920 b^5 (-b d+a e) (a+b x)^5}+\frac {7 e^4 (-10 b B d+A b e+9 a B e) \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{128 b^{11/2} (-b d+a e)^{3/2}} \]
(Sqrt[d + e*x]*(B*(-945*a^5*e^4 + 210*a^4*b*e^3*(2*d - 21*e*x) + 14*a^3*b^ 2*e^2*(14*d^2 + 143*d*e*x - 576*e^2*x^2) + 2*a^2*b^3*e*(64*d^3 + 458*d^2*e *x + 1879*d*e^2*x^2 - 3555*e^3*x^3) + 10*b^5*d*x*(48*d^3 + 200*d^2*e*x + 3 26*d*e^2*x^2 + 279*e^3*x^3) + a*b^4*(96*d^4 + 592*d^3*e*x + 1676*d^2*e^2*x ^2 + 3430*d*e^3*x^3 - 2895*e^4*x^4)) + A*b*(-105*a^4*e^4 - 70*a^3*b*e^3*(d + 7*e*x) - 14*a^2*b^2*e^2*(4*d^2 + 23*d*e*x + 64*e^2*x^2) - 2*a*b^3*e*(24 *d^3 + 128*d^2*e*x + 289*d*e^2*x^2 + 395*e^3*x^3) + b^4*(384*d^4 + 1488*d^ 3*e*x + 2104*d^2*e^2*x^2 + 1210*d*e^3*x^3 + 105*e^4*x^4))))/(1920*b^5*(-(b *d) + a*e)*(a + b*x)^5) + (7*e^4*(-10*b*B*d + A*b*e + 9*a*B*e)*ArcTan[(Sqr t[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(128*b^(11/2)*(-(b*d) + a*e)^(3/2 ))
Time = 0.35 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.77, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1184, 27, 87, 51, 51, 51, 51, 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.
| \(\displaystyle \int \frac {(A+B x) (d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle b^6 \int \frac {(A+B x) (d+e x)^{7/2}}{b^6 (a+b x)^6}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(A+B x) (d+e x)^{7/2}}{(a+b x)^6}dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {(-9 a B e-A b e+10 b B d) \int \frac {(d+e x)^{7/2}}{(a+b x)^5}dx}{10 b (b d-a e)}-\frac {(d+e x)^{9/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {(-9 a B e-A b e+10 b B d) \left (\frac {7 e \int \frac {(d+e x)^{5/2}}{(a+b x)^4}dx}{8 b}-\frac {(d+e x)^{7/2}}{4 b (a+b x)^4}\right )}{10 b (b d-a e)}-\frac {(d+e x)^{9/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {(-9 a B e-A b e+10 b B d) \left (\frac {7 e \left (\frac {5 e \int \frac {(d+e x)^{3/2}}{(a+b x)^3}dx}{6 b}-\frac {(d+e x)^{5/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {(d+e x)^{7/2}}{4 b (a+b x)^4}\right )}{10 b (b d-a e)}-\frac {(d+e x)^{9/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {(-9 a B e-A b e+10 b B d) \left (\frac {7 e \left (\frac {5 e \left (\frac {3 e \int \frac {\sqrt {d+e x}}{(a+b x)^2}dx}{4 b}-\frac {(d+e x)^{3/2}}{2 b (a+b x)^2}\right )}{6 b}-\frac {(d+e x)^{5/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {(d+e x)^{7/2}}{4 b (a+b x)^4}\right )}{10 b (b d-a e)}-\frac {(d+e x)^{9/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {(-9 a B e-A b e+10 b B d) \left (\frac {7 e \left (\frac {5 e \left (\frac {3 e \left (\frac {e \int \frac {1}{(a+b x) \sqrt {d+e x}}dx}{2 b}-\frac {\sqrt {d+e x}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{3/2}}{2 b (a+b x)^2}\right )}{6 b}-\frac {(d+e x)^{5/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {(d+e x)^{7/2}}{4 b (a+b x)^4}\right )}{10 b (b d-a e)}-\frac {(d+e x)^{9/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {(-9 a B e-A b e+10 b B d) \left (\frac {7 e \left (\frac {5 e \left (\frac {3 e \left (\frac {\int \frac {1}{a+\frac {b (d+e x)}{e}-\frac {b d}{e}}d\sqrt {d+e x}}{b}-\frac {\sqrt {d+e x}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{3/2}}{2 b (a+b x)^2}\right )}{6 b}-\frac {(d+e x)^{5/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {(d+e x)^{7/2}}{4 b (a+b x)^4}\right )}{10 b (b d-a e)}-\frac {(d+e x)^{9/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {(-9 a B e-A b e+10 b B d) \left (\frac {7 e \left (\frac {5 e \left (\frac {3 e \left (-\frac {e \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2} \sqrt {b d-a e}}-\frac {\sqrt {d+e x}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{3/2}}{2 b (a+b x)^2}\right )}{6 b}-\frac {(d+e x)^{5/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {(d+e x)^{7/2}}{4 b (a+b x)^4}\right )}{10 b (b d-a e)}-\frac {(d+e x)^{9/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)}\) |
-1/5*((A*b - a*B)*(d + e*x)^(9/2))/(b*(b*d - a*e)*(a + b*x)^5) + ((10*b*B* d - A*b*e - 9*a*B*e)*(-1/4*(d + e*x)^(7/2)/(b*(a + b*x)^4) + (7*e*(-1/3*(d + e*x)^(5/2)/(b*(a + b*x)^3) + (5*e*(-1/2*(d + e*x)^(3/2)/(b*(a + b*x)^2) + (3*e*(-(Sqrt[d + e*x]/(b*(a + b*x))) - (e*ArcTanh[(Sqrt[b]*Sqrt[d + e*x ])/Sqrt[b*d - a*e]])/(b^(3/2)*Sqrt[b*d - a*e])))/(4*b)))/(6*b)))/(8*b)))/( 10*b*(b*d - a*e))
3.19.27.3.1 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] :> 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[1/c^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}, x] && E qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Time = 1.32 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.20
| method | result | size |
| pseudoelliptic | \(-\frac {7 \left (-\left (\left (A e -10 B d \right ) b +9 B a e \right ) \left (b x +a \right )^{5} e^{4} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )+\sqrt {\left (a e -b d \right ) b}\, \left (\left (-A \,e^{4} x^{4}-\frac {242 x^{3} \left (\frac {279 B x}{121}+A \right ) d \,e^{3}}{21}-\frac {2104 \left (\frac {815 B x}{526}+A \right ) x^{2} d^{2} e^{2}}{105}-\frac {496 x \left (\frac {125 B x}{93}+A \right ) d^{3} e}{35}-\frac {128 d^{4} \left (\frac {5 B x}{4}+A \right )}{35}\right ) b^{5}+\frac {16 \left (\left (\frac {965}{16} B \,x^{4}+\frac {395}{24} A \,x^{3}\right ) e^{4}+\frac {289 x^{2} \left (-\frac {1715 B x}{289}+A \right ) d \,e^{3}}{24}+\frac {16 \left (-\frac {419 B x}{64}+A \right ) x \,d^{2} e^{2}}{3}+d^{3} \left (-\frac {37 B x}{3}+A \right ) e -2 B \,d^{4}\right ) a \,b^{4}}{35}+\frac {8 e \left (\left (\frac {3555}{28} B \,x^{3}+16 A \,x^{2}\right ) e^{3}+\frac {23 \left (-\frac {1879 B x}{161}+A \right ) x d \,e^{2}}{4}+d^{2} \left (-\frac {229 B x}{14}+A \right ) e -\frac {16 B \,d^{3}}{7}\right ) a^{2} b^{3}}{15}+\frac {2 \left (\left (\frac {576}{5} B \,x^{2}+7 A x \right ) e^{2}+d \left (-\frac {143 B x}{5}+A \right ) e -\frac {14 B \,d^{2}}{5}\right ) e^{2} a^{3} b^{2}}{3}+\left (\left (42 B x +A \right ) e -4 B d \right ) e^{3} a^{4} b +9 B \,a^{5} e^{4}\right ) \sqrt {e x +d}\right )}{128 \sqrt {\left (a e -b d \right ) b}\, \left (a e -b d \right ) b^{5} \left (b x +a \right )^{5}}\) | \(377\) |
