Integrand size = 20, antiderivative size = 122 \[ \int \frac {A+B x}{x^5 \sqrt {a+c x^2}} \, dx=-\frac {A \sqrt {a+c x^2}}{4 a x^4}-\frac {B \sqrt {a+c x^2}}{3 a x^3}+\frac {3 A c \sqrt {a+c x^2}}{8 a^2 x^2}+\frac {2 B c \sqrt {a+c x^2}}{3 a^2 x}-\frac {3 A c^2 \text {arctanh}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{8 a^{5/2}} \]
-3/8*A*c^2*arctanh((c*x^2+a)^(1/2)/a^(1/2))/a^(5/2)-1/4*A*(c*x^2+a)^(1/2)/ a/x^4-1/3*B*(c*x^2+a)^(1/2)/a/x^3+3/8*A*c*(c*x^2+a)^(1/2)/a^2/x^2+2/3*B*c* (c*x^2+a)^(1/2)/a^2/x
Time = 0.29 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.72 \[ \int \frac {A+B x}{x^5 \sqrt {a+c x^2}} \, dx=\frac {\sqrt {a+c x^2} \left (-2 a (3 A+4 B x)+c x^2 (9 A+16 B x)\right )}{24 a^2 x^4}+\frac {3 A c^2 \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+c x^2}}{\sqrt {a}}\right )}{4 a^{5/2}} \]
(Sqrt[a + c*x^2]*(-2*a*(3*A + 4*B*x) + c*x^2*(9*A + 16*B*x)))/(24*a^2*x^4) + (3*A*c^2*ArcTanh[(Sqrt[c]*x - Sqrt[a + c*x^2])/Sqrt[a]])/(4*a^(5/2))
Time = 0.27 (sec) , antiderivative size = 130, 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 = {539, 25, 539, 27, 539, 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.
\(\displaystyle \int \frac {A+B x}{x^5 \sqrt {a+c x^2}} \, dx\) |
\(\Big \downarrow \) 539 |
\(\displaystyle -\frac {\int -\frac {4 a B-3 A c x}{x^4 \sqrt {c x^2+a}}dx}{4 a}-\frac {A \sqrt {a+c x^2}}{4 a x^4}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {4 a B-3 A c x}{x^4 \sqrt {c x^2+a}}dx}{4 a}-\frac {A \sqrt {a+c x^2}}{4 a x^4}\) |
\(\Big \downarrow \) 539 |
\(\displaystyle \frac {-\frac {\int \frac {a c (9 A+8 B x)}{x^3 \sqrt {c x^2+a}}dx}{3 a}-\frac {4 B \sqrt {a+c x^2}}{3 x^3}}{4 a}-\frac {A \sqrt {a+c x^2}}{4 a x^4}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {-\frac {1}{3} c \int \frac {9 A+8 B x}{x^3 \sqrt {c x^2+a}}dx-\frac {4 B \sqrt {a+c x^2}}{3 x^3}}{4 a}-\frac {A \sqrt {a+c x^2}}{4 a x^4}\) |
\(\Big \downarrow \) 539 |
\(\displaystyle \frac {-\frac {1}{3} c \left (-\frac {\int -\frac {16 a B-9 A c x}{x^2 \sqrt {c x^2+a}}dx}{2 a}-\frac {9 A \sqrt {a+c x^2}}{2 a x^2}\right )-\frac {4 B \sqrt {a+c x^2}}{3 x^3}}{4 a}-\frac {A \sqrt {a+c x^2}}{4 a x^4}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {-\frac {1}{3} c \left (\frac {\int \frac {16 a B-9 A c x}{x^2 \sqrt {c x^2+a}}dx}{2 a}-\frac {9 A \sqrt {a+c x^2}}{2 a x^2}\right )-\frac {4 B \sqrt {a+c x^2}}{3 x^3}}{4 a}-\frac {A \sqrt {a+c x^2}}{4 a x^4}\) |
\(\Big \downarrow \) 534 |
\(\displaystyle \frac {-\frac {1}{3} c \left (\frac {-9 A c \int \frac {1}{x \sqrt {c x^2+a}}dx-\frac {16 B \sqrt {a+c x^2}}{x}}{2 a}-\frac {9 A \sqrt {a+c x^2}}{2 a x^2}\right )-\frac {4 B \sqrt {a+c x^2}}{3 x^3}}{4 a}-\frac {A \sqrt {a+c x^2}}{4 a x^4}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {-\frac {1}{3} c \left (\frac {-\frac {9}{2} A c \int \frac {1}{x^2 \sqrt {c x^2+a}}dx^2-\frac {16 B \sqrt {a+c x^2}}{x}}{2 a}-\frac {9 A \sqrt {a+c x^2}}{2 a x^2}\right )-\frac {4 B \sqrt {a+c x^2}}{3 x^3}}{4 a}-\frac {A \sqrt {a+c x^2}}{4 a x^4}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {-\frac {1}{3} c \left (\frac {-9 A \int \frac {1}{\frac {x^4}{c}-\frac {a}{c}}d\sqrt {c x^2+a}-\frac {16 B \sqrt {a+c x^2}}{x}}{2 a}-\frac {9 A \sqrt {a+c x^2}}{2 a x^2}\right )-\frac {4 B \sqrt {a+c x^2}}{3 x^3}}{4 a}-\frac {A \sqrt {a+c x^2}}{4 a x^4}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {-\frac {1}{3} c \left (\frac {\frac {9 A c \text {arctanh}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {16 B \sqrt {a+c x^2}}{x}}{2 a}-\frac {9 A \sqrt {a+c x^2}}{2 a x^2}\right )-\frac {4 B \sqrt {a+c x^2}}{3 x^3}}{4 a}-\frac {A \sqrt {a+c x^2}}{4 a x^4}\) |
-1/4*(A*Sqrt[a + c*x^2])/(a*x^4) + ((-4*B*Sqrt[a + c*x^2])/(3*x^3) - (c*(( -9*A*Sqrt[a + c*x^2])/(2*a*x^2) + ((-16*B*Sqrt[a + c*x^2])/x + (9*A*c*ArcT anh[Sqrt[a + c*x^2]/Sqrt[a]])/Sqrt[a])/(2*a)))/3)/(4*a)
3.4.63.3.1 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_))*((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[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))), x] + Simp[1/(a*(m + 1)) Int[x^(m + 1)*(a + b*x^2)^p*(a*d*(m + 1) - b*c*(m + 2*p + 3)*x), x], x] /; FreeQ[{a, b, c, d, p}, x] && ILtQ[m, -1] && GtQ[p, -1] && IntegerQ[2*p]
Time = 0.14 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.61
method | result | size |
risch | \(-\frac {\sqrt {c \,x^{2}+a}\, \left (-16 B c \,x^{3}-9 A c \,x^{2}+8 a B x +6 a A \right )}{24 a^{2} x^{4}}-\frac {3 A \,c^{2} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{2}+a}}{x}\right )}{8 a^{\frac {5}{2}}}\) | \(75\) |
default | \(A \left (-\frac {\sqrt {c \,x^{2}+a}}{4 a \,x^{4}}-\frac {3 c \left (-\frac {\sqrt {c \,x^{2}+a}}{2 a \,x^{2}}+\frac {c \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{2}+a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{4 a}\right )+B \left (-\frac {\sqrt {c \,x^{2}+a}}{3 a \,x^{3}}+\frac {2 c \sqrt {c \,x^{2}+a}}{3 a^{2} x}\right )\) | \(113\) |
-1/24*(c*x^2+a)^(1/2)*(-16*B*c*x^3-9*A*c*x^2+8*B*a*x+6*A*a)/a^2/x^4-3/8*A* c^2/a^(5/2)*ln((2*a+2*a^(1/2)*(c*x^2+a)^(1/2))/x)
Time = 0.33 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.40 \[ \int \frac {A+B x}{x^5 \sqrt {a+c x^2}} \, dx=\left [\frac {9 \, A \sqrt {a} c^{2} x^{4} \log \left (-\frac {c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (16 \, B a c x^{3} + 9 \, A a c x^{2} - 8 \, B a^{2} x - 6 \, A a^{2}\right )} \sqrt {c x^{2} + a}}{48 \, a^{3} x^{4}}, \frac {9 \, A \sqrt {-a} c^{2} x^{4} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) + {\left (16 \, B a c x^{3} + 9 \, A a c x^{2} - 8 \, B a^{2} x - 6 \, A a^{2}\right )} \sqrt {c x^{2} + a}}{24 \, a^{3} x^{4}}\right ] \]
[1/48*(9*A*sqrt(a)*c^2*x^4*log(-(c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(a) + 2*a)/ x^2) + 2*(16*B*a*c*x^3 + 9*A*a*c*x^2 - 8*B*a^2*x - 6*A*a^2)*sqrt(c*x^2 + a ))/(a^3*x^4), 1/24*(9*A*sqrt(-a)*c^2*x^4*arctan(sqrt(-a)/sqrt(c*x^2 + a)) + (16*B*a*c*x^3 + 9*A*a*c*x^2 - 8*B*a^2*x - 6*A*a^2)*sqrt(c*x^2 + a))/(a^3 *x^4)]
