Integrand size = 18, antiderivative size = 98 \[ \int \frac {A+B x}{x^2 (a+b x)^{5/2}} \, dx=\frac {-5 A b+2 a B}{3 a^2 (a+b x)^{3/2}}-\frac {A}{a x (a+b x)^{3/2}}-\frac {5 A b-2 a B}{a^3 \sqrt {a+b x}}+\frac {(5 A b-2 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{7/2}} \]
1/3*(-5*A*b+2*B*a)/a^2/(b*x+a)^(3/2)-A/a/x/(b*x+a)^(3/2)+(5*A*b-2*B*a)*arc tanh((b*x+a)^(1/2)/a^(1/2))/a^(7/2)+(-5*A*b+2*B*a)/a^3/(b*x+a)^(1/2)
Time = 0.13 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.88 \[ \int \frac {A+B x}{x^2 (a+b x)^{5/2}} \, dx=\frac {-15 A b^2 x^2+2 a b x (-10 A+3 B x)+a^2 (-3 A+8 B x)}{3 a^3 x (a+b x)^{3/2}}+\frac {(5 A b-2 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{7/2}} \]
(-15*A*b^2*x^2 + 2*a*b*x*(-10*A + 3*B*x) + a^2*(-3*A + 8*B*x))/(3*a^3*x*(a + b*x)^(3/2)) + ((5*A*b - 2*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/a^(7/2)
Time = 0.19 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.96, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {87, 61, 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.
\(\displaystyle \int \frac {A+B x}{x^2 (a+b x)^{5/2}} \, dx\) |
\(\Big \downarrow \) 87 |
\(\displaystyle -\frac {(5 A b-2 a B) \int \frac {1}{x (a+b x)^{5/2}}dx}{2 a}-\frac {A}{a x (a+b x)^{3/2}}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle -\frac {(5 A b-2 a B) \left (\frac {\int \frac {1}{x (a+b x)^{3/2}}dx}{a}+\frac {2}{3 a (a+b x)^{3/2}}\right )}{2 a}-\frac {A}{a x (a+b x)^{3/2}}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle -\frac {(5 A b-2 a B) \left (\frac {\frac {\int \frac {1}{x \sqrt {a+b x}}dx}{a}+\frac {2}{a \sqrt {a+b x}}}{a}+\frac {2}{3 a (a+b x)^{3/2}}\right )}{2 a}-\frac {A}{a x (a+b x)^{3/2}}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {(5 A b-2 a B) \left (\frac {\frac {2 \int \frac {1}{\frac {a+b x}{b}-\frac {a}{b}}d\sqrt {a+b x}}{a b}+\frac {2}{a \sqrt {a+b x}}}{a}+\frac {2}{3 a (a+b x)^{3/2}}\right )}{2 a}-\frac {A}{a x (a+b x)^{3/2}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {(5 A b-2 a B) \left (\frac {\frac {2}{a \sqrt {a+b x}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{3/2}}}{a}+\frac {2}{3 a (a+b x)^{3/2}}\right )}{2 a}-\frac {A}{a x (a+b x)^{3/2}}\) |
-(A/(a*x*(a + b*x)^(3/2))) - ((5*A*b - 2*a*B)*(2/(3*a*(a + b*x)^(3/2)) + ( 2/(a*Sqrt[a + b*x]) - (2*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/a^(3/2))/a))/(2*a )
3.5.50.3.1 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 = 0.55 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.84
method | result | size |
pseudoelliptic | \(-\frac {-5 x \left (b x +a \right )^{\frac {3}{2}} \left (A b -\frac {2 B a}{5}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )+\frac {20 x \left (-\frac {3 B x}{10}+A \right ) b \,a^{\frac {3}{2}}}{3}+\left (-\frac {8 B x}{3}+A \right ) a^{\frac {5}{2}}+5 A \sqrt {a}\, b^{2} x^{2}}{\left (b x +a \right )^{\frac {3}{2}} a^{\frac {7}{2}} x}\) | \(82\) |