| derivativedivides | \(2 e^{4} \left (\frac {\frac {\left (7 A b e -193 B a e +186 B b d \right ) \left (e x +d \right )^{\frac {9}{2}}}{256 \left (a e -b d \right ) b}-\frac {79 \left (A b e +9 B a e -10 B b d \right ) \left (e x +d \right )^{\frac {7}{2}}}{384 b^{2}}-\frac {7 \left (A a b \,e^{2}-A \,b^{2} d e +9 a^{2} B \,e^{2}-19 B a b d e +10 B \,b^{2} d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{30 b^{3}}-\frac {49 \left (A \,a^{2} b \,e^{3}-2 A a \,b^{2} d \,e^{2}+A \,b^{3} d^{2} e +9 B \,e^{3} a^{3}-28 B \,a^{2} b d \,e^{2}+29 B a \,b^{2} d^{2} e -10 B \,b^{3} d^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}}{384 b^{4}}-\frac {7 \left (A \,a^{3} b \,e^{4}-3 A \,a^{2} b^{2} d \,e^{3}+3 A a \,b^{3} d^{2} e^{2}-A \,b^{4} d^{3} e +9 B \,a^{4} e^{4}-37 B \,a^{3} b d \,e^{3}+57 B \,a^{2} b^{2} d^{2} e^{2}-39 B a \,b^{3} d^{3} e +10 b^{4} B \,d^{4}\right ) \sqrt {e x +d}}{256 b^{5}}}{\left (b \left (e x +d \right )+a e -b d \right )^{5}}+\frac {7 \left (A b e +9 B a e -10 B b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{256 \left (a e -b d \right ) b^{5} \sqrt {\left (a e -b d \right ) b}}\right )\) | \(403\) |
| default | \(2 e^{4} \left (\frac {\frac {\left (7 A b e -193 B a e +186 B b d \right ) \left (e x +d \right )^{\frac {9}{2}}}{256 \left (a e -b d \right ) b}-\frac {79 \left (A b e +9 B a e -10 B b d \right ) \left (e x +d \right )^{\frac {7}{2}}}{384 b^{2}}-\frac {7 \left (A a b \,e^{2}-A \,b^{2} d e +9 a^{2} B \,e^{2}-19 B a b d e +10 B \,b^{2} d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{30 b^{3}}-\frac {49 \left (A \,a^{2} b \,e^{3}-2 A a \,b^{2} d \,e^{2}+A \,b^{3} d^{2} e +9 B \,e^{3} a^{3}-28 B \,a^{2} b d \,e^{2}+29 B a \,b^{2} d^{2} e -10 B \,b^{3} d^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}}{384 b^{4}}-\frac {7 \left (A \,a^{3} b \,e^{4}-3 A \,a^{2} b^{2} d \,e^{3}+3 A a \,b^{3} d^{2} e^{2}-A \,b^{4} d^{3} e +9 B \,a^{4} e^{4}-37 B \,a^{3} b d \,e^{3}+57 B \,a^{2} b^{2} d^{2} e^{2}-39 B a \,b^{3} d^{3} e +10 b^{4} B \,d^{4}\right ) \sqrt {e x +d}}{256 b^{5}}}{\left (b \left (e x +d \right )+a e -b d \right )^{5}}+\frac {7 \left (A b e +9 B a e -10 B b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{256 \left (a e -b d \right ) b^{5} \sqrt {\left (a e -b d \right ) b}}\right )\) | \(403\) |
-7/128/((a*e-b*d)*b)^(1/2)*(-((A*e-10*B*d)*b+9*B*a*e)*(b*x+a)^5*e^4*arctan (b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))+((a*e-b*d)*b)^(1/2)*((-A*e^4*x^4-242 /21*x^3*(279/121*B*x+A)*d*e^3-2104/105*(815/526*B*x+A)*x^2*d^2*e^2-496/35* x*(125/93*B*x+A)*d^3*e-128/35*d^4*(5/4*B*x+A))*b^5+16/35*((965/16*B*x^4+39 5/24*A*x^3)*e^4+289/24*x^2*(-1715/289*B*x+A)*d*e^3+16/3*(-419/64*B*x+A)*x* d^2*e^2+d^3*(-37/3*B*x+A)*e-2*B*d^4)*a*b^4+8/15*e*((3555/28*B*x^3+16*A*x^2 )*e^3+23/4*(-1879/161*B*x+A)*x*d*e^2+d^2*(-229/14*B*x+A)*e-16/7*B*d^3)*a^2 *b^3+2/3*((576/5*B*x^2+7*A*x)*e^2+d*(-143/5*B*x+A)*e-14/5*B*d^2)*e^2*a^3*b ^2+((42*B*x+A)*e-4*B*d)*e^3*a^4*b+9*B*a^5*e^4)*(e*x+d)^(1/2))/(a*e-b*d)/b^ 5/(b*x+a)^5
Leaf count of result is larger than twice the leaf count of optimal. 1014 vs. \(2 (281) = 562\).
