Integrand size = 20, antiderivative size = 129 \[ \int \frac {A+B x}{x^3 \left (a+c x^2\right )^{5/2}} \, dx=\frac {A+B x}{3 a x^2 \left (a+c x^2\right )^{3/2}}+\frac {5 A+4 B x}{3 a^2 x^2 \sqrt {a+c x^2}}-\frac {5 A \sqrt {a+c x^2}}{2 a^3 x^2}-\frac {8 B \sqrt {a+c x^2}}{3 a^3 x}+\frac {5 A c \text {arctanh}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{2 a^{7/2}} \]
1/3*(B*x+A)/a/x^2/(c*x^2+a)^(3/2)+5/2*A*c*arctanh((c*x^2+a)^(1/2)/a^(1/2)) /a^(7/2)+1/3*(4*B*x+5*A)/a^2/x^2/(c*x^2+a)^(1/2)-5/2*A*(c*x^2+a)^(1/2)/a^3 /x^2-8/3*B*(c*x^2+a)^(1/2)/a^3/x
Time = 0.42 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.79 \[ \int \frac {A+B x}{x^3 \left (a+c x^2\right )^{5/2}} \, dx=\frac {-3 a^2 (A+2 B x)-4 a c x^2 (5 A+6 B x)-c^2 x^4 (15 A+16 B x)}{6 a^3 x^2 \left (a+c x^2\right )^{3/2}}-\frac {5 A c \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+c x^2}}{\sqrt {a}}\right )}{a^{7/2}} \]
(-3*a^2*(A + 2*B*x) - 4*a*c*x^2*(5*A + 6*B*x) - c^2*x^4*(15*A + 16*B*x))/( 6*a^3*x^2*(a + c*x^2)^(3/2)) - (5*A*c*ArcTanh[(Sqrt[c]*x - Sqrt[a + c*x^2] )/Sqrt[a]])/a^(7/2)
Time = 0.47 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.07, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {532, 25, 2336, 27, 2338, 25, 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.
| \(\displaystyle \int \frac {A+B x}{x^3 \left (a+c x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 532 |
| \(\displaystyle -\frac {\int -\frac {-\frac {2 B c x^3}{a}-\frac {3 A c x^2}{a}+3 B x+3 A}{x^3 \left (c x^2+a\right )^{3/2}}dx}{3 a}-\frac {c (A+B x)}{3 a^2 \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {-\frac {2 B c x^3}{a}-\frac {3 A c x^2}{a}+3 B x+3 A}{x^3 \left (c x^2+a\right )^{3/2}}dx}{3 a}-\frac {c (A+B x)}{3 a^2 \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2336 |
| \(\displaystyle \frac {-\frac {\int -\frac {3 \left (-\frac {2 A c x^2}{a}+B x+A\right )}{x^3 \sqrt {c x^2+a}}dx}{a}-\frac {c (6 A+5 B x)}{a^2 \sqrt {a+c x^2}}}{3 a}-\frac {c (A+B x)}{3 a^2 \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3 \int \frac {-\frac {2 A c x^2}{a}+B x+A}{x^3 \sqrt {c x^2+a}}dx}{a}-\frac {c (6 A+5 B x)}{a^2 \sqrt {a+c x^2}}}{3 a}-\frac {c (A+B x)}{3 a^2 \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2338 |
| \(\displaystyle \frac {\frac {3 \left (-\frac {\int -\frac {2 a B-5 A c x}{x^2 \sqrt {c x^2+a}}dx}{2 a}-\frac {A \sqrt {a+c x^2}}{2 a x^2}\right )}{a}-\frac {c (6 A+5 B x)}{a^2 \sqrt {a+c x^2}}}{3 a}-\frac {c (A+B x)}{3 a^2 \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {3 \left (\frac {\int \frac {2 a B-5 A c x}{x^2 \sqrt {c x^2+a}}dx}{2 a}-\frac {A \sqrt {a+c x^2}}{2 a x^2}\right )}{a}-\frac {c (6 A+5 B x)}{a^2 \sqrt {a+c x^2}}}{3 a}-\frac {c (A+B x)}{3 a^2 \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 534 |
