Integrand size = 28, antiderivative size = 512 \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^5 \left (a d-b d x^3\right )} \, dx=-\frac {\left (a+b x^3\right )^{2/3}}{4 a d x^4}-\frac {3 b \left (a+b x^3\right )^{2/3}}{2 a^2 d x}+\frac {2^{2/3} b^{4/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} a^{5/3} d}+\frac {b^{4/3} \arctan \left (\frac {1+\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3} a^{5/3} d}+\frac {3 b^2 x^2 \sqrt [3]{1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},-\frac {b x^3}{a}\right )}{4 a^2 d \sqrt [3]{a+b x^3}}+\frac {b^{4/3} \log \left (\frac {\left (\sqrt [3]{a}-\sqrt [3]{b} x\right )^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a}\right )}{6 \sqrt [3]{2} a^{5/3} d}+\frac {b^{4/3} \log \left (1+\frac {2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}\right )}{3 \sqrt [3]{2} a^{5/3} d}-\frac {2^{2/3} b^{4/3} \log \left (1+\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}\right )}{3 a^{5/3} d}-\frac {b^{4/3} \log \left (\frac {\sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a}}-\frac {2^{2/3} \sqrt [3]{b} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}\right )}{2 \sqrt [3]{2} a^{5/3} d} \]
-1/4*(b*x^3+a)^(2/3)/a/d/x^4-3/2*b*(b*x^3+a)^(2/3)/a^2/d/x+3/4*b^2*x^2*(1+ b*x^3/a)^(1/3)*hypergeom([1/3, 2/3],[5/3],-b*x^3/a)/a^2/d/(b*x^3+a)^(1/3)+ 1/12*b^(4/3)*ln((a^(1/3)-b^(1/3)*x)^2*(a^(1/3)+b^(1/3)*x)/a)*2^(2/3)/a^(5/ 3)/d+1/6*b^(4/3)*ln(1+2^(2/3)*(a^(1/3)+b^(1/3)*x)^2/(b*x^3+a)^(2/3)-2^(1/3 )*(a^(1/3)+b^(1/3)*x)/(b*x^3+a)^(1/3))*2^(2/3)/a^(5/3)/d-1/3*2^(2/3)*b^(4/ 3)*ln(1+2^(1/3)*(a^(1/3)+b^(1/3)*x)/(b*x^3+a)^(1/3))/a^(5/3)/d-1/4*b^(4/3) *ln(b^(1/3)*(a^(1/3)+b^(1/3)*x)/a^(1/3)-2^(2/3)*b^(1/3)*(b*x^3+a)^(1/3)/a^ (1/3))*2^(2/3)/a^(5/3)/d+1/3*2^(2/3)*b^(4/3)*arctan(1/3*(1-2*2^(1/3)*(a^(1 /3)+b^(1/3)*x)/(b*x^3+a)^(1/3))*3^(1/2))/a^(5/3)/d*3^(1/2)+1/6*b^(4/3)*arc tan(1/3*(1+2^(1/3)*(a^(1/3)+b^(1/3)*x)/(b*x^3+a)^(1/3))*3^(1/2))*2^(2/3)/a ^(5/3)/d*3^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
Time = 11.09 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.29 \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^5 \left (a d-b d x^3\right )} \, dx=\frac {-5 a \left (a^2+7 a b x^3+6 b^2 x^6\right )+35 a b^2 x^6 \sqrt [3]{1+\frac {b x^3}{a}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-\frac {b x^3}{a},\frac {b x^3}{a}\right )-6 b^3 x^9 \sqrt [3]{1+\frac {b x^3}{a}} \operatorname {AppellF1}\left (\frac {5}{3},\frac {1}{3},1,\frac {8}{3},-\frac {b x^3}{a},\frac {b x^3}{a}\right )}{20 a^3 d x^4 \sqrt [3]{a+b x^3}} \]
(-5*a*(a^2 + 7*a*b*x^3 + 6*b^2*x^6) + 35*a*b^2*x^6*(1 + (b*x^3)/a)^(1/3)*A ppellF1[2/3, 1/3, 1, 5/3, -((b*x^3)/a), (b*x^3)/a] - 6*b^3*x^9*(1 + (b*x^3 )/a)^(1/3)*AppellF1[5/3, 1/3, 1, 8/3, -((b*x^3)/a), (b*x^3)/a])/(20*a^3*d* x^4*(a + b*x^3)^(1/3))
Time = 0.71 (sec) , antiderivative size = 496, normalized size of antiderivative = 0.97, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {975, 27, 1053, 25, 27, 1054, 2009}
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.
