Integrand size = 33, antiderivative size = 271 \[ \int \frac {A+B x}{(d+e x)^4 \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=-\frac {(B d-A e) (a+b x)}{3 e (b d-a e) (d+e x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-a B) (a+b x)}{2 (b d-a e)^2 (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b (A b-a B) (a+b x)}{(b d-a e)^3 (d+e x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b^2 (A b-a B) (a+b x) \log (a+b x)}{(b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b^2 (A b-a B) (a+b x) \log (d+e x)}{(b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}} \]
-1/3*(-A*e+B*d)*(b*x+a)/e/(-a*e+b*d)/(e*x+d)^3/((b*x+a)^2)^(1/2)+1/2*(A*b- B*a)*(b*x+a)/(-a*e+b*d)^2/(e*x+d)^2/((b*x+a)^2)^(1/2)+b*(A*b-B*a)*(b*x+a)/ (-a*e+b*d)^3/(e*x+d)/((b*x+a)^2)^(1/2)+b^2*(A*b-B*a)*(b*x+a)*ln(b*x+a)/(-a *e+b*d)^4/((b*x+a)^2)^(1/2)-b^2*(A*b-B*a)*(b*x+a)*ln(e*x+d)/(-a*e+b*d)^4/( (b*x+a)^2)^(1/2)
Time = 1.10 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.62 \[ \int \frac {A+B x}{(d+e x)^4 \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {(a+b x) \left (-2 (b d-a e)^3 (B d-A e)+3 (A b-a B) e (b d-a e)^2 (d+e x)+6 b (A b-a B) e (b d-a e) (d+e x)^2+6 b^2 (A b-a B) e (d+e x)^3 \log (a+b x)-6 b^2 (A b-a B) e (d+e x)^3 \log (d+e x)\right )}{6 e (b d-a e)^4 \sqrt {(a+b x)^2} (d+e x)^3} \]
((a + b*x)*(-2*(b*d - a*e)^3*(B*d - A*e) + 3*(A*b - a*B)*e*(b*d - a*e)^2*( d + e*x) + 6*b*(A*b - a*B)*e*(b*d - a*e)*(d + e*x)^2 + 6*b^2*(A*b - a*B)*e *(d + e*x)^3*Log[a + b*x] - 6*b^2*(A*b - a*B)*e*(d + e*x)^3*Log[d + e*x])) /(6*e*(b*d - a*e)^4*Sqrt[(a + b*x)^2]*(d + e*x)^3)
Time = 0.38 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.63, 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.
| \(\displaystyle \int \frac {A+B x}{\sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^4} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {b (a+b x) \int \frac {A+B x}{b (a+b x) (d+e x)^4}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(a+b x) \int \frac {A+B x}{(a+b x) (d+e x)^4}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \frac {(a+b x) \int \left (\frac {(A b-a B) b^3}{(b d-a e)^4 (a+b x)}-\frac {(A b-a B) e b^2}{(a e-b d)^4 (d+e x)}+\frac {(A b-a B) e b}{(a e-b d)^3 (d+e x)^2}+\frac {(a B-A b) e}{(b d-a e)^2 (d+e x)^3}+\frac {B d-A e}{(b d-a e) (d+e x)^4}\right )dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(a+b x) \left (\frac {b^2 (A b-a B) \log (a+b x)}{(b d-a e)^4}-\frac {b^2 (A b-a B) \log (d+e x)}{(b d-a e)^4}+\frac {b (A b-a B)}{(d+e x) (b d-a e)^3}+\frac {A b-a B}{2 (d+e x)^2 (b d-a e)^2}-\frac {B d-A e}{3 e (d+e x)^3 (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
