Integrand size = 28, antiderivative size = 153 \[ \int \frac {x^2 \left (a+b x^3\right )^{2/3}}{a d-b d x^3} \, dx=-\frac {\left (a+b x^3\right )^{2/3}}{2 b d}-\frac {2^{2/3} a^{2/3} \arctan \left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b d}+\frac {a^{2/3} \log \left (a-b x^3\right )}{3 \sqrt [3]{2} b d}-\frac {a^{2/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} b d} \]
-1/2*(b*x^3+a)^(2/3)/b/d+1/6*a^(2/3)*ln(-b*x^3+a)*2^(2/3)/b/d-1/2*a^(2/3)* ln(2^(1/3)*a^(1/3)-(b*x^3+a)^(1/3))*2^(2/3)/b/d-1/3*2^(2/3)*a^(2/3)*arctan (1/3*(a^(1/3)+2^(2/3)*(b*x^3+a)^(1/3))/a^(1/3)*3^(1/2))/b/d*3^(1/2)
Time = 0.28 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.11 \[ \int \frac {x^2 \left (a+b x^3\right )^{2/3}}{a d-b d x^3} \, dx=-\frac {3 \left (a+b x^3\right )^{2/3}+2\ 2^{2/3} \sqrt {3} a^{2/3} \arctan \left (\frac {1+\frac {2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+2\ 2^{2/3} a^{2/3} \log \left (-2 \sqrt [3]{a}+2^{2/3} \sqrt [3]{a+b x^3}\right )-2^{2/3} a^{2/3} \log \left (2 a^{2/3}+2^{2/3} \sqrt [3]{a} \sqrt [3]{a+b x^3}+\sqrt [3]{2} \left (a+b x^3\right )^{2/3}\right )}{6 b d} \]
-1/6*(3*(a + b*x^3)^(2/3) + 2*2^(2/3)*Sqrt[3]*a^(2/3)*ArcTan[(1 + (2^(2/3) *(a + b*x^3)^(1/3))/a^(1/3))/Sqrt[3]] + 2*2^(2/3)*a^(2/3)*Log[-2*a^(1/3) + 2^(2/3)*(a + b*x^3)^(1/3)] - 2^(2/3)*a^(2/3)*Log[2*a^(2/3) + 2^(2/3)*a^(1 /3)*(a + b*x^3)^(1/3) + 2^(1/3)*(a + b*x^3)^(2/3)])/(b*d)
Time = 0.27 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.98, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {946, 27, 60, 67, 16, 1082, 217}
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.
| \(\displaystyle \int \frac {x^2 \left (a+b x^3\right )^{2/3}}{a d-b d x^3} \, dx\) |
| \(\Big \downarrow \) 946 |
| \(\displaystyle \frac {1}{3} \int \frac {\left (b x^3+a\right )^{2/3}}{d \left (a-b x^3\right )}dx^3\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\left (b x^3+a\right )^{2/3}}{a-b x^3}dx^3}{3 d}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {2 a \int \frac {1}{\left (a-b x^3\right ) \sqrt [3]{b x^3+a}}dx^3-\frac {3 \left (a+b x^3\right )^{2/3}}{2 b}}{3 d}\) |
| \(\Big \downarrow \) 67 |
| \(\displaystyle \frac {2 a \left (-\frac {3 \int \frac {1}{x^6+2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}}{2 b}+\frac {3 \int \frac {1}{\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}}{2 \sqrt [3]{2} \sqrt [3]{a} b}+\frac {\log \left (a-b x^3\right )}{2 \sqrt [3]{2} \sqrt [3]{a} b}\right )-\frac {3 \left (a+b x^3\right )^{2/3}}{2 b}}{3 d}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {2 a \left (-\frac {3 \int \frac {1}{x^6+2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}}{2 b}+\frac {\log \left (a-b x^3\right )}{2 \sqrt [3]{2} \sqrt [3]{a} b}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{2} \sqrt [3]{a} b}\right )-\frac {3 \left (a+b x^3\right )^{2/3}}{2 b}}{3 d}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {2 a \left (\frac {3 \int \frac {1}{-x^6-3}d\left (\frac {2^{2/3} \sqrt [3]{b x^3+a}}{\sqrt [3]{a}}+1\right )}{\sqrt [3]{2} \sqrt [3]{a} b}+\frac {\log \left (a-b x^3\right )}{2 \sqrt [3]{2} \sqrt [3]{a} b}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{2} \sqrt [3]{a} b}\right )-\frac {3 \left (a+b x^3\right )^{2/3}}{2 b}}{3 d}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2 a \left (-\frac {\sqrt {3} \arctan \left (\frac {\frac {2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}+1}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{a} b}+\frac {\log \left (a-b x^3\right )}{2 \sqrt [3]{2} \sqrt [3]{a} b}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{2} \sqrt [3]{a} b}\right )-\frac {3 \left (a+b x^3\right )^{2/3}}{2 b}}{3 d}\) |