| \(\displaystyle \frac {\frac {3 \left (\frac {-5 A c \int \frac {1}{x \sqrt {c x^2+a}}dx-\frac {2 B \sqrt {a+c x^2}}{x}}{2 a}-\frac {A \sqrt {a+c x^2}}{2 a x^2}\right )}{a}-\frac {c (6 A+5 B x)}{a^2 \sqrt {a+c x^2}}}{3 a}-\frac {c (A+B x)}{3 a^2 \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {\frac {3 \left (\frac {-\frac {5}{2} A c \int \frac {1}{x^2 \sqrt {c x^2+a}}dx^2-\frac {2 B \sqrt {a+c x^2}}{x}}{2 a}-\frac {A \sqrt {a+c x^2}}{2 a x^2}\right )}{a}-\frac {c (6 A+5 B x)}{a^2 \sqrt {a+c x^2}}}{3 a}-\frac {c (A+B x)}{3 a^2 \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {3 \left (\frac {-5 A \int \frac {1}{\frac {x^4}{c}-\frac {a}{c}}d\sqrt {c x^2+a}-\frac {2 B \sqrt {a+c x^2}}{x}}{2 a}-\frac {A \sqrt {a+c x^2}}{2 a x^2}\right )}{a}-\frac {c (6 A+5 B x)}{a^2 \sqrt {a+c x^2}}}{3 a}-\frac {c (A+B x)}{3 a^2 \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {3 \left (\frac {\frac {5 A c \text {arctanh}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {2 B \sqrt {a+c x^2}}{x}}{2 a}-\frac {A \sqrt {a+c x^2}}{2 a x^2}\right )}{a}-\frac {c (6 A+5 B x)}{a^2 \sqrt {a+c x^2}}}{3 a}-\frac {c (A+B x)}{3 a^2 \left (a+c x^2\right )^{3/2}}\) |
-1/3*(c*(A + B*x))/(a^2*(a + c*x^2)^(3/2)) + (-((c*(6*A + 5*B*x))/(a^2*Sqr t[a + c*x^2])) + (3*(-1/2*(A*Sqrt[a + c*x^2])/(a*x^2) + ((-2*B*Sqrt[a + c* x^2])/x + (5*A*c*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/Sqrt[a])/(2*a)))/a)/(3* a)
3.4.81.3.1 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]
Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{ Q = PolynomialQuotient[Pq, c*x, x], R = PolynomialRemainder[Pq, c*x, x]}, S imp[R*(c*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] + Simp[1/(a*c*( m + 1)) Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*( m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && Lt Q[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])
Time = 0.18 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.13
| method | result | size |
| default | \(A \left (-\frac {1}{2 a \,x^{2} \left (c \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {5 c \left (\frac {1}{3 a \left (c \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {\frac {1}{a \sqrt {c \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}}{a}\right )}{2 a}\right )+B \left (-\frac {1}{a x \left (c \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {4 c \left (\frac {x}{3 a \left (c \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {c \,x^{2}+a}}\right )}{a}\right )\) | \(146\) |
| risch | \(-\frac {\sqrt {c \,x^{2}+a}\, \left (2 B x +A \right )}{2 a^{3} x^{2}}+\frac {5 c A \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{2}+a}}{x}\right )}{2 a^{\frac {7}{2}}}-\frac {13 c \sqrt {\left (x -\frac {\sqrt {-a c}}{c}\right )^{2} c +2 \sqrt {-a c}\, \left (x -\frac {\sqrt {-a c}}{c}\right )}\, A}{12 a^{3} \sqrt {-a c}\, \left (x -\frac {\sqrt {-a c}}{c}\right )}-\frac {5 \sqrt {\left (x -\frac {\sqrt {-a c}}{c}\right )^{2} c +2 \sqrt {-a c}\, \left (x -\frac {\sqrt {-a c}}{c}\right )}\, B}{6 a^{3} \left (x -\frac {\sqrt {-a c}}{c}\right )}+\frac {13 c \sqrt {\left (x +\frac {\sqrt {-a c}}{c}\right )^{2} c -2 \sqrt {-a c}\, \left (x +\frac {\sqrt {-a c}}{c}\right )}\, A}{12 a^{3} \sqrt {-a c}\, \left (x +\frac {\sqrt {-a c}}{c}\right )}-\frac {5 \sqrt {\left (x +\frac {\sqrt {-a c}}{c}\right )^{2} c -2 \sqrt {-a c}\, \left (x +\frac {\sqrt {-a c}}{c}\right )}\, B}{6 a^{3} \left (x +\frac {\sqrt {-a c}}{c}\right )}+\frac {\sqrt {\left (x +\frac {\sqrt {-a c}}{c}\right )^{2} c -2 \sqrt {-a c}\, \left (x +\frac {\sqrt {-a c}}{c}\right )}\, A}{12 a^{3} \left (x +\frac {\sqrt {-a c}}{c}\right )^{2}}+\frac {\sqrt {\left (x +\frac {\sqrt {-a c}}{c}\right )^{2} c -2 \sqrt {-a c}\, \left (x +\frac {\sqrt {-a c}}{c}\right )}\, B}{12 a^{2} \sqrt {-a c}\, \left (x +\frac {\sqrt {-a c}}{c}\right )^{2}}+\frac {\sqrt {\left (x -\frac {\sqrt {-a c}}{c}\right )^{2} c +2 \sqrt {-a c}\, \left (x -\frac {\sqrt {-a c}}{c}\right )}\, A}{12 a^{3} \left (x -\frac {\sqrt {-a c}}{c}\right )^{2}}-\frac {\sqrt {\left (x -\frac {\sqrt {-a c}}{c}\right )^{2} c +2 \sqrt {-a c}\, \left (x -\frac {\sqrt {-a c}}{c}\right )}\, B}{12 a^{2} \sqrt {-a c}\, \left (x -\frac {\sqrt {-a c}}{c}\right )^{2}}\) | \(565\) |
A*(-1/2/a/x^2/(c*x^2+a)^(3/2)-5/2*c/a*(1/3/a/(c*x^2+a)^(3/2)+1/a*(1/a/(c*x ^2+a)^(1/2)-1/a^(3/2)*ln((2*a+2*a^(1/2)*(c*x^2+a)^(1/2))/x))))+B*(-1/a/x/( c*x^2+a)^(3/2)-4*c/a*(1/3*x/a/(c*x^2+a)^(3/2)+2/3/a^2*x/(c*x^2+a)^(1/2)))
Time = 0.31 (sec) , antiderivative size = 307, normalized size of antiderivative = 2.38 \[ \int \frac {A+B x}{x^3 \left (a+c x^2\right )^{5/2}} \, dx=\left [\frac {15 \, {\left (A c^{3} x^{6} + 2 \, A a c^{2} x^{4} + A a^{2} c x^{2}\right )} \sqrt {a} \log \left (-\frac {c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (16 \, B a c^{2} x^{5} + 15 \, A a c^{2} x^{4} + 24 \, B a^{2} c x^{3} + 20 \, A a^{2} c x^{2} + 6 \, B a^{3} x + 3 \, A a^{3}\right )} \sqrt {c x^{2} + a}}{12 \, {\left (a^{4} c^{2} x^{6} + 2 \, a^{5} c x^{4} + a^{6} x^{2}\right )}}, -\frac {15 \, {\left (A c^{3} x^{6} + 2 \, A a c^{2} x^{4} + A a^{2} c x^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) + {\left (16 \, B a c^{2} x^{5} + 15 \, A a c^{2} x^{4} + 24 \, B a^{2} c x^{3} + 20 \, A a^{2} c x^{2} + 6 \, B a^{3} x + 3 \, A a^{3}\right )} \sqrt {c x^{2} + a}}{6 \, {\left (a^{4} c^{2} x^{6} + 2 \, a^{5} c x^{4} + a^{6} x^{2}\right )}}\right ] \]
[1/12*(15*(A*c^3*x^6 + 2*A*a*c^2*x^4 + A*a^2*c*x^2)*sqrt(a)*log(-(c*x^2 + 2*sqrt(c*x^2 + a)*sqrt(a) + 2*a)/x^2) - 2*(16*B*a*c^2*x^5 + 15*A*a*c^2*x^4 + 24*B*a^2*c*x^3 + 20*A*a^2*c*x^2 + 6*B*a^3*x + 3*A*a^3)*sqrt(c*x^2 + a)) /(a^4*c^2*x^6 + 2*a^5*c*x^4 + a^6*x^2), -1/6*(15*(A*c^3*x^6 + 2*A*a*c^2*x^ 4 + A*a^2*c*x^2)*sqrt(-a)*arctan(sqrt(-a)/sqrt(c*x^2 + a)) + (16*B*a*c^2*x ^5 + 15*A*a*c^2*x^4 + 24*B*a^2*c*x^3 + 20*A*a^2*c*x^2 + 6*B*a^3*x + 3*A*a^ 3)*sqrt(c*x^2 + a))/(a^4*c^2*x^6 + 2*a^5*c*x^4 + a^6*x^2)]
Leaf count of result is larger than twice the leaf count of optimal. 1034 vs. \(2 (122) = 244\).
