Integrand size = 22, antiderivative size = 90 \[ \int \frac {A+B x^2}{x^5 \sqrt {a+b x^2}} \, dx=-\frac {A \sqrt {a+b x^2}}{4 a x^4}+\frac {(3 A b-4 a B) \sqrt {a+b x^2}}{8 a^2 x^2}-\frac {b (3 A b-4 a B) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{8 a^{5/2}} \] Output:
-1/4*A*(b*x^2+a)^(1/2)/a/x^4+1/8*(3*A*b-4*B*a)*(b*x^2+a)^(1/2)/a^2/x^2-1/8 *b*(3*A*b-4*B*a)*arctanh((b*x^2+a)^(1/2)/a^(1/2))/a^(5/2)
Time = 0.14 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.87 \[ \int \frac {A+B x^2}{x^5 \sqrt {a+b x^2}} \, dx=\frac {\sqrt {a+b x^2} \left (-2 a A+3 A b x^2-4 a B x^2\right )}{8 a^2 x^4}+\frac {b (-3 A b+4 a B) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{8 a^{5/2}} \] Input:
Integrate[(A + B*x^2)/(x^5*Sqrt[a + b*x^2]),x]
Output:
(Sqrt[a + b*x^2]*(-2*a*A + 3*A*b*x^2 - 4*a*B*x^2))/(8*a^2*x^4) + (b*(-3*A* b + 4*a*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(8*a^(5/2))
Time = 0.20 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {354, 87, 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.
\(\displaystyle \int \frac {A+B x^2}{x^5 \sqrt {a+b x^2}} \, dx\) |
\(\Big \downarrow \) 354 |
\(\displaystyle \frac {1}{2} \int \frac {B x^2+A}{x^6 \sqrt {b x^2+a}}dx^2\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {1}{2} \left (-\frac {(3 A b-4 a B) \int \frac {1}{x^4 \sqrt {b x^2+a}}dx^2}{4 a}-\frac {A \sqrt {a+b x^2}}{2 a x^4}\right )\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {1}{2} \left (-\frac {(3 A b-4 a B) \left (-\frac {b \int \frac {1}{x^2 \sqrt {b x^2+a}}dx^2}{2 a}-\frac {\sqrt {a+b x^2}}{a x^2}\right )}{4 a}-\frac {A \sqrt {a+b x^2}}{2 a x^4}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{2} \left (-\frac {(3 A b-4 a B) \left (-\frac {\int \frac {1}{\frac {x^4}{b}-\frac {a}{b}}d\sqrt {b x^2+a}}{a}-\frac {\sqrt {a+b x^2}}{a x^2}\right )}{4 a}-\frac {A \sqrt {a+b x^2}}{2 a x^4}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{2} \left (-\frac {(3 A b-4 a B) \left (\frac {b \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {\sqrt {a+b x^2}}{a x^2}\right )}{4 a}-\frac {A \sqrt {a+b x^2}}{2 a x^4}\right )\) |
Input:
Int[(A + B*x^2)/(x^5*Sqrt[a + b*x^2]),x]
Output:
(-1/2*(A*Sqrt[a + b*x^2])/(a*x^4) - ((3*A*b - 4*a*B)*(-(Sqrt[a + b*x^2]/(a *x^2)) + (b*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/a^(3/2)))/(4*a))/2
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_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
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_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
Time = 0.43 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.81
method | result | size |
pseudoelliptic | \(\frac {-\frac {3 b \left (A b -\frac {4 B a}{3}\right ) x^{4} \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{\sqrt {a}}\right )}{8}+\frac {3 \left (\frac {2 \left (-2 x^{2} B -A \right ) a^{\frac {3}{2}}}{3}+A \sqrt {a}\, b \,x^{2}\right ) \sqrt {b \,x^{2}+a}}{8}}{a^{\frac {5}{2}} x^{4}}\) | \(73\) |
risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \left (-3 A b \,x^{2}+4 B a \,x^{2}+2 A a \right )}{8 a^{2} x^{4}}-\frac {\left (3 A b -4 B a \right ) b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{8 a^{\frac {5}{2}}}\) | \(76\) |
default | \(A \left (-\frac {\sqrt {b \,x^{2}+a}}{4 a \,x^{4}}-\frac {3 b \left (-\frac {\sqrt {b \,x^{2}+a}}{2 a \,x^{2}}+\frac {b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{4 a}\right )+B \left (-\frac {\sqrt {b \,x^{2}+a}}{2 a \,x^{2}}+\frac {b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )\) | \(124\) |
Input:
int((B*x^2+A)/x^5/(b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
3/8*(-b*(A*b-4/3*B*a)*x^4*arctanh((b*x^2+a)^(1/2)/a^(1/2))+(2/3*(-2*B*x^2- A)*a^(3/2)+A*a^(1/2)*b*x^2)*(b*x^2+a)^(1/2))/a^(5/2)/x^4
Time = 0.09 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.93 \[ \int \frac {A+B x^2}{x^5 \sqrt {a+b x^2}} \, dx=\left [-\frac {{\left (4 \, B a b - 3 \, A b^{2}\right )} \sqrt {a} x^{4} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (2 \, A a^{2} + {\left (4 \, B a^{2} - 3 \, A a b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{16 \, a^{3} x^{4}}, -\frac {{\left (4 \, B a b - 3 \, A b^{2}\right )} \sqrt {-a} x^{4} \arctan \left (\frac {\sqrt {b x^{2} + a} \sqrt {-a}}{a}\right ) + {\left (2 \, A a^{2} + {\left (4 \, B a^{2} - 3 \, A a b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{8 \, a^{3} x^{4}}\right ] \] Input:
integrate((B*x^2+A)/x^5/(b*x^2+a)^(1/2),x, algorithm="fricas")
Output:
[-1/16*((4*B*a*b - 3*A*b^2)*sqrt(a)*x^4*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sq rt(a) + 2*a)/x^2) + 2*(2*A*a^2 + (4*B*a^2 - 3*A*a*b)*x^2)*sqrt(b*x^2 + a)) /(a^3*x^4), -1/8*((4*B*a*b - 3*A*b^2)*sqrt(-a)*x^4*arctan(sqrt(b*x^2 + a)* sqrt(-a)/a) + (2*A*a^2 + (4*B*a^2 - 3*A*a*b)*x^2)*sqrt(b*x^2 + a))/(a^3*x^ 4)]
Time = 15.72 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.67 \[ \int \frac {A+B x^2}{x^5 \sqrt {a+b x^2}} \, dx=- \frac {A}{4 \sqrt {b} x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {A \sqrt {b}}{8 a x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {3 A b^{\frac {3}{2}}}{8 a^{2} x \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {3 A b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{8 a^{\frac {5}{2}}} - \frac {B \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{2 a x} + \frac {B b \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{2 a^{\frac {3}{2}}} \] Input:
integrate((B*x**2+A)/x**5/(b*x**2+a)**(1/2),x)
Output:
-A/(4*sqrt(b)*x**5*sqrt(a/(b*x**2) + 1)) + A*sqrt(b)/(8*a*x**3*sqrt(a/(b*x **2) + 1)) + 3*A*b**(3/2)/(8*a**2*x*sqrt(a/(b*x**2) + 1)) - 3*A*b**2*asinh (sqrt(a)/(sqrt(b)*x))/(8*a**(5/2)) - B*sqrt(b)*sqrt(a/(b*x**2) + 1)/(2*a*x ) + B*b*asinh(sqrt(a)/(sqrt(b)*x))/(2*a**(3/2))
