Integrand size = 20, antiderivative size = 60 \[ \int \frac {1}{(c x)^{9/2} \sqrt [4]{a-b x^2}} \, dx=-\frac {2 \left (a-b x^2\right )^{3/4}}{7 a c (c x)^{7/2}}-\frac {8 b \left (a-b x^2\right )^{3/4}}{21 a^2 c^3 (c x)^{3/2}} \] Output:
-2/7*(-b*x^2+a)^(3/4)/a/c/(c*x)^(7/2)-8/21*b*(-b*x^2+a)^(3/4)/a^2/c^3/(c*x )^(3/2)
Time = 0.18 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.62 \[ \int \frac {1}{(c x)^{9/2} \sqrt [4]{a-b x^2}} \, dx=-\frac {2 x \left (a-b x^2\right )^{3/4} \left (3 a+4 b x^2\right )}{21 a^2 (c x)^{9/2}} \] Input:
Integrate[1/((c*x)^(9/2)*(a - b*x^2)^(1/4)),x]
Output:
(-2*x*(a - b*x^2)^(3/4)*(3*a + 4*b*x^2))/(21*a^2*(c*x)^(9/2))
Time = 0.17 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.98, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {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.
\(\displaystyle \int \frac {1}{(c x)^{9/2} \sqrt [4]{a-b x^2}} \, dx\) |
\(\Big \downarrow \) 246 |
\(\displaystyle -\frac {4 \int \frac {\left (a-b x^2\right )^{3/4}}{(c x)^{9/2}}dx}{3 a}-\frac {2 \left (a-b x^2\right )^{3/4}}{3 a c (c x)^{7/2}}\) |
\(\Big \downarrow \) 242 |
\(\displaystyle \frac {8 \left (a-b x^2\right )^{7/4}}{21 a^2 c (c x)^{7/2}}-\frac {2 \left (a-b x^2\right )^{3/4}}{3 a c (c x)^{7/2}}\) |
Input:
Int[1/((c*x)^(9/2)*(a - b*x^2)^(1/4)),x]
Output:
(-2*(a - b*x^2)^(3/4))/(3*a*c*(c*x)^(7/2)) + (8*(a - b*x^2)^(7/4))/(21*a^2 *c*(c*x)^(7/2))
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 = 0.29 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.53
method | result | size |
gosper | \(-\frac {2 x \left (-b \,x^{2}+a \right )^{\frac {3}{4}} \left (4 b \,x^{2}+3 a \right )}{21 a^{2} \left (c x \right )^{\frac {9}{2}}}\) | \(32\) |
orering | \(-\frac {2 x \left (-b \,x^{2}+a \right )^{\frac {3}{4}} \left (4 b \,x^{2}+3 a \right )}{21 a^{2} \left (c x \right )^{\frac {9}{2}}}\) | \(32\) |
risch | \(-\frac {2 \left (-b \,x^{2}+a \right )^{\frac {3}{4}} \left (4 b \,x^{2}+3 a \right )}{21 c^{4} \sqrt {c x}\, a^{2} x^{3}}\) | \(37\) |
Input:
int(1/(c*x)^(9/2)/(-b*x^2+a)^(1/4),x,method=_RETURNVERBOSE)
Output:
-2/21*x*(-b*x^2+a)^(3/4)*(4*b*x^2+3*a)/a^2/(c*x)^(9/2)
Time = 0.09 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.60 \[ \int \frac {1}{(c x)^{9/2} \sqrt [4]{a-b x^2}} \, dx=-\frac {2 \, {\left (4 \, b x^{2} + 3 \, a\right )} {\left (-b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {c x}}{21 \, a^{2} c^{5} x^{4}} \] Input:
integrate(1/(c*x)^(9/2)/(-b*x^2+a)^(1/4),x, algorithm="fricas")
Output:
-2/21*(4*b*x^2 + 3*a)*(-b*x^2 + a)^(3/4)*sqrt(c*x)/(a^2*c^5*x^4)
Result contains complex when optimal does not.