Time = 8.22 (sec) , antiderivative size = 1034, normalized size of antiderivative = 8.02 \[ \int \frac {A+B x}{x^3 \left (a+c x^2\right )^{5/2}} \, dx=A \left (- \frac {6 a^{17} \sqrt {1 + \frac {c x^{2}}{a}}}{12 a^{\frac {39}{2}} x^{2} + 36 a^{\frac {37}{2}} c x^{4} + 36 a^{\frac {35}{2}} c^{2} x^{6} + 12 a^{\frac {33}{2}} c^{3} x^{8}} - \frac {46 a^{16} c x^{2} \sqrt {1 + \frac {c x^{2}}{a}}}{12 a^{\frac {39}{2}} x^{2} + 36 a^{\frac {37}{2}} c x^{4} + 36 a^{\frac {35}{2}} c^{2} x^{6} + 12 a^{\frac {33}{2}} c^{3} x^{8}} - \frac {15 a^{16} c x^{2} \log {\left (\frac {c x^{2}}{a} \right )}}{12 a^{\frac {39}{2}} x^{2} + 36 a^{\frac {37}{2}} c x^{4} + 36 a^{\frac {35}{2}} c^{2} x^{6} + 12 a^{\frac {33}{2}} c^{3} x^{8}} + \frac {30 a^{16} c x^{2} \log {\left (\sqrt {1 + \frac {c x^{2}}{a}} + 1 \right )}}{12 a^{\frac {39}{2}} x^{2} + 36 a^{\frac {37}{2}} c x^{4} + 36 a^{\frac {35}{2}} c^{2} x^{6} + 12 a^{\frac {33}{2}} c^{3} x^{8}} - \frac {70 a^{15} c^{2} x^{4} \sqrt {1 + \frac {c x^{2}}{a}}}{12 a^{\frac {39}{2}} x^{2} + 36 a^{\frac {37}{2}} c x^{4} + 36 a^{\frac {35}{2}} c^{2} x^{6} + 12 a^{\frac {33}{2}} c^{3} x^{8}} - \frac {45 a^{15} c^{2} x^{4} \log {\left (\frac {c x^{2}}{a} \right )}}{12 a^{\frac {39}{2}} x^{2} + 36 a^{\frac {37}{2}} c x^{4} + 36 a^{\frac {35}{2}} c^{2} x^{6} + 12 a^{\frac {33}{2}} c^{3} x^{8}} + \frac {90 a^{15} c^{2} x^{4} \log {\left (\sqrt {1 + \frac {c x^{2}}{a}} + 1 \right )}}{12 a^{\frac {39}{2}} x^{2} + 36 a^{\frac {37}{2}} c x^{4} + 36 a^{\frac {35}{2}} c^{2} x^{6} + 12 a^{\frac {33}{2}} c^{3} x^{8}} - \frac {30 a^{14} c^{3} x^{6} \sqrt {1 + \frac {c x^{2}}{a}}}{12 a^{\frac {39}{2}} x^{2} + 36 a^{\frac {37}{2}} c x^{4} + 36 a^{\frac {35}{2}} c^{2} x^{6} + 12 a^{\frac {33}{2}} c^{3} x^{8}} - \frac {45 a^{14} c^{3} x^{6} \log {\left (\frac {c x^{2}}{a} \right )}}{12 a^{\frac {39}{2}} x^{2} + 36 a^{\frac {37}{2}} c x^{4} + 36 a^{\frac {35}{2}} c^{2} x^{6} + 12 a^{\frac {33}{2}} c^{3} x^{8}} + \frac {90 a^{14} c^{3} x^{6} \log {\left (\sqrt {1 + \frac {c x^{2}}{a}} + 1 \right )}}{12 a^{\frac {39}{2}} x^{2} + 36 a^{\frac {37}{2}} c x^{4} + 36 a^{\frac {35}{2}} c^{2} x^{6} + 12 a^{\frac {33}{2}} c^{3} x^{8}} - \frac {15 a^{13} c^{4} x^{8} \log {\left (\frac {c x^{2}}{a} \right )}}{12 a^{\frac {39}{2}} x^{2} + 36 a^{\frac {37}{2}} c x^{4} + 36 a^{\frac {35}{2}} c^{2} x^{6} + 12 a^{\frac {33}{2}} c^{3} x^{8}} + \frac {30 a^{13} c^{4} x^{8} \log {\left (\sqrt {1 + \frac {c x^{2}}{a}} + 1 \right )}}{12 a^{\frac {39}{2}} x^{2} + 36 a^{\frac {37}{2}} c x^{4} + 36 a^{\frac {35}{2}} c^{2} x^{6} + 12 a^{\frac {33}{2}} c^{3} x^{8}}\right ) + B \left (- \frac {3 a^{2} c^{\frac {9}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{3 a^{5} c^{4} + 6 a^{4} c^{5} x^{2} + 3 a^{3} c^{6} x^{4}} - \frac {12 a c^{\frac {11}{2}} x^{2} \sqrt {\frac {a}{c x^{2}} + 1}}{3 a^{5} c^{4} + 6 a^{4} c^{5} x^{2} + 3 a^{3} c^{6} x^{4}} - \frac {8 c^{\frac {13}{2}} x^{4} \sqrt {\frac {a}{c x^{2}} + 1}}{3 a^{5} c^{4} + 6 a^{4} c^{5} x^{2} + 3 a^{3} c^{6} x^{4}}\right ) \]
A*(-6*a**17*sqrt(1 + c*x**2/a)/(12*a**(39/2)*x**2 + 36*a**(37/2)*c*x**4 + 36*a**(35/2)*c**2*x**6 + 12*a**(33/2)*c**3*x**8) - 46*a**16*c*x**2*sqrt(1 + c*x**2/a)/(12*a**(39/2)*x**2 + 36*a**(37/2)*c*x**4 + 36*a**(35/2)*c**2*x **6 + 12*a**(33/2)*c**3*x**8) - 15*a**16*c*x**2*log(c*x**2/a)/(12*a**(39/2 )*x**2 + 36*a**(37/2)*c*x**4 + 36*a**(35/2)*c**2*x**6 + 12*a**(33/2)*c**3* x**8) + 30*a**16*c*x**2*log(sqrt(1 + c*x**2/a) + 1)/(12*a**(39/2)*x**2 + 3 6*a**(37/2)*c*x**4 + 36*a**(35/2)*c**2*x**6 + 12*a**(33/2)*c**3*x**8) - 70 *a**15*c**2*x**4*sqrt(1 + c*x**2/a)/(12*a**(39/2)*x**2 + 36*a**(37/2)*c*x* *4 + 36*a**(35/2)*c**2*x**6 + 12*a**(33/2)*c**3*x**8) - 45*a**15*c**2*x**4 *log(c*x**2/a)/(12*a**(39/2)*x**2 + 36*a**(37/2)*c*x**4 + 36*a**(35/2)*c** 2*x**6 + 12*a**(33/2)*c**3*x**8) + 90*a**15*c**2*x**4*log(sqrt(1 + c*x**2/ a) + 1)/(12*a**(39/2)*x**2 + 36*a**(37/2)*c*x**4 + 36*a**(35/2)*c**2*x**6 + 12*a**(33/2)*c**3*x**8) - 30*a**14*c**3*x**6*sqrt(1 + c*x**2/a)/(12*a**( 39/2)*x**2 + 36*a**(37/2)*c*x**4 + 36*a**(35/2)*c**2*x**6 + 12*a**(33/2)*c **3*x**8) - 45*a**14*c**3*x**6*log(c*x**2/a)/(12*a**(39/2)*x**2 + 36*a**(3 7/2)*c*x**4 + 36*a**(35/2)*c**2*x**6 + 12*a**(33/2)*c**3*x**8) + 90*a**14* c**3*x**6*log(sqrt(1 + c*x**2/a) + 1)/(12*a**(39/2)*x**2 + 36*a**(37/2)*c* x**4 + 36*a**(35/2)*c**2*x**6 + 12*a**(33/2)*c**3*x**8) - 15*a**13*c**4*x* *8*log(c*x**2/a)/(12*a**(39/2)*x**2 + 36*a**(37/2)*c*x**4 + 36*a**(35/2)*c **2*x**6 + 12*a**(33/2)*c**3*x**8) + 30*a**13*c**4*x**8*log(sqrt(1 + c*...
