Integrand size = 17, antiderivative size = 139 \[ \int \frac {x^3}{\left (a+b x^{3/2}\right )^{2/3}} \, dx=-\frac {5 a x \sqrt [3]{a+b x^{3/2}}}{9 b^2}+\frac {x^{5/2} \sqrt [3]{a+b x^{3/2}}}{3 b}-\frac {10 a^2 \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} \sqrt {x}}{\sqrt [3]{a+b x^{3/2}}}}{\sqrt {3}}\right )}{9 \sqrt {3} b^{8/3}}-\frac {5 a^2 \log \left (\sqrt [3]{b} \sqrt {x}-\sqrt [3]{a+b x^{3/2}}\right )}{9 b^{8/3}} \]
-5/9*a*x*(a+b*x^(3/2))^(1/3)/b^2+1/3*x^(5/2)*(a+b*x^(3/2))^(1/3)/b-5/9*a^2 *ln(-(a+b*x^(3/2))^(1/3)+b^(1/3)*x^(1/2))/b^(8/3)-10/27*a^2*arctan(1/3*(1+ 2*b^(1/3)*x^(1/2)/(a+b*x^(3/2))^(1/3))*3^(1/2))/b^(8/3)*3^(1/2)
Time = 0.56 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.40 \[ \int \frac {x^3}{\left (a+b x^{3/2}\right )^{2/3}} \, dx=\frac {\sqrt [3]{a+b x^{3/2}} \left (-5 a x+3 b x^{5/2}\right )}{9 b^2}-\frac {10 a^2 \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} \sqrt {x}}{\sqrt [3]{b} \sqrt {x}+2 \sqrt [3]{a+b x^{3/2}}}\right )}{9 \sqrt {3} b^{8/3}}-\frac {10 a^2 \log \left (-\sqrt [3]{b} \sqrt {x}+\sqrt [3]{a+b x^{3/2}}\right )}{27 b^{8/3}}+\frac {5 a^2 \log \left (b^{2/3} x+\sqrt [3]{b} \sqrt {x} \sqrt [3]{a+b x^{3/2}}+\left (a+b x^{3/2}\right )^{2/3}\right )}{27 b^{8/3}} \]
((a + b*x^(3/2))^(1/3)*(-5*a*x + 3*b*x^(5/2)))/(9*b^2) - (10*a^2*ArcTan[(S qrt[3]*b^(1/3)*Sqrt[x])/(b^(1/3)*Sqrt[x] + 2*(a + b*x^(3/2))^(1/3))])/(9*S qrt[3]*b^(8/3)) - (10*a^2*Log[-(b^(1/3)*Sqrt[x]) + (a + b*x^(3/2))^(1/3)]) /(27*b^(8/3)) + (5*a^2*Log[b^(2/3)*x + b^(1/3)*Sqrt[x]*(a + b*x^(3/2))^(1/ 3) + (a + b*x^(3/2))^(2/3)])/(27*b^(8/3))
Time = 0.24 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.08, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {864, 843, 843, 853}
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.
\(\displaystyle \int \frac {x^3}{\left (a+b x^{3/2}\right )^{2/3}} \, dx\) |
\(\Big \downarrow \) 864 |
\(\displaystyle 2 \int \frac {x^{7/2}}{\left (b x^{3/2}+a\right )^{2/3}}d\sqrt {x}\) |
\(\Big \downarrow \) 843 |
\(\displaystyle 2 \left (\frac {x^{5/2} \sqrt [3]{a+b x^{3/2}}}{6 b}-\frac {5 a \int \frac {x^2}{\left (b x^{3/2}+a\right )^{2/3}}d\sqrt {x}}{6 b}\right )\) |
\(\Big \downarrow \) 843 |
\(\displaystyle 2 \left (\frac {x^{5/2} \sqrt [3]{a+b x^{3/2}}}{6 b}-\frac {5 a \left (\frac {x \sqrt [3]{a+b x^{3/2}}}{3 b}-\frac {2 a \int \frac {\sqrt {x}}{\left (b x^{3/2}+a\right )^{2/3}}d\sqrt {x}}{3 b}\right )}{6 b}\right )\) |
\(\Big \downarrow \) 853 |
\(\displaystyle 2 \left (\frac {x^{5/2} \sqrt [3]{a+b x^{3/2}}}{6 b}-\frac {5 a \left (\frac {x \sqrt [3]{a+b x^{3/2}}}{3 b}-\frac {2 a \left (-\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{b} \sqrt {x}}{\sqrt [3]{a+b x^{3/2}}}+1}{\sqrt {3}}\right )}{\sqrt {3} b^{2/3}}-\frac {\log \left (\sqrt [3]{b} \sqrt {x}-\sqrt [3]{a+b x^{3/2}}\right )}{2 b^{2/3}}\right )}{3 b}\right )}{6 b}\right )\) |
2*((x^(5/2)*(a + b*x^(3/2))^(1/3))/(6*b) - (5*a*((x*(a + b*x^(3/2))^(1/3)) /(3*b) - (2*a*(-(ArcTan[(1 + (2*b^(1/3)*Sqrt[x])/(a + b*x^(3/2))^(1/3))/Sq rt[3]]/(Sqrt[3]*b^(2/3))) - Log[b^(1/3)*Sqrt[x] - (a + b*x^(3/2))^(1/3)]/( 2*b^(2/3))))/(3*b)))/(6*b))
3.23.71.3.1 Defintions of rubi rules used
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Simp[ a*c^n*((m - n + 1)/(b*(m + n*p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^p, x] , x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n* p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]
Int[(x_)/((a_) + (b_.)*(x_)^3)^(2/3), x_Symbol] :> With[{q = Rt[b, 3]}, Sim p[-ArcTan[(1 + 2*q*(x/(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sqrt[3]*q^2), x] - Simp [Log[q*x - (a + b*x^3)^(1/3)]/(2*q^2), x]] /; FreeQ[{a, b}, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denomi nator[n]}, Simp[k Subst[Int[x^(k*(m + 1) - 1)*(a + b*x^(k*n))^p, x], x, x ^(1/k)], x]] /; FreeQ[{a, b, m, p}, x] && FractionQ[n]
\[\int \frac {x^{3}}{\left (a +b \,x^{\frac {3}{2}}\right )^{\frac {2}{3}}}d x\]
Timed out. \[ \int \frac {x^3}{\left (a+b x^{3/2}\right )^{2/3}} \, dx=\text {Timed out} \]
Result contains complex when optimal does not.