risch | \(-\frac {A \sqrt {b x +a}}{a^{3} x}-\frac {-\frac {2 \left (-4 A b +2 B a \right )}{\sqrt {b x +a}}+\frac {4 a \left (A b -B a \right )}{3 \left (b x +a \right )^{\frac {3}{2}}}-\frac {2 \left (5 A b -2 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{\sqrt {a}}}{2 a^{3}}\) | \(86\) |
derivativedivides | \(-\frac {2 \left (2 A b -B a \right )}{a^{3} \sqrt {b x +a}}-\frac {2 \left (A b -B a \right )}{3 a^{2} \left (b x +a \right )^{\frac {3}{2}}}+\frac {-\frac {A \sqrt {b x +a}}{x}+\frac {\left (5 A b -2 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{\sqrt {a}}}{a^{3}}\) | \(88\) |
default | \(-\frac {2 \left (2 A b -B a \right )}{a^{3} \sqrt {b x +a}}-\frac {2 \left (A b -B a \right )}{3 a^{2} \left (b x +a \right )^{\frac {3}{2}}}+\frac {-\frac {A \sqrt {b x +a}}{x}+\frac {\left (5 A b -2 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{\sqrt {a}}}{a^{3}}\) | \(88\) |
-(-5*x*(b*x+a)^(3/2)*(A*b-2/5*B*a)*arctanh((b*x+a)^(1/2)/a^(1/2))+20/3*x*( -3/10*B*x+A)*b*a^(3/2)+(-8/3*B*x+A)*a^(5/2)+5*A*a^(1/2)*b^2*x^2)/(b*x+a)^( 3/2)/a^(7/2)/x
Time = 0.24 (sec) , antiderivative size = 330, normalized size of antiderivative = 3.37 \[ \int \frac {A+B x}{x^2 (a+b x)^{5/2}} \, dx=\left [-\frac {3 \, {\left ({\left (2 \, B a b^{2} - 5 \, A b^{3}\right )} x^{3} + 2 \, {\left (2 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{2} + {\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} x\right )} \sqrt {a} \log \left (\frac {b x + 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (3 \, A a^{3} - 3 \, {\left (2 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{2} - 4 \, {\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} x\right )} \sqrt {b x + a}}{6 \, {\left (a^{4} b^{2} x^{3} + 2 \, a^{5} b x^{2} + a^{6} x\right )}}, \frac {3 \, {\left ({\left (2 \, B a b^{2} - 5 \, A b^{3}\right )} x^{3} + 2 \, {\left (2 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{2} + {\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} x\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) - {\left (3 \, A a^{3} - 3 \, {\left (2 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{2} - 4 \, {\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} x\right )} \sqrt {b x + a}}{3 \, {\left (a^{4} b^{2} x^{3} + 2 \, a^{5} b x^{2} + a^{6} x\right )}}\right ] \]
[-1/6*(3*((2*B*a*b^2 - 5*A*b^3)*x^3 + 2*(2*B*a^2*b - 5*A*a*b^2)*x^2 + (2*B *a^3 - 5*A*a^2*b)*x)*sqrt(a)*log((b*x + 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 2*(3*A*a^3 - 3*(2*B*a^2*b - 5*A*a*b^2)*x^2 - 4*(2*B*a^3 - 5*A*a^2*b)*x)* sqrt(b*x + a))/(a^4*b^2*x^3 + 2*a^5*b*x^2 + a^6*x), 1/3*(3*((2*B*a*b^2 - 5 *A*b^3)*x^3 + 2*(2*B*a^2*b - 5*A*a*b^2)*x^2 + (2*B*a^3 - 5*A*a^2*b)*x)*sqr t(-a)*arctan(sqrt(b*x + a)*sqrt(-a)/a) - (3*A*a^3 - 3*(2*B*a^2*b - 5*A*a*b ^2)*x^2 - 4*(2*B*a^3 - 5*A*a^2*b)*x)*sqrt(b*x + a))/(a^4*b^2*x^3 + 2*a^5*b *x^2 + a^6*x)]
Leaf count of result is larger than twice the leaf count of optimal. 1520 vs. \(2 (90) = 180\).