((-3*(a + b*x^3)^(2/3))/(2*b) + 2*a*(-((Sqrt[3]*ArcTan[(1 + (2^(2/3)*(a + b*x^3)^(1/3))/a^(1/3))/Sqrt[3]])/(2^(1/3)*a^(1/3)*b)) + Log[a - b*x^3]/(2* 2^(1/3)*a^(1/3)*b) - (3*Log[2^(1/3)*a^(1/3) - (a + b*x^3)^(1/3)])/(2*2^(1/ 3)*a^(1/3)*b)))/(3*d)
3.6.89.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, 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_))^(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[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[ {q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q), x ] + (Simp[3/(2*b) Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/3)], x] - Simp[3/(2*b*q) Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] / ; FreeQ[{a, b, c, d}, x] && PosQ[(b*c - a*d)/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. ), x_Symbol] :> Simp[1/n Subst[Int[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m - n + 1, 0]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Time = 4.59 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.82
| method | result | size |
| pseudoelliptic | \(\frac {-2 a^{\frac {2}{3}} \sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\left (a^{\frac {1}{3}}+2^{\frac {2}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right )-2 a^{\frac {2}{3}} 2^{\frac {2}{3}} \ln \left (\left (b \,x^{3}+a \right )^{\frac {1}{3}}-2^{\frac {1}{3}} a^{\frac {1}{3}}\right )+a^{\frac {2}{3}} 2^{\frac {2}{3}} \ln \left (\left (b \,x^{3}+a \right )^{\frac {2}{3}}+2^{\frac {1}{3}} a^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}}+2^{\frac {2}{3}} a^{\frac {2}{3}}\right )-3 \left (b \,x^{3}+a \right )^{\frac {2}{3}}}{6 b d}\) | \(126\) |
1/6*(-2*a^(2/3)*3^(1/2)*2^(2/3)*arctan(1/3*(a^(1/3)+2^(2/3)*(b*x^3+a)^(1/3 ))/a^(1/3)*3^(1/2))-2*a^(2/3)*2^(2/3)*ln((b*x^3+a)^(1/3)-2^(1/3)*a^(1/3))+ a^(2/3)*2^(2/3)*ln((b*x^3+a)^(2/3)+2^(1/3)*a^(1/3)*(b*x^3+a)^(1/3)+2^(2/3) *a^(2/3))-3*(b*x^3+a)^(2/3))/b/d
Time = 0.36 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.09 \[ \int \frac {x^2 \left (a+b x^3\right )^{2/3}}{a d-b d x^3} \, dx=-\frac {2 \cdot 4^{\frac {1}{3}} \sqrt {3} \left (-a^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {4^{\frac {1}{3}} \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-a^{2}\right )^{\frac {1}{3}} - \sqrt {3} a}{3 \, a}\right ) + 4^{\frac {1}{3}} \left (-a^{2}\right )^{\frac {1}{3}} \log \left (4^{\frac {2}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-a^{2}\right )^{\frac {2}{3}} + 2 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} a - 2 \cdot 4^{\frac {1}{3}} \left (-a^{2}\right )^{\frac {1}{3}} a\right ) - 2 \cdot 4^{\frac {1}{3}} \left (-a^{2}\right )^{\frac {1}{3}} \log \left (-4^{\frac {2}{3}} \left (-a^{2}\right )^{\frac {2}{3}} + 2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} a\right ) + 3 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{6 \, b d} \]
-1/6*(2*4^(1/3)*sqrt(3)*(-a^2)^(1/3)*arctan(1/3*(4^(1/3)*sqrt(3)*(b*x^3 + a)^(1/3)*(-a^2)^(1/3) - sqrt(3)*a)/a) + 4^(1/3)*(-a^2)^(1/3)*log(4^(2/3)*( b*x^3 + a)^(1/3)*(-a^2)^(2/3) + 2*(b*x^3 + a)^(2/3)*a - 2*4^(1/3)*(-a^2)^( 1/3)*a) - 2*4^(1/3)*(-a^2)^(1/3)*log(-4^(2/3)*(-a^2)^(2/3) + 2*(b*x^3 + a) ^(1/3)*a) + 3*(b*x^3 + a)^(2/3))/(b*d)
