Integrand size = 19, antiderivative size = 120 \[ \int \frac {\sqrt [3]{a+b x^2}}{(c x)^{29/3}} \, dx=-\frac {3 \left (a+b x^2\right )^{4/3}}{26 a c (c x)^{26/3}}+\frac {27 b \left (a+b x^2\right )^{4/3}}{260 a^2 c^3 (c x)^{20/3}}-\frac {81 b^2 \left (a+b x^2\right )^{4/3}}{910 a^3 c^5 (c x)^{14/3}}+\frac {243 b^3 \left (a+b x^2\right )^{4/3}}{3640 a^4 c^7 (c x)^{8/3}} \] Output:
-3/26*(b*x^2+a)^(4/3)/a/c/(c*x)^(26/3)+27/260*b*(b*x^2+a)^(4/3)/a^2/c^3/(c *x)^(20/3)-81/910*b^2*(b*x^2+a)^(4/3)/a^3/c^5/(c*x)^(14/3)+243/3640*b^3*(b *x^2+a)^(4/3)/a^4/c^7/(c*x)^(8/3)
Time = 4.52 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.48 \[ \int \frac {\sqrt [3]{a+b x^2}}{(c x)^{29/3}} \, dx=-\frac {3 x \left (a+b x^2\right )^{4/3} \left (140 a^3-126 a^2 b x^2+108 a b^2 x^4-81 b^3 x^6\right )}{3640 a^4 (c x)^{29/3}} \] Input:
Integrate[(a + b*x^2)^(1/3)/(c*x)^(29/3),x]
Output:
(-3*x*(a + b*x^2)^(4/3)*(140*a^3 - 126*a^2*b*x^2 + 108*a*b^2*x^4 - 81*b^3* x^6))/(3640*a^4*(c*x)^(29/3))
Time = 0.21 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.08, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {246, 246, 246, 242}
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.
\(\displaystyle \int \frac {\sqrt [3]{a+b x^2}}{(c x)^{29/3}} \, dx\) |
\(\Big \downarrow \) 246 |
\(\displaystyle -\frac {9 \int \frac {\left (b x^2+a\right )^{4/3}}{(c x)^{29/3}}dx}{4 a}-\frac {3 \left (a+b x^2\right )^{4/3}}{8 a c (c x)^{26/3}}\) |
\(\Big \downarrow \) 246 |
\(\displaystyle -\frac {9 \left (-\frac {6 \int \frac {\left (b x^2+a\right )^{7/3}}{(c x)^{29/3}}dx}{7 a}-\frac {3 \left (a+b x^2\right )^{7/3}}{14 a c (c x)^{26/3}}\right )}{4 a}-\frac {3 \left (a+b x^2\right )^{4/3}}{8 a c (c x)^{26/3}}\) |
\(\Big \downarrow \) 246 |
\(\displaystyle -\frac {9 \left (-\frac {6 \left (-\frac {3 \int \frac {\left (b x^2+a\right )^{10/3}}{(c x)^{29/3}}dx}{10 a}-\frac {3 \left (a+b x^2\right )^{10/3}}{20 a c (c x)^{26/3}}\right )}{7 a}-\frac {3 \left (a+b x^2\right )^{7/3}}{14 a c (c x)^{26/3}}\right )}{4 a}-\frac {3 \left (a+b x^2\right )^{4/3}}{8 a c (c x)^{26/3}}\) |
\(\Big \downarrow \) 242 |
\(\displaystyle -\frac {9 \left (-\frac {6 \left (\frac {9 \left (a+b x^2\right )^{13/3}}{260 a^2 c (c x)^{26/3}}-\frac {3 \left (a+b x^2\right )^{10/3}}{20 a c (c x)^{26/3}}\right )}{7 a}-\frac {3 \left (a+b x^2\right )^{7/3}}{14 a c (c x)^{26/3}}\right )}{4 a}-\frac {3 \left (a+b x^2\right )^{4/3}}{8 a c (c x)^{26/3}}\) |
Input:
Int[(a + b*x^2)^(1/3)/(c*x)^(29/3),x]
Output:
(-3*(a + b*x^2)^(4/3))/(8*a*c*(c*x)^(26/3)) - (9*((-3*(a + b*x^2)^(7/3))/( 14*a*c*(c*x)^(26/3)) - (6*((-3*(a + b*x^2)^(10/3))/(20*a*c*(c*x)^(26/3)) + (9*(a + b*x^2)^(13/3))/(260*a^2*c*(c*x)^(26/3))))/(7*a)))/(4*a)
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, p}, x ] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c*x )^(m + 1))*((a + b*x^2)^(p + 1)/(a*c*2*(p + 1))), x] + Simp[(m + 2*p + 3)/( a*2*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m , p}, x] && ILtQ[Simplify[(m + 1)/2 + p + 1], 0] && NeQ[p, -1]
Time = 0.35 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.44
method | result | size |
