Integrand size = 28, antiderivative size = 217 \[ \int \frac {1}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {2 b e}{(b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b}{2 (b d-a e)^2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^2 (a+b x)}{(b d-a e)^3 (d+e x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 b e^2 (a+b x) \log (a+b x)}{(b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 b e^2 (a+b x) \log (d+e x)}{(b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}} \]
2*b*e/(-a*e+b*d)^3/((b*x+a)^2)^(1/2)-1/2*b/(-a*e+b*d)^2/(b*x+a)/((b*x+a)^2 )^(1/2)+e^2*(b*x+a)/(-a*e+b*d)^3/(e*x+d)/((b*x+a)^2)^(1/2)+3*b*e^2*(b*x+a) *ln(b*x+a)/(-a*e+b*d)^4/((b*x+a)^2)^(1/2)-3*b*e^2*(b*x+a)*ln(e*x+d)/(-a*e+ b*d)^4/((b*x+a)^2)^(1/2)
Time = 1.06 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.65 \[ \int \frac {1}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {-\left ((b d-a e) \left (-2 a^2 e^2-a b e (5 d+9 e x)+b^2 \left (d^2-3 d e x-6 e^2 x^2\right )\right )\right )+6 b e^2 (a+b x)^2 (d+e x) \log (a+b x)-6 b e^2 (a+b x)^2 (d+e x) \log (d+e x)}{2 (b d-a e)^4 (a+b x) \sqrt {(a+b x)^2} (d+e x)} \]
(-((b*d - a*e)*(-2*a^2*e^2 - a*b*e*(5*d + 9*e*x) + b^2*(d^2 - 3*d*e*x - 6* e^2*x^2))) + 6*b*e^2*(a + b*x)^2*(d + e*x)*Log[a + b*x] - 6*b*e^2*(a + b*x )^2*(d + e*x)*Log[d + e*x])/(2*(b*d - a*e)^4*(a + b*x)*Sqrt[(a + b*x)^2]*( d + e*x))
Time = 0.32 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.62, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1102, 27, 54, 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.
\(\displaystyle \int \frac {1}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2} (d+e x)^2} \, dx\) |
\(\Big \downarrow \) 1102 |
\(\displaystyle \frac {b^3 (a+b x) \int \frac {1}{b^3 (a+b x)^3 (d+e x)^2}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {(a+b x) \int \frac {1}{(a+b x)^3 (d+e x)^2}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
\(\Big \downarrow \) 54 |
\(\displaystyle \frac {(a+b x) \int \left (-\frac {3 b e^3}{(b d-a e)^4 (d+e x)}-\frac {e^3}{(b d-a e)^3 (d+e x)^2}+\frac {3 b^2 e^2}{(b d-a e)^4 (a+b x)}-\frac {2 b^2 e}{(b d-a e)^3 (a+b x)^2}+\frac {b^2}{(b d-a e)^2 (a+b x)^3}\right )dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {(a+b x) \left (\frac {e^2}{(d+e x) (b d-a e)^3}+\frac {3 b e^2 \log (a+b x)}{(b d-a e)^4}-\frac {3 b e^2 \log (d+e x)}{(b d-a e)^4}+\frac {2 b e}{(a+b x) (b d-a e)^3}-\frac {b}{2 (a+b x)^2 (b d-a e)^2}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
((a + b*x)*(-1/2*b/((b*d - a*e)^2*(a + b*x)^2) + (2*b*e)/((b*d - a*e)^3*(a + b*x)) + e^2/((b*d - a*e)^3*(d + e*x)) + (3*b*e^2*Log[a + b*x])/(b*d - a *e)^4 - (3*b*e^2*Log[d + e*x])/(b*d - a*e)^4))/Sqrt[a^2 + 2*a*b*x + b^2*x^ 2]
3.17.2.3.1 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] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
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. \(329\) vs. \(2(160)=320\).
