Integrand size = 22, antiderivative size = 110 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^3} \, dx=\frac {1}{2} (3 A b+2 a B) \sqrt {a+b x^2}+\frac {(3 A b+2 a B) \left (a+b x^2\right )^{3/2}}{6 a}-\frac {A \left (a+b x^2\right )^{5/2}}{2 a x^2}-\frac {1}{2} \sqrt {a} (3 A b+2 a B) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right ) \]
1/6*(3*A*b+2*B*a)*(b*x^2+a)^(3/2)/a-1/2*A*(b*x^2+a)^(5/2)/a/x^2-1/2*(3*A*b +2*B*a)*arctanh((b*x^2+a)^(1/2)/a^(1/2))*a^(1/2)+1/2*(3*A*b+2*B*a)*(b*x^2+ a)^(1/2)
Time = 0.10 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.74 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^3} \, dx=\frac {\sqrt {a+b x^2} \left (-3 a A+6 A b x^2+8 a B x^2+2 b B x^4\right )}{6 x^2}-\frac {1}{2} \sqrt {a} (3 A b+2 a B) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right ) \]
(Sqrt[a + b*x^2]*(-3*a*A + 6*A*b*x^2 + 8*a*B*x^2 + 2*b*B*x^4))/(6*x^2) - ( Sqrt[a]*(3*A*b + 2*a*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/2
Time = 0.21 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.89, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {354, 87, 60, 60, 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.
\(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^3} \, dx\) |
\(\Big \downarrow \) 354 |
\(\displaystyle \frac {1}{2} \int \frac {\left (b x^2+a\right )^{3/2} \left (B x^2+A\right )}{x^4}dx^2\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {1}{2} \left (\frac {(2 a B+3 A b) \int \frac {\left (b x^2+a\right )^{3/2}}{x^2}dx^2}{2 a}-\frac {A \left (a+b x^2\right )^{5/2}}{a x^2}\right )\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{2} \left (\frac {(2 a B+3 A b) \left (a \int \frac {\sqrt {b x^2+a}}{x^2}dx^2+\frac {2}{3} \left (a+b x^2\right )^{3/2}\right )}{2 a}-\frac {A \left (a+b x^2\right )^{5/2}}{a x^2}\right )\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{2} \left (\frac {(2 a B+3 A b) \left (a \left (a \int \frac {1}{x^2 \sqrt {b x^2+a}}dx^2+2 \sqrt {a+b x^2}\right )+\frac {2}{3} \left (a+b x^2\right )^{3/2}\right )}{2 a}-\frac {A \left (a+b x^2\right )^{5/2}}{a x^2}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{2} \left (\frac {(2 a B+3 A b) \left (a \left (\frac {2 a \int \frac {1}{\frac {x^4}{b}-\frac {a}{b}}d\sqrt {b x^2+a}}{b}+2 \sqrt {a+b x^2}\right )+\frac {2}{3} \left (a+b x^2\right )^{3/2}\right )}{2 a}-\frac {A \left (a+b x^2\right )^{5/2}}{a x^2}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{2} \left (\frac {(2 a B+3 A b) \left (a \left (2 \sqrt {a+b x^2}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )\right )+\frac {2}{3} \left (a+b x^2\right )^{3/2}\right )}{2 a}-\frac {A \left (a+b x^2\right )^{5/2}}{a x^2}\right )\) |
(-((A*(a + b*x^2)^(5/2))/(a*x^2)) + ((3*A*b + 2*a*B)*((2*(a + b*x^2)^(3/2) )/3 + a*(2*Sqrt[a + b*x^2] - 2*Sqrt[a]*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]]))) /(2*a))/2
3.6.29.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, 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_))*((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 = 2.83 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.72
method | result | size |
pseudoelliptic | \(-\frac {3 \left (x^{2} a \left (A b +\frac {2 B a}{3}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{\sqrt {a}}\right )-\frac {2 \left (\left (\frac {4 x^{2} B}{3}-\frac {A}{2}\right ) a^{\frac {3}{2}}+b \,x^{2} \sqrt {a}\, \left (\frac {x^{2} B}{3}+A \right )\right ) \sqrt {b \,x^{2}+a}}{3}\right )}{2 \sqrt {a}\, x^{2}}\) | \(79\) |
risch | \(-\frac {a A \sqrt {b \,x^{2}+a}}{2 x^{2}}+B \,b^{2} \left (\frac {x^{2} \sqrt {b \,x^{2}+a}}{3 b}-\frac {2 a \sqrt {b \,x^{2}+a}}{3 b^{2}}\right )+A b \sqrt {b \,x^{2}+a}+2 B a \sqrt {b \,x^{2}+a}-\frac {\sqrt {a}\, \left (3 A b +2 B a \right ) \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{2}\) | \(118\) |
default | \(B \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )+A \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{2 a \,x^{2}}+\frac {3 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )}{2 a}\right )\) | \(134\) |
