Integrand size = 18, antiderivative size = 73 \[ \int \frac {A+B x}{x^2 (a+b x)^{3/2}} \, dx=-\frac {3 A b-2 a B}{a^2 \sqrt {a+b x}}-\frac {A}{a x \sqrt {a+b x}}+\frac {(3 A b-2 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{5/2}} \]
(3*A*b-2*B*a)*arctanh((b*x+a)^(1/2)/a^(1/2))/a^(5/2)+(-3*A*b+2*B*a)/a^2/(b *x+a)^(1/2)-A/a/x/(b*x+a)^(1/2)
Time = 0.11 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.86 \[ \int \frac {A+B x}{x^2 (a+b x)^{3/2}} \, dx=\frac {-a A-3 A b x+2 a B x}{a^2 x \sqrt {a+b x}}+\frac {(3 A b-2 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{5/2}} \]
(-(a*A) - 3*A*b*x + 2*a*B*x)/(a^2*x*Sqrt[a + b*x]) + ((3*A*b - 2*a*B)*ArcT anh[Sqrt[a + b*x]/Sqrt[a]])/a^(5/2)
Time = 0.19 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {87, 61, 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.
\(\displaystyle \int \frac {A+B x}{x^2 (a+b x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 87 |
\(\displaystyle -\frac {(3 A b-2 a B) \int \frac {1}{x (a+b x)^{3/2}}dx}{2 a}-\frac {A}{a x \sqrt {a+b x}}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle -\frac {(3 A b-2 a B) \left (\frac {\int \frac {1}{x \sqrt {a+b x}}dx}{a}+\frac {2}{a \sqrt {a+b x}}\right )}{2 a}-\frac {A}{a x \sqrt {a+b x}}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {(3 A b-2 a B) \left (\frac {2 \int \frac {1}{\frac {a+b x}{b}-\frac {a}{b}}d\sqrt {a+b x}}{a b}+\frac {2}{a \sqrt {a+b x}}\right )}{2 a}-\frac {A}{a x \sqrt {a+b x}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {(3 A b-2 a B) \left (\frac {2}{a \sqrt {a+b x}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{3/2}}\right )}{2 a}-\frac {A}{a x \sqrt {a+b x}}\) |
-(A/(a*x*Sqrt[a + b*x])) - ((3*A*b - 2*a*B)*(2/(a*Sqrt[a + b*x]) - (2*ArcT anh[Sqrt[a + b*x]/Sqrt[a]])/a^(3/2)))/(2*a)
3.5.40.3.1 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 + 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] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[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]
Time = 0.54 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.84
method | result | size |
pseudoelliptic | \(\frac {-\frac {2 \left (A b -B a \right )}{\sqrt {b x +a}}-\frac {A \sqrt {b x +a}}{x}+\frac {\left (3 A b -2 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{\sqrt {a}}}{a^{2}}\) | \(61\) |
derivativedivides | \(-\frac {2 \left (A b -B a \right )}{a^{2} \sqrt {b x +a}}+\frac {-\frac {A \sqrt {b x +a}}{x}+\frac {\left (3 A b -2 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{\sqrt {a}}}{a^{2}}\) | \(67\) |
default | \(-\frac {2 \left (A b -B a \right )}{a^{2} \sqrt {b x +a}}+\frac {-\frac {A \sqrt {b x +a}}{x}+\frac {\left (3 A b -2 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{\sqrt {a}}}{a^{2}}\) | \(67\) |
risch | \(-\frac {A \sqrt {b x +a}}{a^{2} x}-\frac {-\frac {2 \left (-2 A b +2 B a \right )}{\sqrt {b x +a}}-\frac {2 \left (3 A b -2 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{\sqrt {a}}}{2 a^{2}}\) | \(68\) |
1/a^2*(-2*(A*b-B*a)/(b*x+a)^(1/2)-A*(b*x+a)^(1/2)/x+(3*A*b-2*B*a)/a^(1/2)* arctanh((b*x+a)^(1/2)/a^(1/2)))
Time = 0.24 (sec) , antiderivative size = 211, normalized size of antiderivative = 2.89 \[ \int \frac {A+B x}{x^2 (a+b x)^{3/2}} \, dx=\left [-\frac {{\left ({\left (2 \, B a b - 3 \, A b^{2}\right )} x^{2} + {\left (2 \, B a^{2} - 3 \, A a b\right )} x\right )} \sqrt {a} \log \left (\frac {b x + 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (A a^{2} - {\left (2 \, B a^{2} - 3 \, A a b\right )} x\right )} \sqrt {b x + a}}{2 \, {\left (a^{3} b x^{2} + a^{4} x\right )}}, \frac {{\left ({\left (2 \, B a b - 3 \, A b^{2}\right )} x^{2} + {\left (2 \, B a^{2} - 3 \, A a b\right )} x\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) - {\left (A a^{2} - {\left (2 \, B a^{2} - 3 \, A a b\right )} x\right )} \sqrt {b x + a}}{a^{3} b x^{2} + a^{4} x}\right ] \]
[-1/2*(((2*B*a*b - 3*A*b^2)*x^2 + (2*B*a^2 - 3*A*a*b)*x)*sqrt(a)*log((b*x + 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 2*(A*a^2 - (2*B*a^2 - 3*A*a*b)*x)*sq rt(b*x + a))/(a^3*b*x^2 + a^4*x), (((2*B*a*b - 3*A*b^2)*x^2 + (2*B*a^2 - 3 *A*a*b)*x)*sqrt(-a)*arctan(sqrt(b*x + a)*sqrt(-a)/a) - (A*a^2 - (2*B*a^2 - 3*A*a*b)*x)*sqrt(b*x + a))/(a^3*b*x^2 + a^4*x)]
Leaf count of result is larger than twice the leaf count of optimal. 224 vs. \(2 (65) = 130\).