Time = 0.39 (sec) , antiderivative size = 2041, normalized size of antiderivative = 6.52 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Too large to display} \]
[1/3840*(105*(10*B*a^5*b*d*e^4 - (9*B*a^6 + A*a^5*b)*e^5 + (10*B*b^6*d*e^4 - (9*B*a*b^5 + A*b^6)*e^5)*x^5 + 5*(10*B*a*b^5*d*e^4 - (9*B*a^2*b^4 + A*a *b^5)*e^5)*x^4 + 10*(10*B*a^2*b^4*d*e^4 - (9*B*a^3*b^3 + A*a^2*b^4)*e^5)*x ^3 + 10*(10*B*a^3*b^3*d*e^4 - (9*B*a^4*b^2 + A*a^3*b^3)*e^5)*x^2 + 5*(10*B *a^4*b^2*d*e^4 - (9*B*a^5*b + A*a^4*b^2)*e^5)*x)*sqrt(b^2*d - a*b*e)*log(( b*e*x + 2*b*d - a*e - 2*sqrt(b^2*d - a*b*e)*sqrt(e*x + d))/(b*x + a)) - 2* (96*(B*a*b^6 + 4*A*b^7)*d^5 + 16*(2*B*a^2*b^5 - 27*A*a*b^6)*d^4*e + 4*(17* B*a^3*b^4 - 2*A*a^2*b^5)*d^3*e^2 + 14*(16*B*a^4*b^3 - A*a^3*b^4)*d^2*e^3 - 35*(39*B*a^5*b^2 + A*a^4*b^3)*d*e^4 + 105*(9*B*a^6*b + A*a^5*b^2)*e^5 + 1 5*(186*B*b^7*d^2*e^3 - (379*B*a*b^6 - 7*A*b^7)*d*e^4 + (193*B*a^2*b^5 - 7* A*a*b^6)*e^5)*x^4 + 10*(326*B*b^7*d^3*e^2 + (17*B*a*b^6 + 121*A*b^7)*d^2*e ^3 - 2*(527*B*a^2*b^5 + 100*A*a*b^6)*d*e^4 + 79*(9*B*a^3*b^4 + A*a^2*b^5)* e^5)*x^3 + 2*(1000*B*b^7*d^4*e - 2*(81*B*a*b^6 - 526*A*b^7)*d^3*e^2 + 3*(3 47*B*a^2*b^5 - 447*A*a*b^6)*d^2*e^3 - (5911*B*a^3*b^4 + 159*A*a^2*b^5)*d*e ^4 + 448*(9*B*a^4*b^3 + A*a^3*b^4)*e^5)*x^2 + 2*(240*B*b^7*d^5 + 8*(7*B*a* b^6 + 93*A*b^7)*d^4*e + 2*(81*B*a^2*b^5 - 436*A*a*b^6)*d^3*e^2 + 3*(181*B* a^3*b^4 - 11*A*a^2*b^5)*d^2*e^3 - 14*(229*B*a^4*b^3 + 6*A*a^3*b^4)*d*e^4 + 245*(9*B*a^5*b^2 + A*a^4*b^3)*e^5)*x)*sqrt(e*x + d))/(a^5*b^8*d^2 - 2*a^6 *b^7*d*e + a^7*b^6*e^2 + (b^13*d^2 - 2*a*b^12*d*e + a^2*b^11*e^2)*x^5 + 5* (a*b^12*d^2 - 2*a^2*b^11*d*e + a^3*b^10*e^2)*x^4 + 10*(a^2*b^11*d^2 - 2...
