Integrand size = 33, antiderivative size = 110 \[ \int \frac {(a+b x) \sqrt {d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {\sqrt {d+e x}}{2 b (a+b x)^2}-\frac {e \sqrt {d+e x}}{4 b (b d-a e) (a+b x)}+\frac {e^2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{3/2} (b d-a e)^{3/2}} \] Output:
-1/2*(e*x+d)^(1/2)/b/(b*x+a)^2-1/4*e*(e*x+d)^(1/2)/b/(-a*e+b*d)/(b*x+a)+1/ 4*e^2*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))/b^(3/2)/(-a*e+b*d)^( 3/2)
Time = 0.67 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.90 \[ \int \frac {(a+b x) \sqrt {d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {\frac {\sqrt {b} \sqrt {d+e x} (2 b d-a e+b e x)}{(-b d+a e) (a+b x)^2}+\frac {e^2 \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{(-b d+a e)^{3/2}}}{4 b^{3/2}} \] Input:
Integrate[((a + b*x)*Sqrt[d + e*x])/(a^2 + 2*a*b*x + b^2*x^2)^2,x]
Output:
((Sqrt[b]*Sqrt[d + e*x]*(2*b*d - a*e + b*e*x))/((-(b*d) + a*e)*(a + b*x)^2 ) + (e^2*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(-(b*d) + a*e )^(3/2))/(4*b^(3/2))
Time = 0.42 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1184, 27, 51, 52, 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.
\(\displaystyle \int \frac {(a+b x) \sqrt {d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 1184 |
\(\displaystyle b^4 \int \frac {\sqrt {d+e x}}{b^4 (a+b x)^3}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {\sqrt {d+e x}}{(a+b x)^3}dx\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {e \int \frac {1}{(a+b x)^2 \sqrt {d+e x}}dx}{4 b}-\frac {\sqrt {d+e x}}{2 b (a+b x)^2}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {e \left (-\frac {e \int \frac {1}{(a+b x) \sqrt {d+e x}}dx}{2 (b d-a e)}-\frac {\sqrt {d+e x}}{(a+b x) (b d-a e)}\right )}{4 b}-\frac {\sqrt {d+e x}}{2 b (a+b x)^2}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {e \left (-\frac {\int \frac {1}{a+\frac {b (d+e x)}{e}-\frac {b d}{e}}d\sqrt {d+e x}}{b d-a e}-\frac {\sqrt {d+e x}}{(a+b x) (b d-a e)}\right )}{4 b}-\frac {\sqrt {d+e x}}{2 b (a+b x)^2}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {e \left (\frac {e \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{\sqrt {b} (b d-a e)^{3/2}}-\frac {\sqrt {d+e x}}{(a+b x) (b d-a e)}\right )}{4 b}-\frac {\sqrt {d+e x}}{2 b (a+b x)^2}\) |
Input:
Int[((a + b*x)*Sqrt[d + e*x])/(a^2 + 2*a*b*x + b^2*x^2)^2,x]
Output:
-1/2*Sqrt[d + e*x]/(b*(a + b*x)^2) + (e*(-(Sqrt[d + e*x]/((b*d - a*e)*(a + b*x))) + (e*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(Sqrt[b]*(b *d - a*e)^(3/2))))/(4*b)
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] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[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_)^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 = 1.55 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.95
method | result | size |
pseudoelliptic | \(\frac {\arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) \left (b x +a \right )^{2} e^{2}-\sqrt {e x +d}\, \sqrt {b \left (a e -b d \right )}\, \left (\left (-e x -2 d \right ) b +a e \right )}{4 \sqrt {b \left (a e -b d \right )}\, b \left (a e -b d \right ) \left (b x +a \right )^{2}}\) | \(104\) |
derivativedivides | \(2 e^{2} \left (\frac {\frac {\left (e x +d \right )^{\frac {3}{2}}}{8 a e -8 b d}-\frac {\sqrt {e x +d}}{8 b}}{\left (b \left (e x +d \right )+a e -b d \right )^{2}}+\frac {\arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{8 \left (a e -b d \right ) b \sqrt {b \left (a e -b d \right )}}\right )\) | \(106\) |
default | \(2 e^{2} \left (\frac {\frac {\left (e x +d \right )^{\frac {3}{2}}}{8 a e -8 b d}-\frac {\sqrt {e x +d}}{8 b}}{\left (b \left (e x +d \right )+a e -b d \right )^{2}}+\frac {\arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{8 \left (a e -b d \right ) b \sqrt {b \left (a e -b d \right )}}\right )\) | \(106\) |
Input:
int((b*x+a)*(e*x+d)^(1/2)/(b^2*x^2+2*a*b*x+a^2)^2,x,method=_RETURNVERBOSE)
Output:
1/4/(b*(a*e-b*d))^(1/2)*(arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*(b*x+ a)^2*e^2-(e*x+d)^(1/2)*(b*(a*e-b*d))^(1/2)*((-e*x-2*d)*b+a*e))/b/(a*e-b*d) /(b*x+a)^2
Leaf count of result is larger than twice the leaf count of optimal. 221 vs. \(2 (90) = 180\).
