Integrand size = 24, antiderivative size = 370 \[ \int \frac {\sqrt [3]{a+b x^3}}{x^7 \left (c+d x^3\right )} \, dx=\frac {(b c+3 a d) \sqrt [3]{a+b x^3}}{9 a c^2 x^3}-\frac {\left (a+b x^3\right )^{4/3}}{6 a c x^6}+\frac {\left (b^2 c^2+3 a b c d-9 a^2 d^2\right ) \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x^3}}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{5/3} c^3}-\frac {d^{5/3} \sqrt [3]{b c-a d} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{\sqrt {3} c^3}+\frac {\left (b^2 c^2+3 a b c d-9 a^2 d^2\right ) \log (x)}{18 a^{5/3} c^3}-\frac {d^{5/3} \sqrt [3]{b c-a d} \log \left (c+d x^3\right )}{6 c^3}-\frac {\left (b^2 c^2+3 a b c d-9 a^2 d^2\right ) \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{18 a^{5/3} c^3}+\frac {d^{5/3} \sqrt [3]{b c-a d} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 c^3} \]
1/9*(3*a*d+b*c)*(b*x^3+a)^(1/3)/a/c^2/x^3-1/6*(b*x^3+a)^(4/3)/a/c/x^6+1/18 *(-9*a^2*d^2+3*a*b*c*d+b^2*c^2)*ln(x)/a^(5/3)/c^3-1/6*d^(5/3)*(-a*d+b*c)^( 1/3)*ln(d*x^3+c)/c^3-1/18*(-9*a^2*d^2+3*a*b*c*d+b^2*c^2)*ln(a^(1/3)-(b*x^3 +a)^(1/3))/a^(5/3)/c^3+1/2*d^(5/3)*(-a*d+b*c)^(1/3)*ln((-a*d+b*c)^(1/3)+d^ (1/3)*(b*x^3+a)^(1/3))/c^3+1/27*(-9*a^2*d^2+3*a*b*c*d+b^2*c^2)*arctan(1/3* (a^(1/3)+2*(b*x^3+a)^(1/3))/a^(1/3)*3^(1/2))/a^(5/3)/c^3*3^(1/2)-1/3*d^(5/ 3)*(-a*d+b*c)^(1/3)*arctan(1/3*(1-2*d^(1/3)*(b*x^3+a)^(1/3)/(-a*d+b*c)^(1/ 3))*3^(1/2))/c^3*3^(1/2)
Time = 1.27 (sec) , antiderivative size = 413, normalized size of antiderivative = 1.12 \[ \int \frac {\sqrt [3]{a+b x^3}}{x^7 \left (c+d x^3\right )} \, dx=\frac {\frac {3 c \sqrt [3]{a+b x^3} \left (-3 a c-b c x^3+6 a d x^3\right )}{a x^6}+\frac {2 \sqrt {3} \left (b^2 c^2+3 a b c d-9 a^2 d^2\right ) \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{5/3}}-18 \sqrt {3} d^{5/3} \sqrt [3]{b c-a d} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )-\frac {2 \left (b^2 c^2+3 a b c d-9 a^2 d^2\right ) \log \left (-\sqrt [3]{a}+\sqrt [3]{a+b x^3}\right )}{a^{5/3}}+18 d^{5/3} \sqrt [3]{b c-a d} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )+\frac {\left (b^2 c^2+3 a b c d-9 a^2 d^2\right ) \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )}{a^{5/3}}-9 d^{5/3} \sqrt [3]{b c-a d} \log \left ((b c-a d)^{2/3}-\sqrt [3]{d} \sqrt [3]{b c-a d} \sqrt [3]{a+b x^3}+d^{2/3} \left (a+b x^3\right )^{2/3}\right )}{54 c^3} \]
