Integrand size = 18, antiderivative size = 73 \[ \int \frac {\left (a+b x^n\right )^{3/2}}{c x} \, dx=\frac {2 a \sqrt {a+b x^n}}{c n}+\frac {2 \left (a+b x^n\right )^{3/2}}{3 c n}-\frac {2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{c n} \] Output:
2*a*(a+b*x^n)^(1/2)/c/n+2/3*(a+b*x^n)^(3/2)/c/n-2*a^(3/2)*arctanh((a+b*x^n )^(1/2)/a^(1/2))/c/n
Time = 0.02 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.79 \[ \int \frac {\left (a+b x^n\right )^{3/2}}{c x} \, dx=\frac {2 \sqrt {a+b x^n} \left (4 a+b x^n\right )-6 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{3 c n} \] Input:
Integrate[(a + b*x^n)^(3/2)/(c*x),x]
Output:
(2*Sqrt[a + b*x^n]*(4*a + b*x^n) - 6*a^(3/2)*ArcTanh[Sqrt[a + b*x^n]/Sqrt[ a]])/(3*c*n)
Time = 0.30 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.88, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {27, 798, 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.
\(\displaystyle \int \frac {\left (a+b x^n\right )^{3/2}}{c x} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {\left (b x^n+a\right )^{3/2}}{x}dx}{c}\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {\int x^{-n} \left (b x^n+a\right )^{3/2}dx^n}{c n}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {a \int x^{-n} \sqrt {b x^n+a}dx^n+\frac {2}{3} \left (a+b x^n\right )^{3/2}}{c n}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {a \left (a \int \frac {x^{-n}}{\sqrt {b x^n+a}}dx^n+2 \sqrt {a+b x^n}\right )+\frac {2}{3} \left (a+b x^n\right )^{3/2}}{c n}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {a \left (\frac {2 a \int \frac {1}{\frac {x^{2 n}}{b}-\frac {a}{b}}d\sqrt {b x^n+a}}{b}+2 \sqrt {a+b x^n}\right )+\frac {2}{3} \left (a+b x^n\right )^{3/2}}{c n}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {a \left (2 \sqrt {a+b x^n}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )\right )+\frac {2}{3} \left (a+b x^n\right )^{3/2}}{c n}\) |
Input:
Int[(a + b*x^n)^(3/2)/(c*x),x]
Output:
((2*(a + b*x^n)^(3/2))/3 + a*(2*Sqrt[a + b*x^n] - 2*Sqrt[a]*ArcTanh[Sqrt[a + b*x^n]/Sqrt[a]]))/(c*n)
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_))^(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_)^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_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Time = 0.74 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.70
method | result | size |
derivativedivides | \(\frac {\frac {2 \left (a +b \,x^{n}\right )^{\frac {3}{2}}}{3}+2 a \sqrt {a +b \,x^{n}}-2 a^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {a +b \,x^{n}}}{\sqrt {a}}\right )}{c n}\) | \(51\) |
default | \(\frac {\frac {2 \left (a +b \,x^{n}\right )^{\frac {3}{2}}}{3}+2 a \sqrt {a +b \,x^{n}}-2 a^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {a +b \,x^{n}}}{\sqrt {a}}\right )}{c n}\) | \(51\) |
risch | \(\frac {2 \left (b \,{\mathrm e}^{n \ln \left (x \right )}+4 a \right ) \sqrt {a +b \,{\mathrm e}^{n \ln \left (x \right )}}}{3 n c}-\frac {2 a^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {a +b \,{\mathrm e}^{n \ln \left (x \right )}}}{\sqrt {a}}\right )}{n c}\) | \(59\) |