-3/2/a^(1/2)*(x^2*a*(A*b+2/3*B*a)*arctanh((b*x^2+a)^(1/2)/a^(1/2))-2/3*((4 /3*x^2*B-1/2*A)*a^(3/2)+b*x^2*a^(1/2)*(1/3*x^2*B+A))*(b*x^2+a)^(1/2))/x^2
Time = 0.29 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.52 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^3} \, dx=\left [\frac {3 \, {\left (2 \, B a + 3 \, A b\right )} \sqrt {a} x^{2} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (2 \, B b x^{4} + 2 \, {\left (4 \, B a + 3 \, A b\right )} x^{2} - 3 \, A a\right )} \sqrt {b x^{2} + a}}{12 \, x^{2}}, \frac {3 \, {\left (2 \, B a + 3 \, A b\right )} \sqrt {-a} x^{2} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + {\left (2 \, B b x^{4} + 2 \, {\left (4 \, B a + 3 \, A b\right )} x^{2} - 3 \, A a\right )} \sqrt {b x^{2} + a}}{6 \, x^{2}}\right ] \]
[1/12*(3*(2*B*a + 3*A*b)*sqrt(a)*x^2*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt( a) + 2*a)/x^2) + 2*(2*B*b*x^4 + 2*(4*B*a + 3*A*b)*x^2 - 3*A*a)*sqrt(b*x^2 + a))/x^2, 1/6*(3*(2*B*a + 3*A*b)*sqrt(-a)*x^2*arctan(sqrt(-a)/sqrt(b*x^2 + a)) + (2*B*b*x^4 + 2*(4*B*a + 3*A*b)*x^2 - 3*A*a)*sqrt(b*x^2 + a))/x^2]
Time = 14.58 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.83 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^3} \, dx=- \frac {3 A \sqrt {a} b \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{2} - \frac {A a \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{2 x} + \frac {A a \sqrt {b}}{x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {A b^{\frac {3}{2}} x}{\sqrt {\frac {a}{b x^{2}} + 1}} - B a^{\frac {3}{2}} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )} + \frac {B a^{2}}{\sqrt {b} x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {B a \sqrt {b} x}{\sqrt {\frac {a}{b x^{2}} + 1}} + B b \left (\begin {cases} \frac {a \sqrt {a + b x^{2}}}{3 b} + \frac {x^{2} \sqrt {a + b x^{2}}}{3} & \text {for}\: b \neq 0 \\\frac {\sqrt {a} x^{2}}{2} & \text {otherwise} \end {cases}\right ) \]
-3*A*sqrt(a)*b*asinh(sqrt(a)/(sqrt(b)*x))/2 - A*a*sqrt(b)*sqrt(a/(b*x**2) + 1)/(2*x) + A*a*sqrt(b)/(x*sqrt(a/(b*x**2) + 1)) + A*b**(3/2)*x/sqrt(a/(b *x**2) + 1) - B*a**(3/2)*asinh(sqrt(a)/(sqrt(b)*x)) + B*a**2/(sqrt(b)*x*sq rt(a/(b*x**2) + 1)) + B*a*sqrt(b)*x/sqrt(a/(b*x**2) + 1) + B*b*Piecewise(( a*sqrt(a + b*x**2)/(3*b) + x**2*sqrt(a + b*x**2)/3, Ne(b, 0)), (sqrt(a)*x* *2/2, True))
Time = 0.21 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^3} \, dx=-B a^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right ) - \frac {3}{2} \, A \sqrt {a} b \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right ) + \frac {1}{3} \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B + \sqrt {b x^{2} + a} B a + \frac {3}{2} \, \sqrt {b x^{2} + a} A b + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A b}{2 \, a} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A}{2 \, a x^{2}} \]
-B*a^(3/2)*arcsinh(a/(sqrt(a*b)*abs(x))) - 3/2*A*sqrt(a)*b*arcsinh(a/(sqrt (a*b)*abs(x))) + 1/3*(b*x^2 + a)^(3/2)*B + sqrt(b*x^2 + a)*B*a + 3/2*sqrt( b*x^2 + a)*A*b + 1/2*(b*x^2 + a)^(3/2)*A*b/a - 1/2*(b*x^2 + a)^(5/2)*A/(a* x^2)
Time = 0.32 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^3} \, dx=\frac {2 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B b + 6 \, \sqrt {b x^{2} + a} B a b + 6 \, \sqrt {b x^{2} + a} A b^{2} - \frac {3 \, \sqrt {b x^{2} + a} A a b}{x^{2}} + \frac {3 \, {\left (2 \, B a^{2} b + 3 \, A a b^{2}\right )} \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}}}{6 \, b} \]
1/6*(2*(b*x^2 + a)^(3/2)*B*b + 6*sqrt(b*x^2 + a)*B*a*b + 6*sqrt(b*x^2 + a) *A*b^2 - 3*sqrt(b*x^2 + a)*A*a*b/x^2 + 3*(2*B*a^2*b + 3*A*a*b^2)*arctan(sq rt(b*x^2 + a)/sqrt(-a))/sqrt(-a))/b
Time = 5.98 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^3} \, dx=\frac {B\,{\left (b\,x^2+a\right )}^{3/2}}{3}-B\,a^{3/2}\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )+A\,b\,\sqrt {b\,x^2+a}+B\,a\,\sqrt {b\,x^2+a}-\frac {A\,a\,\sqrt {b\,x^2+a}}{2\,x^2}-\frac {3\,A\,\sqrt {a}\,b\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{2} \]