((3*c*(a + b*x^3)^(1/3)*(-3*a*c - b*c*x^3 + 6*a*d*x^3))/(a*x^6) + (2*Sqrt[ 3]*(b^2*c^2 + 3*a*b*c*d - 9*a^2*d^2)*ArcTan[(1 + (2*(a + b*x^3)^(1/3))/a^( 1/3))/Sqrt[3]])/a^(5/3) - 18*Sqrt[3]*d^(5/3)*(b*c - a*d)^(1/3)*ArcTan[(1 - (2*d^(1/3)*(a + b*x^3)^(1/3))/(b*c - a*d)^(1/3))/Sqrt[3]] - (2*(b^2*c^2 + 3*a*b*c*d - 9*a^2*d^2)*Log[-a^(1/3) + (a + b*x^3)^(1/3)])/a^(5/3) + 18*d^ (5/3)*(b*c - a*d)^(1/3)*Log[(b*c - a*d)^(1/3) + d^(1/3)*(a + b*x^3)^(1/3)] + ((b^2*c^2 + 3*a*b*c*d - 9*a^2*d^2)*Log[a^(2/3) + a^(1/3)*(a + b*x^3)^(1 /3) + (a + b*x^3)^(2/3)])/a^(5/3) - 9*d^(5/3)*(b*c - a*d)^(1/3)*Log[(b*c - a*d)^(2/3) - d^(1/3)*(b*c - a*d)^(1/3)*(a + b*x^3)^(1/3) + d^(2/3)*(a + b *x^3)^(2/3)])/(54*c^3)
Time = 0.49 (sec) , antiderivative size = 345, normalized size of antiderivative = 0.93, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {948, 114, 27, 166, 27, 174, 69, 16, 70, 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.
| \(\displaystyle \int \frac {\sqrt [3]{a+b x^3}}{x^7 \left (c+d x^3\right )} \, dx\) |
| \(\Big \downarrow \) 948 |
| \(\displaystyle \frac {1}{3} \int \frac {\sqrt [3]{b x^3+a}}{x^9 \left (d x^3+c\right )}dx^3\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle \frac {1}{3} \left (-\frac {\int \frac {2 \sqrt [3]{b x^3+a} \left (b d x^3+b c+3 a d\right )}{3 x^6 \left (d x^3+c\right )}dx^3}{2 a c}-\frac {\left (a+b x^3\right )^{4/3}}{2 a c x^6}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (-\frac {\int \frac {\sqrt [3]{b x^3+a} \left (b d x^3+b c+3 a d\right )}{x^6 \left (d x^3+c\right )}dx^3}{3 a c}-\frac {\left (a+b x^3\right )^{4/3}}{2 a c x^6}\right )\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{3} \left (-\frac {\frac {\int \frac {b d (b c-6 a d) x^3+b^2 c^2-9 a^2 d^2+3 a b c d}{3 x^3 \left (b x^3+a\right )^{2/3} \left (d x^3+c\right )}dx^3}{c}-\frac {\sqrt [3]{a+b x^3} (3 a d+b c)}{c x^3}}{3 a c}-\frac {\left (a+b x^3\right )^{4/3}}{2 a c x^6}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (-\frac {\frac {\int \frac {b d (b c-6 a d) x^3+b^2 c^2-9 a^2 d^2+3 a b c d}{x^3 \left (b x^3+a\right )^{2/3} \left (d x^3+c\right )}dx^3}{3 c}-\frac {\sqrt [3]{a+b x^3} (3 a d+b c)}{c x^3}}{3 a c}-\frac {\left (a+b x^3\right )^{4/3}}{2 a c x^6}\right )\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {1}{3} \left (-\frac {\frac {\frac {\left (-9 a^2 d^2+3 a b c d+b^2 c^2\right ) \int \frac {1}{x^3 \left (b x^3+a\right )^{2/3}}dx^3}{c}-\frac {9 a d^2 (b c-a d) \int \frac {1}{\left (b x^3+a\right )^{2/3} \left (d x^3+c\right )}dx^3}{c}}{3 c}-\frac {\sqrt [3]{a+b x^3} (3 a d+b c)}{c x^3}}{3 a c}-\frac {\left (a+b x^3\right )^{4/3}}{2 a c x^6}\right )\) |
| \(\Big \downarrow \) 69 |
