Integrand size = 19, antiderivative size = 89 \[ \int \frac {1}{(c x)^{13/2} \sqrt [4]{a+b x^2}} \, dx=-\frac {2 \left (a+b x^2\right )^{3/4}}{11 a c (c x)^{11/2}}+\frac {16 b \left (a+b x^2\right )^{3/4}}{77 a^2 c^3 (c x)^{7/2}}-\frac {64 b^2 \left (a+b x^2\right )^{3/4}}{231 a^3 c^5 (c x)^{3/2}} \] Output:
-2/11*(b*x^2+a)^(3/4)/a/c/(c*x)^(11/2)+16/77*b*(b*x^2+a)^(3/4)/a^2/c^3/(c* x)^(7/2)-64/231*b^2*(b*x^2+a)^(3/4)/a^3/c^5/(c*x)^(3/2)
Time = 0.19 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.53 \[ \int \frac {1}{(c x)^{13/2} \sqrt [4]{a+b x^2}} \, dx=-\frac {2 x \left (a+b x^2\right )^{3/4} \left (21 a^2-24 a b x^2+32 b^2 x^4\right )}{231 a^3 (c x)^{13/2}} \] Input:
Integrate[1/((c*x)^(13/2)*(a + b*x^2)^(1/4)),x]
Output:
(-2*x*(a + b*x^2)^(3/4)*(21*a^2 - 24*a*b*x^2 + 32*b^2*x^4))/(231*a^3*(c*x) ^(13/2))
Time = 0.19 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.04, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {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.
\(\displaystyle \int \frac {1}{(c x)^{13/2} \sqrt [4]{a+b x^2}} \, dx\) |
\(\Big \downarrow \) 246 |
\(\displaystyle -\frac {8 \int \frac {\left (b x^2+a\right )^{3/4}}{(c x)^{13/2}}dx}{3 a}-\frac {2 \left (a+b x^2\right )^{3/4}}{3 a c (c x)^{11/2}}\) |
\(\Big \downarrow \) 246 |
\(\displaystyle -\frac {8 \left (-\frac {4 \int \frac {\left (b x^2+a\right )^{7/4}}{(c x)^{13/2}}dx}{7 a}-\frac {2 \left (a+b x^2\right )^{7/4}}{7 a c (c x)^{11/2}}\right )}{3 a}-\frac {2 \left (a+b x^2\right )^{3/4}}{3 a c (c x)^{11/2}}\) |
\(\Big \downarrow \) 242 |
\(\displaystyle -\frac {8 \left (\frac {8 \left (a+b x^2\right )^{11/4}}{77 a^2 c (c x)^{11/2}}-\frac {2 \left (a+b x^2\right )^{7/4}}{7 a c (c x)^{11/2}}\right )}{3 a}-\frac {2 \left (a+b x^2\right )^{3/4}}{3 a c (c x)^{11/2}}\) |
Input:
Int[1/((c*x)^(13/2)*(a + b*x^2)^(1/4)),x]
Output:
(-2*(a + b*x^2)^(3/4))/(3*a*c*(c*x)^(11/2)) - (8*((-2*(a + b*x^2)^(7/4))/( 7*a*c*(c*x)^(11/2)) + (8*(a + b*x^2)^(11/4))/(77*a^2*c*(c*x)^(11/2))))/(3* 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 = 0.26 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.47
method | result | size |
gosper | \(-\frac {2 x \left (b \,x^{2}+a \right )^{\frac {3}{4}} \left (32 b^{2} x^{4}-24 a b \,x^{2}+21 a^{2}\right )}{231 a^{3} \left (c x \right )^{\frac {13}{2}}}\) | \(42\) |
orering | \(-\frac {2 x \left (b \,x^{2}+a \right )^{\frac {3}{4}} \left (32 b^{2} x^{4}-24 a b \,x^{2}+21 a^{2}\right )}{231 a^{3} \left (c x \right )^{\frac {13}{2}}}\) | \(42\) |
risch | \(-\frac {2 \left (b \,x^{2}+a \right )^{\frac {3}{4}} \left (32 b^{2} x^{4}-24 a b \,x^{2}+21 a^{2}\right )}{231 c^{6} \sqrt {c x}\, a^{3} x^{5}}\) | \(47\) |
Input:
int(1/(c*x)^(13/2)/(b*x^2+a)^(1/4),x,method=_RETURNVERBOSE)
Output:
-2/231*x*(b*x^2+a)^(3/4)*(32*b^2*x^4-24*a*b*x^2+21*a^2)/a^3/(c*x)^(13/2)
Time = 0.15 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.52 \[ \int \frac {1}{(c x)^{13/2} \sqrt [4]{a+b x^2}} \, dx=-\frac {2 \, {\left (32 \, b^{2} x^{4} - 24 \, a b x^{2} + 21 \, a^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {c x}}{231 \, a^{3} c^{7} x^{6}} \] Input:
integrate(1/(c*x)^(13/2)/(b*x^2+a)^(1/4),x, algorithm="fricas")
Output:
-2/231*(32*b^2*x^4 - 24*a*b*x^2 + 21*a^2)*(b*x^2 + a)^(3/4)*sqrt(c*x)/(a^3 *c^7*x^6)
Leaf count of result is larger than twice the leaf count of optimal. 483 vs. \(2 (82) = 164\).