Time = 0.09 (sec) , antiderivative size = 456, normalized size of antiderivative = 4.15 \[ \int \frac {(a+b x) \sqrt {d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\left [-\frac {{\left (b^{2} e^{2} x^{2} + 2 \, a b e^{2} x + a^{2} e^{2}\right )} \sqrt {b^{2} d - a b e} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {b^{2} d - a b e} \sqrt {e x + d}}{b x + a}\right ) + 2 \, {\left (2 \, b^{3} d^{2} - 3 \, a b^{2} d e + a^{2} b e^{2} + {\left (b^{3} d e - a b^{2} e^{2}\right )} x\right )} \sqrt {e x + d}}{8 \, {\left (a^{2} b^{4} d^{2} - 2 \, a^{3} b^{3} d e + a^{4} b^{2} e^{2} + {\left (b^{6} d^{2} - 2 \, a b^{5} d e + a^{2} b^{4} e^{2}\right )} x^{2} + 2 \, {\left (a b^{5} d^{2} - 2 \, a^{2} b^{4} d e + a^{3} b^{3} e^{2}\right )} x\right )}}, -\frac {{\left (b^{2} e^{2} x^{2} + 2 \, a b e^{2} x + a^{2} e^{2}\right )} \sqrt {-b^{2} d + a b e} \arctan \left (\frac {\sqrt {-b^{2} d + a b e} \sqrt {e x + d}}{b e x + b d}\right ) + {\left (2 \, b^{3} d^{2} - 3 \, a b^{2} d e + a^{2} b e^{2} + {\left (b^{3} d e - a b^{2} e^{2}\right )} x\right )} \sqrt {e x + d}}{4 \, {\left (a^{2} b^{4} d^{2} - 2 \, a^{3} b^{3} d e + a^{4} b^{2} e^{2} + {\left (b^{6} d^{2} - 2 \, a b^{5} d e + a^{2} b^{4} e^{2}\right )} x^{2} + 2 \, {\left (a b^{5} d^{2} - 2 \, a^{2} b^{4} d e + a^{3} b^{3} e^{2}\right )} x\right )}}\right ] \] Input:
integrate((b*x+a)*(e*x+d)^(1/2)/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fric as")
Output:
[-1/8*((b^2*e^2*x^2 + 2*a*b*e^2*x + a^2*e^2)*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*(2*b ^3*d^2 - 3*a*b^2*d*e + a^2*b*e^2 + (b^3*d*e - a*b^2*e^2)*x)*sqrt(e*x + d)) /(a^2*b^4*d^2 - 2*a^3*b^3*d*e + a^4*b^2*e^2 + (b^6*d^2 - 2*a*b^5*d*e + a^2 *b^4*e^2)*x^2 + 2*(a*b^5*d^2 - 2*a^2*b^4*d*e + a^3*b^3*e^2)*x), -1/4*((b^2 *e^2*x^2 + 2*a*b*e^2*x + a^2*e^2)*sqrt(-b^2*d + a*b*e)*arctan(sqrt(-b^2*d + a*b*e)*sqrt(e*x + d)/(b*e*x + b*d)) + (2*b^3*d^2 - 3*a*b^2*d*e + a^2*b*e ^2 + (b^3*d*e - a*b^2*e^2)*x)*sqrt(e*x + d))/(a^2*b^4*d^2 - 2*a^3*b^3*d*e + a^4*b^2*e^2 + (b^6*d^2 - 2*a*b^5*d*e + a^2*b^4*e^2)*x^2 + 2*(a*b^5*d^2 - 2*a^2*b^4*d*e + a^3*b^3*e^2)*x)]
Timed out. \[ \int \frac {(a+b x) \sqrt {d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\text {Timed out} \] Input:
integrate((b*x+a)*(e*x+d)**(1/2)/(b**2*x**2+2*a*b*x+a**2)**2,x)
Output:
Timed out
Exception generated. \[ \int \frac {(a+b x) \sqrt {d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((b*x+a)*(e*x+d)^(1/2)/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxi ma")
Output:
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
Time = 0.15 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.15 \[ \int \frac {(a+b x) \sqrt {d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {e^{2} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{4 \, {\left (b^{2} d - a b e\right )} \sqrt {-b^{2} d + a b e}} - \frac {{\left (e x + d\right )}^{\frac {3}{2}} b e^{2} + \sqrt {e x + d} b d e^{2} - \sqrt {e x + d} a e^{3}}{4 \, {\left (b^{2} d - a b e\right )} {\left ({\left (e x + d\right )} b - b d + a e\right )}^{2}} \] Input:
integrate((b*x+a)*(e*x+d)^(1/2)/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac ")
Output:
-1/4*e^2*arctan(sqrt(e*x + d)*b/sqrt(-b^2*d + a*b*e))/((b^2*d - a*b*e)*sqr t(-b^2*d + a*b*e)) - 1/4*((e*x + d)^(3/2)*b*e^2 + sqrt(e*x + d)*b*d*e^2 - sqrt(e*x + d)*a*e^3)/((b^2*d - a*b*e)*((e*x + d)*b - b*d + a*e)^2)
Time = 11.64 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.23 \[ \int \frac {(a+b x) \sqrt {d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {e^2\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}}{\sqrt {a\,e-b\,d}}\right )}{4\,b^{3/2}\,{\left (a\,e-b\,d\right )}^{3/2}}-\frac {\frac {e^2\,\sqrt {d+e\,x}}{4\,b}-\frac {e^2\,{\left (d+e\,x\right )}^{3/2}}{4\,\left (a\,e-b\,d\right )}}{b^2\,{\left (d+e\,x\right )}^2-\left (2\,b^2\,d-2\,a\,b\,e\right )\,\left (d+e\,x\right )+a^2\,e^2+b^2\,d^2-2\,a\,b\,d\,e} \] Input:
int(((a + b*x)*(d + e*x)^(1/2))/(a^2 + b^2*x^2 + 2*a*b*x)^2,x)
Output:
(e^2*atan((b^(1/2)*(d + e*x)^(1/2))/(a*e - b*d)^(1/2)))/(4*b^(3/2)*(a*e - b*d)^(3/2)) - ((e^2*(d + e*x)^(1/2))/(4*b) - (e^2*(d + e*x)^(3/2))/(4*(a*e - b*d)))/(b^2*(d + e*x)^2 - (2*b^2*d - 2*a*b*e)*(d + e*x) + a^2*e^2 + b^2 *d^2 - 2*a*b*d*e)
Time = 0.27 (sec) , antiderivative size = 301, normalized size of antiderivative = 2.74 \[ \int \frac {(a+b x) \sqrt {d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {\sqrt {b}\, \sqrt {a e -b d}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, b}{\sqrt {b}\, \sqrt {a e -b d}}\right ) a^{2} e^{2}+2 \sqrt {b}\, \sqrt {a e -b d}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, b}{\sqrt {b}\, \sqrt {a e -b d}}\right ) a b \,e^{2} x +\sqrt {b}\, \sqrt {a e -b d}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, b}{\sqrt {b}\, \sqrt {a e -b d}}\right ) b^{2} e^{2} x^{2}-\sqrt {e x +d}\, a^{2} b \,e^{2}+3 \sqrt {e x +d}\, a \,b^{2} d e +\sqrt {e x +d}\, a \,b^{2} e^{2} x -2 \sqrt {e x +d}\, b^{3} d^{2}-\sqrt {e x +d}\, b^{3} d e x}{4 b^{2} \left (a^{2} b^{2} e^{2} x^{2}-2 a \,b^{3} d e \,x^{2}+b^{4} d^{2} x^{2}+2 a^{3} b \,e^{2} x -4 a^{2} b^{2} d e x +2 a \,b^{3} d^{2} x +a^{4} e^{2}-2 a^{3} b d e +a^{2} b^{2} d^{2}\right )} \] Input:
int((b*x+a)*(e*x+d)^(1/2)/(b^2*x^2+2*a*b*x+a^2)^2,x)
Output:
(sqrt(b)*sqrt(a*e - b*d)*atan((sqrt(d + e*x)*b)/(sqrt(b)*sqrt(a*e - b*d))) *a**2*e**2 + 2*sqrt(b)*sqrt(a*e - b*d)*atan((sqrt(d + e*x)*b)/(sqrt(b)*sqr t(a*e - b*d)))*a*b*e**2*x + sqrt(b)*sqrt(a*e - b*d)*atan((sqrt(d + e*x)*b) /(sqrt(b)*sqrt(a*e - b*d)))*b**2*e**2*x**2 - sqrt(d + e*x)*a**2*b*e**2 + 3 *sqrt(d + e*x)*a*b**2*d*e + sqrt(d + e*x)*a*b**2*e**2*x - 2*sqrt(d + e*x)* b**3*d**2 - sqrt(d + e*x)*b**3*d*e*x)/(4*b**2*(a**4*e**2 - 2*a**3*b*d*e + 2*a**3*b*e**2*x + a**2*b**2*d**2 - 4*a**2*b**2*d*e*x + a**2*b**2*e**2*x**2 + 2*a*b**3*d**2*x - 2*a*b**3*d*e*x**2 + b**4*d**2*x**2))