Integrand size = 19, antiderivative size = 89 \[ \int \frac {1}{(c x)^{11/2} \left (a+b x^2\right )^{3/4}} \, dx=-\frac {2 \sqrt [4]{a+b x^2}}{9 a c (c x)^{9/2}}+\frac {16 b \sqrt [4]{a+b x^2}}{45 a^2 c^3 (c x)^{5/2}}-\frac {64 b^2 \sqrt [4]{a+b x^2}}{45 a^3 c^5 \sqrt {c x}} \] Output:
-2/9*(b*x^2+a)^(1/4)/a/c/(c*x)^(9/2)+16/45*b*(b*x^2+a)^(1/4)/a^2/c^3/(c*x) ^(5/2)-64/45*b^2*(b*x^2+a)^(1/4)/a^3/c^5/(c*x)^(1/2)
Time = 0.27 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.53 \[ \int \frac {1}{(c x)^{11/2} \left (a+b x^2\right )^{3/4}} \, dx=-\frac {2 x \sqrt [4]{a+b x^2} \left (5 a^2-8 a b x^2+32 b^2 x^4\right )}{45 a^3 (c x)^{11/2}} \] Input:
Integrate[1/((c*x)^(11/2)*(a + b*x^2)^(3/4)),x]
Output:
(-2*x*(a + b*x^2)^(1/4)*(5*a^2 - 8*a*b*x^2 + 32*b^2*x^4))/(45*a^3*(c*x)^(1 1/2))
Time = 0.19 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, 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.
\(\displaystyle \int \frac {1}{(c x)^{11/2} \left (a+b x^2\right )^{3/4}} \, dx\) |
\(\Big \downarrow \) 246 |
\(\displaystyle -\frac {8 \int \frac {\sqrt [4]{b x^2+a}}{(c x)^{11/2}}dx}{a}-\frac {2 \sqrt [4]{a+b x^2}}{a c (c x)^{9/2}}\) |
\(\Big \downarrow \) 246 |
\(\displaystyle -\frac {8 \left (-\frac {4 \int \frac {\left (b x^2+a\right )^{5/4}}{(c x)^{11/2}}dx}{5 a}-\frac {2 \left (a+b x^2\right )^{5/4}}{5 a c (c x)^{9/2}}\right )}{a}-\frac {2 \sqrt [4]{a+b x^2}}{a c (c x)^{9/2}}\) |
\(\Big \downarrow \) 242 |
\(\displaystyle -\frac {8 \left (\frac {8 \left (a+b x^2\right )^{9/4}}{45 a^2 c (c x)^{9/2}}-\frac {2 \left (a+b x^2\right )^{5/4}}{5 a c (c x)^{9/2}}\right )}{a}-\frac {2 \sqrt [4]{a+b x^2}}{a c (c x)^{9/2}}\) |
Input:
Int[1/((c*x)^(11/2)*(a + b*x^2)^(3/4)),x]
Output:
(-2*(a + b*x^2)^(1/4))/(a*c*(c*x)^(9/2)) - (8*((-2*(a + b*x^2)^(5/4))/(5*a *c*(c*x)^(9/2)) + (8*(a + b*x^2)^(9/4))/(45*a^2*c*(c*x)^(9/2))))/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 = 0.28 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.47
method | result | size |
gosper | \(-\frac {2 x \left (b \,x^{2}+a \right )^{\frac {1}{4}} \left (32 b^{2} x^{4}-8 a b \,x^{2}+5 a^{2}\right )}{45 a^{3} \left (c x \right )^{\frac {11}{2}}}\) | \(42\) |
orering | \(-\frac {2 x \left (b \,x^{2}+a \right )^{\frac {1}{4}} \left (32 b^{2} x^{4}-8 a b \,x^{2}+5 a^{2}\right )}{45 a^{3} \left (c x \right )^{\frac {11}{2}}}\) | \(42\) |
risch | \(-\frac {2 \left (b \,x^{2}+a \right )^{\frac {1}{4}} \left (32 b^{2} x^{4}-8 a b \,x^{2}+5 a^{2}\right )}{45 c^{5} \sqrt {c x}\, a^{3} x^{4}}\) | \(47\) |
Input:
int(1/(c*x)^(11/2)/(b*x^2+a)^(3/4),x,method=_RETURNVERBOSE)
Output:
-2/45*x*(b*x^2+a)^(1/4)*(32*b^2*x^4-8*a*b*x^2+5*a^2)/a^3/(c*x)^(11/2)
Time = 0.09 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.52 \[ \int \frac {1}{(c x)^{11/2} \left (a+b x^2\right )^{3/4}} \, dx=-\frac {2 \, {\left (32 \, b^{2} x^{4} - 8 \, a b x^{2} + 5 \, a^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {1}{4}} \sqrt {c x}}{45 \, a^{3} c^{6} x^{5}} \] Input:
integrate(1/(c*x)^(11/2)/(b*x^2+a)^(3/4),x, algorithm="fricas")
Output:
-2/45*(32*b^2*x^4 - 8*a*b*x^2 + 5*a^2)*(b*x^2 + a)^(1/4)*sqrt(c*x)/(a^3*c^ 6*x^5)
Leaf count of result is larger than twice the leaf count of optimal. 483 vs. \(2 (82) = 164\).
