Integrand size = 21, antiderivative size = 139 \[ \int x^m \left (a+b x^{-\frac {3}{2} (1+m)}\right )^{2/3} \, dx=\frac {x^{1+m} \left (a+b x^{-\frac {3}{2} (1+m)}\right )^{2/3}}{1+m}-\frac {2 b^{2/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x^{\frac {1}{2} (-1-m)}}{\sqrt [3]{a+b x^{-\frac {3}{2} (1+m)}}}}{\sqrt {3}}\right )}{\sqrt {3} (1+m)}+\frac {b^{2/3} \log \left (\sqrt [3]{b} x^{\frac {1}{2} (-1-m)}-\sqrt [3]{a+b x^{-\frac {3}{2} (1+m)}}\right )}{1+m} \] Output:
x^(1+m)*(a+b/(x^(3/2+3/2*m)))^(2/3)/(1+m)-2/3*b^(2/3)*arctan(1/3*(1+2*b^(1 /3)*x^(-1/2-1/2*m)/(a+b/(x^(3/2+3/2*m)))^(1/3))*3^(1/2))*3^(1/2)/(1+m)+b^( 2/3)*ln(b^(1/3)*x^(-1/2-1/2*m)-(a+b/(x^(3/2+3/2*m)))^(1/3))/(1+m)
Leaf count is larger than twice the leaf count of optimal. \(347\) vs. \(2(139)=278\).
Time = 0.67 (sec) , antiderivative size = 347, normalized size of antiderivative = 2.50 \[ \int x^m \left (a+b x^{-\frac {3}{2} (1+m)}\right )^{2/3} \, dx=-\frac {x^{\frac {1}{2} (-1-m)} \left (a+b x^{-\frac {3}{2} (1+m)}\right )^{2/3} \left (\sqrt [3]{b}-\sqrt [3]{b+a x^{\frac {3 (1+m)}{2}}}\right ) \left (b^{2/3}+\sqrt [3]{b} \sqrt [3]{b+a x^{\frac {3 (1+m)}{2}}}+\left (b+a x^{\frac {3 (1+m)}{2}}\right )^{2/3}\right )^2 \left (3 \left (b+a x^{\frac {3 (1+m)}{2}}\right )^{2/3}+2 \sqrt {3} b^{2/3} \arctan \left (\frac {\sqrt [3]{b}+2 \sqrt [3]{b+a x^{\frac {3 (1+m)}{2}}}}{\sqrt {3} \sqrt [3]{b}}\right )+2 b^{2/3} \log \left (\sqrt [3]{b}-\sqrt [3]{b+a x^{\frac {3 (1+m)}{2}}}\right )-b^{2/3} \log \left (b^{2/3}+\sqrt [3]{b} \sqrt [3]{b+a x^{\frac {3 (1+m)}{2}}}+\left (b+a x^{\frac {3 (1+m)}{2}}\right )^{2/3}\right )\right )}{3 a (1+m) \sqrt [3]{b+a x^{\frac {3 (1+m)}{2}}} \left (b+a x^{\frac {3 (1+m)}{2}}+b^{2/3} \sqrt [3]{b+a x^{\frac {3 (1+m)}{2}}}+\sqrt [3]{b} \left (b+a x^{\frac {3 (1+m)}{2}}\right )^{2/3}\right )} \] Input:
Integrate[x^m*(a + b/x^((3*(1 + m))/2))^(2/3),x]
Output:
-1/3*(x^((-1 - m)/2)*(a + b/x^((3*(1 + m))/2))^(2/3)*(b^(1/3) - (b + a*x^( (3*(1 + m))/2))^(1/3))*(b^(2/3) + b^(1/3)*(b + a*x^((3*(1 + m))/2))^(1/3) + (b + a*x^((3*(1 + m))/2))^(2/3))^2*(3*(b + a*x^((3*(1 + m))/2))^(2/3) + 2*Sqrt[3]*b^(2/3)*ArcTan[(b^(1/3) + 2*(b + a*x^((3*(1 + m))/2))^(1/3))/(Sq rt[3]*b^(1/3))] + 2*b^(2/3)*Log[b^(1/3) - (b + a*x^((3*(1 + m))/2))^(1/3)] - b^(2/3)*Log[b^(2/3) + b^(1/3)*(b + a*x^((3*(1 + m))/2))^(1/3) + (b + a* x^((3*(1 + m))/2))^(2/3)]))/(a*(1 + m)*(b + a*x^((3*(1 + m))/2))^(1/3)*(b + a*x^((3*(1 + m))/2) + b^(2/3)*(b + a*x^((3*(1 + m))/2))^(1/3) + b^(1/3)* (b + a*x^((3*(1 + m))/2))^(2/3)))
Time = 0.44 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.03, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {872, 868, 769}
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.