\(\displaystyle \int \frac {\left (a+b x^3\right )^{2/3}}{x^5 \left (a d-b d x^3\right )} \, dx\) |
\(\Big \downarrow \) 975 |
\(\displaystyle \frac {\int \frac {2 b \left (b x^3+3 a\right )}{x^2 \left (a-b x^3\right ) \sqrt [3]{b x^3+a}}dx}{4 a d}-\frac {\left (a+b x^3\right )^{2/3}}{4 a d x^4}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {b \int \frac {b x^3+3 a}{x^2 \left (a-b x^3\right ) \sqrt [3]{b x^3+a}}dx}{2 a d}-\frac {\left (a+b x^3\right )^{2/3}}{4 a d x^4}\) |
\(\Big \downarrow \) 1053 |
\(\displaystyle \frac {b \left (-\frac {\int -\frac {a b x \left (7 a-3 b x^3\right )}{\left (a-b x^3\right ) \sqrt [3]{b x^3+a}}dx}{a^2}-\frac {3 \left (a+b x^3\right )^{2/3}}{a x}\right )}{2 a d}-\frac {\left (a+b x^3\right )^{2/3}}{4 a d x^4}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {b \left (\frac {\int \frac {a b x \left (7 a-3 b x^3\right )}{\left (a-b x^3\right ) \sqrt [3]{b x^3+a}}dx}{a^2}-\frac {3 \left (a+b x^3\right )^{2/3}}{a x}\right )}{2 a d}-\frac {\left (a+b x^3\right )^{2/3}}{4 a d x^4}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {b \left (\frac {b \int \frac {x \left (7 a-3 b x^3\right )}{\left (a-b x^3\right ) \sqrt [3]{b x^3+a}}dx}{a}-\frac {3 \left (a+b x^3\right )^{2/3}}{a x}\right )}{2 a d}-\frac {\left (a+b x^3\right )^{2/3}}{4 a d x^4}\) |
\(\Big \downarrow \) 1054 |
\(\displaystyle \frac {b \left (\frac {b \int \left (\frac {4 a x}{\left (a-b x^3\right ) \sqrt [3]{b x^3+a}}+\frac {3 x}{\sqrt [3]{b x^3+a}}\right )dx}{a}-\frac {3 \left (a+b x^3\right )^{2/3}}{a x}\right )}{2 a d}-\frac {\left (a+b x^3\right )^{2/3}}{4 a d x^4}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {b \left (\frac {b \left (\frac {2\ 2^{2/3} \sqrt [3]{a} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} b^{2/3}}+\frac {2^{2/3} \sqrt [3]{a} \arctan \left (\frac {\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} b^{2/3}}+\frac {2^{2/3} \sqrt [3]{a} \log \left (\frac {2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{3 b^{2/3}}-\frac {2\ 2^{2/3} \sqrt [3]{a} \log \left (\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{3 b^{2/3}}-\frac {\sqrt [3]{a} \log \left (\frac {\sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a}}-\frac {2^{2/3} \sqrt [3]{b} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}\right )}{\sqrt [3]{2} b^{2/3}}+\frac {\sqrt [3]{a} \log \left (\frac {\left (\sqrt [3]{a}-\sqrt [3]{b} x\right )^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a}\right )}{3 \sqrt [3]{2} b^{2/3}}+\frac {3 x^2 \sqrt [3]{\frac {b x^3}{a}+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},-\frac {b x^3}{a}\right )}{2 \sqrt [3]{a+b x^3}}\right )}{a}-\frac {3 \left (a+b x^3\right )^{2/3}}{a x}\right )}{2 a d}-\frac {\left (a+b x^3\right )^{2/3}}{4 a d x^4}\) |