Time = 20.01 (sec) , antiderivative size = 224, normalized size of antiderivative = 3.07 \[ \int \frac {A+B x}{x^2 (a+b x)^{3/2}} \, dx=A \left (- \frac {1}{a \sqrt {b} x^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {3 \sqrt {b}}{a^{2} \sqrt {x} \sqrt {\frac {a}{b x} + 1}} + \frac {3 b \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{a^{\frac {5}{2}}}\right ) + B \left (\frac {2 a^{3} \sqrt {1 + \frac {b x}{a}}}{a^{\frac {9}{2}} + a^{\frac {7}{2}} b x} + \frac {a^{3} \log {\left (\frac {b x}{a} \right )}}{a^{\frac {9}{2}} + a^{\frac {7}{2}} b x} - \frac {2 a^{3} \log {\left (\sqrt {1 + \frac {b x}{a}} + 1 \right )}}{a^{\frac {9}{2}} + a^{\frac {7}{2}} b x} + \frac {a^{2} b x \log {\left (\frac {b x}{a} \right )}}{a^{\frac {9}{2}} + a^{\frac {7}{2}} b x} - \frac {2 a^{2} b x \log {\left (\sqrt {1 + \frac {b x}{a}} + 1 \right )}}{a^{\frac {9}{2}} + a^{\frac {7}{2}} b x}\right ) \]
A*(-1/(a*sqrt(b)*x**(3/2)*sqrt(a/(b*x) + 1)) - 3*sqrt(b)/(a**2*sqrt(x)*sqr t(a/(b*x) + 1)) + 3*b*asinh(sqrt(a)/(sqrt(b)*sqrt(x)))/a**(5/2)) + B*(2*a* *3*sqrt(1 + b*x/a)/(a**(9/2) + a**(7/2)*b*x) + a**3*log(b*x/a)/(a**(9/2) + a**(7/2)*b*x) - 2*a**3*log(sqrt(1 + b*x/a) + 1)/(a**(9/2) + a**(7/2)*b*x) + a**2*b*x*log(b*x/a)/(a**(9/2) + a**(7/2)*b*x) - 2*a**2*b*x*log(sqrt(1 + b*x/a) + 1)/(a**(9/2) + a**(7/2)*b*x))
Time = 0.28 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.47 \[ \int \frac {A+B x}{x^2 (a+b x)^{3/2}} \, dx=-\frac {1}{2} \, b {\left (\frac {2 \, {\left (2 \, B a^{2} - 2 \, A a b - {\left (2 \, B a - 3 \, A b\right )} {\left (b x + a\right )}\right )}}{{\left (b x + a\right )}^{\frac {3}{2}} a^{2} b - \sqrt {b x + a} a^{3} b} - \frac {{\left (2 \, B a - 3 \, A b\right )} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{a^{\frac {5}{2}} b}\right )} \]
-1/2*b*(2*(2*B*a^2 - 2*A*a*b - (2*B*a - 3*A*b)*(b*x + a))/((b*x + a)^(3/2) *a^2*b - sqrt(b*x + a)*a^3*b) - (2*B*a - 3*A*b)*log((sqrt(b*x + a) - sqrt( a))/(sqrt(b*x + a) + sqrt(a)))/(a^(5/2)*b))
Time = 0.28 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.19 \[ \int \frac {A+B x}{x^2 (a+b x)^{3/2}} \, dx=\frac {{\left (2 \, B a - 3 \, A b\right )} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2}} + \frac {2 \, {\left (b x + a\right )} B a - 2 \, B a^{2} - 3 \, {\left (b x + a\right )} A b + 2 \, A a b}{{\left ({\left (b x + a\right )}^{\frac {3}{2}} - \sqrt {b x + a} a\right )} a^{2}} \]
(2*B*a - 3*A*b)*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*a^2) + (2*(b*x + a)*B*a - 2*B*a^2 - 3*(b*x + a)*A*b + 2*A*a*b)/(((b*x + a)^(3/2) - sqrt(b*x + a)*a)*a^2)
Time = 0.51 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.12 \[ \int \frac {A+B x}{x^2 (a+b x)^{3/2}} \, dx=\frac {\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )\,\left (3\,A\,b-2\,B\,a\right )}{a^{5/2}}-\frac {\frac {2\,\left (A\,b-B\,a\right )}{a}-\frac {\left (3\,A\,b-2\,B\,a\right )\,\left (a+b\,x\right )}{a^2}}{a\,\sqrt {a+b\,x}-{\left (a+b\,x\right )}^{3/2}} \]