Integrand size = 19, antiderivative size = 120 \[ \int \frac {1}{(c x)^{23/3} \left (a+b x^2\right )^{2/3}} \, dx=-\frac {3 \sqrt [3]{a+b x^2}}{20 a c (c x)^{20/3}}+\frac {27 b \sqrt [3]{a+b x^2}}{140 a^2 c^3 (c x)^{14/3}}-\frac {81 b^2 \sqrt [3]{a+b x^2}}{280 a^3 c^5 (c x)^{8/3}}+\frac {243 b^3 \sqrt [3]{a+b x^2}}{280 a^4 c^7 (c x)^{2/3}} \] Output:
-3/20*(b*x^2+a)^(1/3)/a/c/(c*x)^(20/3)+27/140*b*(b*x^2+a)^(1/3)/a^2/c^3/(c *x)^(14/3)-81/280*b^2*(b*x^2+a)^(1/3)/a^3/c^5/(c*x)^(8/3)+243/280*b^3*(b*x ^2+a)^(1/3)/a^4/c^7/(c*x)^(2/3)
Time = 5.09 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.48 \[ \int \frac {1}{(c x)^{23/3} \left (a+b x^2\right )^{2/3}} \, dx=-\frac {3 x \sqrt [3]{a+b x^2} \left (14 a^3-18 a^2 b x^2+27 a b^2 x^4-81 b^3 x^6\right )}{280 a^4 (c x)^{23/3}} \] Input:
Integrate[1/((c*x)^(23/3)*(a + b*x^2)^(2/3)),x]
Output:
(-3*x*(a + b*x^2)^(1/3)*(14*a^3 - 18*a^2*b*x^2 + 27*a*b^2*x^4 - 81*b^3*x^6 ))/(280*a^4*(c*x)^(23/3))
Time = 0.21 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.06, 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.
\(\displaystyle \int \frac {1}{(c x)^{23/3} \left (a+b x^2\right )^{2/3}} \, dx\) |
\(\Big \downarrow \) 246 |
\(\displaystyle -\frac {9 \int \frac {\sqrt [3]{b x^2+a}}{(c x)^{23/3}}dx}{a}-\frac {3 \sqrt [3]{a+b x^2}}{2 a c (c x)^{20/3}}\) |
\(\Big \downarrow \) 246 |
\(\displaystyle -\frac {9 \left (-\frac {3 \int \frac {\left (b x^2+a\right )^{4/3}}{(c x)^{23/3}}dx}{2 a}-\frac {3 \left (a+b x^2\right )^{4/3}}{8 a c (c x)^{20/3}}\right )}{a}-\frac {3 \sqrt [3]{a+b x^2}}{2 a c (c x)^{20/3}}\) |
\(\Big \downarrow \) 246 |
\(\displaystyle -\frac {9 \left (-\frac {3 \left (-\frac {3 \int \frac {\left (b x^2+a\right )^{7/3}}{(c x)^{23/3}}dx}{7 a}-\frac {3 \left (a+b x^2\right )^{7/3}}{14 a c (c x)^{20/3}}\right )}{2 a}-\frac {3 \left (a+b x^2\right )^{4/3}}{8 a c (c x)^{20/3}}\right )}{a}-\frac {3 \sqrt [3]{a+b x^2}}{2 a c (c x)^{20/3}}\) |
\(\Big \downarrow \) 242 |
\(\displaystyle -\frac {9 \left (-\frac {3 \left (\frac {9 \left (a+b x^2\right )^{10/3}}{140 a^2 c (c x)^{20/3}}-\frac {3 \left (a+b x^2\right )^{7/3}}{14 a c (c x)^{20/3}}\right )}{2 a}-\frac {3 \left (a+b x^2\right )^{4/3}}{8 a c (c x)^{20/3}}\right )}{a}-\frac {3 \sqrt [3]{a+b x^2}}{2 a c (c x)^{20/3}}\) |
Input:
Int[1/((c*x)^(23/3)*(a + b*x^2)^(2/3)),x]
Output:
(-3*(a + b*x^2)^(1/3))/(2*a*c*(c*x)^(20/3)) - (9*((-3*(a + b*x^2)^(4/3))/( 8*a*c*(c*x)^(20/3)) - (3*((-3*(a + b*x^2)^(7/3))/(14*a*c*(c*x)^(20/3)) + ( 9*(a + b*x^2)^(10/3))/(140*a^2*c*(c*x)^(20/3))))/(2*a)))/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 | \(-\frac {3 x \left (b \,x^{2}+a \right )^{\frac {1}{3}} \left (-81 b^{3} x^{6}+27 a \,b^{2} x^{4}-18 a^{2} b \,x^{2}+14 a^{3}\right )}{280 a^{4} \left (c x \right )^{\frac {23}{3}}}\) | \(53\) |