Time = 1.84 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.29 \[ \int \frac {x^3}{\left (a+b x^{3/2}\right )^{2/3}} \, dx=\frac {2 x^{4} \Gamma \left (\frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {2}{3}, \frac {8}{3} \\ \frac {11}{3} \end {matrix}\middle | {\frac {b x^{\frac {3}{2}} e^{i \pi }}{a}} \right )}}{3 a^{\frac {2}{3}} \Gamma \left (\frac {11}{3}\right )} \]
2*x**4*gamma(8/3)*hyper((2/3, 8/3), (11/3,), b*x**(3/2)*exp_polar(I*pi)/a) /(3*a**(2/3)*gamma(11/3))
Time = 0.29 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.32 \[ \int \frac {x^3}{\left (a+b x^{3/2}\right )^{2/3}} \, dx=\frac {10 \, \sqrt {3} a^{2} \arctan \left (\frac {\sqrt {3} {\left (b^{\frac {1}{3}} + \frac {2 \, {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {1}{3}}}{\sqrt {x}}\right )}}{3 \, b^{\frac {1}{3}}}\right )}{27 \, b^{\frac {8}{3}}} + \frac {5 \, a^{2} \log \left (b^{\frac {2}{3}} + \frac {{\left (b x^{\frac {3}{2}} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}}}{\sqrt {x}} + \frac {{\left (b x^{\frac {3}{2}} + a\right )}^{\frac {2}{3}}}{x}\right )}{27 \, b^{\frac {8}{3}}} - \frac {10 \, a^{2} \log \left (-b^{\frac {1}{3}} + \frac {{\left (b x^{\frac {3}{2}} + a\right )}^{\frac {1}{3}}}{\sqrt {x}}\right )}{27 \, b^{\frac {8}{3}}} + \frac {\frac {8 \, {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {1}{3}} a^{2} b}{\sqrt {x}} - \frac {5 \, {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {4}{3}} a^{2}}{x^{2}}}{9 \, {\left (b^{4} - \frac {2 \, {\left (b x^{\frac {3}{2}} + a\right )} b^{3}}{x^{\frac {3}{2}}} + \frac {{\left (b x^{\frac {3}{2}} + a\right )}^{2} b^{2}}{x^{3}}\right )}} \]
10/27*sqrt(3)*a^2*arctan(1/3*sqrt(3)*(b^(1/3) + 2*(b*x^(3/2) + a)^(1/3)/sq rt(x))/b^(1/3))/b^(8/3) + 5/27*a^2*log(b^(2/3) + (b*x^(3/2) + a)^(1/3)*b^( 1/3)/sqrt(x) + (b*x^(3/2) + a)^(2/3)/x)/b^(8/3) - 10/27*a^2*log(-b^(1/3) + (b*x^(3/2) + a)^(1/3)/sqrt(x))/b^(8/3) + 1/9*(8*(b*x^(3/2) + a)^(1/3)*a^2 *b/sqrt(x) - 5*(b*x^(3/2) + a)^(4/3)*a^2/x^2)/(b^4 - 2*(b*x^(3/2) + a)*b^3 /x^(3/2) + (b*x^(3/2) + a)^2*b^2/x^3)
\[ \int \frac {x^3}{\left (a+b x^{3/2}\right )^{2/3}} \, dx=\int { \frac {x^{3}}{{\left (b x^{\frac {3}{2}} + a\right )}^{\frac {2}{3}}} \,d x } \]
Timed out. \[ \int \frac {x^3}{\left (a+b x^{3/2}\right )^{2/3}} \, dx=\int \frac {x^3}{{\left (a+b\,x^{3/2}\right )}^{2/3}} \,d x \]