Integrand size = 20, antiderivative size = 104 \[ \int \frac {A+B x}{x^2 \left (a+b x^2\right )^{5/2}} \, dx=\frac {A+B x}{3 a x \left (a+b x^2\right )^{3/2}}+\frac {4 A+3 B x}{3 a^2 x \sqrt {a+b x^2}}-\frac {8 A \sqrt {a+b x^2}}{3 a^3 x}-\frac {B \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{5/2}} \]
1/3*(B*x+A)/a/x/(b*x^2+a)^(3/2)-B*arctanh((b*x^2+a)^(1/2)/a^(1/2))/a^(5/2) +1/3*(3*B*x+4*A)/a^2/x/(b*x^2+a)^(1/2)-8/3*A*(b*x^2+a)^(1/2)/a^3/x
Time = 0.36 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int \frac {A+B x}{x^2 \left (a+b x^2\right )^{5/2}} \, dx=\frac {-8 A b^2 x^4+3 a b x^2 (-4 A+B x)+a^2 (-3 A+4 B x)}{3 a^3 x \left (a+b x^2\right )^{3/2}}+\frac {2 B \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{5/2}} \]
(-8*A*b^2*x^4 + 3*a*b*x^2*(-4*A + B*x) + a^2*(-3*A + 4*B*x))/(3*a^3*x*(a + b*x^2)^(3/2)) + (2*B*ArcTanh[(Sqrt[b]*x - Sqrt[a + b*x^2])/Sqrt[a]])/a^(5 /2)
Time = 0.34 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.09, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {532, 25, 2336, 27, 534, 243, 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.
| \(\displaystyle \int \frac {A+B x}{x^2 \left (a+b x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 532 |
| \(\displaystyle \frac {a B-A b x}{3 a^2 \left (a+b x^2\right )^{3/2}}-\frac {\int -\frac {-\frac {2 A b x^2}{a}+3 B x+3 A}{x^2 \left (b x^2+a\right )^{3/2}}dx}{3 a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {-\frac {2 A b x^2}{a}+3 B x+3 A}{x^2 \left (b x^2+a\right )^{3/2}}dx}{3 a}+\frac {a B-A b x}{3 a^2 \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2336 |
| \(\displaystyle \frac {\frac {3 a B-5 A b x}{a^2 \sqrt {a+b x^2}}-\frac {\int -\frac {3 (A+B x)}{x^2 \sqrt {b x^2+a}}dx}{a}}{3 a}+\frac {a B-A b x}{3 a^2 \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3 \int \frac {A+B x}{x^2 \sqrt {b x^2+a}}dx}{a}+\frac {3 a B-5 A b x}{a^2 \sqrt {a+b x^2}}}{3 a}+\frac {a B-A b x}{3 a^2 \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 534 |
| \(\displaystyle \frac {\frac {3 \left (B \int \frac {1}{x \sqrt {b x^2+a}}dx-\frac {A \sqrt {a+b x^2}}{a x}\right )}{a}+\frac {3 a B-5 A b x}{a^2 \sqrt {a+b x^2}}}{3 a}+\frac {a B-A b x}{3 a^2 \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {\frac {3 \left (\frac {1}{2} B \int \frac {1}{x^2 \sqrt {b x^2+a}}dx^2-\frac {A \sqrt {a+b x^2}}{a x}\right )}{a}+\frac {3 a B-5 A b x}{a^2 \sqrt {a+b x^2}}}{3 a}+\frac {a B-A b x}{3 a^2 \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {3 \left (\frac {B \int \frac {1}{\frac {x^4}{b}-\frac {a}{b}}d\sqrt {b x^2+a}}{b}-\frac {A \sqrt {a+b x^2}}{a x}\right )}{a}+\frac {3 a B-5 A b x}{a^2 \sqrt {a+b x^2}}}{3 a}+\frac {a B-A b x}{3 a^2 \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {3 a B-5 A b x}{a^2 \sqrt {a+b x^2}}+\frac {3 \left (-\frac {A \sqrt {a+b x^2}}{a x}-\frac {B \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {a}}\right )}{a}}{3 a}+\frac {a B-A b x}{3 a^2 \left (a+b x^2\right )^{3/2}}\) |
(a*B - A*b*x)/(3*a^2*(a + b*x^2)^(3/2)) + ((3*a*B - 5*A*b*x)/(a^2*Sqrt[a + b*x^2]) + (3*(-((A*Sqrt[a + b*x^2])/(a*x)) - (B*ArcTanh[Sqrt[a + b*x^2]/S qrt[a]])/Sqrt[a]))/a)/(3*a)
3.1.41.3.1 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] :> 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[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Coe ff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Pol ynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x) *((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) Int[x^m *(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*(Qx/x^m) + e*((2*p + 3)/x^m), x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && ILtQ[m, 0] && LtQ[p, -1] && IntegerQ[2*p]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-c)*x^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1))), x] + Simp[d Int[ x^(m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, m, p}, x] && ILtQ[m, 0] && GtQ[p, -1] && EqQ[m + 2*p + 3, 0]