orering | \(-\frac {3 x \left (b \,x^{2}+a \right )^{\frac {1}{3}} \left (-81 b^{3} x^{6}+27 a \,b^{2} x^{4}-18 a^{2} b \,x^{2}+14 a^{3}\right )}{280 a^{4} \left (c x \right )^{\frac {23}{3}}}\) | \(53\) |
risch | \(-\frac {3 \left (b \,x^{2}+a \right )^{\frac {1}{3}} \left (-81 b^{3} x^{6}+27 a \,b^{2} x^{4}-18 a^{2} b \,x^{2}+14 a^{3}\right )}{280 c^{7} \left (c x \right )^{\frac {2}{3}} a^{4} x^{6}}\) | \(58\) |
Input:
int(1/(c*x)^(23/3)/(b*x^2+a)^(2/3),x,method=_RETURNVERBOSE)
Output:
-3/280*x*(b*x^2+a)^(1/3)*(-81*b^3*x^6+27*a*b^2*x^4-18*a^2*b*x^2+14*a^3)/a^ 4/(c*x)^(23/3)
Time = 0.12 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.48 \[ \int \frac {1}{(c x)^{23/3} \left (a+b x^2\right )^{2/3}} \, dx=\frac {3 \, {\left (81 \, b^{3} x^{6} - 27 \, a b^{2} x^{4} + 18 \, a^{2} b x^{2} - 14 \, a^{3}\right )} {\left (b x^{2} + a\right )}^{\frac {1}{3}} \left (c x\right )^{\frac {1}{3}}}{280 \, a^{4} c^{8} x^{7}} \] Input:
integrate(1/(c*x)^(23/3)/(b*x^2+a)^(2/3),x, algorithm="fricas")
Output:
3/280*(81*b^3*x^6 - 27*a*b^2*x^4 + 18*a^2*b*x^2 - 14*a^3)*(b*x^2 + a)^(1/3 )*(c*x)^(1/3)/(a^4*c^8*x^7)
Timed out. \[ \int \frac {1}{(c x)^{23/3} \left (a+b x^2\right )^{2/3}} \, dx=\text {Timed out} \] Input:
integrate(1/(c*x)**(23/3)/(b*x**2+a)**(2/3),x)
Output:
Timed out
Time = 0.05 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.53 \[ \int \frac {1}{(c x)^{23/3} \left (a+b x^2\right )^{2/3}} \, dx=\frac {3 \, {\left (81 \, b^{4} x^{9} + 54 \, a b^{3} x^{7} - 9 \, a^{2} b^{2} x^{5} + 4 \, a^{3} b x^{3} - 14 \, a^{4} x\right )}}{280 \, {\left (b x^{2} + a\right )}^{\frac {2}{3}} a^{4} c^{\frac {23}{3}} x^{\frac {23}{3}}} \] Input:
integrate(1/(c*x)^(23/3)/(b*x^2+a)^(2/3),x, algorithm="maxima")
Output:
3/280*(81*b^4*x^9 + 54*a*b^3*x^7 - 9*a^2*b^2*x^5 + 4*a^3*b*x^3 - 14*a^4*x) /((b*x^2 + a)^(2/3)*a^4*c^(23/3)*x^(23/3))
\[ \int \frac {1}{(c x)^{23/3} \left (a+b x^2\right )^{2/3}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {2}{3}} \left (c x\right )^{\frac {23}{3}}} \,d x } \] Input:
integrate(1/(c*x)^(23/3)/(b*x^2+a)^(2/3),x, algorithm="giac")
Output:
integrate(1/((b*x^2 + a)^(2/3)*(c*x)^(23/3)), x)
Timed out. \[ \int \frac {1}{(c x)^{23/3} \left (a+b x^2\right )^{2/3}} \, dx=\int \frac {1}{{\left (c\,x\right )}^{23/3}\,{\left (b\,x^2+a\right )}^{2/3}} \,d x \] Input:
int(1/((c*x)^(23/3)*(a + b*x^2)^(2/3)),x)
Output:
int(1/((c*x)^(23/3)*(a + b*x^2)^(2/3)), x)
\[ \int \frac {1}{(c x)^{23/3} \left (a+b x^2\right )^{2/3}} \, dx=\frac {\int \frac {1}{x^{\frac {23}{3}} \left (b \,x^{2}+a \right )^{\frac {2}{3}}}d x}{c^{\frac {23}{3}}} \] Input:
int(1/(c*x)^(23/3)/(b*x^2+a)^(2/3),x)
Output:
int(1/(x**(2/3)*(a + b*x**2)**(2/3)*x**7),x)/(c**(2/3)*c**7)