Timed out. \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*e-b*d>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 777 vs. \(2 (281) = 562\).
Time = 0.32 (sec) , antiderivative size = 777, normalized size of antiderivative = 2.48 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {7 \, {\left (10 \, B b d e^{4} - 9 \, B a e^{5} - A b e^{5}\right )} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{128 \, {\left (b^{6} d - a b^{5} e\right )} \sqrt {-b^{2} d + a b e}} - \frac {2790 \, {\left (e x + d\right )}^{\frac {9}{2}} B b^{5} d e^{4} - 7900 \, {\left (e x + d\right )}^{\frac {7}{2}} B b^{5} d^{2} e^{4} + 8960 \, {\left (e x + d\right )}^{\frac {5}{2}} B b^{5} d^{3} e^{4} - 4900 \, {\left (e x + d\right )}^{\frac {3}{2}} B b^{5} d^{4} e^{4} + 1050 \, \sqrt {e x + d} B b^{5} d^{5} e^{4} - 2895 \, {\left (e x + d\right )}^{\frac {9}{2}} B a b^{4} e^{5} + 105 \, {\left (e x + d\right )}^{\frac {9}{2}} A b^{5} e^{5} + 15010 \, {\left (e x + d\right )}^{\frac {7}{2}} B a b^{4} d e^{5} + 790 \, {\left (e x + d\right )}^{\frac {7}{2}} A b^{5} d e^{5} - 25984 \, {\left (e x + d\right )}^{\frac {5}{2}} B a b^{4} d^{2} e^{5} - 896 \, {\left (e x + d\right )}^{\frac {5}{2}} A b^{5} d^{2} e^{5} + 19110 \, {\left (e x + d\right )}^{\frac {3}{2}} B a b^{4} d^{3} e^{5} + 490 \, {\left (e x + d\right )}^{\frac {3}{2}} A b^{5} d^{3} e^{5} - 5145 \, \sqrt {e x + d} B a b^{4} d^{4} e^{5} - 105 \, \sqrt {e x + d} A b^{5} d^{4} e^{5} - 7110 \, {\left (e x + d\right )}^{\frac {7}{2}} B a^{2} b^{3} e^{6} - 790 \, {\left (e x + d\right )}^{\frac {7}{2}} A a b^{4} e^{6} + 25088 \, {\left (e x + d\right )}^{\frac {5}{2}} B a^{2} b^{3} d e^{6} + 1792 \, {\left (e x + d\right )}^{\frac {5}{2}} A a b^{4} d e^{6} - 27930 \, {\left (e x + d\right )}^{\frac {3}{2}} B a^{2} b^{3} d^{2} e^{6} - 1470 \, {\left (e x + d\right )}^{\frac {3}{2}} A a b^{4} d^{2} e^{6} + 10080 \, \sqrt {e x + d} B a^{2} b^{3} d^{3} e^{6} + 420 \, \sqrt {e x + d} A a b^{4} d^{3} e^{6} - 8064 \, {\left (e x + d\right )}^{\frac {5}{2}} B a^{3} b^{2} e^{7} - 896 \, {\left (e x + d\right )}^{\frac {5}{2}} A a^{2} b^{3} e^{7} + 18130 \, {\left (e x + d\right )}^{\frac {3}{2}} B a^{3} b^{2} d e^{7} + 1470 \, {\left (e x + d\right )}^{\frac {3}{2}} A a^{2} b^{3} d e^{7} - 9870 \, \sqrt {e x + d} B a^{3} b^{2} d^{2} e^{7} - 630 \, \sqrt {e x + d} A a^{2} b^{3} d^{2} e^{7} - 4410 \, {\left (e x + d\right )}^{\frac {3}{2}} B a^{4} b e^{8} - 490 \, {\left (e x + d\right )}^{\frac {3}{2}} A a^{3} b^{2} e^{8} + 4830 \, \sqrt {e x + d} B a^{4} b d e^{8} + 420 \, \sqrt {e x + d} A a^{3} b^{2} d e^{8} - 945 \, \sqrt {e x + d} B a^{5} e^{9} - 105 \, \sqrt {e x + d} A a^{4} b e^{9}}{1920 \, {\left (b^{6} d - a b^{5} e\right )} {\left ({\left (e x + d\right )} b - b d + a e\right )}^{5}} \]
7/128*(10*B*b*d*e^4 - 9*B*a*e^5 - A*b*e^5)*arctan(sqrt(e*x + d)*b/sqrt(-b^ 2*d + a*b*e))/((b^6*d - a*b^5*e)*sqrt(-b^2*d + a*b*e)) - 1/1920*(2790*(e*x + d)^(9/2)*B*b^5*d*e^4 - 7900*(e*x + d)^(7/2)*B*b^5*d^2*e^4 + 8960*(e*x + d)^(5/2)*B*b^5*d^3*e^4 - 4900*(e*x + d)^(3/2)*B*b^5*d^4*e^4 + 1050*sqrt(e *x + d)*B*b^5*d^5*e^4 - 2895*(e*x + d)^(9/2)*B*a*b^4*e^5 + 105*(e*x + d)^( 9/2)*A*b^5*e^5 + 15010*(e*x + d)^(7/2)*B*a*b^4*d*e^5 + 790*(e*x + d)^(7/2) *A*b^5*d*e^5 - 25984*(e*x + d)^(5/2)*B*a*b^4*d^2*e^5 - 896*(e*x + d)^(5/2) *A*b^5*d^2*e^5 + 19110*(e*x + d)^(3/2)*B*a*b^4*d^3*e^5 + 490*(e*x + d)^(3/ 2)*A*b^5*d^3*e^5 - 5145*sqrt(e*x + d)*B*a*b^4*d^4*e^5 - 105*sqrt(e*x + d)* A*b^5*d^4*e^5 - 7110*(e*x + d)^(7/2)*B*a^2*b^3*e^6 - 790*(e*x + d)^(7/2)*A *a*b^4*e^6 + 25088*(e*x + d)^(5/2)*B*a^2*b^3*d*e^6 + 1792*(e*x + d)^(5/2)* A*a*b^4*d*e^6 - 27930*(e*x + d)^(3/2)*B*a^2*b^3*d^2*e^6 - 1470*(e*x + d)^( 3/2)*A*a*b^4*d^2*e^6 + 10080*sqrt(e*x + d)*B*a^2*b^3*d^3*e^6 + 420*sqrt(e* x + d)*A*a*b^4*d^3*e^6 - 8064*(e*x + d)^(5/2)*B*a^3*b^2*e^7 - 896*(e*x + d )^(5/2)*A*a^2*b^3*e^7 + 18130*(e*x + d)^(3/2)*B*a^3*b^2*d*e^7 + 1470*(e*x + d)^(3/2)*A*a^2*b^3*d*e^7 - 9870*sqrt(e*x + d)*B*a^3*b^2*d^2*e^7 - 630*sq rt(e*x + d)*A*a^2*b^3*d^2*e^7 - 4410*(e*x + d)^(3/2)*B*a^4*b*e^8 - 490*(e* x + d)^(3/2)*A*a^3*b^2*e^8 + 4830*sqrt(e*x + d)*B*a^4*b*d*e^8 + 420*sqrt(e *x + d)*A*a^3*b^2*d*e^8 - 945*sqrt(e*x + d)*B*a^5*e^9 - 105*sqrt(e*x + d)* A*a^4*b*e^9)/((b^6*d - a*b^5*e)*((e*x + d)*b - b*d + a*e)^5)
Time = 0.47 (sec) , antiderivative size = 594, normalized size of antiderivative = 1.90 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {7\,e^4\,\mathrm {atan}\left (\frac {\sqrt {b}\,e^4\,\sqrt {d+e\,x}\,\left (A\,b\,e+9\,B\,a\,e-10\,B\,b\,d\right )}{\sqrt {a\,e-b\,d}\,\left (A\,b\,e^5+9\,B\,a\,e^5-10\,B\,b\,d\,e^4\right )}\right )\,\left (A\,b\,e+9\,B\,a\,e-10\,B\,b\,d\right )}{128\,b^{11/2}\,{\left (a\,e-b\,d\right )}^{3/2}}-\frac {\frac {79\,{\left (d+e\,x\right )}^{7/2}\,\left (A\,b\,e^5+9\,B\,a\,e^5-10\,B\,b\,d\,e^4\right )}{192\,b^2}+\frac {7\,\sqrt {d+e\,x}\,\left (A\,b\,e^5+9\,B\,a\,e^5-10\,B\,b\,d\,e^4\right )\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}{128\,b^5}+\frac {7\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{5/2}\,\left (A\,b\,e^5+9\,B\,a\,e^5-10\,B\,b\,d\,e^4\right )}{15\,b^3}-\frac {{\left (d+e\,x\right )}^{9/2}\,\left (7\,A\,b\,e^5-193\,B\,a\,e^5+186\,B\,b\,d\,e^4\right )}{128\,b\,\left (a\,e-b\,d\right )}+\frac {49\,{\left (d+e\,x\right )}^{3/2}\,\left (a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2\right )\,\left (A\,b\,e^5+9\,B\,a\,e^5-10\,B\,b\,d\,e^4\right )}{192\,b^4}}{\left (d+e\,x\right )\,\left (5\,a^4\,b\,e^4-20\,a^3\,b^2\,d\,e^3+30\,a^2\,b^3\,d^2\,e^2-20\,a\,b^4\,d^3\,e+5\,b^5\,d^4\right )-{\left (d+e\,x\right )}^2\,\left (-10\,a^3\,b^2\,e^3+30\,a^2\,b^3\,d\,e^2-30\,a\,b^4\,d^2\,e+10\,b^5\,d^3\right )+b^5\,{\left (d+e\,x\right )}^5-\left (5\,b^5\,d-5\,a\,b^4\,e\right )\,{\left (d+e\,x\right )}^4+a^5\,e^5-b^5\,d^5+{\left (d+e\,x\right )}^3\,\left (10\,a^2\,b^3\,e^2-20\,a\,b^4\,d\,e+10\,b^5\,d^2\right )-10\,a^2\,b^3\,d^3\,e^2+10\,a^3\,b^2\,d^2\,e^3+5\,a\,b^4\,d^4\,e-5\,a^4\,b\,d\,e^4} \]
(7*e^4*atan((b^(1/2)*e^4*(d + e*x)^(1/2)*(A*b*e + 9*B*a*e - 10*B*b*d))/((a *e - b*d)^(1/2)*(A*b*e^5 + 9*B*a*e^5 - 10*B*b*d*e^4)))*(A*b*e + 9*B*a*e - 10*B*b*d))/(128*b^(11/2)*(a*e - b*d)^(3/2)) - ((79*(d + e*x)^(7/2)*(A*b*e^ 5 + 9*B*a*e^5 - 10*B*b*d*e^4))/(192*b^2) + (7*(d + e*x)^(1/2)*(A*b*e^5 + 9 *B*a*e^5 - 10*B*b*d*e^4)*(a^3*e^3 - b^3*d^3 + 3*a*b^2*d^2*e - 3*a^2*b*d*e^ 2))/(128*b^5) + (7*(a*e - b*d)*(d + e*x)^(5/2)*(A*b*e^5 + 9*B*a*e^5 - 10*B *b*d*e^4))/(15*b^3) - ((d + e*x)^(9/2)*(7*A*b*e^5 - 193*B*a*e^5 + 186*B*b* d*e^4))/(128*b*(a*e - b*d)) + (49*(d + e*x)^(3/2)*(a^2*e^2 + b^2*d^2 - 2*a *b*d*e)*(A*b*e^5 + 9*B*a*e^5 - 10*B*b*d*e^4))/(192*b^4))/((d + e*x)*(5*b^5 *d^4 + 5*a^4*b*e^4 - 20*a^3*b^2*d*e^3 + 30*a^2*b^3*d^2*e^2 - 20*a*b^4*d^3* e) - (d + e*x)^2*(10*b^5*d^3 - 10*a^3*b^2*e^3 + 30*a^2*b^3*d*e^2 - 30*a*b^ 4*d^2*e) + b^5*(d + e*x)^5 - (5*b^5*d - 5*a*b^4*e)*(d + e*x)^4 + a^5*e^5 - b^5*d^5 + (d + e*x)^3*(10*b^5*d^2 + 10*a^2*b^3*e^2 - 20*a*b^4*d*e) - 10*a ^2*b^3*d^3*e^2 + 10*a^3*b^2*d^2*e^3 + 5*a*b^4*d^4*e - 5*a^4*b*d*e^4)