((a + b*x)*(-1/3*(B*d - A*e)/(e*(b*d - a*e)*(d + e*x)^3) + (A*b - a*B)/(2* (b*d - a*e)^2*(d + e*x)^2) + (b*(A*b - a*B))/((b*d - a*e)^3*(d + e*x)) + ( b^2*(A*b - a*B)*Log[a + b*x])/(b*d - a*e)^4 - (b^2*(A*b - a*B)*Log[d + e*x ])/(b*d - a*e)^4))/Sqrt[a^2 + 2*a*b*x + b^2*x^2]
3.18.66.3.1 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]
Time = 0.50 (sec) , antiderivative size = 413, normalized size of antiderivative = 1.52
| method | result | size |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {b \,e^{2} \left (A b -B a \right ) x^{2}}{a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}}+\frac {\left (a e -5 b d \right ) e \left (A b -B a \right ) x}{2 a^{3} e^{3}-6 a^{2} b d \,e^{2}+6 a \,b^{2} d^{2} e -2 b^{3} d^{3}}-\frac {2 A \,a^{2} e^{3}-7 A a b d \,e^{2}+11 A \,b^{2} d^{2} e +B \,a^{2} d \,e^{2}-5 B a b \,d^{2} e -2 B \,b^{2} d^{3}}{6 e \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}\right )}{\left (b x +a \right ) \left (e x +d \right )^{3}}-\frac {\sqrt {\left (b x +a \right )^{2}}\, b^{2} \left (A b -B a \right ) \ln \left (e x +d \right )}{\left (b x +a \right ) \left (e^{4} a^{4}-4 b d \,e^{3} a^{3}+6 b^{2} d^{2} e^{2} a^{2}-4 b^{3} d^{3} e a +b^{4} d^{4}\right )}+\frac {\sqrt {\left (b x +a \right )^{2}}\, b^{2} \left (A b -B a \right ) \ln \left (-b x -a \right )}{\left (b x +a \right ) \left (e^{4} a^{4}-4 b d \,e^{3} a^{3}+6 b^{2} d^{2} e^{2} a^{2}-4 b^{3} d^{3} e a +b^{4} d^{4}\right )}\) | \(413\) |
| default | \(\frac {\left (b x +a \right ) \left (-2 B \,b^{3} d^{4}-2 A \,a^{3} e^{4}-6 B \ln \left (b x +a \right ) a \,b^{2} d^{3} e +18 A \ln \left (b x +a \right ) b^{3} d^{2} e^{2} x +18 A \ln \left (b x +a \right ) b^{3} d \,e^{3} x^{2}-6 B \ln \left (b x +a \right ) a \,b^{2} e^{4} x^{3}-18 A \ln \left (e x +d \right ) b^{3} d \,e^{3} x^{2}-18 A \ln \left (e x +d \right ) b^{3} d^{2} e^{2} x +6 B \ln \left (e x +d \right ) a \,b^{2} e^{4} x^{3}+6 B \ln \left (e x +d \right ) a \,b^{2} d^{3} e -3 B \,a^{3} e^{4} x +18 B \,a^{2} b d \,e^{3} x -15 