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ {Q = PolynomialQuotient[(c*x)^m*Pq, a + b*x^2, x], f = Coeff[PolynomialRema inder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[(c*x) ^m*Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b*f*x)*((a + b*x^2)^(p + 1)/(2*a* b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1)*Ex pandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x], x]] /; F reeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]
Time = 3.44 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.17
| method | result | size |
| default | \(B \left (\frac {1}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {\frac {1}{a \sqrt {b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}}{a}\right )+A \left (-\frac {1}{a x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {4 b \left (\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}\right )}{a}\right )\) | \(122\) |
| risch | \(-\frac {A \sqrt {b \,x^{2}+a}}{a^{3} x}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right ) B}{a^{\frac {5}{2}}}-\frac {5 \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}\, A}{6 a^{3} \left (x -\frac {\sqrt {-a b}}{b}\right )}+\frac {7 \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}\, B}{12 a^{2} \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}-\frac {5 \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}\, A}{6 a^{3} \left (x +\frac {\sqrt {-a b}}{b}\right )}-\frac {7 \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}\, B}{12 a^{2} \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}+\frac {\sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}\, A}{12 a^{2} \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {\sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}\, B}{12 a^{2} b \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {\sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}\, A}{12 a^{2} \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {\sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}\, B}{12 a^{2} b \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}}\) | \(563\) |
B*(1/3/a/(b*x^2+a)^(3/2)+1/a*(1/a/(b*x^2+a)^(1/2)-1/a^(3/2)*ln((2*a+2*a^(1 /2)*(b*x^2+a)^(1/2))/x)))+A*(-1/a/x/(b*x^2+a)^(3/2)-4*b/a*(1/3*x/a/(b*x^2+ a)^(3/2)+2/3*x/a^2/(b*x^2+a)^(1/2)))
Time = 0.27 (sec) , antiderivative size = 264, normalized size of antiderivative = 2.54 \[ \int \frac {A+B x}{x^2 \left (a+b x^2\right )^{5/2}} \, dx=\left [\frac {3 \, {\left (B b^{2} x^{5} + 2 \, B a b x^{3} + B a^{2} x\right )} \sqrt {a} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (8 \, A b^{2} x^{4} - 3 \, B a b x^{3} + 12 \, A a b x^{2} - 4 \, B a^{2} x + 3 \, A a^{2}\right )} \sqrt {b x^{2} + a}}{6 \, {\left (a^{3} b^{2} x^{5} + 2 \, a^{4} b x^{3} + a^{5} x\right )}}, \frac {3 \, {\left (B b^{2} x^{5} + 2 \, B a b x^{3} + B a^{2} x\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) - {\left (8 \, A b^{2} x^{4} - 3 \, B a b x^{3} + 12 \, A a b x^{2} - 4 \, B a^{2} x + 3 \, A a^{2}\right )} \sqrt {b x^{2} + a}}{3 \, {\left (a^{3} b^{2} x^{5} + 2 \, a^{4} b x^{3} + a^{5} x\right )}}\right ] \]
[1/6*(3*(B*b^2*x^5 + 2*B*a*b*x^3 + B*a^2*x)*sqrt(a)*log(-(b*x^2 - 2*sqrt(b *x^2 + a)*sqrt(a) + 2*a)/x^2) - 2*(8*A*b^2*x^4 - 3*B*a*b*x^3 + 12*A*a*b*x^ 2 - 4*B*a^2*x + 3*A*a^2)*sqrt(b*x^2 + a))/(a^3*b^2*x^5 + 2*a^4*b*x^3 + a^5 *x), 1/3*(3*(B*b^2*x^5 + 2*B*a*b*x^3 + B*a^2*x)*sqrt(-a)*arctan(sqrt(-a)/s qrt(b*x^2 + a)) - (8*A*b^2*x^4 - 3*B*a*b*x^3 + 12*A*a*b*x^2 - 4*B*a^2*x + 3*A*a^2)*sqrt(b*x^2 + a))/(a^3*b^2*x^5 + 2*a^4*b*x^3 + a^5*x)]
Leaf count of result is larger than twice the leaf count of optimal. 910 vs. \(2 (88) = 176\).