| \(\displaystyle \frac {1}{3} \left (-\frac {\frac {\frac {\left (-9 a^2 d^2+3 a b c d+b^2 c^2\right ) \left (-\frac {3 \int \frac {1}{\sqrt [3]{a}-\sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}}{2 a^{2/3}}-\frac {3 \int \frac {1}{x^6+a^{2/3}+\sqrt [3]{a} \sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}}{2 \sqrt [3]{a}}-\frac {\log \left (x^3\right )}{2 a^{2/3}}\right )}{c}-\frac {9 a d^2 (b c-a d) \int \frac {1}{\left (b x^3+a\right )^{2/3} \left (d x^3+c\right )}dx^3}{c}}{3 c}-\frac {\sqrt [3]{a+b x^3} (3 a d+b c)}{c x^3}}{3 a c}-\frac {\left (a+b x^3\right )^{4/3}}{2 a c x^6}\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {1}{3} \left (-\frac {\frac {\frac {\left (-9 a^2 d^2+3 a b c d+b^2 c^2\right ) \left (-\frac {3 \int \frac {1}{x^6+a^{2/3}+\sqrt [3]{a} \sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}}{2 \sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 a^{2/3}}-\frac {\log \left (x^3\right )}{2 a^{2/3}}\right )}{c}-\frac {9 a d^2 (b c-a d) \int \frac {1}{\left (b x^3+a\right )^{2/3} \left (d x^3+c\right )}dx^3}{c}}{3 c}-\frac {\sqrt [3]{a+b x^3} (3 a d+b c)}{c x^3}}{3 a c}-\frac {\left (a+b x^3\right )^{4/3}}{2 a c x^6}\right )\) |
| \(\Big \downarrow \) 70 |
| \(\displaystyle \frac {1}{3} \left (-\frac {\frac {\frac {\left (-9 a^2 d^2+3 a b c d+b^2 c^2\right ) \left (-\frac {3 \int \frac {1}{x^6+a^{2/3}+\sqrt [3]{a} \sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}}{2 \sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 a^{2/3}}-\frac {\log \left (x^3\right )}{2 a^{2/3}}\right )}{c}-\frac {9 a d^2 (b c-a d) \left (\frac {3 \int \frac {1}{x^6+\frac {(b c-a d)^{2/3}}{d^{2/3}}-\frac {\sqrt [3]{b c-a d} \sqrt [3]{b x^3+a}}{\sqrt [3]{d}}}d\sqrt [3]{b x^3+a}}{2 d^{2/3} \sqrt [3]{b c-a d}}+\frac {3 \int \frac {1}{\frac {\sqrt [3]{b c-a d}}{\sqrt [3]{d}}+\sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}}{2 \sqrt [3]{d} (b c-a d)^{2/3}}-\frac {\log \left (c+d x^3\right )}{2 \sqrt [3]{d} (b c-a d)^{2/3}}\right )}{c}}{3 c}-\frac {\sqrt [3]{a+b x^3} (3 a d+b c)}{c x^3}}{3 a c}-\frac {\left (a+b x^3\right )^{4/3}}{2 a c x^6}\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {1}{3} \left (-\frac {\frac {\frac {\left (-9 a^2 d^2+3 a b c d+b^2 c^2\right ) \left (-\frac {3 \int \frac {1}{x^6+a^{2/3}+\sqrt [3]{a} \sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}}{2 \sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 a^{2/3}}-\frac {\log \left (x^3\right )}{2 a^{2/3}}\right )}{c}-\frac {9 a d^2 (b c-a d) \left (\frac {3 \int \frac {1}{x^6+\frac {(b c-a d)^{2/3}}{d^{2/3}}-\frac {\sqrt [3]{b c-a d} \sqrt [3]{b x^3+a}}{\sqrt [3]{d}}}d\sqrt [3]{b x^3+a}}{2 d^{2/3} \sqrt [3]{b c-a d}}-\frac {\log \left (c+d x^3\right )}{2 \sqrt [3]{d} (b c-a d)^{2/3}}+\frac {3 \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{d} (b c-a d)^{2/3}}\right )}{c}}{3 c}-\frac {\sqrt [3]{a+b x^3} (3 a d+b c)}{c x^3}}{3 a c}-\frac {\left (a+b x^3\right )^{4/3}}{2 