Time = 3.08 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.25 \[ \int \frac {A+B x}{x^5 \sqrt {a+c x^2}} \, dx=- \frac {A}{4 \sqrt {c} x^{5} \sqrt {\frac {a}{c x^{2}} + 1}} + \frac {A \sqrt {c}}{8 a x^{3} \sqrt {\frac {a}{c x^{2}} + 1}} + \frac {3 A c^{\frac {3}{2}}}{8 a^{2} x \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {3 A c^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {c} x} \right )}}{8 a^{\frac {5}{2}}} - \frac {B \sqrt {c} \sqrt {\frac {a}{c x^{2}} + 1}}{3 a x^{2}} + \frac {2 B c^{\frac {3}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{3 a^{2}} \]
-A/(4*sqrt(c)*x**5*sqrt(a/(c*x**2) + 1)) + A*sqrt(c)/(8*a*x**3*sqrt(a/(c*x **2) + 1)) + 3*A*c**(3/2)/(8*a**2*x*sqrt(a/(c*x**2) + 1)) - 3*A*c**2*asinh (sqrt(a)/(sqrt(c)*x))/(8*a**(5/2)) - B*sqrt(c)*sqrt(a/(c*x**2) + 1)/(3*a*x **2) + 2*B*c**(3/2)*sqrt(a/(c*x**2) + 1)/(3*a**2)
Time = 0.20 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.79 \[ \int \frac {A+B x}{x^5 \sqrt {a+c x^2}} \, dx=-\frac {3 \, A c^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a c} {\left | x \right |}}\right )}{8 \, a^{\frac {5}{2}}} + \frac {2 \, \sqrt {c x^{2} + a} B c}{3 \, a^{2} x} + \frac {3 \, \sqrt {c x^{2} + a} A c}{8 \, a^{2} x^{2}} - \frac {\sqrt {c x^{2} + a} B}{3 \, a x^{3}} - \frac {\sqrt {c x^{2} + a} A}{4 \, a x^{4}} \]
-3/8*A*c^2*arcsinh(a/(sqrt(a*c)*abs(x)))/a^(5/2) + 2/3*sqrt(c*x^2 + a)*B*c /(a^2*x) + 3/8*sqrt(c*x^2 + a)*A*c/(a^2*x^2) - 1/3*sqrt(c*x^2 + a)*B/(a*x^ 3) - 1/4*sqrt(c*x^2 + a)*A/(a*x^4)
Leaf count of result is larger than twice the leaf count of optimal. 241 vs. \(2 (98) = 196\).
Time = 0.28 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.98 \[ \int \frac {A+B x}{x^5 \sqrt {a+c x^2}} \, dx=\frac {3 \, A c^{2} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + a}}{\sqrt {-a}}\right )}{4 \, \sqrt {-a} a^{2}} - \frac {9 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{7} A c^{2} - 33 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} A a c^{2} - 48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} B a^{2} c^{\frac {3}{2}} - 33 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} A a^{2} c^{2} + 64 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} B a^{3} c^{\frac {3}{2}} + 9 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} A a^{3} c^{2} - 16 \, B a^{4} c^{\frac {3}{2}}}{12 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} - a\right )}^{4} a^{2}} \]
3/4*A*c^2*arctan(-(sqrt(c)*x - sqrt(c*x^2 + a))/sqrt(-a))/(sqrt(-a)*a^2) - 1/12*(9*(sqrt(c)*x - sqrt(c*x^2 + a))^7*A*c^2 - 33*(sqrt(c)*x - sqrt(c*x^ 2 + a))^5*A*a*c^2 - 48*(sqrt(c)*x - sqrt(c*x^2 + a))^4*B*a^2*c^(3/2) - 33* (sqrt(c)*x - sqrt(c*x^2 + a))^3*A*a^2*c^2 + 64*(sqrt(c)*x - sqrt(c*x^2 + a ))^2*B*a^3*c^(3/2) + 9*(sqrt(c)*x - sqrt(c*x^2 + a))*A*a^3*c^2 - 16*B*a^4* c^(3/2))/(((sqrt(c)*x - sqrt(c*x^2 + a))^2 - a)^4*a^2)
Time = 10.54 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.70 \[ \int \frac {A+B x}{x^5 \sqrt {a+c x^2}} \, dx=\frac {3\,A\,{\left (c\,x^2+a\right )}^{3/2}}{8\,a^2\,x^4}-\frac {5\,A\,\sqrt {c\,x^2+a}}{8\,a\,x^4}-\frac {3\,A\,c^2\,\mathrm {atanh}\left (\frac {\sqrt {c\,x^2+a}}{\sqrt {a}}\right )}{8\,a^{5/2}}-\frac {B\,\sqrt {c\,x^2+a}\,\left (a-2\,c\,x^2\right )}{3\,a^2\,x^3} \]