gosper | \(-\frac {3 x \left (b \,x^{2}+a \right )^{\frac {4}{3}} \left (-81 b^{3} x^{6}+108 a \,b^{2} x^{4}-126 a^{2} b \,x^{2}+140 a^{3}\right )}{3640 a^{4} \left (c x \right )^{\frac {29}{3}}}\) | \(53\) |
orering | \(-\frac {3 x \left (b \,x^{2}+a \right )^{\frac {4}{3}} \left (-81 b^{3} x^{6}+108 a \,b^{2} x^{4}-126 a^{2} b \,x^{2}+140 a^{3}\right )}{3640 a^{4} \left (c x \right )^{\frac {29}{3}}}\) | \(53\) |
risch | \(-\frac {3 \left (b \,x^{2}+a \right )^{\frac {1}{3}} \left (-81 b^{4} x^{8}+27 a \,b^{3} x^{6}-18 a^{2} b^{2} x^{4}+14 a^{3} b \,x^{2}+140 a^{4}\right )}{3640 c^{9} \left (c x \right )^{\frac {2}{3}} x^{8} a^{4}}\) | \(69\) |
Input:
int((b*x^2+a)^(1/3)/(c*x)^(29/3),x,method=_RETURNVERBOSE)
Output:
-3/3640*x*(b*x^2+a)^(4/3)*(-81*b^3*x^6+108*a*b^2*x^4-126*a^2*b*x^2+140*a^3 )/a^4/(c*x)^(29/3)
Time = 0.08 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.57 \[ \int \frac {\sqrt [3]{a+b x^2}}{(c x)^{29/3}} \, dx=\frac {3 \, {\left (81 \, b^{4} x^{8} - 27 \, a b^{3} x^{6} + 18 \, a^{2} b^{2} x^{4} - 14 \, a^{3} b x^{2} - 140 \, a^{4}\right )} {\left (b x^{2} + a\right )}^{\frac {1}{3}} \left (c x\right )^{\frac {1}{3}}}{3640 \, a^{4} c^{10} x^{9}} \] Input:
integrate((b*x^2+a)^(1/3)/(c*x)^(29/3),x, algorithm="fricas")
Output:
3/3640*(81*b^4*x^8 - 27*a*b^3*x^6 + 18*a^2*b^2*x^4 - 14*a^3*b*x^2 - 140*a^ 4)*(b*x^2 + a)^(1/3)*(c*x)^(1/3)/(a^4*c^10*x^9)
Timed out. \[ \int \frac {\sqrt [3]{a+b x^2}}{(c x)^{29/3}} \, dx=\text {Timed out} \] Input:
integrate((b*x**2+a)**(1/3)/(c*x)**(29/3),x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.53 \[ \int \frac {\sqrt [3]{a+b x^2}}{(c x)^{29/3}} \, dx=\frac {3 \, {\left (81 \, b^{4} x^{9} - 27 \, a b^{3} x^{7} + 18 \, a^{2} b^{2} x^{5} - 14 \, a^{3} b x^{3} - 140 \, a^{4} x\right )} {\left (b x^{2} + a\right )}^{\frac {1}{3}}}{3640 \, a^{4} c^{\frac {29}{3}} x^{\frac {29}{3}}} \] Input:
integrate((b*x^2+a)^(1/3)/(c*x)^(29/3),x, algorithm="maxima")
Output:
3/3640*(81*b^4*x^9 - 27*a*b^3*x^7 + 18*a^2*b^2*x^5 - 14*a^3*b*x^3 - 140*a^ 4*x)*(b*x^2 + a)^(1/3)/(a^4*c^(29/3)*x^(29/3))
\[ \int \frac {\sqrt [3]{a+b x^2}}{(c x)^{29/3}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {1}{3}}}{\left (c x\right )^{\frac {29}{3}}} \,d x } \] Input:
integrate((b*x^2+a)^(1/3)/(c*x)^(29/3),x, algorithm="giac")
Output:
integrate((b*x^2 + a)^(1/3)/(c*x)^(29/3), x)
Timed out. \[ \int \frac {\sqrt [3]{a+b x^2}}{(c x)^{29/3}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{1/3}}{{\left (c\,x\right )}^{29/3}} \,d x \] Input:
int((a + b*x^2)^(1/3)/(c*x)^(29/3),x)
Output:
int((a + b*x^2)^(1/3)/(c*x)^(29/3), x)
Time = 0.28 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.52 \[ \int \frac {\sqrt [3]{a+b x^2}}{(c x)^{29/3}} \, dx=\frac {3 \left (b \,x^{2}+a \right )^{\frac {1}{3}} \left (81 b^{4} x^{8}-27 a \,b^{3} x^{6}+18 a^{2} b^{2} x^{4}-14 a^{3} b \,x^{2}-140 a^{4}\right )}{3640 x^{\frac {26}{3}} c^{\frac {29}{3}} a^{4}} \] Input:
int((b*x^2+a)^(1/3)/(c*x)^(29/3),x)
Output:
(3*(a + b*x**2)**(1/3)*( - 140*a**4 - 14*a**3*b*x**2 + 18*a**2*b**2*x**4 - 27*a*b**3*x**6 + 81*b**4*x**8))/(3640*x**(2/3)*c**(2/3)*a**4*c**9*x**8)