Integrand size = 33, antiderivative size = 460 \[ \int \frac {A+B x}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {2 b e^2 (3 b B d-5 A b e+2 a B e)}{(b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b (A b-a B)}{4 (b d-a e)^3 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b (b B d-3 A b e+2 a B e)}{3 (b d-a e)^4 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 b e (b B d-2 A b e+a B e)}{2 (b d-a e)^5 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e^3 (B d-A e) (a+b x)}{2 (b d-a e)^5 (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e^3 (4 b B d-5 A b e+a B e) (a+b x)}{(b d-a e)^6 (d+e x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 b e^3 (2 b B d-3 A b e+a B e) (a+b x) \log (a+b x)}{(b d-a e)^7 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 b e^3 (2 b B d-3 A b e+a B e) (a+b x) \log (d+e x)}{(b d-a e)^7 \sqrt {a^2+2 a b x+b^2 x^2}} \]
-2*b*e^2*(-5*A*b*e+2*B*a*e+3*B*b*d)/(-a*e+b*d)^6/((b*x+a)^2)^(1/2)-1/4*b*( A*b-B*a)/(-a*e+b*d)^3/(b*x+a)^3/((b*x+a)^2)^(1/2)-1/3*b*(-3*A*b*e+2*B*a*e+ B*b*d)/(-a*e+b*d)^4/(b*x+a)^2/((b*x+a)^2)^(1/2)+3/2*b*e*(-2*A*b*e+B*a*e+B* b*d)/(-a*e+b*d)^5/(b*x+a)/((b*x+a)^2)^(1/2)-1/2*e^3*(-A*e+B*d)*(b*x+a)/(-a *e+b*d)^5/(e*x+d)^2/((b*x+a)^2)^(1/2)-e^3*(-5*A*b*e+B*a*e+4*B*b*d)*(b*x+a) /(-a*e+b*d)^6/(e*x+d)/((b*x+a)^2)^(1/2)-5*b*e^3*(-3*A*b*e+B*a*e+2*B*b*d)*( b*x+a)*ln(b*x+a)/(-a*e+b*d)^7/((b*x+a)^2)^(1/2)+5*b*e^3*(-3*A*b*e+B*a*e+2* B*b*d)*(b*x+a)*ln(e*x+d)/(-a*e+b*d)^7/((b*x+a)^2)^(1/2)
Time = 1.21 (sec) , antiderivative size = 302, normalized size of antiderivative = 0.66 \[ \int \frac {A+B x}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {-4 b (b d-a e)^3 (b B d-3 A b e+2 a B e)-\frac {3 b (A b-a B) (b d-a e)^4}{a+b x}-18 b e (b d-a e)^2 (-b B d+2 A b e-a B e) (a+b x)+24 b e^2 (b d-a e) (-3 b B d+5 A b e-2 a B e) (a+b x)^2+\frac {6 e^3 (b d-a e)^2 (-B d+A e) (a+b x)^3}{(d+e x)^2}+\frac {12 e^3 (b d-a e) (-4 b B d+5 A b e-a B e) (a+b x)^3}{d+e x}-60 b e^3 (2 b B d-3 A b e+a B e) (a+b x)^3 \log (a+b x)+60 b e^3 (2 b B d-3 A b e+a B e) (a+b x)^3 \log (d+e x)}{12 (b d-a e)^7 \left ((a+b x)^2\right )^{3/2}} \]
(-4*b*(b*d - a*e)^3*(b*B*d - 3*A*b*e + 2*a*B*e) - (3*b*(A*b - a*B)*(b*d - a*e)^4)/(a + b*x) - 18*b*e*(b*d - a*e)^2*(-(b*B*d) + 2*A*b*e - a*B*e)*(a + b*x) + 24*b*e^2*(b*d - a*e)*(-3*b*B*d + 5*A*b*e - 2*a*B*e)*(a + b*x)^2 + (6*e^3*(b*d - a*e)^2*(-(B*d) + A*e)*(a + b*x)^3)/(d + e*x)^2 + (12*e^3*(b* d - a*e)*(-4*b*B*d + 5*A*b*e - a*B*e)*(a + b*x)^3)/(d + e*x) - 60*b*e^3*(2 *b*B*d - 3*A*b*e + a*B*e)*(a + b*x)^3*Log[a + b*x] + 60*b*e^3*(2*b*B*d - 3 *A*b*e + a*B*e)*(a + b*x)^3*Log[d + e*x])/(12*(b*d - a*e)^7*((a + b*x)^2)^ (3/2))
Time = 0.72 (sec) , antiderivative size = 313, normalized size of antiderivative = 0.68, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {1187, 27, 86, 2009}
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.