Time = 0.06 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.07 \[ \int \frac {A+B x^2}{x^5 \sqrt {a+b x^2}} \, dx=\frac {B b \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{2 \, a^{\frac {3}{2}}} - \frac {3 \, A b^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{8 \, a^{\frac {5}{2}}} - \frac {\sqrt {b x^{2} + a} B}{2 \, a x^{2}} + \frac {3 \, \sqrt {b x^{2} + a} A b}{8 \, a^{2} x^{2}} - \frac {\sqrt {b x^{2} + a} A}{4 \, a x^{4}} \] Input:
integrate((B*x^2+A)/x^5/(b*x^2+a)^(1/2),x, algorithm="maxima")
Output:
1/2*B*b*arcsinh(a/(sqrt(a*b)*abs(x)))/a^(3/2) - 3/8*A*b^2*arcsinh(a/(sqrt( a*b)*abs(x)))/a^(5/2) - 1/2*sqrt(b*x^2 + a)*B/(a*x^2) + 3/8*sqrt(b*x^2 + a )*A*b/(a^2*x^2) - 1/4*sqrt(b*x^2 + a)*A/(a*x^4)
Time = 0.12 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.34 \[ \int \frac {A+B x^2}{x^5 \sqrt {a+b x^2}} \, dx=-\frac {\frac {{\left (4 \, B a b^{2} - 3 \, A b^{3}\right )} \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2}} + \frac {4 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B a b^{2} - 4 \, \sqrt {b x^{2} + a} B a^{2} b^{2} - 3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b^{3} + 5 \, \sqrt {b x^{2} + a} A a b^{3}}{a^{2} b^{2} x^{4}}}{8 \, b} \] Input:
integrate((B*x^2+A)/x^5/(b*x^2+a)^(1/2),x, algorithm="giac")
Output:
-1/8*((4*B*a*b^2 - 3*A*b^3)*arctan(sqrt(b*x^2 + a)/sqrt(-a))/(sqrt(-a)*a^2 ) + (4*(b*x^2 + a)^(3/2)*B*a*b^2 - 4*sqrt(b*x^2 + a)*B*a^2*b^2 - 3*(b*x^2 + a)^(3/2)*A*b^3 + 5*sqrt(b*x^2 + a)*A*a*b^3)/(a^2*b^2*x^4))/b
Time = 1.26 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.10 \[ \int \frac {A+B x^2}{x^5 \sqrt {a+b x^2}} \, dx=\frac {3\,A\,{\left (b\,x^2+a\right )}^{3/2}}{8\,a^2\,x^4}-\frac {5\,A\,\sqrt {b\,x^2+a}}{8\,a\,x^4}-\frac {3\,A\,b^2\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{8\,a^{5/2}}-\frac {B\,\sqrt {b\,x^2+a}}{2\,a\,x^2}+\frac {B\,b\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{2\,a^{3/2}} \] Input:
int((A + B*x^2)/(x^5*(a + b*x^2)^(1/2)),x)
Output:
(3*A*(a + b*x^2)^(3/2))/(8*a^2*x^4) - (5*A*(a + b*x^2)^(1/2))/(8*a*x^4) - (3*A*b^2*atanh((a + b*x^2)^(1/2)/a^(1/2)))/(8*a^(5/2)) - (B*(a + b*x^2)^(1 /2))/(2*a*x^2) + (B*b*atanh((a + b*x^2)^(1/2)/a^(1/2)))/(2*a^(3/2))
Time = 0.21 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.11 \[ \int \frac {A+B x^2}{x^5 \sqrt {a+b x^2}} \, dx=\frac {-2 \sqrt {b \,x^{2}+a}\, a^{2}-\sqrt {b \,x^{2}+a}\, a b \,x^{2}-\sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} x^{4}+\sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} x^{4}}{8 a^{2} x^{4}} \] Input:
int((B*x^2+A)/x^5/(b*x^2+a)^(1/2),x)
Output:
( - 2*sqrt(a + b*x**2)*a**2 - sqrt(a + b*x**2)*a*b*x**2 - sqrt(a)*log((sqr t(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*b**2*x**4 + sqrt(a)*log((sqr t(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*b**2*x**4)/(8*a**2*x**4)