-1/4*(a + b*x^3)^(2/3)/(a*d*x^4) + (b*((-3*(a + b*x^3)^(2/3))/(a*x) + (b*( (2*2^(2/3)*a^(1/3)*ArcTan[(1 - (2*2^(1/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^ 3)^(1/3))/Sqrt[3]])/(Sqrt[3]*b^(2/3)) + (2^(2/3)*a^(1/3)*ArcTan[(1 + (2^(1 /3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3))/Sqrt[3]])/(Sqrt[3]*b^(2/3)) + (3*x^2*(1 + (b*x^3)/a)^(1/3)*Hypergeometric2F1[1/3, 2/3, 5/3, -((b*x^3)/ a)])/(2*(a + b*x^3)^(1/3)) + (a^(1/3)*Log[((a^(1/3) - b^(1/3)*x)^2*(a^(1/3 ) + b^(1/3)*x))/a])/(3*2^(1/3)*b^(2/3)) + (2^(2/3)*a^(1/3)*Log[1 + (2^(2/3 )*(a^(1/3) + b^(1/3)*x)^2)/(a + b*x^3)^(2/3) - (2^(1/3)*(a^(1/3) + b^(1/3) *x))/(a + b*x^3)^(1/3)])/(3*b^(2/3)) - (2*2^(2/3)*a^(1/3)*Log[1 + (2^(1/3) *(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3)])/(3*b^(2/3)) - (a^(1/3)*Log[(b^ (1/3)*(a^(1/3) + b^(1/3)*x))/a^(1/3) - (2^(2/3)*b^(1/3)*(a + b*x^3)^(1/3)) /a^(1/3)])/(2^(1/3)*b^(2/3))))/a))/(2*a*d)
3.7.4.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_) )^(q_), x_Symbol] :> Simp[(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/ (a*e*(m + 1))), x] - Simp[1/(a*e^n*(m + 1)) Int[(e*x)^(m + n)*(a + b*x^n) ^p*(c + d*x^n)^(q - 1)*Simp[c*b*(m + 1) + n*(b*c*(p + 1) + a*d*q) + d*(b*(m + 1) + b*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[0, q, 1] && LtQ[m, -1] && IntBinomi alQ[a, b, c, d, e, m, n, p, q, x]
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_ ))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b *x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^n*( m + 1)) Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2 ) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && IGtQ[n, 0] && LtQ[m, -1]
Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n _)))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*((e + f*x^n)/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && IGtQ[n, 0]
\[\int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{5} \left (-b d \,x^{3}+a d \right )}d x\]
Timed out. \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^5 \left (a d-b d x^3\right )} \, dx=\text {Timed out} \]
\[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^5 \left (a d-b d x^3\right )} \, dx=- \frac {\int \frac {\left (a + b x^{3}\right )^{\frac {2}{3}}}{- a x^{5} + b x^{8}}\, dx}{d} \]
\[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^5 \left (a d-b d x^3\right )} \, dx=\int { -\frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{{\left (b d x^{3} - a d\right )} x^{5}} \,d x } \]
\[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^5 \left (a d-b d x^3\right )} \, dx=\int { -\frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{{\left (b d x^{3} - a d\right )} x^{5}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^5 \left (a d-b d x^3\right )} \, dx=\int \frac {{\left (b\,x^3+a\right )}^{2/3}}{x^5\,\left (a\,d-b\,d\,x^3\right )} \,d x \]