\(\displaystyle \int x^m \left (a+b x^{-\frac {3}{2} (m+1)}\right )^{2/3} \, dx\) |
\(\Big \downarrow \) 872 |
\(\displaystyle b \int \frac {x^{m-\frac {3 (m+1)}{2}}}{\sqrt [3]{b x^{-\frac {3}{2} (m+1)}+a}}dx+\frac {x^{m+1} \left (a+b x^{-\frac {3}{2} (m+1)}\right )^{2/3}}{m+1}\) |
\(\Big \downarrow \) 868 |
\(\displaystyle \frac {x^{m+1} \left (a+b x^{-\frac {3}{2} (m+1)}\right )^{2/3}}{m+1}-\frac {2 b \int \frac {1}{\sqrt [3]{b x^{\frac {3}{2} (-m-1)}+a}}dx^{\frac {1}{2} (-m-1)}}{m+1}\) |
\(\Big \downarrow \) 769 |
\(\displaystyle \frac {x^{m+1} \left (a+b x^{-\frac {3}{2} (m+1)}\right )^{2/3}}{m+1}-\frac {2 b \left (\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{b} x^{\frac {1}{2} (-m-1)}}{\sqrt [3]{a+b x^{\frac {3}{2} (-m-1)}}}+1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a+b x^{\frac {3}{2} (-m-1)}}-\sqrt [3]{b} x^{\frac {1}{2} (-m-1)}\right )}{2 \sqrt [3]{b}}\right )}{m+1}\) |
Input:
Int[x^m*(a + b/x^((3*(1 + m))/2))^(2/3),x]
Output:
(x^(1 + m)*(a + b/x^((3*(1 + m))/2))^(2/3))/(1 + m) - (2*b*(ArcTan[(1 + (2 *b^(1/3)*x^((-1 - m)/2))/(a + b*x^((3*(-1 - m))/2))^(1/3))/Sqrt[3]]/(Sqrt[ 3]*b^(1/3)) - Log[-(b^(1/3)*x^((-1 - m)/2)) + (a + b*x^((3*(-1 - m))/2))^( 1/3)]/(2*b^(1/3))))/(1 + m)
Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + 2*Rt[b, 3]* (x/(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sqrt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^ 3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/(m + 1) Subst[Int[(a + b*x^Simplify[n/(m + 1)])^p, x], x, x^(m + 1)], x] /; FreeQ[ {a, b, m, n, p}, x] && IntegerQ[Simplify[n/(m + 1)]] && !IntegerQ[n]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m + 1)*( (a + b*x^n)^p/(m + 1)), x] - Simp[b*n*(p/(m + 1)) Int[x^(m + n)*(a + b*x^ n)^(p - 1), x], x] /; FreeQ[{a, b, m, n}, x] && EqQ[(m + 1)/n + p, 0] && Gt Q[p, 0]
\[\int x^{m} \left (a +b \,x^{-\frac {3 m}{2}-\frac {3}{2}}\right )^{\frac {2}{3}}d x\]
Input:
int(x^m*(a+b/(x^(3/2+3/2*m)))^(2/3),x)
Output:
int(x^m*(a+b/(x^(3/2+3/2*m)))^(2/3),x)
Exception generated. \[ \int x^m \left (a+b x^{-\frac {3}{2} (1+m)}\right )^{2/3} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^m*(a+b/(x^(3/2+3/2*m)))^(2/3),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (has polynomial part)
Result contains complex when optimal does not.