a c x^6}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{3} \left (-\frac {\frac {\frac {\left (-9 a^2 d^2+3 a b c d+b^2 c^2\right ) \left (\frac {3 \int \frac {1}{-x^6-3}d\left (\frac {2 \sqrt [3]{b x^3+a}}{\sqrt [3]{a}}+1\right )}{a^{2/3}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 a^{2/3}}-\frac {\log \left (x^3\right )}{2 a^{2/3}}\right )}{c}-\frac {9 a d^2 (b c-a d) \left (\frac {3 \int \frac {1}{-x^6-3}d\left (1-\frac {2 \sqrt [3]{d} \sqrt [3]{b x^3+a}}{\sqrt [3]{b c-a d}}\right )}{\sqrt [3]{d} (b c-a d)^{2/3}}-\frac {\log \left (c+d x^3\right )}{2 \sqrt [3]{d} (b c-a d)^{2/3}}+\frac {3 \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{d} (b c-a d)^{2/3}}\right )}{c}}{3 c}-\frac {\sqrt [3]{a+b x^3} (3 a d+b c)}{c x^3}}{3 a c}-\frac {\left (a+b x^3\right )^{4/3}}{2 a c x^6}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{3} \left (-\frac {\frac {\frac {\left (-9 a^2 d^2+3 a b c d+b^2 c^2\right ) \left (-\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}+1}{\sqrt {3}}\right )}{a^{2/3}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 a^{2/3}}-\frac {\log \left (x^3\right )}{2 a^{2/3}}\right )}{c}-\frac {9 a d^2 (b c-a d) \left (-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{\sqrt [3]{d} (b c-a d)^{2/3}}-\frac {\log \left (c+d x^3\right )}{2 \sqrt [3]{d} (b c-a d)^{2/3}}+\frac {3 \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{d} (b c-a d)^{2/3}}\right )}{c}}{3 c}-\frac {\sqrt [3]{a+b x^3} (3 a d+b c)}{c x^3}}{3 a c}-\frac {\left (a+b x^3\right )^{4/3}}{2 a c x^6}\right )\) |
(-1/2*(a + b*x^3)^(4/3)/(a*c*x^6) - (-(((b*c + 3*a*d)*(a + b*x^3)^(1/3))/( c*x^3)) + (((b^2*c^2 + 3*a*b*c*d - 9*a^2*d^2)*(-((Sqrt[3]*ArcTan[(1 + (2*( a + b*x^3)^(1/3))/a^(1/3))/Sqrt[3]])/a^(2/3)) - Log[x^3]/(2*a^(2/3)) + (3* Log[a^(1/3) - (a + b*x^3)^(1/3)])/(2*a^(2/3))))/c - (9*a*d^2*(b*c - a*d)*( -((Sqrt[3]*ArcTan[(1 - (2*d^(1/3)*(a + b*x^3)^(1/3))/(b*c - a*d)^(1/3))/Sq rt[3]])/(d^(1/3)*(b*c - a*d)^(2/3))) - Log[c + d*x^3]/(2*d^(1/3)*(b*c - a* d)^(2/3)) + (3*Log[(b*c - a*d)^(1/3) + d^(1/3)*(a + b*x^3)^(1/3)])/(2*d^(1 /3)*(b*c - a*d)^(2/3))))/c)/(3*c))/(3*a*c))/3
3.7.64.3.1 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[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[ {q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q^2), x] + (-Simp[3/(2*b*q) Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1 /3)], x] - Simp[3/(2*b*q^2) 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[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[ {q = Rt[-(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q^2) , x] + (Simp[3/(2*b*q) Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d*x)^(1 /3)], x] + Simp[3/(2*b*q^2) Subst[Int[1/(q + x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && NegQ[(b*c - a*d)/b]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b* c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h )*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]
Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* ((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d) Int[(e + f*x)^ p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d) Int[(e + f*x)^p/(c + d *x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
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[x^(Simplify[(m + 1)/n] - 1)*(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] && IntegerQ[Simplify[(m + 1)/n]]
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 = 5.11 (sec) , antiderivative size = 428, normalized size of antiderivative = 1.16
| method | result | size |
| pseudoelliptic | \(-\frac {-\frac {\left (a^{\frac {11}{3}} d -a^{\frac {8}{3}} b c \right ) x^{6} d \ln \left (\left (b \,x^{3}+a \right )^{\frac {2}{3}}+\left (\frac {a d -b c}{d}\right )^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{d}\right )^{\frac {2}{3}}\right )}{2}-\left (a^{\frac {11}{3}} d -a^{\frac {8}{3}} b c \right ) x^{6} \sqrt {3}\, d \arctan \left (\frac {\sqrt {3}\, \left (2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{d}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a d -b c}{d}\right )^{\frac {1}{3}}}\right )+\frac {\left (\frac {a d -b c}{d}\right )^{\frac {2}{3}} x^{6} \left (a^{2} d^{2}-\frac {1}{3} a b c d -\frac {1}{9} b^{2} c^{2}\right ) a \ln \left (\left (b \,x^{3}+a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{2}+\left (a^{\frac {11}{3}} d -a^{\frac {8}{3}} b c \right ) x^{6} d \ln \left (\left (b \,x^{3}+a \right )^{\frac {1}{3}}-\left (\frac {a d -b c}{d}\right )^{\frac {1}{3}}\right )-\left (\frac {a d -b c}{d}\right )^{\frac {2}{3}} \left (-x^{6} \sqrt {3}\, \left (a^{2} d^{2}-\frac {1}{3} a b c d -\frac {1}{9} b^{2} c^{2}\right ) a \arctan \left (\frac {\left (a^{\frac {1}{3}}+2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right )+x^{6} \left (a^{2} d^{2}-\frac {1}{3} a b c d -\frac {1}{9} b^{2} c^{2}\right ) a \ln \left (\left (b \,x^{3}+a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )-\frac {c \left (b \,x^{3}+a \right )^{\frac {1}{3}} \left (\left (-6 d \,x^{3}+3 c \right ) a^{\frac {8}{3}}+a^{\frac {5}{3}} b c \,x^{3}\right )}{6}\right )}{3 \left (\frac {a d -b c}{d}\right )^{\frac {2}{3}} a^{\frac {8}{3}} c^{3} x^{6}}\) | \(428\) |