Time = 106.18 (sec) , antiderivative size = 483, normalized size of antiderivative = 5.43 \[ \int \frac {1}{(c x)^{11/2} \left (a+b x^2\right )^{3/4}} \, dx=\frac {5 a^{4} b^{\frac {17}{4}} \sqrt [4]{\frac {a}{b x^{2}} + 1} \Gamma \left (- \frac {9}{4}\right )}{32 a^{5} b^{4} c^{\frac {11}{2}} x^{4} \Gamma \left (\frac {3}{4}\right ) + 64 a^{4} b^{5} c^{\frac {11}{2}} x^{6} \Gamma \left (\frac {3}{4}\right ) + 32 a^{3} b^{6} c^{\frac {11}{2}} x^{8} \Gamma \left (\frac {3}{4}\right )} + \frac {2 a^{3} b^{\frac {21}{4}} x^{2} \sqrt [4]{\frac {a}{b x^{2}} + 1} \Gamma \left (- \frac {9}{4}\right )}{32 a^{5} b^{4} c^{\frac {11}{2}} x^{4} \Gamma \left (\frac {3}{4}\right ) + 64 a^{4} b^{5} c^{\frac {11}{2}} x^{6} \Gamma \left (\frac {3}{4}\right ) + 32 a^{3} b^{6} c^{\frac {11}{2}} x^{8} \Gamma \left (\frac {3}{4}\right )} + \frac {21 a^{2} b^{\frac {25}{4}} x^{4} \sqrt [4]{\frac {a}{b x^{2}} + 1} \Gamma \left (- \frac {9}{4}\right )}{32 a^{5} b^{4} c^{\frac {11}{2}} x^{4} \Gamma \left (\frac {3}{4}\right ) + 64 a^{4} b^{5} c^{\frac {11}{2}} x^{6} \Gamma \left (\frac {3}{4}\right ) + 32 a^{3} b^{6} c^{\frac {11}{2}} x^{8} \Gamma \left (\frac {3}{4}\right )} + \frac {56 a b^{\frac {29}{4}} x^{6} \sqrt [4]{\frac {a}{b x^{2}} + 1} \Gamma \left (- \frac {9}{4}\right )}{32 a^{5} b^{4} c^{\frac {11}{2}} x^{4} \Gamma \left (\frac {3}{4}\right ) + 64 a^{4} b^{5} c^{\frac {11}{2}} x^{6} \Gamma \left (\frac {3}{4}\right ) + 32 a^{3} b^{6} c^{\frac {11}{2}} x^{8} \Gamma \left (\frac {3}{4}\right )} + \frac {32 b^{\frac {33}{4}} x^{8} \sqrt [4]{\frac {a}{b x^{2}} + 1} \Gamma \left (- \frac {9}{4}\right )}{32 a^{5} b^{4} c^{\frac {11}{2}} x^{4} \Gamma \left (\frac {3}{4}\right ) + 64 a^{4} b^{5} c^{\frac {11}{2}} x^{6} \Gamma \left (\frac {3}{4}\right ) + 32 a^{3} b^{6} c^{\frac {11}{2}} x^{8} \Gamma \left (\frac {3}{4}\right )} \] Input:
integrate(1/(c*x)**(11/2)/(b*x**2+a)**(3/4),x)
Output:
5*a**4*b**(17/4)*(a/(b*x**2) + 1)**(1/4)*gamma(-9/4)/(32*a**5*b**4*c**(11/ 2)*x**4*gamma(3/4) + 64*a**4*b**5*c**(11/2)*x**6*gamma(3/4) + 32*a**3*b**6 *c**(11/2)*x**8*gamma(3/4)) + 2*a**3*b**(21/4)*x**2*(a/(b*x**2) + 1)**(1/4 )*gamma(-9/4)/(32*a**5*b**4*c**(11/2)*x**4*gamma(3/4) + 64*a**4*b**5*c**(1 1/2)*x**6*gamma(3/4) + 32*a**3*b**6*c**(11/2)*x**8*gamma(3/4)) + 21*a**2*b **(25/4)*x**4*(a/(b*x**2) + 1)**(1/4)*gamma(-9/4)/(32*a**5*b**4*c**(11/2)* x**4*gamma(3/4) + 64*a**4*b**5*c**(11/2)*x**6*gamma(3/4) + 32*a**3*b**6*c* *(11/2)*x**8*gamma(3/4)) + 56*a*b**(29/4)*x**6*(a/(b*x**2) + 1)**(1/4)*gam ma(-9/4)/(32*a**5*b**4*c**(11/2)*x**4*gamma(3/4) + 64*a**4*b**5*c**(11/2)* x**6*gamma(3/4) + 32*a**3*b**6*c**(11/2)*x**8*gamma(3/4)) + 32*b**(33/4)*x **8*(a/(b*x**2) + 1)**(1/4)*gamma(-9/4)/(32*a**5*b**4*c**(11/2)*x**4*gamma (3/4) + 64*a**4*b**5*c**(11/2)*x**6*gamma(3/4) + 32*a**3*b**6*c**(11/2)*x* *8*gamma(3/4))
\[ \int \frac {1}{(c x)^{11/2} \left (a+b x^2\right )^{3/4}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} \left (c x\right )^{\frac {11}{2}}} \,d x } \] Input:
integrate(1/(c*x)^(11/2)/(b*x^2+a)^(3/4),x, algorithm="maxima")
Output:
integrate(1/((b*x^2 + a)^(3/4)*(c*x)^(11/2)), x)
\[ \int \frac {1}{(c x)^{11/2} \left (a+b x^2\right )^{3/4}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} \left (c x\right )^{\frac {11}{2}}} \,d x } \] Input:
integrate(1/(c*x)^(11/2)/(b*x^2+a)^(3/4),x, algorithm="giac")
Output:
integrate(1/((b*x^2 + a)^(3/4)*(c*x)^(11/2)), x)
Time = 0.54 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.61 \[ \int \frac {1}{(c x)^{11/2} \left (a+b x^2\right )^{3/4}} \, dx=-\frac {{\left (b\,x^2+a\right )}^{1/4}\,\left (\frac {2}{9\,a\,c^5}-\frac {16\,b\,x^2}{45\,a^2\,c^5}+\frac {64\,b^2\,x^4}{45\,a^3\,c^5}\right )}{x^4\,\sqrt {c\,x}} \] Input:
int(1/((c*x)^(11/2)*(a + b*x^2)^(3/4)),x)
Output:
-((a + b*x^2)^(1/4)*(2/(9*a*c^5) - (16*b*x^2)/(45*a^2*c^5) + (64*b^2*x^4)/ (45*a^3*c^5)))/(x^4*(c*x)^(1/2))
Time = 0.24 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.53 \[ \int \frac {1}{(c x)^{11/2} \left (a+b x^2\right )^{3/4}} \, dx=\frac {2 \sqrt {c}\, \left (b \,x^{2}+a \right )^{\frac {1}{4}} \left (-32 b^{2} x^{4}+8 a b \,x^{2}-5 a^{2}\right )}{45 \sqrt {x}\, a^{3} c^{6} x^{4}} \] Input:
int(1/(c*x)^(11/2)/(b*x^2+a)^(3/4),x)
Output:
(2*sqrt(c)*(a + b*x**2)**(1/4)*( - 5*a**2 + 8*a*b*x**2 - 32*b**2*x**4))/(4 5*sqrt(x)*a**3*c**6*x**4)