Integrand size = 33, antiderivative size = 114 \[ \int \frac {(a+b x) (d+e x)^2}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=-\frac {(b d-a e)^2}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^2 x (a+b x)}{b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 e (b d-a e) (a+b x) \log (a+b x)}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}} \] Output:
-(-a*e+b*d)^2/b^3/((b*x+a)^2)^(1/2)+e^2*x*(b*x+a)/b^2/((b*x+a)^2)^(1/2)+2* e*(-a*e+b*d)*(b*x+a)*ln(b*x+a)/b^3/((b*x+a)^2)^(1/2)
Time = 1.05 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.65 \[ \int \frac {(a+b x) (d+e x)^2}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {-a^2 e^2+a b e (2 d+e x)+b^2 \left (-d^2+e^2 x^2\right )-2 e (-b d+a e) (a+b x) \log (a+b x)}{b^3 \sqrt {(a+b x)^2}} \] Input:
Integrate[((a + b*x)*(d + e*x)^2)/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]
Output:
(-(a^2*e^2) + a*b*e*(2*d + e*x) + b^2*(-d^2 + e^2*x^2) - 2*e*(-(b*d) + a*e )*(a + b*x)*Log[a + b*x])/(b^3*Sqrt[(a + b*x)^2])
Time = 0.44 (sec) , antiderivative size = 77, 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, 49, 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.
\(\displaystyle \int \frac {(a+b x) (d+e x)^2}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 1187 |
\(\displaystyle \frac {b^3 (a+b x) \int \frac {(d+e x)^2}{b^3 (a+b x)^2}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {(a+b x) \int \frac {(d+e x)^2}{(a+b x)^2}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {(a+b x) \int \left (\frac {e^2}{b^2}+\frac {2 (b d-a e) e}{b^2 (a+b x)}+\frac {(b d-a e)^2}{b^2 (a+b x)^2}\right )dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {(a+b x) \left (-\frac {(b d-a e)^2}{b^3 (a+b x)}+\frac {2 e (b d-a e) \log (a+b x)}{b^3}+\frac {e^2 x}{b^2}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
Input:
Int[((a + b*x)*(d + e*x)^2)/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]
Output:
((a + b*x)*((e^2*x)/b^2 - (b*d - a*e)^2/(b^3*(a + b*x)) + (2*e*(b*d - a*e) *Log[a + b*x])/b^3))/Sqrt[a^2 + 2*a*b*x + b^2*x^2]
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 [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
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 = 1.38 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.91
method | result | size |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, e^{2} x}{\left (b x +a \right ) b^{2}}-\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )}{\left (b x +a \right )^{2} b^{3}}-\frac {2 \sqrt {\left (b x +a \right )^{2}}\, e \left (a e -b d \right ) \ln \left (b x +a \right )}{\left (b x +a \right ) b^{3}}\) | \(104\) |
default | \(-\frac {\left (2 \ln \left (b x +a \right ) a b \,e^{2} x -2 \ln \left (b x +a \right ) b^{2} d e x -b^{2} e^{2} x^{2}+2 \ln \left (b x +a \right ) a^{2} e^{2}-2 \ln \left (b x +a \right ) a b d e -x a b \,e^{2}+e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (b x +a \right )^{2}}{b^{3} \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}\) | \(116\) |
Input:
int((b*x+a)*(e*x+d)^2/(b^2*x^2+2*a*b*x+a^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
((b*x+a)^2)^(1/2)/(b*x+a)*e^2*x/b^2-((b*x+a)^2)^(1/2)/(b*x+a)^2/b^3*(a^2*e ^2-2*a*b*d*e+b^2*d^2)-2*((b*x+a)^2)^(1/2)/(b*x+a)/b^3*e*(a*e-b*d)*ln(b*x+a )
Time = 0.08 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.81 \[ \int \frac {(a+b x) (d+e x)^2}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {b^{2} e^{2} x^{2} + a b e^{2} x - b^{2} d^{2} + 2 \, a b d e - a^{2} e^{2} + 2 \, {\left (a b d e - a^{2} e^{2} + {\left (b^{2} d e - a b e^{2}\right )} x\right )} \log \left (b x + a\right )}{b^{4} x + a b^{3}} \] Input:
integrate((b*x+a)*(e*x+d)^2/(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="fric as")
Output:
(b^2*e^2*x^2 + a*b*e^2*x - b^2*d^2 + 2*a*b*d*e - a^2*e^2 + 2*(a*b*d*e - a^ 2*e^2 + (b^2*d*e - a*b*e^2)*x)*log(b*x + a))/(b^4*x + a*b^3)
\[ \int \frac {(a+b x) (d+e x)^2}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int \frac {\left (a + b x\right ) \left (d + e x\right )^{2}}{\left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((b*x+a)*(e*x+d)**2/(b**2*x**2+2*a*b*x+a**2)**(3/2),x)
Output:
Integral((a + b*x)*(d + e*x)**2/((a + b*x)**2)**(3/2), x)
Leaf count of result is larger than twice the leaf count of optimal. 272 vs. \(2 (81) = 162\).