Time = 6.91 (sec) , antiderivative size = 910, normalized size of antiderivative = 8.75 \[ \int \frac {A+B x}{x^2 \left (a+b x^2\right )^{5/2}} \, dx=A \left (- \frac {3 a^{2} b^{\frac {9}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{5} b^{4} + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{4}} - \frac {12 a b^{\frac {11}{2}} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{5} b^{4} + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{4}} - \frac {8 b^{\frac {13}{2}} x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{5} b^{4} + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{4}}\right ) + B \left (\frac {8 a^{7} \sqrt {1 + \frac {b x^{2}}{a}}}{6 a^{\frac {19}{2}} + 18 a^{\frac {17}{2}} b x^{2} + 18 a^{\frac {15}{2}} b^{2} x^{4} + 6 a^{\frac {13}{2}} b^{3} x^{6}} + \frac {3 a^{7} \log {\left (\frac {b x^{2}}{a} \right )}}{6 a^{\frac {19}{2}} + 18 a^{\frac {17}{2}} b x^{2} + 18 a^{\frac {15}{2}} b^{2} x^{4} + 6 a^{\frac {13}{2}} b^{3} x^{6}} - \frac {6 a^{7} \log {\left (\sqrt {1 + \frac {b x^{2}}{a}} + 1 \right )}}{6 a^{\frac {19}{2}} + 18 a^{\frac {17}{2}} b x^{2} + 18 a^{\frac {15}{2}} b^{2} x^{4} + 6 a^{\frac {13}{2}} b^{3} x^{6}} + \frac {14 a^{6} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}}}{6 a^{\frac {19}{2}} + 18 a^{\frac {17}{2}} b x^{2} + 18 a^{\frac {15}{2}} b^{2} x^{4} + 6 a^{\frac {13}{2}} b^{3} x^{6}} + \frac {9 a^{6} b x^{2} \log {\left (\frac {b x^{2}}{a} \right )}}{6 a^{\frac {19}{2}} + 18 a^{\frac {17}{2}} b x^{2} + 18 a^{\frac {15}{2}} b^{2} x^{4} + 6 a^{\frac {13}{2}} b^{3} x^{6}} - \frac {18 a^{6} b x^{2} \log {\left (\sqrt {1 + \frac {b x^{2}}{a}} + 1 \right )}}{6 a^{\frac {19}{2}} + 18 a^{\frac {17}{2}} b x^{2} + 18 a^{\frac {15}{2}} b^{2} x^{4} + 6 a^{\frac {13}{2}} b^{3} x^{6}} + \frac {6 a^{5} b^{2} x^{4} \sqrt {1 + \frac {b x^{2}}{a}}}{6 a^{\frac {19}{2}} + 18 a^{\frac {17}{2}} b x^{2} + 18 a^{\frac {15}{2}} b^{2} x^{4} + 6 a^{\frac {13}{2}} b^{3} x^{6}} + \frac {9 a^{5} b^{2} x^{4} \log {\left (\frac {b x^{2}}{a} \right )}}{6 a^{\frac {19}{2}} + 18 a^{\frac {17}{2}} b x^{2} + 18 a^{\frac {15}{2}} b^{2} x^{4} + 6 a^{\frac {13}{2}} b^{3} x^{6}} - \frac {18 a^{5} b^{2} x^{4} \log {\left (\sqrt {1 + \frac {b x^{2}}{a}} + 1 \right )}}{6 a^{\frac {19}{2}} + 18 a^{\frac {17}{2}} b x^{2} + 18 a^{\frac {15}{2}} b^{2} x^{4} + 6 a^{\frac {13}{2}} b^{3} x^{6}} + \frac {3 a^{4} b^{3} x^{6} \log {\left (\frac {b x^{2}}{a} \right )}}{6 a^{\frac {19}{2}} + 18 a^{\frac {17}{2}} b x^{2} + 18 a^{\frac {15}{2}} b^{2} x^{4} + 6 a^{\frac {13}{2}} b^{3} x^{6}} - \frac {6 a^{4} b^{3} x^{6} \log {\left (\sqrt {1 + \frac {b x^{2}}{a}} + 1 \right )}}{6 a^{\frac {19}{2}} + 18 a^{\frac {17}{2}} b x^{2} + 18 a^{\frac {15}{2}} b^{2} x^{4} + 6 a^{\frac {13}{2}} b^{3} x^{6}}\right ) \]
