Integrand size = 20, antiderivative size = 95 \[ \int \frac {A+B x}{x^3 \left (a+b x^2\right )^{3/2}} \, dx=\frac {A+B x}{a x^2 \sqrt {a+b x^2}}-\frac {3 A \sqrt {a+b x^2}}{2 a^2 x^2}-\frac {2 B \sqrt {a+b x^2}}{a^2 x}+\frac {3 A b \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{5/2}} \]
3/2*A*b*arctanh((b*x^2+a)^(1/2)/a^(1/2))/a^(5/2)+(B*x+A)/a/x^2/(b*x^2+a)^( 1/2)-3/2*A*(b*x^2+a)^(1/2)/a^2/x^2-2*B*(b*x^2+a)^(1/2)/a^2/x
Time = 0.30 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.87 \[ \int \frac {A+B x}{x^3 \left (a+b x^2\right )^{3/2}} \, dx=\frac {-a (A+2 B x)-b x^2 (3 A+4 B x)}{2 a^2 x^2 \sqrt {a+b x^2}}-\frac {3 A b \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{5/2}} \]
(-(a*(A + 2*B*x)) - b*x^2*(3*A + 4*B*x))/(2*a^2*x^2*Sqrt[a + b*x^2]) - (3* A*b*ArcTanh[(Sqrt[b]*x - Sqrt[a + b*x^2])/Sqrt[a]])/a^(5/2)
Time = 0.33 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.07, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {532, 25, 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.
\(\displaystyle \int \frac {A+B x}{x^3 \left (a+b x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 532 |
\(\displaystyle -\frac {\int -\frac {-\frac {A b x^2}{a}+B x+A}{x^3 \sqrt {b x^2+a}}dx}{a}-\frac {b (A+B x)}{a^2 \sqrt {a+b x^2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {-\frac {A b x^2}{a}+B x+A}{x^3 \sqrt {b x^2+a}}dx}{a}-\frac {b (A+B x)}{a^2 \sqrt {a+b x^2}}\) |
\(\Big \downarrow \) 2338 |
\(\displaystyle \frac {-\frac {\int -\frac {2 a B-3 A b x}{x^2 \sqrt {b x^2+a}}dx}{2 a}-\frac {A \sqrt {a+b x^2}}{2 a x^2}}{a}-\frac {b (A+B x)}{a^2 \sqrt {a+b x^2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {\int \frac {2 a B-3 A b x}{x^2 \sqrt {b x^2+a}}dx}{2 a}-\frac {A \sqrt {a+b x^2}}{2 a x^2}}{a}-\frac {b (A+B x)}{a^2 \sqrt {a+b x^2}}\) |
\(\Big \downarrow \) 534 |
\(\displaystyle \frac {\frac {-3 A b \int \frac {1}{x \sqrt {b x^2+a}}dx-\frac {2 B \sqrt {a+b x^2}}{x}}{2 a}-\frac {A \sqrt {a+b x^2}}{2 a x^2}}{a}-\frac {b (A+B x)}{a^2 \sqrt {a+b x^2}}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {\frac {-\frac {3}{2} A b \int \frac {1}{x^2 \sqrt {b x^2+a}}dx^2-\frac {2 B \sqrt {a+b x^2}}{x}}{2 a}-\frac {A \sqrt {a+b x^2}}{2 a x^2}}{a}-\frac {b (A+B x)}{a^2 \sqrt {a+b x^2}}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {\frac {-3 A \int \frac {1}{\frac {x^4}{b}-\frac {a}{b}}d\sqrt {b x^2+a}-\frac {2 B \sqrt {a+b x^2}}{x}}{2 a}-\frac {A \sqrt {a+b x^2}}{2 a x^2}}{a}-\frac {b (A+B x)}{a^2 \sqrt {a+b x^2}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {\frac {\frac {3 A b \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {2 B \sqrt {a+b x^2}}{x}}{2 a}-\frac {A \sqrt {a+b x^2}}{2 a x^2}}{a}-\frac {b (A+B x)}{a^2 \sqrt {a+b x^2}}\) |
-((b*(A + B*x))/(a^2*Sqrt[a + b*x^2])) + (-1/2*(A*Sqrt[a + b*x^2])/(a*x^2) + ((-2*B*Sqrt[a + b*x^2])/x + (3*A*b*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/Sq rt[a])/(2*a))/a
3.1.35.3.1 Defintions of rubi rules used
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[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 = 3.52 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.93
method | result | size |
risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \left (2 B x +A \right )}{2 a^{2} x^{2}}-\frac {b B x}{a^{2} \sqrt {b \,x^{2}+a}}-\frac {A b}{a^{2} \sqrt {b \,x^{2}+a}}+\frac {3 b A \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{2 a^{\frac {5}{2}}}\) | \(88\) |
default | \(B \left (-\frac {1}{a x \sqrt {b \,x^{2}+a}}-\frac {2 b x}{a^{2} \sqrt {b \,x^{2}+a}}\right )+A \left (-\frac {1}{2 a \,x^{2} \sqrt {b \,x^{2}+a}}-\frac {3 b \left (\frac {1}{a \sqrt {b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}\right )}{2 a}\right )\) | \(106\) |
-1/2*(b*x^2+a)^(1/2)*(2*B*x+A)/a^2/x^2-b/a^2*B*x/(b*x^2+a)^(1/2)-1/a^2*A/( b*x^2+a)^(1/2)*b+3/2*b/a^(5/2)*A*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)