Time = 20.14 (sec) , antiderivative size = 343, normalized size of antiderivative = 5.72 \[ \int \frac {1}{(c x)^{9/2} \sqrt [4]{a-b x^2}} \, dx=\begin {cases} - \frac {3 b^{\frac {3}{4}} \left (\frac {a}{b x^{2}} - 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {7}{4}\right )}{8 a c^{\frac {9}{2}} x^{2} \Gamma \left (\frac {1}{4}\right )} - \frac {b^{\frac {7}{4}} \left (\frac {a}{b x^{2}} - 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {7}{4}\right )}{2 a^{2} c^{\frac {9}{2}} \Gamma \left (\frac {1}{4}\right )} & \text {for}\: \left |{\frac {a}{b x^{2}}}\right | > 1 \\- \frac {3 a^{2} b^{\frac {7}{4}} \left (- \frac {a}{b x^{2}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {7}{4}\right )}{- 8 a^{3} b c^{\frac {9}{2}} x^{2} e^{\frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right ) + 8 a^{2} b^{2} c^{\frac {9}{2}} x^{4} e^{\frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right )} - \frac {a b^{\frac {11}{4}} x^{2} \left (- \frac {a}{b x^{2}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {7}{4}\right )}{- 8 a^{3} b c^{\frac {9}{2}} x^{2} e^{\frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right ) + 8 a^{2} b^{2} c^{\frac {9}{2}} x^{4} e^{\frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right )} + \frac {4 b^{\frac {15}{4}} x^{4} \left (- \frac {a}{b x^{2}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {7}{4}\right )}{- 8 a^{3} b c^{\frac {9}{2}} x^{2} e^{\frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right ) + 8 a^{2} b^{2} c^{\frac {9}{2}} x^{4} e^{\frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right )} & \text {otherwise} \end {cases} \] Input:
integrate(1/(c*x)**(9/2)/(-b*x**2+a)**(1/4),x)
Output:
Piecewise((-3*b**(3/4)*(a/(b*x**2) - 1)**(3/4)*gamma(-7/4)/(8*a*c**(9/2)*x **2*gamma(1/4)) - b**(7/4)*(a/(b*x**2) - 1)**(3/4)*gamma(-7/4)/(2*a**2*c** (9/2)*gamma(1/4)), Abs(a/(b*x**2)) > 1), (-3*a**2*b**(7/4)*(-a/(b*x**2) + 1)**(3/4)*gamma(-7/4)/(-8*a**3*b*c**(9/2)*x**2*exp(I*pi/4)*gamma(1/4) + 8* a**2*b**2*c**(9/2)*x**4*exp(I*pi/4)*gamma(1/4)) - a*b**(11/4)*x**2*(-a/(b* x**2) + 1)**(3/4)*gamma(-7/4)/(-8*a**3*b*c**(9/2)*x**2*exp(I*pi/4)*gamma(1 /4) + 8*a**2*b**2*c**(9/2)*x**4*exp(I*pi/4)*gamma(1/4)) + 4*b**(15/4)*x**4 *(-a/(b*x**2) + 1)**(3/4)*gamma(-7/4)/(-8*a**3*b*c**(9/2)*x**2*exp(I*pi/4) *gamma(1/4) + 8*a**2*b**2*c**(9/2)*x**4*exp(I*pi/4)*gamma(1/4)), True))
\[ \int \frac {1}{(c x)^{9/2} \sqrt [4]{a-b x^2}} \, dx=\int { \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {1}{4}} \left (c x\right )^{\frac {9}{2}}} \,d x } \] Input:
integrate(1/(c*x)^(9/2)/(-b*x^2+a)^(1/4),x, algorithm="maxima")
Output:
integrate(1/((-b*x^2 + a)^(1/4)*(c*x)^(9/2)), x)
\[ \int \frac {1}{(c x)^{9/2} \sqrt [4]{a-b x^2}} \, dx=\int { \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {1}{4}} \left (c x\right )^{\frac {9}{2}}} \,d x } \] Input:
integrate(1/(c*x)^(9/2)/(-b*x^2+a)^(1/4),x, algorithm="giac")
Output:
integrate(1/((-b*x^2 + a)^(1/4)*(c*x)^(9/2)), x)
Time = 0.52 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.68 \[ \int \frac {1}{(c x)^{9/2} \sqrt [4]{a-b x^2}} \, dx=-\frac {{\left (a-b\,x^2\right )}^{3/4}\,\left (\frac {2}{7\,a\,c^4}+\frac {8\,b\,x^2}{21\,a^2\,c^4}\right )}{x^3\,\sqrt {c\,x}} \] Input:
int(1/((c*x)^(9/2)*(a - b*x^2)^(1/4)),x)
Output:
-((a - b*x^2)^(3/4)*(2/(7*a*c^4) + (8*b*x^2)/(21*a^2*c^4)))/(x^3*(c*x)^(1/ 2))
Time = 0.25 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.62 \[ \int \frac {1}{(c x)^{9/2} \sqrt [4]{a-b x^2}} \, dx=\frac {\sqrt {c}\, \left (3 \left (-b \,x^{2}+a \right ) a^{2}+8 \left (-b \,x^{2}+a \right ) a b \,x^{2}-32 \left (-b \,x^{2}+a \right ) b^{2} x^{4}-21 a^{3}+21 a^{2} b \,x^{2}\right )}{63 \left (-b \,x^{2}+a \right )^{\frac {3}{4}} \sqrt {x}\, \sqrt {-b \,x^{2}+a}\, a^{2} c^{5} x^{3}} \] Input:
int(1/(c*x)^(9/2)/(-b*x^2+a)^(1/4),x)
Output:
(sqrt(c)*(a - b*x**2)**(1/4)*(3*(a - b*x**2)*a**2 + 8*(a - b*x**2)*a*b*x** 2 - 32*(a - b*x**2)*b**2*x**4 - 21*a**3 + 21*a**2*b*x**2))/(63*sqrt(x)*sqr t(a - b*x**2)*a**2*c**5*x**3*(a - b*x**2))