Input:
int((a+b*x^n)^(3/2)/c/x,x,method=_RETURNVERBOSE)
Output:
1/c/n*(2/3*(a+b*x^n)^(3/2)+2*a*(a+b*x^n)^(1/2)-2*a^(3/2)*arctanh((a+b*x^n) ^(1/2)/a^(1/2)))
Time = 0.09 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.60 \[ \int \frac {\left (a+b x^n\right )^{3/2}}{c x} \, dx=\left [\frac {3 \, a^{\frac {3}{2}} \log \left (\frac {b x^{n} - 2 \, \sqrt {b x^{n} + a} \sqrt {a} + 2 \, a}{x^{n}}\right ) + 2 \, {\left (b x^{n} + 4 \, a\right )} \sqrt {b x^{n} + a}}{3 \, c n}, \frac {2 \, {\left (3 \, \sqrt {-a} a \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{n} + a}}\right ) + {\left (b x^{n} + 4 \, a\right )} \sqrt {b x^{n} + a}\right )}}{3 \, c n}\right ] \] Input:
integrate((a+b*x^n)^(3/2)/c/x,x, algorithm="fricas")
Output:
[1/3*(3*a^(3/2)*log((b*x^n - 2*sqrt(b*x^n + a)*sqrt(a) + 2*a)/x^n) + 2*(b* x^n + 4*a)*sqrt(b*x^n + a))/(c*n), 2/3*(3*sqrt(-a)*a*arctan(sqrt(-a)/sqrt( b*x^n + a)) + (b*x^n + 4*a)*sqrt(b*x^n + a))/(c*n)]
Time = 1.53 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.21 \[ \int \frac {\left (a+b x^n\right )^{3/2}}{c x} \, dx=\frac {\frac {8 a^{\frac {3}{2}} \sqrt {1 + \frac {b x^{n}}{a}}}{3 n} + \frac {a^{\frac {3}{2}} \log {\left (\frac {b x^{n}}{a} \right )}}{n} - \frac {2 a^{\frac {3}{2}} \log {\left (\sqrt {1 + \frac {b x^{n}}{a}} + 1 \right )}}{n} + \frac {2 \sqrt {a} b x^{n} \sqrt {1 + \frac {b x^{n}}{a}}}{3 n}}{c} \] Input:
integrate((a+b*x**n)**(3/2)/c/x,x)
Output:
(8*a**(3/2)*sqrt(1 + b*x**n/a)/(3*n) + a**(3/2)*log(b*x**n/a)/n - 2*a**(3/ 2)*log(sqrt(1 + b*x**n/a) + 1)/n + 2*sqrt(a)*b*x**n*sqrt(1 + b*x**n/a)/(3* n))/c
Time = 0.12 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^n\right )^{3/2}}{c x} \, dx=\frac {\frac {3 \, a^{\frac {3}{2}} \log \left (\frac {\sqrt {b x^{n} + a} - \sqrt {a}}{\sqrt {b x^{n} + a} + \sqrt {a}}\right )}{n} + \frac {2 \, {\left ({\left (b x^{n} + a\right )}^{\frac {3}{2}} + 3 \, \sqrt {b x^{n} + a} a\right )}}{n}}{3 \, c} \] Input:
integrate((a+b*x^n)^(3/2)/c/x,x, algorithm="maxima")
Output:
1/3*(3*a^(3/2)*log((sqrt(b*x^n + a) - sqrt(a))/(sqrt(b*x^n + a) + sqrt(a)) )/n + 2*((b*x^n + a)^(3/2) + 3*sqrt(b*x^n + a)*a)/n)/c
\[ \int \frac {\left (a+b x^n\right )^{3/2}}{c x} \, dx=\int { \frac {{\left (b x^{n} + a\right )}^{\frac {3}{2}}}{c x} \,d x } \] Input:
integrate((a+b*x^n)^(3/2)/c/x,x, algorithm="giac")
Output:
integrate((b*x^n + a)^(3/2)/(c*x), x)
Timed out. \[ \int \frac {\left (a+b x^n\right )^{3/2}}{c x} \, dx=\int \frac {{\left (a+b\,x^n\right )}^{3/2}}{c\,x} \,d x \] Input:
int((a + b*x^n)^(3/2)/(c*x),x)
Output:
int((a + b*x^n)^(3/2)/(c*x), x)
\[ \int \frac {\left (a+b x^n\right )^{3/2}}{c x} \, dx=\frac {2 x^{n} \sqrt {x^{n} b +a}\, b +8 \sqrt {x^{n} b +a}\, a +3 \left (\int \frac {\sqrt {x^{n} b +a}}{x^{n} b x +a x}d x \right ) a^{2} n}{3 c n} \] Input:
int((a+b*x^n)^(3/2)/c/x,x)
Output:
(2*x**n*sqrt(x**n*b + a)*b + 8*sqrt(x**n*b + a)*a + 3*int(sqrt(x**n*b + a) /(x**n*b*x + a*x),x)*a**2*n)/(3*c*n)