\(\displaystyle \int \frac {A+B x}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2} (d+e x)^3} \, dx\) |
\(\Big \downarrow \) 1187 |
\(\displaystyle \frac {b^5 (a+b x) \int \frac {A+B x}{b^5 (a+b x)^5 (d+e x)^3}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {(a+b x) \int \frac {A+B x}{(a+b x)^5 (d+e x)^3}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
\(\Big \downarrow \) 86 |
\(\displaystyle \frac {(a+b x) \int \left (-\frac {5 b (-2 b B d+3 A b e-a B e) e^4}{(b d-a e)^7 (d+e x)}-\frac {(-4 b B d+5 A b e-a B e) e^4}{(b d-a e)^6 (d+e x)^2}-\frac {(A e-B d) e^4}{(b d-a e)^5 (d+e x)^3}+\frac {5 b^2 (-2 b B d+3 A b e-a B e) e^3}{(b d-a e)^7 (a+b x)}-\frac {2 b^2 (-3 b B d+5 A b e-2 a B e) e^2}{(b d-a e)^6 (a+b x)^2}+\frac {3 b^2 (-b B d+2 A b e-a B e) e}{(b d-a e)^5 (a+b x)^3}+\frac {b^2 (b B d-3 A b e+2 a B e)}{(b d-a e)^4 (a+b x)^4}+\frac {b^2 (A b-a B)}{(b d-a e)^3 (a+b x)^5}\right )dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {(a+b x) \left (-\frac {e^3 (a B e-5 A b e+4 b B d)}{(d+e x) (b d-a e)^6}-\frac {e^3 (B d-A e)}{2 (d+e x)^2 (b d-a e)^5}-\frac {5 b e^3 \log (a+b x) (a B e-3 A b e+2 b B d)}{(b d-a e)^7}+\frac {5 b e^3 \log (d+e x) (a B e-3 A b e+2 b B d)}{(b d-a e)^7}-\frac {2 b e^2 (2 a B e-5 A b e+3 b B d)}{(a+b x) (b d-a e)^6}+\frac {3 b e (a B e-2 A b e+b B d)}{2 (a+b x)^2 (b d-a e)^5}-\frac {b (2 a B e-3 A b e+b B d)}{3 (a+b x)^3 (b d-a e)^4}-\frac {b (A b-a B)}{4 (a+b x)^4 (b d-a e)^3}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
((a + b*x)*(-1/4*(b*(A*b - a*B))/((b*d - a*e)^3*(a + b*x)^4) - (b*(b*B*d - 3*A*b*e + 2*a*B*e))/(3*(b*d - a*e)^4*(a + b*x)^3) + (3*b*e*(b*B*d - 2*A*b *e + a*B*e))/(2*(b*d - a*e)^5*(a + b*x)^2) - (2*b*e^2*(3*b*B*d - 5*A*b*e + 2*a*B*e))/((b*d - a*e)^6*(a + b*x)) - (e^3*(B*d - A*e))/(2*(b*d - a*e)^5* (d + e*x)^2) - (e^3*(4*b*B*d - 5*A*b*e + a*B*e))/((b*d - a*e)^6*(d + e*x)) - (5*b*e^3*(2*b*B*d - 3*A*b*e + a*B*e)*Log[a + b*x])/(b*d - a*e)^7 + (5*b *e^3*(2*b*B*d - 3*A*b*e + a*B*e)*Log[d + e*x])/(b*d - a*e)^7))/Sqrt[a^2 + 2*a*b*x + b^2*x^2]
3.18.83.3.1 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_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
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]
Leaf count of result is larger than twice the leaf count of optimal. \(1302\) vs. \(2(364)=728\).