Time = 2.49 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.52
method | result | size |
default | \(\frac {\left (6 \ln \left (b x +a \right ) x^{3} b^{3} e^{3}-6 \ln \left (e x +d \right ) b^{3} e^{3} x^{3}+12 \ln \left (b x +a \right ) x^{2} a \,b^{2} e^{3}+6 \ln \left (b x +a \right ) b^{3} d \,e^{2} x^{2}-12 \ln \left (e x +d \right ) a \,b^{2} e^{3} x^{2}-6 \ln \left (e x +d \right ) b^{3} d \,e^{2} x^{2}+6 \ln \left (b x +a \right ) x \,a^{2} b \,e^{3}+12 \ln \left (b x +a \right ) x a \,b^{2} d \,e^{2}-6 \ln \left (e x +d \right ) a^{2} b \,e^{3} x -12 \ln \left (e x +d \right ) a \,b^{2} d \,e^{2} x -6 x^{2} a \,b^{2} e^{3}+6 x^{2} b^{3} d \,e^{2}+6 \ln \left (b x +a \right ) a^{2} b d \,e^{2}-6 \ln \left (e x +d \right ) a^{2} b d \,e^{2}-9 a^{2} b \,e^{3} x +6 x a \,b^{2} d \,e^{2}+3 b^{3} d^{2} e x -2 a^{3} e^{3}-3 a^{2} b d \,e^{2}+6 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \left (b x +a \right )}{2 \left (e x +d \right ) \left (a e -b d \right )^{4} \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}\) | \(330\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {3 b^{2} e^{2} x^{2}}{a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}}-\frac {3 \left (3 a e +b d \right ) e b x}{2 \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}-\frac {2 a^{2} e^{2}+5 a b d e -b^{2} d^{2}}{2 \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}\right )}{\left (b x +a \right )^{3} \left (e x +d \right )}-\frac {3 \sqrt {\left (b x +a \right )^{2}}\, e^{2} b \ln \left (e x +d \right )}{\left (b x +a \right ) \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}+\frac {3 \sqrt {\left (b x +a \right )^{2}}\, e^{2} b \ln \left (-b x -a \right )}{\left (b x +a \right ) \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}\) | \(351\) |
1/2*(6*ln(b*x+a)*x^3*b^3*e^3-6*ln(e*x+d)*b^3*e^3*x^3+12*ln(b*x+a)*x^2*a*b^ 2*e^3+6*ln(b*x+a)*b^3*d*e^2*x^2-12*ln(e*x+d)*a*b^2*e^3*x^2-6*ln(e*x+d)*b^3 *d*e^2*x^2+6*ln(b*x+a)*x*a^2*b*e^3+12*ln(b*x+a)*x*a*b^2*d*e^2-6*ln(e*x+d)* a^2*b*e^3*x-12*ln(e*x+d)*a*b^2*d*e^2*x-6*x^2*a*b^2*e^3+6*x^2*b^3*d*e^2+6*l n(b*x+a)*a^2*b*d*e^2-6*ln(e*x+d)*a^2*b*d*e^2-9*a^2*b*e^3*x+6*x*a*b^2*d*e^2 +3*b^3*d^2*e*x-2*a^3*e^3-3*a^2*b*d*e^2+6*a*b^2*d^2*e-b^3*d^3)*(b*x+a)/(e*x +d)/(a*e-b*d)^4/((b*x+a)^2)^(3/2)
Leaf count of result is larger than twice the leaf count of optimal. 494 vs. \(2 (160) = 320\).
Time = 0.25 (sec) , antiderivative size = 494, normalized size of antiderivative = 2.28 \[ \int \frac {1}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=-\frac {b^{3} d^{3} - 6 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} + 2 \, a^{3} e^{3} - 6 \, {\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} - 3 \, {\left (b^{3} d^{2} e + 2 \, a b^{2} d e^{2} - 3 \, a^{2} b e^{3}\right )} x - 6 \, {\left (b^{3} e^{3} x^{3} + a^{2} b d e^{2} + {\left (b^{3} d e^{2} + 2 \, a b^{2} e^{3}\right )} x^{2} + {\left (2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x\right )} \log \left (b x + a\right ) + 6 \, {\left (b^{3} e^{3} x^{3} + a^{2} b d e^{2} + {\left (b^{3} d e^{2} + 2 \, a b^{2} e^{3}\right )} x^{2} + {\left (2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \, {\left (a^{2} b^{4} d^{5} - 4 \, a^{3} b^{3} d^{4} e + 6 \, a^{4} b^{2} d^{3} e^{2} - 4 \, a^{5} b d^{2} e^{3} + a^{6} d e^{4} + {\left (b^{6} d^{4} e - 4 \, a b^{5} d^{3} e^{2} + 6 \, a^{2} b^{4} d^{2} e^{3} - 4 \, a^{3} b^{3} d e^{4} + a^{4} b^{2} e^{5}\right )} x^{3} + {\left (b^{6} d^{5} - 2 \, a b^{5} d^{4} e - 2 \, a^{2} b^{4} d^{3} e^{2} + 8 \, a^{3} b^{3} d^{2} e^{3} - 7 \, a^{4} b^{2} d e^{4} + 2 \, a^{5} b e^{5}\right )} x^{2} + {\left (2 \, a b^{5} d^{5} - 7 \, a^{2} b^{4} d^{4} e + 8 \, a^{3} b^{3} d^{3} e^{2} - 2 \, a^{4} b^{2} d^{2} e^{3} - 2 \, a^{5} b d e^{4} + a^{6} e^{5}\right )} x\right )}} \]