A*(-3*a**2*b**(9/2)*sqrt(a/(b*x**2) + 1)/(3*a**5*b**4 + 6*a**4*b**5*x**2 + 3*a**3*b**6*x**4) - 12*a*b**(11/2)*x**2*sqrt(a/(b*x**2) + 1)/(3*a**5*b**4 + 6*a**4*b**5*x**2 + 3*a**3*b**6*x**4) - 8*b**(13/2)*x**4*sqrt(a/(b*x**2) + 1)/(3*a**5*b**4 + 6*a**4*b**5*x**2 + 3*a**3*b**6*x**4)) + B*(8*a**7*sqr t(1 + b*x**2/a)/(6*a**(19/2) + 18*a**(17/2)*b*x**2 + 18*a**(15/2)*b**2*x** 4 + 6*a**(13/2)*b**3*x**6) + 3*a**7*log(b*x**2/a)/(6*a**(19/2) + 18*a**(17 /2)*b*x**2 + 18*a**(15/2)*b**2*x**4 + 6*a**(13/2)*b**3*x**6) - 6*a**7*log( sqrt(1 + b*x**2/a) + 1)/(6*a**(19/2) + 18*a**(17/2)*b*x**2 + 18*a**(15/2)* b**2*x**4 + 6*a**(13/2)*b**3*x**6) + 14*a**6*b*x**2*sqrt(1 + b*x**2/a)/(6* a**(19/2) + 18*a**(17/2)*b*x**2 + 18*a**(15/2)*b**2*x**4 + 6*a**(13/2)*b** 3*x**6) + 9*a**6*b*x**2*log(b*x**2/a)/(6*a**(19/2) + 18*a**(17/2)*b*x**2 + 18*a**(15/2)*b**2*x**4 + 6*a**(13/2)*b**3*x**6) - 18*a**6*b*x**2*log(sqrt (1 + b*x**2/a) + 1)/(6*a**(19/2) + 18*a**(17/2)*b*x**2 + 18*a**(15/2)*b**2 *x**4 + 6*a**(13/2)*b**3*x**6) + 6*a**5*b**2*x**4*sqrt(1 + b*x**2/a)/(6*a* *(19/2) + 18*a**(17/2)*b*x**2 + 18*a**(15/2)*b**2*x**4 + 6*a**(13/2)*b**3* x**6) + 9*a**5*b**2*x**4*log(b*x**2/a)/(6*a**(19/2) + 18*a**(17/2)*b*x**2 + 18*a**(15/2)*b**2*x**4 + 6*a**(13/2)*b**3*x**6) - 18*a**5*b**2*x**4*log( sqrt(1 + b*x**2/a) + 1)/(6*a**(19/2) + 18*a**(17/2)*b*x**2 + 18*a**(15/2)* b**2*x**4 + 6*a**(13/2)*b**3*x**6) + 3*a**4*b**3*x**6*log(b*x**2/a)/(6*a** (19/2) + 18*a**(17/2)*b*x**2 + 18*a**(15/2)*b**2*x**4 + 6*a**(13/2)*b**...
Time = 0.18 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.96 \[ \int \frac {A+B x}{x^2 \left (a+b x^2\right )^{5/2}} \, dx=-\frac {8 \, A b x}{3 \, \sqrt {b x^{2} + a} a^{3}} - \frac {4 \, A b x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2}} - \frac {B \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{a^{\frac {5}{2}}} + \frac {B}{\sqrt {b x^{2} + a} a^{2}} + \frac {B}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a} - \frac {A}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} a x} \]
-8/3*A*b*x/(sqrt(b*x^2 + a)*a^3) - 4/3*A*b*x/((b*x^2 + a)^(3/2)*a^2) - B*a rcsinh(a/(sqrt(a*b)*abs(x)))/a^(5/2) + B/(sqrt(b*x^2 + a)*a^2) + 1/3*B/((b *x^2 + a)^(3/2)*a) - A/((b*x^2 + a)^(3/2)*a*x)
Time = 0.30 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.14 \[ \int \frac {A+B x}{x^2 \left (a+b x^2\right )^{5/2}} \, dx=-\frac {{\left ({\left (\frac {5 \, A b^{2} x}{a^{3}} - \frac {3 \, B b}{a^{2}}\right )} x + \frac {6 \, A b}{a^{2}}\right )} x - \frac {4 \, B}{a}}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}}} + \frac {2 \, B \arctan \left (-\frac {\sqrt {b} x - \sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2}} + \frac {2 \, A \sqrt {b}}{{\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )} a^{2}} \]
-1/3*(((5*A*b^2*x/a^3 - 3*B*b/a^2)*x + 6*A*b/a^2)*x - 4*B/a)/(b*x^2 + a)^( 3/2) + 2*B*arctan(-(sqrt(b)*x - sqrt(b*x^2 + a))/sqrt(-a))/(sqrt(-a)*a^2) + 2*A*sqrt(b)/(((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)*a^2)
Time = 6.65 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.92 \[ \int \frac {A+B x}{x^2 \left (a+b x^2\right )^{5/2}} \, dx=\frac {\frac {B}{3\,a}+\frac {B\,\left (b\,x^2+a\right )}{a^2}}{{\left (b\,x^2+a\right )}^{3/2}}-\frac {B\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{a^{5/2}}+\frac {A\,a^2-8\,A\,{\left (b\,x^2+a\right )}^2+4\,A\,a\,\left (b\,x^2+a\right )}{3\,a^3\,x\,{\left (b\,x^2+a\right )}^{3/2}} \]