Time = 23.35 (sec) , antiderivative size = 1520, normalized size of antiderivative = 15.51 \[ \int \frac {A+B x}{x^2 (a+b x)^{5/2}} \, dx=\text {Too large to display} \]
A*(-6*a**17*sqrt(1 + b*x/a)/(6*a**(39/2)*x + 18*a**(37/2)*b*x**2 + 18*a**( 35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) - 46*a**16*b*x*sqrt(1 + b*x/a)/(6 *a**(39/2)*x + 18*a**(37/2)*b*x**2 + 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)* b**3*x**4) - 15*a**16*b*x*log(b*x/a)/(6*a**(39/2)*x + 18*a**(37/2)*b*x**2 + 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) + 30*a**16*b*x*log(sqrt( 1 + b*x/a) + 1)/(6*a**(39/2)*x + 18*a**(37/2)*b*x**2 + 18*a**(35/2)*b**2*x **3 + 6*a**(33/2)*b**3*x**4) - 70*a**15*b**2*x**2*sqrt(1 + b*x/a)/(6*a**(3 9/2)*x + 18*a**(37/2)*b*x**2 + 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x **4) - 45*a**15*b**2*x**2*log(b*x/a)/(6*a**(39/2)*x + 18*a**(37/2)*b*x**2 + 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) + 90*a**15*b**2*x**2*log (sqrt(1 + b*x/a) + 1)/(6*a**(39/2)*x + 18*a**(37/2)*b*x**2 + 18*a**(35/2)* b**2*x**3 + 6*a**(33/2)*b**3*x**4) - 30*a**14*b**3*x**3*sqrt(1 + b*x/a)/(6 *a**(39/2)*x + 18*a**(37/2)*b*x**2 + 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)* b**3*x**4) - 45*a**14*b**3*x**3*log(b*x/a)/(6*a**(39/2)*x + 18*a**(37/2)*b *x**2 + 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) + 90*a**14*b**3*x* *3*log(sqrt(1 + b*x/a) + 1)/(6*a**(39/2)*x + 18*a**(37/2)*b*x**2 + 18*a**( 35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) - 15*a**13*b**4*x**4*log(b*x/a)/( 6*a**(39/2)*x + 18*a**(37/2)*b*x**2 + 18*a**(35/2)*b**2*x**3 + 6*a**(33/2) *b**3*x**4) + 30*a**13*b**4*x**4*log(sqrt(1 + b*x/a) + 1)/(6*a**(39/2)*x + 18*a**(37/2)*b*x**2 + 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4))...
Time = 0.29 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.33 \[ \int \frac {A+B x}{x^2 (a+b x)^{5/2}} \, dx=-\frac {1}{6} \, b {\left (\frac {2 \, {\left (2 \, B a^{3} - 2 \, A a^{2} b - 3 \, {\left (2 \, B a - 5 \, A b\right )} {\left (b x + a\right )}^{2} + 2 \, {\left (2 \, B a^{2} - 5 \, A a b\right )} {\left (b x + a\right )}\right )}}{{\left (b x + a\right )}^{\frac {5}{2}} a^{3} b - {\left (b x + a\right )}^{\frac {3}{2}} a^{4} b} - \frac {3 \, {\left (2 \, B a - 5 \, A b\right )} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{a^{\frac {7}{2}} b}\right )} \]
-1/6*b*(2*(2*B*a^3 - 2*A*a^2*b - 3*(2*B*a - 5*A*b)*(b*x + a)^2 + 2*(2*B*a^ 2 - 5*A*a*b)*(b*x + a))/((b*x + a)^(5/2)*a^3*b - (b*x + a)^(3/2)*a^4*b) - 3*(2*B*a - 5*A*b)*log((sqrt(b*x + a) - sqrt(a))/(sqrt(b*x + a) + sqrt(a))) /(a^(7/2)*b))
Time = 0.30 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.92 \[ \int \frac {A+B x}{x^2 (a+b x)^{5/2}} \, dx=\frac {{\left (2 \, B a - 5 \, A b\right )} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{3}} - \frac {\sqrt {b x + a} A}{a^{3} x} + \frac {2 \, {\left (3 \, {\left (b x + a\right )} B a + B a^{2} - 6 \, {\left (b x + a\right )} A b - A a b\right )}}{3 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3}} \]
(2*B*a - 5*A*b)*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*a^3) - sqrt(b*x + a)*A/(a^3*x) + 2/3*(3*(b*x + a)*B*a + B*a^2 - 6*(b*x + a)*A*b - A*a*b)/(( b*x + a)^(3/2)*a^3)
Time = 0.48 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.05 \[ \int \frac {A+B x}{x^2 (a+b x)^{5/2}} \, dx=\frac {\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )\,\left (5\,A\,b-2\,B\,a\right )}{a^{7/2}}-\frac {\frac {2\,\left (A\,b-B\,a\right )}{3\,a}+\frac {2\,\left (5\,A\,b-2\,B\,a\right )\,\left (a+b\,x\right )}{3\,a^2}-\frac {\left (5\,A\,b-2\,B\,a\right )\,{\left (a+b\,x\right )}^2}{a^3}}{a\,{\left (a+b\,x\right )}^{3/2}-{\left (a+b\,x\right )}^{5/2}} \]