-1/3/(1/d*(a*d-b*c))^(2/3)*(-1/2*(a^(11/3)*d-a^(8/3)*b*c)*x^6*d*ln((b*x^3+ a)^(2/3)+(1/d*(a*d-b*c))^(1/3)*(b*x^3+a)^(1/3)+(1/d*(a*d-b*c))^(2/3))-(a^( 11/3)*d-a^(8/3)*b*c)*x^6*3^(1/2)*d*arctan(1/3*3^(1/2)*(2*(b*x^3+a)^(1/3)+( 1/d*(a*d-b*c))^(1/3))/(1/d*(a*d-b*c))^(1/3))+1/2*(1/d*(a*d-b*c))^(2/3)*x^6 *(a^2*d^2-1/3*a*b*c*d-1/9*b^2*c^2)*a*ln((b*x^3+a)^(2/3)+a^(1/3)*(b*x^3+a)^ (1/3)+a^(2/3))+(a^(11/3)*d-a^(8/3)*b*c)*x^6*d*ln((b*x^3+a)^(1/3)-(1/d*(a*d -b*c))^(1/3))-(1/d*(a*d-b*c))^(2/3)*(-x^6*3^(1/2)*(a^2*d^2-1/3*a*b*c*d-1/9 *b^2*c^2)*a*arctan(1/3*(a^(1/3)+2*(b*x^3+a)^(1/3))/a^(1/3)*3^(1/2))+x^6*(a ^2*d^2-1/3*a*b*c*d-1/9*b^2*c^2)*a*ln((b*x^3+a)^(1/3)-a^(1/3))-1/6*c*(b*x^3 +a)^(1/3)*((-6*d*x^3+3*c)*a^(8/3)+a^(5/3)*b*c*x^3)))/a^(8/3)/c^3/x^6
Time = 1.12 (sec) , antiderivative size = 472, normalized size of antiderivative = 1.28 \[ \int \frac {\sqrt [3]{a+b x^3}}{x^7 \left (c+d x^3\right )} \, dx=-\frac {18 \, \sqrt {3} {\left (b c d^{2} - a d^{3}\right )}^{\frac {1}{3}} a^{3} d x^{6} \arctan \left (-\frac {2 \, \sqrt {3} {\left (b c d^{2} - a d^{3}\right )}^{\frac {2}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}} - \sqrt {3} {\left (b c d - a d^{2}\right )}}{3 \, {\left (b c d - a d^{2}\right )}}\right ) + 9 \, {\left (b c d^{2} - a d^{3}\right )}^{\frac {1}{3}} a^{3} d x^{6} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} d^{2} - {\left (b c d^{2} - a d^{3}\right )}^{\frac {1}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}} d + {\left (b c d^{2} - a d^{3}\right )}^{\frac {2}{3}}\right ) - 18 \, {\left (b c d^{2} - a d^{3}\right )}^{\frac {1}{3}} a^{3} d x^{6} \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} d + {\left (b c d^{2} - a d^{3}\right )}^{\frac {1}{3}}\right ) - 2 \, \sqrt {3} {\left (a b^{2} c^{2} + 3 \, a^{2} b c d - 9 \, a^{3} d^{2}\right )} {\left (a^{2}\right )}^{\frac {1}{6}} x^{6} \arctan \left (\frac {{\left (a^{2}\right )}^{\frac {1}{6}} {\left (\sqrt {3} {\left (a^{2}\right )}^{\frac {1}{3}} a + 2 \, \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (a^{2}\right )}^{\frac {2}{3}}\right )}}{3 \, a^{2}}\right ) - {\left (b^{2} c^{2} + 3 \, a b c d - 9 \, a^{2} d^{2}\right )} {\left (a^{2}\right )}^{\frac {2}{3}} x^{6} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} a + {\left (a^{2}\right )}^{\frac {1}{3}} a + {\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (a^{2}\right )}^{\frac {2}{3}}\right ) + 2 \, {\left (b^{2} c^{2} + 3 \, a b c d - 9 \, a^{2} d^{2}\right )} {\left (a^{2}\right )}^{\frac {2}{3}} x^{6} \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} a - {\left (a^{2}\right )}^{\frac {2}{3}}\right ) + 3 \, {\left (3 \, a^{3} c^{2} + {\left (a^{2} b c^{2} - 6 \, a^{3} c d\right )} x^{3}\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{54 \, a^{3} c^{3} x^{6}} \]