Time = 0.46 (sec) , antiderivative size = 1303, normalized size of antiderivative = 2.83
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1303\) |
default | \(\text {Expression too large to display}\) | \(2420\) |
((b*x+a)^2)^(1/2)/(b*x+a)^5*(5*b^4*e^4*(3*A*b*e-B*a*e-2*B*b*d)/(a^6*e^6-6* a^5*b*d*e^5+15*a^4*b^2*d^2*e^4-20*a^3*b^3*d^3*e^3+15*a^2*b^4*d^4*e^2-6*a*b ^5*d^5*e+b^6*d^6)*x^5+5/2*b^3*e^3*(7*a*e+3*b*d)*(3*A*b*e-B*a*e-2*B*b*d)/(a ^6*e^6-6*a^5*b*d*e^5+15*a^4*b^2*d^2*e^4-20*a^3*b^3*d^3*e^3+15*a^2*b^4*d^4* e^2-6*a*b^5*d^5*e+b^6*d^6)*x^4+5/3*b^2*e^2*(39*A*a^2*b*e^3+48*A*a*b^2*d*e^ 2+3*A*b^3*d^2*e-13*B*a^3*e^3-42*B*a^2*b*d*e^2-33*B*a*b^2*d^2*e-2*B*b^3*d^3 )/(a^6*e^6-6*a^5*b*d*e^5+15*a^4*b^2*d^2*e^4-20*a^3*b^3*d^3*e^3+15*a^2*b^4* d^4*e^2-6*a*b^5*d^5*e+b^6*d^6)*x^3+5/12*b*e*(75*A*a^3*b*e^4+243*A*a^2*b^2* d*e^3+45*A*a*b^3*d^2*e^2-3*A*b^4*d^3*e-25*B*a^4*e^4-131*B*a^3*b*d*e^3-177* B*a^2*b^2*d^2*e^2-29*B*a*b^3*d^3*e+2*B*b^4*d^4)/(a^6*e^6-6*a^5*b*d*e^5+15* a^4*b^2*d^2*e^4-20*a^3*b^3*d^3*e^3+15*a^2*b^4*d^4*e^2-6*a*b^5*d^5*e+b^6*d^ 6)*x^2+1/6*(18*A*a^4*b*e^5+303*A*a^3*b^2*d*e^4+153*A*a^2*b^3*d^2*e^3-27*A* a*b^4*d^3*e^2+3*A*b^5*d^4*e-6*B*a^5*e^5-113*B*a^4*b*d*e^4-253*B*a^3*b^2*d^ 2*e^3-93*B*a^2*b^3*d^3*e^2+17*B*a*b^4*d^4*e-2*B*b^5*d^5)/(a^6*e^6-6*a^5*b* d*e^5+15*a^4*b^2*d^2*e^4-20*a^3*b^3*d^3*e^3+15*a^2*b^4*d^4*e^2-6*a*b^5*d^5 *e+b^6*d^6)*x-1/12*(6*A*a^5*e^5-66*A*a^4*b*d*e^4-171*A*a^3*b^2*d^2*e^3+69* A*a^2*b^3*d^3*e^2-21*A*a*b^4*d^4*e+3*A*b^5*d^5+6*B*a^5*d*e^4+131*B*a^4*b*d ^2*e^3+51*B*a^3*b^2*d^3*e^2-9*B*a^2*b^3*d^4*e+B*a*b^4*d^5)/(a^6*e^6-6*a^5* b*d*e^5+15*a^4*b^2*d^2*e^4-20*a^3*b^3*d^3*e^3+15*a^2*b^4*d^4*e^2-6*a*b^5*d ^5*e+b^6*d^6))/(e*x+d)^2+5*((b*x+a)^2)^(1/2)/(b*x+a)*b*e^3*(3*A*b*e-B*a...