Time = 4.62 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.94 \[ \int x^m \left (a+b x^{-\frac {3}{2} (1+m)}\right )^{2/3} \, dx=- \frac {2 a^{\frac {2 m}{3 m + 3} + \frac {2}{3} + \frac {2}{3 m + 3}} x^{m + 1} \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {2}{3} \\ - \frac {2 m}{3 m + 3} + 1 - \frac {2}{3 m + 3} \end {matrix}\middle | {\frac {b x^{- \frac {3 m}{2} - \frac {3}{2}} e^{i \pi }}{a}} \right )}}{3 a^{\frac {2}{3}} m \Gamma \left (- \frac {2 m}{3 m + 3} + 1 - \frac {2}{3 m + 3}\right ) + 3 a^{\frac {2}{3}} \Gamma \left (- \frac {2 m}{3 m + 3} + 1 - \frac {2}{3 m + 3}\right )} \] Input:
integrate(x**m*(a+b/(x**(3/2+3/2*m)))**(2/3),x)
Output:
-2*a**(2*m/(3*m + 3) + 2/3 + 2/(3*m + 3))*x**(m + 1)*gamma(-2/3)*hyper((-2 /3, -2/3), (-2*m/(3*m + 3) + 1 - 2/(3*m + 3),), b*x**(-3*m/2 - 3/2)*exp_po lar(I*pi)/a)/(3*a**(2/3)*m*gamma(-2*m/(3*m + 3) + 1 - 2/(3*m + 3)) + 3*a** (2/3)*gamma(-2*m/(3*m + 3) + 1 - 2/(3*m + 3)))
\[ \int x^m \left (a+b x^{-\frac {3}{2} (1+m)}\right )^{2/3} \, dx=\int { {\left (a + \frac {b}{x^{\frac {3}{2} \, m + \frac {3}{2}}}\right )}^{\frac {2}{3}} x^{m} \,d x } \] Input:
integrate(x^m*(a+b/(x^(3/2+3/2*m)))^(2/3),x, algorithm="maxima")
Output:
integrate((b*x^(-3/2*m - 3/2) + a)^(2/3)*x^m, x)
\[ \int x^m \left (a+b x^{-\frac {3}{2} (1+m)}\right )^{2/3} \, dx=\int { {\left (a + \frac {b}{x^{\frac {3}{2} \, m + \frac {3}{2}}}\right )}^{\frac {2}{3}} x^{m} \,d x } \] Input:
integrate(x^m*(a+b/(x^(3/2+3/2*m)))^(2/3),x, algorithm="giac")
Output:
integrate((a + b/x^(3/2*m + 3/2))^(2/3)*x^m, x)
Timed out. \[ \int x^m \left (a+b x^{-\frac {3}{2} (1+m)}\right )^{2/3} \, dx=\int x^m\,{\left (a+\frac {b}{x^{\frac {3\,m}{2}+\frac {3}{2}}}\right )}^{2/3} \,d x \] Input:
int(x^m*(a + b/x^((3*m)/2 + 3/2))^(2/3),x)
Output:
int(x^m*(a + b/x^((3*m)/2 + 3/2))^(2/3), x)
\[ \int x^m \left (a+b x^{-\frac {3}{2} (1+m)}\right )^{2/3} \, dx=\frac {\left (x^{\frac {3 m}{2}+\frac {1}{2}} a x +b \right )^{\frac {2}{3}}+\left (\int \frac {\left (x^{\frac {3 m}{2}+\frac {1}{2}} a x +b \right )^{\frac {2}{3}}}{x^{\frac {3 m}{2}+\frac {1}{2}} a \,x^{2}+b x}d x \right ) b m +\left (\int \frac {\left (x^{\frac {3 m}{2}+\frac {1}{2}} a x +b \right )^{\frac {2}{3}}}{x^{\frac {3 m}{2}+\frac {1}{2}} a \,x^{2}+b x}d x \right ) b}{m +1} \] Input:
int(x^m*(a+b/(x^(3/2+3/2*m)))^(2/3),x)
Output:
((x**((3*m + 1)/2)*a*x + b)**(2/3) + int((x**((3*m + 1)/2)*a*x + b)**(2/3) /(x**((3*m + 1)/2)*a*x**2 + b*x),x)*b*m + int((x**((3*m + 1)/2)*a*x + b)** (2/3)/(x**((3*m + 1)/2)*a*x**2 + b*x),x)*b)/(m + 1)