-1/54*(18*sqrt(3)*(b*c*d^2 - a*d^3)^(1/3)*a^3*d*x^6*arctan(-1/3*(2*sqrt(3) *(b*c*d^2 - a*d^3)^(2/3)*(b*x^3 + a)^(1/3) - sqrt(3)*(b*c*d - a*d^2))/(b*c *d - a*d^2)) + 9*(b*c*d^2 - a*d^3)^(1/3)*a^3*d*x^6*log((b*x^3 + a)^(2/3)*d ^2 - (b*c*d^2 - a*d^3)^(1/3)*(b*x^3 + a)^(1/3)*d + (b*c*d^2 - a*d^3)^(2/3) ) - 18*(b*c*d^2 - a*d^3)^(1/3)*a^3*d*x^6*log((b*x^3 + a)^(1/3)*d + (b*c*d^ 2 - a*d^3)^(1/3)) - 2*sqrt(3)*(a*b^2*c^2 + 3*a^2*b*c*d - 9*a^3*d^2)*(a^2)^ (1/6)*x^6*arctan(1/3*(a^2)^(1/6)*(sqrt(3)*(a^2)^(1/3)*a + 2*sqrt(3)*(b*x^3 + a)^(1/3)*(a^2)^(2/3))/a^2) - (b^2*c^2 + 3*a*b*c*d - 9*a^2*d^2)*(a^2)^(2 /3)*x^6*log((b*x^3 + a)^(2/3)*a + (a^2)^(1/3)*a + (b*x^3 + a)^(1/3)*(a^2)^ (2/3)) + 2*(b^2*c^2 + 3*a*b*c*d - 9*a^2*d^2)*(a^2)^(2/3)*x^6*log((b*x^3 + a)^(1/3)*a - (a^2)^(2/3)) + 3*(3*a^3*c^2 + (a^2*b*c^2 - 6*a^3*c*d)*x^3)*(b *x^3 + a)^(1/3))/(a^3*c^3*x^6)
\[ \int \frac {\sqrt [3]{a+b x^3}}{x^7 \left (c+d x^3\right )} \, dx=\int \frac {\sqrt [3]{a + b x^{3}}}{x^{7} \left (c + d x^{3}\right )}\, dx \]
\[ \int \frac {\sqrt [3]{a+b x^3}}{x^7 \left (c+d x^3\right )} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{{\left (d x^{3} + c\right )} x^{7}} \,d x } \]
Time = 0.56 (sec) , antiderivative size = 455, normalized size of antiderivative = 1.23 \[ \int \frac {\sqrt [3]{a+b x^3}}{x^7 \left (c+d x^3\right )} \, dx=-\frac {{\left (b c d^{2} - a d^{3}\right )} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} \log \left ({\left | {\left (b x^{3} + a\right )}^{\frac {1}{3}} - \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} \right |}\right )}{3 \, {\left (b c^{4} - a c^{3} d\right )}} + \frac {\sqrt {3} {\left (-b c d^{2} + a d^{3}\right )}^{\frac {1}{3}} d \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}}}\right )}{3 \, c^{3}} + \frac {{\left (-b c d^{2} + a d^{3}\right )}^{\frac {1}{3}} d \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} + \left (-\frac {b c - a d}{d}\right )^{\frac {2}{3}}\right )}{6 \, c^{3}} + \frac {\sqrt {3} {\left (b^{2} c^{2} + 3 \, a b c d - 9 \, a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{27 \, a^{\frac {5}{3}} c^{3}} + \frac {{\left (b^{2} c^{2} + 3 \, a b c d - 9 \, a^{2} d^{2}\right )} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{54 \, a^{\frac {5}{3}} c^{3}} - \frac {{\left (b^{2} c^{2} + 3 \, a b c d - 9 \, a^{2} d^{2}\right )} \log \left ({\left | {\left (b x^{3} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{27 \, a^{\frac {5}{3}} c^{3}} - \frac {{\left (b x^{3} + a\right )}^{\frac {4}{3}} b^{2} c + 2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} a b^{2} c - 6 \, {\left (b x^{3} + a\right )}^{\frac {4}{3}} a b d + 6 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{2} b d}{18 \, a b^{2} c^{2} x^{6}} \]