\[ \int \frac {x^2 \left (a+b x^3\right )^{2/3}}{a d-b d x^3} \, dx=- \frac {\int \frac {x^{2} \left (a + b x^{3}\right )^{\frac {2}{3}}}{- a + b x^{3}}\, dx}{d} \]
Time = 0.29 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.92 \[ \int \frac {x^2 \left (a+b x^3\right )^{2/3}}{a d-b d x^3} \, dx=-\frac {\frac {2 \, \sqrt {3} 2^{\frac {2}{3}} a^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} a^{\frac {1}{3}} + 2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}\right )}}{6 \, a^{\frac {1}{3}}}\right )}{d} - \frac {2^{\frac {2}{3}} a^{\frac {2}{3}} \log \left (2^{\frac {2}{3}} a^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (b x^{3} + a\right )}^{\frac {2}{3}}\right )}{d} + \frac {2 \cdot 2^{\frac {2}{3}} a^{\frac {2}{3}} \log \left (-2^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}}\right )}{d} + \frac {3 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{d}}{6 \, b} \]
-1/6*(2*sqrt(3)*2^(2/3)*a^(2/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3)*a^(1/3 ) + 2*(b*x^3 + a)^(1/3))/a^(1/3))/d - 2^(2/3)*a^(2/3)*log(2^(2/3)*a^(2/3) + 2^(1/3)*(b*x^3 + a)^(1/3)*a^(1/3) + (b*x^3 + a)^(2/3))/d + 2*2^(2/3)*a^( 2/3)*log(-2^(1/3)*a^(1/3) + (b*x^3 + a)^(1/3))/d + 3*(b*x^3 + a)^(2/3)/d)/ b
Time = 0.76 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.97 \[ \int \frac {x^2 \left (a+b x^3\right )^{2/3}}{a d-b d x^3} \, dx=-\frac {\sqrt {3} 2^{\frac {2}{3}} a^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} a^{\frac {1}{3}} + 2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}\right )}}{6 \, a^{\frac {1}{3}}}\right )}{3 \, b d} + \frac {2^{\frac {2}{3}} a^{\frac {2}{3}} \log \left (2^{\frac {2}{3}} a^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (b x^{3} + a\right )}^{\frac {2}{3}}\right )}{6 \, b d} - \frac {2^{\frac {2}{3}} a^{\frac {2}{3}} \log \left ({\left | -2^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} \right |}\right )}{3 \, b d} - \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{2 \, b d} \]
-1/3*sqrt(3)*2^(2/3)*a^(2/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3)*a^(1/3) + 2*(b*x^3 + a)^(1/3))/a^(1/3))/(b*d) + 1/6*2^(2/3)*a^(2/3)*log(2^(2/3)*a^( 2/3) + 2^(1/3)*(b*x^3 + a)^(1/3)*a^(1/3) + (b*x^3 + a)^(2/3))/(b*d) - 1/3* 2^(2/3)*a^(2/3)*log(abs(-2^(1/3)*a^(1/3) + (b*x^3 + a)^(1/3)))/(b*d) - 1/2 *(b*x^3 + a)^(2/3)/(b*d)
Time = 8.66 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.22 \[ \int \frac {x^2 \left (a+b x^3\right )^{2/3}}{a d-b d x^3} \, dx=-\frac {{\left (b\,x^3+a\right )}^{2/3}}{2\,b\,d}-\frac {4^{1/3}\,a^{2/3}\,\ln \left ({\left (b\,x^3+a\right )}^{1/3}-2^{1/3}\,a^{1/3}\right )}{3\,b\,d}-\frac {4^{1/3}\,a^{2/3}\,\ln \left (\frac {4\,a^2\,{\left (b\,x^3+a\right )}^{1/3}}{b^2\,d^2}-\frac {2\,4^{2/3}\,a^{7/3}\,{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2}{b^2\,d^2}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,b\,d}+\frac {4^{1/3}\,a^{2/3}\,\ln \left (\frac {4\,a^2\,{\left (b\,x^3+a\right )}^{1/3}}{b^2\,d^2}-\frac {18\,4^{2/3}\,a^{7/3}\,{\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}^2}{b^2\,d^2}\right )\,\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}{b\,d} \]
(4^(1/3)*a^(2/3)*log((4*a^2*(a + b*x^3)^(1/3))/(b^2*d^2) - (18*4^(2/3)*a^( 7/3)*((3^(1/2)*1i)/6 + 1/6)^2)/(b^2*d^2))*((3^(1/2)*1i)/6 + 1/6))/(b*d) - (4^(1/3)*a^(2/3)*log((a + b*x^3)^(1/3) - 2^(1/3)*a^(1/3)))/(3*b*d) - (4^(1 /3)*a^(2/3)*log((4*a^2*(a + b*x^3)^(1/3))/(b^2*d^2) - (2*4^(2/3)*a^(7/3)*( (3^(1/2)*1i)/2 - 1/2)^2)/(b^2*d^2))*((3^(1/2)*1i)/2 - 1/2))/(3*b*d) - (a + b*x^3)^(2/3)/(2*b*d)