Time = 0.28 (sec) , antiderivative size = 211, normalized size of antiderivative = 2.22 \[ \int \frac {A+B x}{x^3 \left (a+b x^2\right )^{3/2}} \, dx=\left [\frac {3 \, {\left (A b^{2} x^{4} + A a b x^{2}\right )} \sqrt {a} \log \left (-\frac {b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (4 \, B a b x^{3} + 3 \, A a b x^{2} + 2 \, B a^{2} x + A a^{2}\right )} \sqrt {b x^{2} + a}}{4 \, {\left (a^{3} b x^{4} + a^{4} x^{2}\right )}}, -\frac {3 \, {\left (A b^{2} x^{4} + A a b x^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + {\left (4 \, B a b x^{3} + 3 \, A a b x^{2} + 2 \, B a^{2} x + A a^{2}\right )} \sqrt {b x^{2} + a}}{2 \, {\left (a^{3} b x^{4} + a^{4} x^{2}\right )}}\right ] \]
[1/4*(3*(A*b^2*x^4 + A*a*b*x^2)*sqrt(a)*log(-(b*x^2 + 2*sqrt(b*x^2 + a)*sq rt(a) + 2*a)/x^2) - 2*(4*B*a*b*x^3 + 3*A*a*b*x^2 + 2*B*a^2*x + A*a^2)*sqrt (b*x^2 + a))/(a^3*b*x^4 + a^4*x^2), -1/2*(3*(A*b^2*x^4 + A*a*b*x^2)*sqrt(- a)*arctan(sqrt(-a)/sqrt(b*x^2 + a)) + (4*B*a*b*x^3 + 3*A*a*b*x^2 + 2*B*a^2 *x + A*a^2)*sqrt(b*x^2 + a))/(a^3*b*x^4 + a^4*x^2)]
Time = 3.96 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.31 \[ \int \frac {A+B x}{x^3 \left (a+b x^2\right )^{3/2}} \, dx=A \left (- \frac {1}{2 a \sqrt {b} x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {3 \sqrt {b}}{2 a^{2} x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {3 b \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{2 a^{\frac {5}{2}}}\right ) + B \left (- \frac {1}{a \sqrt {b} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {2 \sqrt {b}}{a^{2} \sqrt {\frac {a}{b x^{2}} + 1}}\right ) \]
A*(-1/(2*a*sqrt(b)*x**3*sqrt(a/(b*x**2) + 1)) - 3*sqrt(b)/(2*a**2*x*sqrt(a /(b*x**2) + 1)) + 3*b*asinh(sqrt(a)/(sqrt(b)*x))/(2*a**(5/2))) + B*(-1/(a* sqrt(b)*x**2*sqrt(a/(b*x**2) + 1)) - 2*sqrt(b)/(a**2*sqrt(a/(b*x**2) + 1)) )
Time = 0.19 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.94 \[ \int \frac {A+B x}{x^3 \left (a+b x^2\right )^{3/2}} \, dx=-\frac {2 \, B b x}{\sqrt {b x^{2} + a} a^{2}} + \frac {3 \, A b \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{2 \, a^{\frac {5}{2}}} - \frac {3 \, A b}{2 \, \sqrt {b x^{2} + a} a^{2}} - \frac {B}{\sqrt {b x^{2} + a} a x} - \frac {A}{2 \, \sqrt {b x^{2} + a} a x^{2}} \]
-2*B*b*x/(sqrt(b*x^2 + a)*a^2) + 3/2*A*b*arcsinh(a/(sqrt(a*b)*abs(x)))/a^( 5/2) - 3/2*A*b/(sqrt(b*x^2 + a)*a^2) - B/(sqrt(b*x^2 + a)*a*x) - 1/2*A/(sq rt(b*x^2 + a)*a*x^2)
Leaf count of result is larger than twice the leaf count of optimal. 171 vs. \(2 (79) = 158\).
Time = 0.30 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.80 \[ \int \frac {A+B x}{x^3 \left (a+b x^2\right )^{3/2}} \, dx=-\frac {\frac {B b x}{a^{2}} + \frac {A b}{a^{2}}}{\sqrt {b x^{2} + a}} - \frac {3 \, A b \arctan \left (-\frac {\sqrt {b} x - \sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2}} + \frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} A b + 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a \sqrt {b} + {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} A a b - 2 \, B a^{2} \sqrt {b}}{{\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{2} a^{2}} \]
-(B*b*x/a^2 + A*b/a^2)/sqrt(b*x^2 + a) - 3*A*b*arctan(-(sqrt(b)*x - sqrt(b *x^2 + a))/sqrt(-a))/(sqrt(-a)*a^2) + ((sqrt(b)*x - sqrt(b*x^2 + a))^3*A*b + 2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*a*sqrt(b) + (sqrt(b)*x - sqrt(b*x^2 + a))*A*a*b - 2*B*a^2*sqrt(b))/(((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^2*a ^2)
Time = 5.94 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.99 \[ \int \frac {A+B x}{x^3 \left (a+b x^2\right )^{3/2}} \, dx=\frac {3\,A\,b\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{2\,a^{5/2}}-\frac {3\,A\,b}{2\,a^2\,\sqrt {b\,x^2+a}}-\frac {A}{2\,a\,x^2\,\sqrt {b\,x^2+a}}-\frac {\sqrt {b\,x^2+a}\,\left (\frac {B}{a}+\frac {2\,B\,b\,x^2}{a^2}\right )}{b\,x^3+a\,x} \]