-1/3*(b*c*d^2 - a*d^3)*(-(b*c - a*d)/d)^(1/3)*log(abs((b*x^3 + a)^(1/3) - (-(b*c - a*d)/d)^(1/3)))/(b*c^4 - a*c^3*d) + 1/3*sqrt(3)*(-b*c*d^2 + a*d^3 )^(1/3)*d*arctan(1/3*sqrt(3)*(2*(b*x^3 + a)^(1/3) + (-(b*c - a*d)/d)^(1/3) )/(-(b*c - a*d)/d)^(1/3))/c^3 + 1/6*(-b*c*d^2 + a*d^3)^(1/3)*d*log((b*x^3 + a)^(2/3) + (b*x^3 + a)^(1/3)*(-(b*c - a*d)/d)^(1/3) + (-(b*c - a*d)/d)^( 2/3))/c^3 + 1/27*sqrt(3)*(b^2*c^2 + 3*a*b*c*d - 9*a^2*d^2)*arctan(1/3*sqrt (3)*(2*(b*x^3 + a)^(1/3) + a^(1/3))/a^(1/3))/(a^(5/3)*c^3) + 1/54*(b^2*c^2 + 3*a*b*c*d - 9*a^2*d^2)*log((b*x^3 + a)^(2/3) + (b*x^3 + a)^(1/3)*a^(1/3 ) + a^(2/3))/(a^(5/3)*c^3) - 1/27*(b^2*c^2 + 3*a*b*c*d - 9*a^2*d^2)*log(ab s((b*x^3 + a)^(1/3) - a^(1/3)))/(a^(5/3)*c^3) - 1/18*((b*x^3 + a)^(4/3)*b^ 2*c + 2*(b*x^3 + a)^(1/3)*a*b^2*c - 6*(b*x^3 + a)^(4/3)*a*b*d + 6*(b*x^3 + a)^(1/3)*a^2*b*d)/(a*b^2*c^2*x^6)
Time = 16.50 (sec) , antiderivative size = 2767, normalized size of antiderivative = 7.48 \[ \int \frac {\sqrt [3]{a+b x^3}}{x^7 \left (c+d x^3\right )} \, dx=\text {Too large to display} \]
log(((((81*a*b^4*c^4*d^3*(2*a^2*d^2 + b^2*c^2 - 3*a*b*c*d)*(-(d^5*(a*d - b *c))/c^9)^(1/3) + (9*b^5*c^2*d^3*(a + b*x^3)^(1/3)*(12*a^3*d^3 + b^3*c^3 + a*b^2*c^2*d - 14*a^2*b*c*d^2))/a)*(-(d^5*(a*d - b*c))/c^9)^(2/3))/9 - (b^ 5*d^4*(729*a^6*d^6 + b^6*c^6 - 9*a^2*b^4*c^4*d^2 - 135*a^3*b^3*c^3*d^3 + 8 64*a^4*b^2*c^2*d^4 + 8*a*b^5*c^5*d - 1458*a^5*b*c*d^5))/(81*a^3*c^4))*(-(d ^5*(a*d - b*c))/c^9)^(1/3))/3 - (b^4*d^6*(a + b*x^3)^(1/3)*(1458*a^7*d^7 + b^7*c^7 + 72*a^2*b^5*c^5*d^2 - 135*a^3*b^4*c^4*d^3 - 1080*a^4*b^3*c^3*d^4 + 3564*a^5*b^2*c^2*d^5 + 8*a*b^6*c^6*d - 3888*a^6*b*c*d^6))/(243*a^3*c^8) )*(-(a*d^6 - b*c*d^5)/(27*c^9))^(1/3) + log((((((9*b^5*c^2*d^3*(a + b*x^3) ^(1/3)*(12*a^3*d^3 + b^3*c^3 + a*b^2*c^2*d - 14*a^2*b*c*d^2))/a + 9*a*b^4* c^4*d^3*(2*a^2*d^2 + b^2*c^2 - 3*a*b*c*d)*(-(b^2*c^2 - 9*a^2*d^2 + 3*a*b*c *d)^3/(a^5*c^9))^(1/3))*(-(b^2*c^2 - 9*a^2*d^2 + 3*a*b*c*d)^3/(a^5*c^9))^( 2/3))/729 - (b^5*d^4*(729*a^6*d^6 + b^6*c^6 - 9*a^2*b^4*c^4*d^2 - 135*a^3* b^3*c^3*d^3 + 864*a^4*b^2*c^2*d^4 + 8*a*b^5*c^5*d - 1458*a^5*b*c*d^5))/(81 *a^3*c^4))*(-(b^2*c^2 - 9*a^2*d^2 + 3*a*b*c*d)^3/(a^5*c^9))^(1/3))/27 - (b ^4*d^6*(a + b*x^3)^(1/3)*(1458*a^7*d^7 + b^7*c^7 + 72*a^2*b^5*c^5*d^2 - 13 5*a^3*b^4*c^4*d^3 - 1080*a^4*b^3*c^3*d^4 + 3564*a^5*b^2*c^2*d^5 + 8*a*b^6* c^6*d - 3888*a^6*b*c*d^6))/(243*a^3*c^8))*(-(b^6*c^6 - 729*a^6*d^6 - 135*a ^3*b^3*c^3*d^3 + 9*a*b^5*c^5*d + 729*a^5*b*c*d^5)/(19683*a^5*c^9))^(1/3) - (((a + b*x^3)^(1/3)*(b^2*c + 3*a*b*d))/(9*c^2) - (b*(a + b*x^3)^(4/3)*...