-1/2*(b^3*d^3 - 6*a*b^2*d^2*e + 3*a^2*b*d*e^2 + 2*a^3*e^3 - 6*(b^3*d*e^2 - a*b^2*e^3)*x^2 - 3*(b^3*d^2*e + 2*a*b^2*d*e^2 - 3*a^2*b*e^3)*x - 6*(b^3*e ^3*x^3 + a^2*b*d*e^2 + (b^3*d*e^2 + 2*a*b^2*e^3)*x^2 + (2*a*b^2*d*e^2 + a^ 2*b*e^3)*x)*log(b*x + a) + 6*(b^3*e^3*x^3 + a^2*b*d*e^2 + (b^3*d*e^2 + 2*a *b^2*e^3)*x^2 + (2*a*b^2*d*e^2 + a^2*b*e^3)*x)*log(e*x + d))/(a^2*b^4*d^5 - 4*a^3*b^3*d^4*e + 6*a^4*b^2*d^3*e^2 - 4*a^5*b*d^2*e^3 + a^6*d*e^4 + (b^6 *d^4*e - 4*a*b^5*d^3*e^2 + 6*a^2*b^4*d^2*e^3 - 4*a^3*b^3*d*e^4 + a^4*b^2*e ^5)*x^3 + (b^6*d^5 - 2*a*b^5*d^4*e - 2*a^2*b^4*d^3*e^2 + 8*a^3*b^3*d^2*e^3 - 7*a^4*b^2*d*e^4 + 2*a^5*b*e^5)*x^2 + (2*a*b^5*d^5 - 7*a^2*b^4*d^4*e + 8 *a^3*b^3*d^3*e^2 - 2*a^4*b^2*d^2*e^3 - 2*a^5*b*d*e^4 + a^6*e^5)*x)
\[ \int \frac {1}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (d + e x\right )^{2} \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}\, dx \]
Exception generated. \[ \int \frac {1}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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. 321 vs. \(2 (160) = 320\).
Time = 0.28 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.48 \[ \int \frac {1}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {3 \, b^{2} e^{2} \log \left ({\left | b x + a \right |}\right )}{b^{5} d^{4} \mathrm {sgn}\left (b x + a\right ) - 4 \, a b^{4} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b^{3} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 4 \, a^{3} b^{2} d e^{3} \mathrm {sgn}\left (b x + a\right ) + a^{4} b e^{4} \mathrm {sgn}\left (b x + a\right )} - \frac {3 \, b e^{3} \log \left ({\left | e x + d \right |}\right )}{b^{4} d^{4} e \mathrm {sgn}\left (b x + a\right ) - 4 \, a b^{3} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b^{2} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) - 4 \, a^{3} b d e^{4} \mathrm {sgn}\left (b x + a\right ) + a^{4} e^{5} \mathrm {sgn}\left (b x + a\right )} - \frac {b^{3} d^{3} - 6 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} + 2 \, a^{3} e^{3} - 6 \, {\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} - 3 \, {\left (b^{3} d^{2} e + 2 \, a b^{2} d e^{2} - 3 \, a^{2} b e^{3}\right )} x}{2 \, {\left (b d - a e\right )}^{4} {\left (b x + a\right )}^{2} {\left (e x + d\right )} \mathrm {sgn}\left (b x + a\right )} \]
3*b^2*e^2*log(abs(b*x + a))/(b^5*d^4*sgn(b*x + a) - 4*a*b^4*d^3*e*sgn(b*x + a) + 6*a^2*b^3*d^2*e^2*sgn(b*x + a) - 4*a^3*b^2*d*e^3*sgn(b*x + a) + a^4 *b*e^4*sgn(b*x + a)) - 3*b*e^3*log(abs(e*x + d))/(b^4*d^4*e*sgn(b*x + a) - 4*a*b^3*d^3*e^2*sgn(b*x + a) + 6*a^2*b^2*d^2*e^3*sgn(b*x + a) - 4*a^3*b*d *e^4*sgn(b*x + a) + a^4*e^5*sgn(b*x + a)) - 1/2*(b^3*d^3 - 6*a*b^2*d^2*e + 3*a^2*b*d*e^2 + 2*a^3*e^3 - 6*(b^3*d*e^2 - a*b^2*e^3)*x^2 - 3*(b^3*d^2*e + 2*a*b^2*d*e^2 - 3*a^2*b*e^3)*x)/((b*d - a*e)^4*(b*x + a)^2*(e*x + d)*sgn (b*x + a))
Timed out. \[ \int \frac {1}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int \frac {1}{{\left (d+e\,x\right )}^2\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}} \,d x \]