Time = 168.84 (sec) , antiderivative size = 483, normalized size of antiderivative = 5.43 \[ \int \frac {1}{(c x)^{13/2} \sqrt [4]{a+b x^2}} \, dx=\frac {21 a^{4} b^{\frac {19}{4}} \left (\frac {a}{b x^{2}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {11}{4}\right )}{32 a^{5} b^{4} c^{\frac {13}{2}} x^{4} \Gamma \left (\frac {1}{4}\right ) + 64 a^{4} b^{5} c^{\frac {13}{2}} x^{6} \Gamma \left (\frac {1}{4}\right ) + 32 a^{3} b^{6} c^{\frac {13}{2}} x^{8} \Gamma \left (\frac {1}{4}\right )} + \frac {18 a^{3} b^{\frac {23}{4}} x^{2} \left (\frac {a}{b x^{2}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {11}{4}\right )}{32 a^{5} b^{4} c^{\frac {13}{2}} x^{4} \Gamma \left (\frac {1}{4}\right ) + 64 a^{4} b^{5} c^{\frac {13}{2}} x^{6} \Gamma \left (\frac {1}{4}\right ) + 32 a^{3} b^{6} c^{\frac {13}{2}} x^{8} \Gamma \left (\frac {1}{4}\right )} + \frac {5 a^{2} b^{\frac {27}{4}} x^{4} \left (\frac {a}{b x^{2}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {11}{4}\right )}{32 a^{5} b^{4} c^{\frac {13}{2}} x^{4} \Gamma \left (\frac {1}{4}\right ) + 64 a^{4} b^{5} c^{\frac {13}{2}} x^{6} \Gamma \left (\frac {1}{4}\right ) + 32 a^{3} b^{6} c^{\frac {13}{2}} x^{8} \Gamma \left (\frac {1}{4}\right )} + \frac {40 a b^{\frac {31}{4}} x^{6} \left (\frac {a}{b x^{2}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {11}{4}\right )}{32 a^{5} b^{4} c^{\frac {13}{2}} x^{4} \Gamma \left (\frac {1}{4}\right ) + 64 a^{4} b^{5} c^{\frac {13}{2}} x^{6} \Gamma \left (\frac {1}{4}\right ) + 32 a^{3} b^{6} c^{\frac {13}{2}} x^{8} \Gamma \left (\frac {1}{4}\right )} + \frac {32 b^{\frac {35}{4}} x^{8} \left (\frac {a}{b x^{2}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {11}{4}\right )}{32 a^{5} b^{4} c^{\frac {13}{2}} x^{4} \Gamma \left (\frac {1}{4}\right ) + 64 a^{4} b^{5} c^{\frac {13}{2}} x^{6} \Gamma \left (\frac {1}{4}\right ) + 32 a^{3} b^{6} c^{\frac {13}{2}} x^{8} \Gamma \left (\frac {1}{4}\right )} \] Input:
integrate(1/(c*x)**(13/2)/(b*x**2+a)**(1/4),x)
Output:
21*a**4*b**(19/4)*(a/(b*x**2) + 1)**(3/4)*gamma(-11/4)/(32*a**5*b**4*c**(1 3/2)*x**4*gamma(1/4) + 64*a**4*b**5*c**(13/2)*x**6*gamma(1/4) + 32*a**3*b* *6*c**(13/2)*x**8*gamma(1/4)) + 18*a**3*b**(23/4)*x**2*(a/(b*x**2) + 1)**( 3/4)*gamma(-11/4)/(32*a**5*b**4*c**(13/2)*x**4*gamma(1/4) + 64*a**4*b**5*c **(13/2)*x**6*gamma(1/4) + 32*a**3*b**6*c**(13/2)*x**8*gamma(1/4)) + 5*a** 2*b**(27/4)*x**4*(a/(b*x**2) + 1)**(3/4)*gamma(-11/4)/(32*a**5*b**4*c**(13 /2)*x**4*gamma(1/4) + 64*a**4*b**5*c**(13/2)*x**6*gamma(1/4) + 32*a**3*b** 6*c**(13/2)*x**8*gamma(1/4)) + 40*a*b**(31/4)*x**6*(a/(b*x**2) + 1)**(3/4) *gamma(-11/4)/(32*a**5*b**4*c**(13/2)*x**4*gamma(1/4) + 64*a**4*b**5*c**(1 3/2)*x**6*gamma(1/4) + 32*a**3*b**6*c**(13/2)*x**8*gamma(1/4)) + 32*b**(35 /4)*x**8*(a/(b*x**2) + 1)**(3/4)*gamma(-11/4)/(32*a**5*b**4*c**(13/2)*x**4 *gamma(1/4) + 64*a**4*b**5*c**(13/2)*x**6*gamma(1/4) + 32*a**3*b**6*c**(13 /2)*x**8*gamma(1/4))
\[ \int \frac {1}{(c x)^{13/2} \sqrt [4]{a+b x^2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {1}{4}} \left (c x\right )^{\frac {13}{2}}} \,d x } \] Input:
integrate(1/(c*x)^(13/2)/(b*x^2+a)^(1/4),x, algorithm="maxima")
Output:
integrate(1/((b*x^2 + a)^(1/4)*(c*x)^(13/2)), x)
\[ \int \frac {1}{(c x)^{13/2} \sqrt [4]{a+b x^2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {1}{4}} \left (c x\right )^{\frac {13}{2}}} \,d x } \] Input:
integrate(1/(c*x)^(13/2)/(b*x^2+a)^(1/4),x, algorithm="giac")
Output:
integrate(1/((b*x^2 + a)^(1/4)*(c*x)^(13/2)), x)
Time = 0.52 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.61 \[ \int \frac {1}{(c x)^{13/2} \sqrt [4]{a+b x^2}} \, dx=-\frac {{\left (b\,x^2+a\right )}^{3/4}\,\left (\frac {2}{11\,a\,c^6}-\frac {16\,b\,x^2}{77\,a^2\,c^6}+\frac {64\,b^2\,x^4}{231\,a^3\,c^6}\right )}{x^5\,\sqrt {c\,x}} \] Input:
int(1/((c*x)^(13/2)*(a + b*x^2)^(1/4)),x)
Output:
-((a + b*x^2)^(3/4)*(2/(11*a*c^6) - (16*b*x^2)/(77*a^2*c^6) + (64*b^2*x^4) /(231*a^3*c^6)))/(x^5*(c*x)^(1/2))
Time = 0.28 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.24 \[ \int \frac {1}{(c x)^{13/2} \sqrt [4]{a+b x^2}} \, dx=\frac {\sqrt {c}\, \left (7 \left (b \,x^{2}+a \right ) a^{3}-12 \left (b \,x^{2}+a \right ) a^{2} b \,x^{2}+32 \left (b \,x^{2}+a \right ) a \,b^{2} x^{4}+128 \left (b \,x^{2}+a \right ) b^{3} x^{6}-77 a^{4}-77 a^{3} b \,x^{2}\right )}{385 \left (b \,x^{2}+a \right )^{\frac {3}{4}} \sqrt {x}\, \sqrt {b \,x^{2}+a}\, a^{3} c^{7} x^{5}} \] Input:
int(1/(c*x)^(13/2)/(b*x^2+a)^(1/4),x)
Output:
(sqrt(c)*(a + b*x**2)**(1/4)*(7*(a + b*x**2)*a**3 - 12*(a + b*x**2)*a**2*b *x**2 + 32*(a + b*x**2)*a*b**2*x**4 + 128*(a + b*x**2)*b**3*x**6 - 77*a**4 - 77*a**3*b*x**2))/(385*sqrt(x)*sqrt(a + b*x**2)*a**3*c**7*x**5*(a + b*x* *2))