Integrand size = 23, antiderivative size = 45 \[ \int \frac {(1+a x)^{3/2}}{x (1-a x)^{3/2}} \, dx=\frac {4 (1+a x)}{\sqrt {1-a^2 x^2}}-\arcsin (a x)-\text {arctanh}\left (\sqrt {1-a^2 x^2}\right ) \] Output:
4*(a*x+1)/(-a^2*x^2+1)^(1/2)-arcsin(a*x)-arctanh((-a^2*x^2+1)^(1/2))
Time = 0.49 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.82 \[ \int \frac {(1+a x)^{3/2}}{x (1-a x)^{3/2}} \, dx=-\frac {4 \sqrt {1-a^2 x^2}}{-1+a x}+4 \arctan \left (\frac {\sqrt {1+a x}}{\sqrt {2}-\sqrt {1-a x}}\right )+2 \text {arctanh}\left (\frac {\sqrt {1-a^2 x^2}}{-1+a x}\right ) \] Input:
Integrate[(1 + a*x)^(3/2)/(x*(1 - a*x)^(3/2)),x]
Output:
(-4*Sqrt[1 - a^2*x^2])/(-1 + a*x) + 4*ArcTan[Sqrt[1 + a*x]/(Sqrt[2] - Sqrt [1 - a*x])] + 2*ArcTanh[Sqrt[1 - a^2*x^2]/(-1 + a*x)]
Time = 0.31 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.13, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {109, 27, 140, 39, 103, 221, 223}
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.
\(\displaystyle \int \frac {(a x+1)^{3/2}}{x (1-a x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 109 |
\(\displaystyle \frac {4 \sqrt {a x+1}}{\sqrt {1-a x}}-\frac {2 \int -\frac {a \sqrt {1-a x}}{2 x \sqrt {a x+1}}dx}{a}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {\sqrt {1-a x}}{x \sqrt {a x+1}}dx+\frac {4 \sqrt {a x+1}}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 140 |
\(\displaystyle -a \int \frac {1}{\sqrt {1-a x} \sqrt {a x+1}}dx+\int \frac {1}{x \sqrt {1-a x} \sqrt {a x+1}}dx+\frac {4 \sqrt {a x+1}}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 39 |
\(\displaystyle -a \int \frac {1}{\sqrt {1-a^2 x^2}}dx+\int \frac {1}{x \sqrt {1-a x} \sqrt {a x+1}}dx+\frac {4 \sqrt {a x+1}}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 103 |
\(\displaystyle -a \int \frac {1}{\sqrt {1-a^2 x^2}}dx-a \int \frac {1}{a-a (1-a x) (a x+1)}d\left (\sqrt {1-a x} \sqrt {a x+1}\right )+\frac {4 \sqrt {a x+1}}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -a \int \frac {1}{\sqrt {1-a^2 x^2}}dx-\text {arctanh}\left (\sqrt {1-a x} \sqrt {a x+1}\right )+\frac {4 \sqrt {a x+1}}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle -\arcsin (a x)-\text {arctanh}\left (\sqrt {1-a x} \sqrt {a x+1}\right )+\frac {4 \sqrt {a x+1}}{\sqrt {1-a x}}\) |
Input:
Int[(1 + a*x)^(3/2)/(x*(1 - a*x)^(3/2)),x]
Output:
(4*Sqrt[1 + a*x])/Sqrt[1 - a*x] - ArcSin[a*x] - ArcTanh[Sqrt[1 - a*x]*Sqrt [1 + a*x]]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[( a*c + b*d*x^2)^m, x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[b*c + a*d, 0] && ( IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))
Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_ ))), x_] :> Simp[b*f Subst[Int[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sq rt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[2*b*d *e - f*(b*c + a*d), 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f *x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*d^(m + n)*f^p Int[(a + b*x)^(m - 1)/(c + d*x)^m, x] , x] + Int[(a + b*x)^(m - 1)*((e + f*x)^p/(c + d*x)^m)*ExpandToSum[(a + b*x )*(c + d*x)^(-p - 1) - (b*d^(-p - 1)*f^p)/(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[m + n + p + 1, 0] && ILtQ[p, 0] && (GtQ[m, 0] || SumSimplerQ[m, -1] || !(GtQ[n, 0] || SumSimplerQ[n, -1]))