B a \,b^{2} d^{2} e^{2} x -6 B a \,b^{2} d \,e^{3} x^{2}-18 A a \,b^{2} d \,e^{3} x -B \,a^{3} d \,e^{3}+11 A \,b^{3} d^{3} e +6 A \,b^{3} d \,e^{3} x^{2}+15 A \,b^{3} d^{2} e^{2} x +6 A \ln \left (b x +a \right ) b^{3} e^{4} x^{3}+6 A \ln \left (b x +a \right ) b^{3} d^{3} e -6 A \ln \left (e x +d \right ) b^{3} e^{4} x^{3}-6 A \ln \left (e x +d \right ) b^{3} d^{3} e +18 B \ln \left (e x +d \right ) a \,b^{2} d^{2} e^{2} x -18 B \ln \left (b x +a \right ) a \,b^{2} d \,e^{3} x^{2}-18 B \ln \left (b x +a \right ) a \,b^{2} d^{2} e^{2} x +18 B \ln \left (e x +d \right ) a \,b^{2} d \,e^{3} x^{2}-6 A a \,b^{2} e^{4} x^{2}+6 B \,a^{2} b \,e^{4} x^{2}+3 A \,a^{2} b \,e^{4} x +6 B \,a^{2} b \,d^{2} e^{2}-3 B a \,b^{2} d^{3} e +9 A \,a^{2} b d \,e^{3}-18 A a \,b^{2} d^{2} e^{2}\right )}{6 \sqrt {\left (b x +a \right )^{2}}\, \left (a e -b d \right )^{4} e \left (e x +d \right )^{3}}\) | \(545\) |
((b*x+a)^2)^(1/2)/(b*x+a)*(-b*e^2*(A*b-B*a)/(a^3*e^3-3*a^2*b*d*e^2+3*a*b^2 *d^2*e-b^3*d^3)*x^2+1/2*(a*e-5*b*d)*e*(A*b-B*a)/(a^3*e^3-3*a^2*b*d*e^2+3*a *b^2*d^2*e-b^3*d^3)*x-1/6/e*(2*A*a^2*e^3-7*A*a*b*d*e^2+11*A*b^2*d^2*e+B*a^ 2*d*e^2-5*B*a*b*d^2*e-2*B*b^2*d^3)/(a^3*e^3-3*a^2*b*d*e^2+3*a*b^2*d^2*e-b^ 3*d^3))/(e*x+d)^3-((b*x+a)^2)^(1/2)/(b*x+a)*b^2*(A*b-B*a)/(a^4*e^4-4*a^3*b *d*e^3+6*a^2*b^2*d^2*e^2-4*a*b^3*d^3*e+b^4*d^4)*ln(e*x+d)+((b*x+a)^2)^(1/2 )/(b*x+a)*b^2*(A*b-B*a)/(a^4*e^4-4*a^3*b*d*e^3+6*a^2*b^2*d^2*e^2-4*a*b^3*d ^3*e+b^4*d^4)*ln(-b*x-a)
Leaf count of result is larger than twice the leaf count of optimal. 608 vs. \(2 (213) = 426\).
Time = 0.37 (sec) , antiderivative size = 608, normalized size of antiderivative = 2.24 \[ \int \frac {A+B x}{(d+e x)^4 \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=-\frac {2 \, B b^{3} d^{4} + 2 \, A a^{3} e^{4} + {\left (3 \, B a b^{2} - 11 \, A b^{3}\right )} d^{3} e - 6 \, {\left (B a^{2} b - 3 \, A a b^{2}\right )} d^{2} e^{2} + {\left (B a^{3} - 9 \, A a^{2} b\right )} d e^{3} + 6 \, {\left ({\left (B a b^{2} - A b^{3}\right )} d e^{3} - {\left (B a^{2} b - A a b^{2}\right )} e^{4}\right )} x^{2} + 3 \, {\left (5 \, {\left (B a b^{2} - A b^{3}\right )} d^{2} e^{2} - 6 \, {\left (B a^{2} b - A a b^{2}\right )} d e^{3} + {\left (B a^{3} - A a^{2} b\right )} e^{4}\right )} x + 6 \, {\left ({\left (B a b^{2} - A b^{3}\right )} e^{4} x^{3} + 3 \, {\left (B a