Time = 0.22 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.95 \[ \int \frac {A+B x}{x^3 \left (a+c x^2\right )^{5/2}} \, dx=-\frac {8 \, B c x}{3 \, \sqrt {c x^{2} + a} a^{3}} - \frac {4 \, B c x}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a^{2}} + \frac {5 \, A c \operatorname {arsinh}\left (\frac {a}{\sqrt {a c} {\left | x \right |}}\right )}{2 \, a^{\frac {7}{2}}} - \frac {5 \, A c}{2 \, \sqrt {c x^{2} + a} a^{3}} - \frac {5 \, A c}{6 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a^{2}} - \frac {B}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} a x} - \frac {A}{2 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a x^{2}} \]
-8/3*B*c*x/(sqrt(c*x^2 + a)*a^3) - 4/3*B*c*x/((c*x^2 + a)^(3/2)*a^2) + 5/2 *A*c*arcsinh(a/(sqrt(a*c)*abs(x)))/a^(7/2) - 5/2*A*c/(sqrt(c*x^2 + a)*a^3) - 5/6*A*c/((c*x^2 + a)^(3/2)*a^2) - B/((c*x^2 + a)^(3/2)*a*x) - 1/2*A/((c *x^2 + a)^(3/2)*a*x^2)
Time = 0.32 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.53 \[ \int \frac {A+B x}{x^3 \left (a+c x^2\right )^{5/2}} \, dx=-\frac {{\left ({\left (\frac {5 \, B c^{2} x}{a^{3}} + \frac {6 \, A c^{2}}{a^{3}}\right )} x + \frac {6 \, B c}{a^{2}}\right )} x + \frac {7 \, A c}{a^{2}}}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}}} - \frac {5 \, A c \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{3}} + \frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} A c + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} B a \sqrt {c} + {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} A a c - 2 \, B a^{2} \sqrt {c}}{{\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} - a\right )}^{2} a^{3}} \]
-1/3*(((5*B*c^2*x/a^3 + 6*A*c^2/a^3)*x + 6*B*c/a^2)*x + 7*A*c/a^2)/(c*x^2 + a)^(3/2) - 5*A*c*arctan(-(sqrt(c)*x - sqrt(c*x^2 + a))/sqrt(-a))/(sqrt(- a)*a^3) + ((sqrt(c)*x - sqrt(c*x^2 + a))^3*A*c + 2*(sqrt(c)*x - sqrt(c*x^2 + a))^2*B*a*sqrt(c) + (sqrt(c)*x - sqrt(c*x^2 + a))*A*a*c - 2*B*a^2*sqrt( c))/(((sqrt(c)*x - sqrt(c*x^2 + a))^2 - a)^2*a^3)
Time = 10.54 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.95 \[ \int \frac {A+B x}{x^3 \left (a+c x^2\right )^{5/2}} \, dx=\frac {B\,a^2-8\,B\,{\left (c\,x^2+a\right )}^2+4\,B\,a\,\left (c\,x^2+a\right )}{3\,a^3\,x\,{\left (c\,x^2+a\right )}^{3/2}}-\frac {10\,A\,c}{3\,a^2\,{\left (c\,x^2+a\right )}^{3/2}}-\frac {A}{2\,a\,x^2\,{\left (c\,x^2+a\right )}^{3/2}}+\frac {5\,A\,c\,\mathrm {atanh}\left (\frac {\sqrt {c\,x^2+a}}{\sqrt {a}}\right )}{2\,a^{7/2}}-\frac {5\,A\,c^2\,x^2}{2\,a^3\,{\left (c\,x^2+a\right )}^{3/2}} \]