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[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.09 (sec) , antiderivative size = 128, normalized size of antiderivative = 2.84
method | result | size |
default | \(\frac {\left (\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (a \right ) a x +\arctan \left (\frac {\operatorname {csgn}\left (a \right ) a x}{\sqrt {-\left (a x +1\right ) \left (a x -1\right )}}\right ) a x -\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (a \right )+4 \sqrt {-a^{2} x^{2}+1}\, \operatorname {csgn}\left (a \right )-\arctan \left (\frac {\operatorname {csgn}\left (a \right ) a x}{\sqrt {-\left (a x +1\right ) \left (a x -1\right )}}\right )\right ) \sqrt {a x +1}\, \operatorname {csgn}\left (a \right )}{\sqrt {-a x +1}\, \sqrt {-a^{2} x^{2}+1}}\) | \(128\) |
Input:
int((a*x+1)^(3/2)/x/(-a*x+1)^(3/2),x,method=_RETURNVERBOSE)
Output:
(arctanh(1/(-a^2*x^2+1)^(1/2))*csgn(a)*a*x+arctan(csgn(a)*a*x/(-(a*x+1)*(a *x-1))^(1/2))*a*x-arctanh(1/(-a^2*x^2+1)^(1/2))*csgn(a)+4*(-a^2*x^2+1)^(1/ 2)*csgn(a)-arctan(csgn(a)*a*x/(-(a*x+1)*(a*x-1))^(1/2)))*(a*x+1)^(1/2)*csg n(a)/(-a*x+1)^(1/2)/(-a^2*x^2+1)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 93 vs. \(2 (41) = 82\).
Time = 0.09 (sec) , antiderivative size = 93, normalized size of antiderivative = 2.07 \[ \int \frac {(1+a x)^{3/2}}{x (1-a x)^{3/2}} \, dx=\frac {4 \, a x + 2 \, {\left (a x - 1\right )} \arctan \left (\frac {\sqrt {a x + 1} \sqrt {-a x + 1} - 1}{a x}\right ) + {\left (a x - 1\right )} \log \left (\frac {\sqrt {a x + 1} \sqrt {-a x + 1} - 1}{x}\right ) - 4 \, \sqrt {a x + 1} \sqrt {-a x + 1} - 4}{a x - 1} \] Input:
integrate((a*x+1)^(3/2)/x/(-a*x+1)^(3/2),x, algorithm="fricas")
Output:
(4*a*x + 2*(a*x - 1)*arctan((sqrt(a*x + 1)*sqrt(-a*x + 1) - 1)/(a*x)) + (a *x - 1)*log((sqrt(a*x + 1)*sqrt(-a*x + 1) - 1)/x) - 4*sqrt(a*x + 1)*sqrt(- a*x + 1) - 4)/(a*x - 1)
\[ \int \frac {(1+a x)^{3/2}}{x (1-a x)^{3/2}} \, dx=\int \frac {\left (a x + 1\right )^{\frac {3}{2}}}{x \left (- a x + 1\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((a*x+1)**(3/2)/x/(-a*x+1)**(3/2),x)
Output:
Integral((a*x + 1)**(3/2)/(x*(-a*x + 1)**(3/2)), x)
Time = 0.11 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.44 \[ \int \frac {(1+a x)^{3/2}}{x (1-a x)^{3/2}} \, dx=\frac {4 \, a x}{\sqrt {-a^{2} x^{2} + 1}} + \frac {4}{\sqrt {-a^{2} x^{2} + 1}} - \arcsin \left (a x\right ) - \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) \] Input:
integrate((a*x+1)^(3/2)/x/(-a*x+1)^(3/2),x, algorithm="maxima")
Output:
4*a*x/sqrt(-a^2*x^2 + 1) + 4/sqrt(-a^2*x^2 + 1) - arcsin(a*x) - log(2*sqrt (-a^2*x^2 + 1)/abs(x) + 2/abs(x))
Leaf count of result is larger than twice the leaf count of optimal. 196 vs. \(2 (41) = 82\).
