Integrand size = 16, antiderivative size = 103 \[ \int \frac {\left (a-b x^2\right )^{3/4}}{x^4} \, dx=-\frac {\left (a-b x^2\right )^{3/4}}{3 x^3}+\frac {b \left (a-b x^2\right )^{3/4}}{2 a x}+\frac {b^{3/2} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{2 \sqrt {a} \sqrt [4]{a-b x^2}} \] Output:
-1/3*(-b*x^2+a)^(3/4)/x^3+1/2*b*(-b*x^2+a)^(3/4)/a/x+1/2*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^(1/2)/(-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 = 52, normalized size of antiderivative = 0.50 \[ \int \frac {\left (a-b x^2\right )^{3/4}}{x^4} \, dx=-\frac {\left (a-b x^2\right )^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {3}{4},-\frac {1}{2},\frac {b x^2}{a}\right )}{3 x^3 \left (1-\frac {b x^2}{a}\right )^{3/4}} \] Input:
Integrate[(a - b*x^2)^(3/4)/x^4,x]
Output:
-1/3*((a - b*x^2)^(3/4)*Hypergeometric2F1[-3/2, -3/4, -1/2, (b*x^2)/a])/(x ^3*(1 - (b*x^2)/a)^(3/4))
Time = 0.19 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {247, 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.
\(\displaystyle \int \frac {\left (a-b x^2\right )^{3/4}}{x^4} \, dx\) |
\(\Big \downarrow \) 247 |
\(\displaystyle -\frac {1}{2} b \int \frac {1}{x^2 \sqrt [4]{a-b x^2}}dx-\frac {\left (a-b x^2\right )^{3/4}}{3 x^3}\) |
\(\Big \downarrow \) 264 |
\(\displaystyle -\frac {1}{2} b \left (-\frac {b \int \frac {1}{\sqrt [4]{a-b x^2}}dx}{2 a}-\frac {\left (a-b x^2\right )^{3/4}}{a x}\right )-\frac {\left (a-b x^2\right )^{3/4}}{3 x^3}\) |
\(\Big \downarrow \) 227 |
\(\displaystyle -\frac {1}{2} b \left (-\frac {b \sqrt [4]{1-\frac {b x^2}{a}} \int \frac {1}{\sqrt [4]{1-\frac {b x^2}{a}}}dx}{2 a \sqrt [4]{a-b x^2}}-\frac {\left (a-b x^2\right )^{3/4}}{a x}\right )-\frac {\left (a-b x^2\right )^{3/4}}{3 x^3}\) |
\(\Big \downarrow \) 226 |
\(\displaystyle -\frac {1}{2} b \left (-\frac {\sqrt {b} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {a} \sqrt [4]{a-b x^2}}-\frac {\left (a-b x^2\right )^{3/4}}{a x}\right )-\frac {\left (a-b x^2\right )^{3/4}}{3 x^3}\) |
Input:
Int[(a - b*x^2)^(3/4)/x^4,x]
Output:
-1/3*(a - b*x^2)^(3/4)/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
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/(c*(m + 1))), x] - Simp[2*b*(p/(c^2*(m + 1))) Int[ (c*x)^(m + 2)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && LtQ[m, -1] && !ILtQ[(m + 2*p + 3)/2, 0] && 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]
\[\int \frac {\left (-b \,x^{2}+a \right )^{\frac {3}{4}}}{x^{4}}d x\]
Input:
int((-b*x^2+a)^(3/4)/x^4,x)
Output:
int((-b*x^2+a)^(3/4)/x^4,x)
\[ \int \frac {\left (a-b x^2\right )^{3/4}}{x^4} \, dx=\int { \frac {{\left (-b x^{2} + a\right )}^{\frac {3}{4}}}{x^{4}} \,d x } \] Input:
integrate((-b*x^2+a)^(3/4)/x^4,x, algorithm="fricas")
Output:
integral((-b*x^2 + a)^(3/4)/x^4, x)
Result contains complex when optimal does not.
Time = 0.56 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.35 \[ \int \frac {\left (a-b x^2\right )^{3/4}}{x^4} \, dx=- \frac {a^{\frac {3}{4}} {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, - \frac {3}{4} \\ - \frac {1}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{3 x^{3}} \] Input:
integrate((-b*x**2+a)**(3/4)/x**4,x)
Output:
-a**(3/4)*hyper((-3/2, -3/4), (-1/2,), b*x**2*exp_polar(2*I*pi)/a)/(3*x**3 )
\[ \int \frac {\left (a-b x^2\right )^{3/4}}{x^4} \, dx=\int { \frac {{\left (-b x^{2} + a\right )}^{\frac {3}{4}}}{x^{4}} \,d x } \] Input:
integrate((-b*x^2+a)^(3/4)/x^4,x, algorithm="maxima")
Output:
integrate((-b*x^2 + a)^(3/4)/x^4, x)
\[ \int \frac {\left (a-b x^2\right )^{3/4}}{x^4} \, dx=\int { \frac {{\left (-b x^{2} + a\right )}^{\frac {3}{4}}}{x^{4}} \,d x } \] Input:
integrate((-b*x^2+a)^(3/4)/x^4,x, algorithm="giac")
Output:
integrate((-b*x^2 + a)^(3/4)/x^4, x)
Timed out. \[ \int \frac {\left (a-b x^2\right )^{3/4}}{x^4} \, dx=\int \frac {{\left (a-b\,x^2\right )}^{3/4}}{x^4} \,d x \] Input:
int((a - b*x^2)^(3/4)/x^4,x)
Output:
int((a - b*x^2)^(3/4)/x^4, x)
\[ \int \frac {\left (a-b x^2\right )^{3/4}}{x^4} \, dx=\frac {-2 \left (-b \,x^{2}+a \right )^{\frac {3}{4}}-3 \left (\int \frac {\left (-b \,x^{2}+a \right )^{\frac {3}{4}}}{-b \,x^{6}+a \,x^{4}}d x \right ) a \,x^{3}}{3 x^{3}} \] Input:
int((-b*x^2+a)^(3/4)/x^4,x)
Output:
( - 2*(a - b*x**2)**(3/4) - 3*int((a - b*x**2)**(3/4)/(a*x**4 - b*x**6),x) *a*x**3)/(3*x**3)