Time = 0.04 (sec) , antiderivative size = 272, normalized size of antiderivative = 2.39 \[ \int \frac {(a+b x) (d+e x)^2}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {e^{2} x^{2}}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b} - \frac {3 \, a e^{2} \log \left (x + \frac {a}{b}\right )}{b^{3}} + \frac {2 \, a^{2} e^{2}}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{3}} - \frac {6 \, a^{2} e^{2} x}{b^{4} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {{\left (2 \, b d e + a e^{2}\right )} \log \left (x + \frac {a}{b}\right )}{b^{3}} - \frac {b d^{2} + 2 \, a d e}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} - \frac {a d^{2}}{2 \, b^{3} {\left (x + \frac {a}{b}\right )}^{2}} - \frac {11 \, a^{3} e^{2}}{2 \, b^{5} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {2 \, {\left (2 \, b d e + a e^{2}\right )} a x}{b^{4} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {3 \, {\left (2 \, b d e + a e^{2}\right )} a^{2}}{2 \, b^{5} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {{\left (b d^{2} + 2 \, a d e\right )} a}{2 \, b^{4} {\left (x + \frac {a}{b}\right )}^{2}} \] Input:
integrate((b*x+a)*(e*x+d)^2/(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="maxi ma")
Output:
e^2*x^2/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*b) - 3*a*e^2*log(x + a/b)/b^3 + 2*a ^2*e^2/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*b^3) - 6*a^2*e^2*x/(b^4*(x + a/b)^2) + (2*b*d*e + a*e^2)*log(x + a/b)/b^3 - (b*d^2 + 2*a*d*e)/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*b^2) - 1/2*a*d^2/(b^3*(x + a/b)^2) - 11/2*a^3*e^2/(b^5*(x + a/b)^2) + 2*(2*b*d*e + a*e^2)*a*x/(b^4*(x + a/b)^2) + 3/2*(2*b*d*e + a*e^ 2)*a^2/(b^5*(x + a/b)^2) + 1/2*(b*d^2 + 2*a*d*e)*a/(b^4*(x + a/b)^2)
Time = 0.14 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.78 \[ \int \frac {(a+b x) (d+e x)^2}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {e^{2} x}{b^{2} \mathrm {sgn}\left (b x + a\right )} + \frac {2 \, {\left (b d e - a e^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{3} \mathrm {sgn}\left (b x + a\right )} - \frac {b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}}{{\left (b x + a\right )} b^{3} \mathrm {sgn}\left (b x + a\right )} \] Input:
integrate((b*x+a)*(e*x+d)^2/(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="giac ")
Output:
e^2*x/(b^2*sgn(b*x + a)) + 2*(b*d*e - a*e^2)*log(abs(b*x + a))/(b^3*sgn(b* x + a)) - (b^2*d^2 - 2*a*b*d*e + a^2*e^2)/((b*x + a)*b^3*sgn(b*x + a))
Timed out. \[ \int \frac {(a+b x) (d+e x)^2}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int \frac {\left (a+b\,x\right )\,{\left (d+e\,x\right )}^2}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}} \,d x \] Input:
int(((a + b*x)*(d + e*x)^2)/(a^2 + b^2*x^2 + 2*a*b*x)^(3/2),x)
Output:
int(((a + b*x)*(d + e*x)^2)/(a^2 + b^2*x^2 + 2*a*b*x)^(3/2), x)
Time = 0.23 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.98 \[ \int \frac {(a+b x) (d+e x)^2}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {-2 \,\mathrm {log}\left (b x +a \right ) a^{3} e^{2}+2 \,\mathrm {log}\left (b x +a \right ) a^{2} b d e -2 \,\mathrm {log}\left (b x +a \right ) a^{2} b \,e^{2} x +2 \,\mathrm {log}\left (b x +a \right ) a \,b^{2} d e x +2 a^{2} b \,e^{2} x -2 a \,b^{2} d e x +a \,b^{2} e^{2} x^{2}+b^{3} d^{2} x}{a \,b^{3} \left (b x +a \right )} \] Input:
int((b*x+a)*(e*x+d)^2/(b^2*x^2+2*a*b*x+a^2)^(3/2),x)
Output:
( - 2*log(a + b*x)*a**3*e**2 + 2*log(a + b*x)*a**2*b*d*e - 2*log(a + b*x)* a**2*b*e**2*x + 2*log(a + b*x)*a*b**2*d*e*x + 2*a**2*b*e**2*x - 2*a*b**2*d *e*x + a*b**2*e**2*x**2 + b**3*d**2*x)/(a*b**3*(a + b*x))