Time = 0.22 (sec) , antiderivative size = 196, normalized size of antiderivative = 4.36 \[ \int \frac {(1+a x)^{3/2}}{x (1-a x)^{3/2}} \, dx=-\frac {{\left (\pi + 2 \, \arctan \left (\frac {\sqrt {a x + 1} {\left (\frac {{\left (\sqrt {2} - \sqrt {-a x + 1}\right )}^{2}}{a x + 1} - 1\right )}}{2 \, {\left (\sqrt {2} - \sqrt {-a x + 1}\right )}}\right )\right )} a + a \log \left ({\left | -\frac {\sqrt {2} - \sqrt {-a x + 1}}{\sqrt {a x + 1}} + \frac {\sqrt {a x + 1}}{\sqrt {2} - \sqrt {-a x + 1}} + 2 \right |}\right ) - a \log \left ({\left | -\frac {\sqrt {2} - \sqrt {-a x + 1}}{\sqrt {a x + 1}} + \frac {\sqrt {a x + 1}}{\sqrt {2} - \sqrt {-a x + 1}} - 2 \right |}\right ) + \frac {4 \, \sqrt {a x + 1} \sqrt {-a x + 1} a}{a x - 1}}{a} \] Input:
integrate((a*x+1)^(3/2)/x/(-a*x+1)^(3/2),x, algorithm="giac")
Output:
-((pi + 2*arctan(1/2*sqrt(a*x + 1)*((sqrt(2) - sqrt(-a*x + 1))^2/(a*x + 1) - 1)/(sqrt(2) - sqrt(-a*x + 1))))*a + a*log(abs(-(sqrt(2) - sqrt(-a*x + 1 ))/sqrt(a*x + 1) + sqrt(a*x + 1)/(sqrt(2) - sqrt(-a*x + 1)) + 2)) - a*log( abs(-(sqrt(2) - sqrt(-a*x + 1))/sqrt(a*x + 1) + sqrt(a*x + 1)/(sqrt(2) - s qrt(-a*x + 1)) - 2)) + 4*sqrt(a*x + 1)*sqrt(-a*x + 1)*a/(a*x - 1))/a
Timed out. \[ \int \frac {(1+a x)^{3/2}}{x (1-a x)^{3/2}} \, dx=\int \frac {{\left (a\,x+1\right )}^{3/2}}{x\,{\left (1-a\,x\right )}^{3/2}} \,d x \] Input:
int((a*x + 1)^(3/2)/(x*(1 - a*x)^(3/2)),x)
Output:
int((a*x + 1)^(3/2)/(x*(1 - a*x)^(3/2)), x)
Time = 0.20 (sec) , antiderivative size = 163, normalized size of antiderivative = 3.62 \[ \int \frac {(1+a x)^{3/2}}{x (1-a x)^{3/2}} \, dx=\frac {2 \sqrt {-a x +1}\, \mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )-\sqrt {-a x +1}\, \mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )-1\right )+\sqrt {-a x +1}\, \mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )+1\right )-\sqrt {-a x +1}\, \mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )-1\right )+\sqrt {-a x +1}\, \mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )+1\right )+4 \sqrt {a x +1}}{\sqrt {-a x +1}} \] Input:
int((a*x+1)^(3/2)/x/(-a*x+1)^(3/2),x)
Output:
(2*sqrt( - a*x + 1)*asin(sqrt( - a*x + 1)/sqrt(2)) - sqrt( - a*x + 1)*log( - sqrt(2) + tan(asin(sqrt( - a*x + 1)/sqrt(2))/2) - 1) + sqrt( - a*x + 1) *log( - sqrt(2) + tan(asin(sqrt( - a*x + 1)/sqrt(2))/2) + 1) - sqrt( - a*x + 1)*log(sqrt(2) + tan(asin(sqrt( - a*x + 1)/sqrt(2))/2) - 1) + sqrt( - a *x + 1)*log(sqrt(2) + tan(asin(sqrt( - a*x + 1)/sqrt(2))/2) + 1) + 4*sqrt( a*x + 1))/sqrt( - a*x + 1)