Leaf count of result is larger than twice the leaf count of optimal. 2466 vs. \(2 (364) = 728\).
Time = 0.55 (sec) , antiderivative size = 2466, normalized size of antiderivative = 5.36 \[ \int \frac {A+B x}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\text {Too large to display} \]
1/12*(6*A*a^6*e^6 - (B*a*b^5 + 3*A*b^6)*d^6 + 2*(5*B*a^2*b^4 + 12*A*a*b^5) *d^5*e - 30*(2*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^2 - 80*(B*a^4*b^2 - 3*A*a^3* b^3)*d^3*e^3 + 5*(25*B*a^5*b - 21*A*a^4*b^2)*d^2*e^4 + 6*(B*a^6 - 12*A*a^5 *b)*d*e^5 - 60*(2*B*b^6*d^2*e^4 - (B*a*b^5 + 3*A*b^6)*d*e^5 - (B*a^2*b^4 - 3*A*a*b^5)*e^6)*x^5 - 30*(6*B*b^6*d^3*e^3 + (11*B*a*b^5 - 9*A*b^6)*d^2*e^ 4 - 2*(5*B*a^2*b^4 + 6*A*a*b^5)*d*e^5 - 7*(B*a^3*b^3 - 3*A*a^2*b^4)*e^6)*x ^4 - 20*(2*B*b^6*d^4*e^2 + (31*B*a*b^5 - 3*A*b^6)*d^3*e^3 + 9*(B*a^2*b^4 - 5*A*a*b^5)*d^2*e^4 - (29*B*a^3*b^3 - 9*A*a^2*b^4)*d*e^5 - 13*(B*a^4*b^2 - 3*A*a^3*b^3)*e^6)*x^3 + 5*(2*B*b^6*d^5*e - (31*B*a*b^5 + 3*A*b^6)*d^4*e^2 - 4*(37*B*a^2*b^4 - 12*A*a*b^5)*d^3*e^3 + 2*(23*B*a^3*b^3 + 99*A*a^2*b^4) *d^2*e^4 + 2*(53*B*a^4*b^2 - 84*A*a^3*b^3)*d*e^5 + 25*(B*a^5*b - 3*A*a^4*b ^2)*e^6)*x^2 - 2*(2*B*b^6*d^6 - (19*B*a*b^5 + 3*A*b^6)*d^5*e + 10*(11*B*a^ 2*b^4 + 3*A*a*b^5)*d^4*e^2 + 20*(8*B*a^3*b^3 - 9*A*a^2*b^4)*d^3*e^3 - 10*( 14*B*a^4*b^2 + 15*A*a^3*b^3)*d^2*e^4 - (107*B*a^5*b - 285*A*a^4*b^2)*d*e^5 - 6*(B*a^6 - 3*A*a^5*b)*e^6)*x - 60*(2*B*a^4*b^2*d^3*e^3 + (B*a^5*b - 3*A *a^4*b^2)*d^2*e^4 + (2*B*b^6*d*e^5 + (B*a*b^5 - 3*A*b^6)*e^6)*x^6 + 2*(2*B *b^6*d^2*e^4 + (5*B*a*b^5 - 3*A*b^6)*d*e^5 + 2*(B*a^2*b^4 - 3*A*a*b^5)*e^6 )*x^5 + (2*B*b^6*d^3*e^3 + (17*B*a*b^5 - 3*A*b^6)*d^2*e^4 + 4*(5*B*a^2*b^4 - 6*A*a*b^5)*d*e^5 + 6*(B*a^3*b^3 - 3*A*a^2*b^4)*e^6)*x^4 + 4*(2*B*a*b^5* d^3*e^3 + (7*B*a^2*b^4 - 3*A*a*b^5)*d^2*e^4 + (5*B*a^3*b^3 - 9*A*a^2*b^...
