Integrand size = 16, antiderivative size = 126 \[ \int \frac {1}{x^4 \left (a-b x^2\right )^{5/4}} \, dx=\frac {2}{a x^3 \sqrt [4]{a-b x^2}}-\frac {7 \left (a-b x^2\right )^{3/4}}{3 a^2 x^3}-\frac {7 b \left (a-b x^2\right )^{3/4}}{2 a^3 x}-\frac {7 b^{3/2} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{2 a^{5/2} \sqrt [4]{a-b x^2}} \] Output:
2/a/x^3/(-b*x^2+a)^(1/4)-7/3*(-b*x^2+a)^(3/4)/a^2/x^3-7/2*b*(-b*x^2+a)^(3/ 4)/a^3/x-7/2*b^(3/2)*(1-b*x^2/a)^(1/4)*EllipticE(sin(1/2*arcsin(b^(1/2)*x/ a^(1/2))),2^(1/2))/a^(5/2)/(-b*x^2+a)^(1/4)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.01 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.44 \[ \int \frac {1}{x^4 \left (a-b x^2\right )^{5/4}} \, dx=-\frac {\sqrt [4]{1-\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {5}{4},-\frac {1}{2},\frac {b x^2}{a}\right )}{3 a x^3 \sqrt [4]{a-b x^2}} \] Input:
Integrate[1/(x^4*(a - b*x^2)^(5/4)),x]
Output:
-1/3*((1 - (b*x^2)/a)^(1/4)*Hypergeometric2F1[-3/2, 5/4, -1/2, (b*x^2)/a]) /(a*x^3*(a - b*x^2)^(1/4))
Time = 0.22 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.08, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {253, 264, 264, 227, 226}
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.
\(\displaystyle \int \frac {1}{x^4 \left (a-b x^2\right )^{5/4}} \, dx\) |
\(\Big \downarrow \) 253 |
\(\displaystyle \frac {7 \int \frac {1}{x^4 \sqrt [4]{a-b x^2}}dx}{a}+\frac {2}{a x^3 \sqrt [4]{a-b x^2}}\) |
\(\Big \downarrow \) 264 |
\(\displaystyle \frac {7 \left (\frac {b \int \frac {1}{x^2 \sqrt [4]{a-b x^2}}dx}{2 a}-\frac {\left (a-b x^2\right )^{3/4}}{3 a x^3}\right )}{a}+\frac {2}{a x^3 \sqrt [4]{a-b x^2}}\) |
\(\Big \downarrow \) 264 |
\(\displaystyle \frac {7 \left (\frac {b \left (-\frac {b \int \frac {1}{\sqrt [4]{a-b x^2}}dx}{2 a}-\frac {\left (a-b x^2\right )^{3/4}}{a x}\right )}{2 a}-\frac {\left (a-b x^2\right )^{3/4}}{3 a x^3}\right )}{a}+\frac {2}{a x^3 \sqrt [4]{a-b x^2}}\) |
\(\Big \downarrow \) 227 |
\(\displaystyle \frac {7 \left (\frac {b \left (-\frac {b \sqrt [4]{1-\frac {b x^2}{a}} \int \frac {1}{\sqrt [4]{1-\frac {b x^2}{a}}}dx}{2 a \sqrt [4]{a-b x^2}}-\frac {\left (a-b x^2\right )^{3/4}}{a x}\right )}{2 a}-\frac {\left (a-b x^2\right )^{3/4}}{3 a x^3}\right )}{a}+\frac {2}{a x^3 \sqrt [4]{a-b x^2}}\) |
\(\Big \downarrow \) 226 |
\(\displaystyle \frac {7 \left (\frac {b \left (-\frac {\sqrt {b} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {a} \sqrt [4]{a-b x^2}}-\frac {\left (a-b x^2\right )^{3/4}}{a x}\right )}{2 a}-\frac {\left (a-b x^2\right )^{3/4}}{3 a x^3}\right )}{a}+\frac {2}{a x^3 \sqrt [4]{a-b x^2}}\) |
Input:
Int[1/(x^4*(a - b*x^2)^(5/4)),x]
Output:
2/(a*x^3*(a - b*x^2)^(1/4)) + (7*(-1/3*(a - b*x^2)^(3/4)/(a*x^3) + (b*(-(( a - b*x^2)^(3/4)/(a*x)) - (Sqrt[b]*(1 - (b*x^2)/a)^(1/4)*EllipticE[ArcSin[ (Sqrt[b]*x)/Sqrt[a]]/2, 2])/(Sqrt[a]*(a - b*x^2)^(1/4))))/(2*a)))/a
Int[((a_) + (b_.)*(x_)^2)^(-1/4), x_Symbol] :> Simp[(2/(a^(1/4)*Rt[-b/a, 2] ))*EllipticE[(1/2)*ArcSin[Rt[-b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ [a, 0] && NegQ[b/a]
Int[((a_) + (b_.)*(x_)^2)^(-1/4), x_Symbol] :> Simp[(1 + b*(x^2/a))^(1/4)/( a + b*x^2)^(1/4) Int[1/(1 + b*(x^2/a))^(1/4), x], x] /; FreeQ[{a, b}, x] && PosQ[a]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c*x )^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 2*a*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m }, x] && LtQ[p, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
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] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
\[\int \frac {1}{x^{4} \left (-b \,x^{2}+a \right )^{\frac {5}{4}}}d x\]
Input:
int(1/x^4/(-b*x^2+a)^(5/4),x)
Output:
int(1/x^4/(-b*x^2+a)^(5/4),x)
\[ \int \frac {1}{x^4 \left (a-b x^2\right )^{5/4}} \, dx=\int { \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {5}{4}} x^{4}} \,d x } \] Input:
integrate(1/x^4/(-b*x^2+a)^(5/4),x, algorithm="fricas")
Output:
integral((-b*x^2 + a)^(3/4)/(b^2*x^8 - 2*a*b*x^6 + a^2*x^4), x)
Result contains complex when optimal does not.
Time = 0.64 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.27 \[ \int \frac {1}{x^4 \left (a-b x^2\right )^{5/4}} \, dx=- \frac {{{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, \frac {5}{4} \\ - \frac {1}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{3 a^{\frac {5}{4}} x^{3}} \] Input:
integrate(1/x**4/(-b*x**2+a)**(5/4),x)
Output:
-hyper((-3/2, 5/4), (-1/2,), b*x**2*exp_polar(2*I*pi)/a)/(3*a**(5/4)*x**3)
\[ \int \frac {1}{x^4 \left (a-b x^2\right )^{5/4}} \, dx=\int { \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {5}{4}} x^{4}} \,d x } \] Input:
integrate(1/x^4/(-b*x^2+a)^(5/4),x, algorithm="maxima")
Output:
integrate(1/((-b*x^2 + a)^(5/4)*x^4), x)
\[ \int \frac {1}{x^4 \left (a-b x^2\right )^{5/4}} \, dx=\int { \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {5}{4}} x^{4}} \,d x } \] Input:
integrate(1/x^4/(-b*x^2+a)^(5/4),x, algorithm="giac")
Output:
integrate(1/((-b*x^2 + a)^(5/4)*x^4), x)
Timed out. \[ \int \frac {1}{x^4 \left (a-b x^2\right )^{5/4}} \, dx=\int \frac {1}{x^4\,{\left (a-b\,x^2\right )}^{5/4}} \,d x \] Input:
int(1/(x^4*(a - b*x^2)^(5/4)),x)
Output:
int(1/(x^4*(a - b*x^2)^(5/4)), x)
\[ \int \frac {1}{x^4 \left (a-b x^2\right )^{5/4}} \, dx=\int \frac {1}{\left (-b \,x^{2}+a \right )^{\frac {1}{4}} a \,x^{4}-\left (-b \,x^{2}+a \right )^{\frac {1}{4}} b \,x^{6}}d x \] Input:
int(1/x^4/(-b*x^2+a)^(5/4),x)
Output:
int(1/((a - b*x**2)**(1/4)*a*x**4 - (a - b*x**2)**(1/4)*b*x**6),x)