b^{2} - A b^{3}\right )} d e^{3} x^{2} + 3 \, {\left (B a b^{2} - A b^{3}\right )} d^{2} e^{2} x + {\left (B a b^{2} - A b^{3}\right )} d^{3} e\right )} \log \left (b x + a\right ) - 6 \, {\left ({\left (B a b^{2} - A b^{3}\right )} e^{4} x^{3} + 3 \, {\left (B a b^{2} - A b^{3}\right )} d e^{3} x^{2} + 3 \, {\left (B a b^{2} - A b^{3}\right )} d^{2} e^{2} x + {\left (B a b^{2} - A b^{3}\right )} d^{3} e\right )} \log \left (e x + d\right )}{6 \, {\left (b^{4} d^{7} e - 4 \, a b^{3} d^{6} e^{2} + 6 \, a^{2} b^{2} d^{5} e^{3} - 4 \, a^{3} b d^{4} e^{4} + a^{4} d^{3} e^{5} + {\left (b^{4} d^{4} e^{4} - 4 \, a b^{3} d^{3} e^{5} + 6 \, a^{2} b^{2} d^{2} e^{6} - 4 \, a^{3} b d e^{7} + a^{4} e^{8}\right )} x^{3} + 3 \, {\left (b^{4} d^{5} e^{3} - 4 \, a b^{3} d^{4} e^{4} + 6 \, a^{2} b^{2} d^{3} e^{5} - 4 \, a^{3} b d^{2} e^{6} + a^{4} d e^{7}\right )} x^{2} + 3 \, {\left (b^{4} d^{6} e^{2} - 4 \, a b^{3} d^{5} e^{3} + 6 \, a^{2} b^{2} d^{4} e^{4} - 4 \, a^{3} b d^{3} e^{5} + a^{4} d^{2} e^{6}\right )} x\right )}} \]
-1/6*(2*B*b^3*d^4 + 2*A*a^3*e^4 + (3*B*a*b^2 - 11*A*b^3)*d^3*e - 6*(B*a^2* b - 3*A*a*b^2)*d^2*e^2 + (B*a^3 - 9*A*a^2*b)*d*e^3 + 6*((B*a*b^2 - A*b^3)* d*e^3 - (B*a^2*b - A*a*b^2)*e^4)*x^2 + 3*(5*(B*a*b^2 - A*b^3)*d^2*e^2 - 6* (B*a^2*b - A*a*b^2)*d*e^3 + (B*a^3 - A*a^2*b)*e^4)*x + 6*((B*a*b^2 - A*b^3 )*e^4*x^3 + 3*(B*a*b^2 - A*b^3)*d*e^3*x^2 + 3*(B*a*b^2 - A*b^3)*d^2*e^2*x + (B*a*b^2 - A*b^3)*d^3*e)*log(b*x + a) - 6*((B*a*b^2 - A*b^3)*e^4*x^3 + 3 *(B*a*b^2 - A*b^3)*d*e^3*x^2 + 3*(B*a*b^2 - A*b^3)*d^2*e^2*x + (B*a*b^2 - A*b^3)*d^3*e)*log(e*x + d))/(b^4*d^7*e - 4*a*b^3*d^6*e^2 + 6*a^2*b^2*d^5*e ^3 - 4*a^3*b*d^4*e^4 + a^4*d^3*e^5 + (b^4*d^4*e^4 - 4*a*b^3*d^3*e^5 + 6*a^ 2*b^2*d^2*e^6 - 4*a^3*b*d*e^7 + a^4*e^8)*x^3 + 3*(b^4*d^5*e^3 - 4*a*b^3*d^ 4*e^4 + 6*a^2*b^2*d^3*e^5 - 4*a^3*b*d^2*e^6 + a^4*d*e^7)*x^2 + 3*(b^4*d^6* e^2 - 4*a*b^3*d^5*e^3 + 6*a^2*b^2*d^4*e^4 - 4*a^3*b*d^3*e^5 + a^4*d^2*e^6) *x)
\[ \int \frac {A+B x}{(d+e x)^4 \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\int \frac {A + B x}{\left (d + e x\right )^{4} \sqrt {\left (a + b x\right )^{2}}}\, dx \]
Exception generated. \[ \int \frac {A+B x}{(d+e x)^4 \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\text {Exception raised: ValueError} \]
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. 508 vs. \(2 (213) = 426\).
Time = 0.32 (sec) , antiderivative size = 508, normalized size of antiderivative = 1.87 \[ \int \frac {A+B x}{(d+e x)^4 \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=-\frac {{\left (B a b^{3} \mathrm {sgn}\left (b x + a\right ) - A b^{4} \mathrm {sgn}\left (b x + a\right )\right )} \log \left ({\left | b x + a \right |}\right )}{b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}} + \frac {{\left (B a b^{2} e \mathrm {sgn}\left (b x + a\right ) - A b^{3} e \mathrm {sgn}\left (b x + a\right )\right )} \log \left ({\left | e x + d \right |}\right )}{b^{4} d^{4} e - 4 \, a b^{3} d^{3} e^{2} + 6 \, a^{2} b^{2} d^{2} e^{3} - 4 \, a^{3} b d e^{4} + a^{4} e^{5}} - \frac {2 \, B b^{3} d^{4} \mathrm {sgn}\left (b x + a\right ) + 3 \, B a b^{2} d^{3} e \mathrm {sgn}\left (b x + a\right ) - 11 \, A b^{3} d^{3} e \mathrm {sgn}\left (b x + a\right ) - 6 \, B a^{2} b d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 18 \, A a b^{2} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + B a^{3} d e^{3} \mathrm {sgn}\left (b x + a\right ) - 9 \, A a^{2} b d e^{3} \mathrm {sgn}\left (b x + a\right ) + 2 \, A a^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) + 6 \, {\left (B a b^{2} d e^{3} \mathrm {sgn}\left (b x + a\right ) - A b^{3} d e^{3} \mathrm {sgn}\left (b x + a\right ) - B a^{2} b e^{4} \mathrm {sgn}\left (b x + a\right ) + A a b^{2} e^{4} \mathrm {sgn}\left (b x + a\right )\right )} x^{2} + 3 \, {\left (5 \, B a b^{2} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 5 \, A b^{3} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 6 \, B a^{2} b d e^{3} \mathrm {sgn}\left (b x + a\right ) + 6 \, A a b^{2} d e^{3} \mathrm {sgn}\left (b x + a\right ) + B a^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) - A a^{2} b e^{4} \mathrm {sgn}\left (b x + a\right )\right )} x}{6 \, {\left (b d - a e\right )}^{4} {\left (e x + d\right )}^{3} e} \]
-(B*a*b^3*sgn(b*x + a) - A*b^4*sgn(b*x + a))*log(abs(b*x + a))/(b^5*d^4 - 4*a*b^4*d^3*e + 6*a^2*b^3*d^2*e^2 - 4*a^3*b^2*d*e^3 + a^4*b*e^4) + (B*a*b^ 2*e*sgn(b*x + a) - A*b^3*e*sgn(b*x + a))*log(abs(e*x + d))/(b^4*d^4*e - 4* a*b^3*d^3*e^2 + 6*a^2*b^2*d^2*e^3 - 4*a^3*b*d*e^4 + a^4*e^5) - 1/6*(2*B*b^ 3*d^4*sgn(b*x + a) + 3*B*a*b^2*d^3*e*sgn(b*x + a) - 11*A*b^3*d^3*e*sgn(b*x + a) - 6*B*a^2*b*d^2*e^2*sgn(b*x + a) + 18*A*a*b^2*d^2*e^2*sgn(b*x + a) + B*a^3*d*e^3*sgn(b*x + a) - 9*A*a^2*b*d*e^3*sgn(b*x + a) + 2*A*a^3*e^4*sgn (b*x + a) + 6*(B*a*b^2*d*e^3*sgn(b*x + a) - A*b^3*d*e^3*sgn(b*x + a) - B*a ^2*b*e^4*sgn(b*x + a) + A*a*b^2*e^4*sgn(b*x + a))*x^2 + 3*(5*B*a*b^2*d^2*e ^2*sgn(b*x + a) - 5*A*b^3*d^2*e^2*sgn(b*x + a) - 6*B*a^2*b*d*e^3*sgn(b*x + a) + 6*A*a*b^2*d*e^3*sgn(b*x + a) + B*a^3*e^4*sgn(b*x + a) - A*a^2*b*e^4* sgn(b*x + a))*x)/((b*d - a*e)^4*(e*x + d)^3*e)
Timed out. \[ \int \frac {A+B x}{(d+e x)^4 \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\int \frac {A+B\,x}{\sqrt {{\left (a+b\,x\right )}^2}\,{\left (d+e\,x\right )}^4} \,d x \]