Timed out. \[ \int \frac {A+B x}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {A+B x}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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. 1145 vs. \(2 (364) = 728\).
Time = 0.37 (sec) , antiderivative size = 1145, normalized size of antiderivative = 2.49 \[ \int \frac {A+B x}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\text {Too large to display} \]
-5*(2*B*b^3*d*e^3 + B*a*b^2*e^4 - 3*A*b^3*e^4)*log(abs(b*x + a))/(b^8*d^7* sgn(b*x + a) - 7*a*b^7*d^6*e*sgn(b*x + a) + 21*a^2*b^6*d^5*e^2*sgn(b*x + a ) - 35*a^3*b^5*d^4*e^3*sgn(b*x + a) + 35*a^4*b^4*d^3*e^4*sgn(b*x + a) - 21 *a^5*b^3*d^2*e^5*sgn(b*x + a) + 7*a^6*b^2*d*e^6*sgn(b*x + a) - a^7*b*e^7*s gn(b*x + a)) + 5*(2*B*b^2*d*e^4 + B*a*b*e^5 - 3*A*b^2*e^5)*log(abs(e*x + d ))/(b^7*d^7*e*sgn(b*x + a) - 7*a*b^6*d^6*e^2*sgn(b*x + a) + 21*a^2*b^5*d^5 *e^3*sgn(b*x + a) - 35*a^3*b^4*d^4*e^4*sgn(b*x + a) + 35*a^4*b^3*d^3*e^5*s gn(b*x + a) - 21*a^5*b^2*d^2*e^6*sgn(b*x + a) + 7*a^6*b*d*e^7*sgn(b*x + a) - a^7*e^8*sgn(b*x + a)) - 1/12*(B*a*b^5*d^6 + 3*A*b^6*d^6 - 10*B*a^2*b^4* d^5*e - 24*A*a*b^5*d^5*e + 60*B*a^3*b^3*d^4*e^2 + 90*A*a^2*b^4*d^4*e^2 + 8 0*B*a^4*b^2*d^3*e^3 - 240*A*a^3*b^3*d^3*e^3 - 125*B*a^5*b*d^2*e^4 + 105*A* a^4*b^2*d^2*e^4 - 6*B*a^6*d*e^5 + 72*A*a^5*b*d*e^5 - 6*A*a^6*e^6 + 60*(2*B *b^6*d^2*e^4 - B*a*b^5*d*e^5 - 3*A*b^6*d*e^5 - B*a^2*b^4*e^6 + 3*A*a*b^5*e ^6)*x^5 + 30*(6*B*b^6*d^3*e^3 + 11*B*a*b^5*d^2*e^4 - 9*A*b^6*d^2*e^4 - 10* B*a^2*b^4*d*e^5 - 12*A*a*b^5*d*e^5 - 7*B*a^3*b^3*e^6 + 21*A*a^2*b^4*e^6)*x ^4 + 20*(2*B*b^6*d^4*e^2 + 31*B*a*b^5*d^3*e^3 - 3*A*b^6*d^3*e^3 + 9*B*a^2* b^4*d^2*e^4 - 45*A*a*b^5*d^2*e^4 - 29*B*a^3*b^3*d*e^5 + 9*A*a^2*b^4*d*e^5 - 13*B*a^4*b^2*e^6 + 39*A*a^3*b^3*e^6)*x^3 - 5*(2*B*b^6*d^5*e - 31*B*a*b^5 *d^4*e^2 - 3*A*b^6*d^4*e^2 - 148*B*a^2*b^4*d^3*e^3 + 48*A*a*b^5*d^3*e^3 + 46*B*a^3*b^3*d^2*e^4 + 198*A*a^2*b^4*d^2*e^4 + 106*B*a^4*b^2*d*e^5 - 16...
Timed out. \[ \int \frac {A+B x}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int \frac {A+B\,x}{{\left (d+e\,x\right )}^3\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \]