Integrand size = 15, antiderivative size = 99 \[ \int \frac {\left (a+b x^4\right )^{3/4}}{x^6} \, dx=-\frac {3 b}{5 x \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{5 x^5}+\frac {3 b^{3/2} \sqrt [4]{1+\frac {a}{b x^4}} x E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{5 \sqrt {a} \sqrt [4]{a+b x^4}} \] Output:
-3/5*b/x/(b*x^4+a)^(1/4)-1/5*(b*x^4+a)^(3/4)/x^5+3/5*b^(3/2)*(1+a/b/x^4)^( 1/4)*x*EllipticE(sin(1/2*arccot(b^(1/2)*x^2/a^(1/2))),2^(1/2))/a^(1/2)/(b* x^4+a)^(1/4)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.01 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.52 \[ \int \frac {\left (a+b x^4\right )^{3/4}}{x^6} \, dx=-\frac {\left (a+b x^4\right )^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},-\frac {3}{4},-\frac {1}{4},-\frac {b x^4}{a}\right )}{5 x^5 \left (1+\frac {b x^4}{a}\right )^{3/4}} \] Input:
Integrate[(a + b*x^4)^(3/4)/x^6,x]
Output:
-1/5*((a + b*x^4)^(3/4)*Hypergeometric2F1[-5/4, -3/4, -1/4, -((b*x^4)/a)]) /(x^5*(1 + (b*x^4)/a)^(3/4))
Time = 0.43 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {809, 841, 813, 858, 807, 212}
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.
\(\displaystyle \int \frac {\left (a+b x^4\right )^{3/4}}{x^6} \, dx\) |
\(\Big \downarrow \) 809 |
\(\displaystyle \frac {3}{5} b \int \frac {1}{x^2 \sqrt [4]{b x^4+a}}dx-\frac {\left (a+b x^4\right )^{3/4}}{5 x^5}\) |
\(\Big \downarrow \) 841 |
\(\displaystyle \frac {3}{5} b \left (-b \int \frac {x^2}{\left (b x^4+a\right )^{5/4}}dx-\frac {1}{x \sqrt [4]{a+b x^4}}\right )-\frac {\left (a+b x^4\right )^{3/4}}{5 x^5}\) |
\(\Big \downarrow \) 813 |
\(\displaystyle \frac {3}{5} b \left (-\frac {x \sqrt [4]{\frac {a}{b x^4}+1} \int \frac {1}{\left (\frac {a}{b x^4}+1\right )^{5/4} x^3}dx}{\sqrt [4]{a+b x^4}}-\frac {1}{x \sqrt [4]{a+b x^4}}\right )-\frac {\left (a+b x^4\right )^{3/4}}{5 x^5}\) |
\(\Big \downarrow \) 858 |
\(\displaystyle \frac {3}{5} b \left (\frac {x \sqrt [4]{\frac {a}{b x^4}+1} \int \frac {1}{\left (\frac {a}{b x^4}+1\right )^{5/4} x}d\frac {1}{x}}{\sqrt [4]{a+b x^4}}-\frac {1}{x \sqrt [4]{a+b x^4}}\right )-\frac {\left (a+b x^4\right )^{3/4}}{5 x^5}\) |
\(\Big \downarrow \) 807 |
\(\displaystyle \frac {3}{5} b \left (\frac {x \sqrt [4]{\frac {a}{b x^4}+1} \int \frac {1}{\left (\frac {a}{b x^2}+1\right )^{5/4}}d\frac {1}{x^2}}{2 \sqrt [4]{a+b x^4}}-\frac {1}{x \sqrt [4]{a+b x^4}}\right )-\frac {\left (a+b x^4\right )^{3/4}}{5 x^5}\) |
\(\Big \downarrow \) 212 |
\(\displaystyle \frac {3}{5} b \left (\frac {\sqrt {b} x \sqrt [4]{\frac {a}{b x^4}+1} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {a}}{\sqrt {b} x^2}\right )\right |2\right )}{\sqrt {a} \sqrt [4]{a+b x^4}}-\frac {1}{x \sqrt [4]{a+b x^4}}\right )-\frac {\left (a+b x^4\right )^{3/4}}{5 x^5}\) |
Input:
Int[(a + b*x^4)^(3/4)/x^6,x]
Output:
-1/5*(a + b*x^4)^(3/4)/x^5 + (3*b*(-(1/(x*(a + b*x^4)^(1/4))) + (Sqrt[b]*( 1 + a/(b*x^4))^(1/4)*x*EllipticE[ArcTan[Sqrt[a]/(Sqrt[b]*x^2)]/2, 2])/(Sqr t[a]*(a + b*x^4)^(1/4))))/5
Int[((a_) + (b_.)*(x_)^2)^(-5/4), x_Symbol] :> Simp[(2/(a^(5/4)*Rt[b/a, 2]) )*EllipticE[(1/2)*ArcTan[Rt[b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a , 0] && PosQ[b/a]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* x)^(m + 1)*((a + b*x^n)^p/(c*(m + 1))), x] - Simp[b*n*(p/(c^n*(m + 1))) I nt[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && IGtQ [n, 0] && GtQ[p, 0] && LtQ[m, -1] && !ILtQ[(m + n*p + n + 1)/n, 0] && IntB inomialQ[a, b, c, n, m, p, x]
Int[(x_)^2/((a_) + (b_.)*(x_)^4)^(5/4), x_Symbol] :> Simp[x*((1 + a/(b*x^4) )^(1/4)/(b*(a + b*x^4)^(1/4))) Int[1/(x^3*(1 + a/(b*x^4))^(5/4)), x], x] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[1/((x_)^2*((a_) + (b_.)*(x_)^4)^(1/4)), x_Symbol] :> -Simp[(x*(a + b*x^ 4)^(1/4))^(-1), x] - Simp[b Int[x^2/(a + b*x^4)^(5/4), x], x] /; FreeQ[{a , b}, x] && PosQ[b/a]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /; FreeQ[{a, b, p}, x] && ILtQ[n, 0] && Int egerQ[m]
\[\int \frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}}}{x^{6}}d x\]
Input:
int((b*x^4+a)^(3/4)/x^6,x)
Output:
int((b*x^4+a)^(3/4)/x^6,x)
\[ \int \frac {\left (a+b x^4\right )^{3/4}}{x^6} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {3}{4}}}{x^{6}} \,d x } \] Input:
integrate((b*x^4+a)^(3/4)/x^6,x, algorithm="fricas")
Output:
integral((b*x^4 + a)^(3/4)/x^6, x)
Result contains complex when optimal does not.
Time = 0.63 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.31 \[ \int \frac {\left (a+b x^4\right )^{3/4}}{x^6} \, dx=- \frac {b^{\frac {3}{4}} {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{4}}} \right )}}{2 x^{2}} \] Input:
integrate((b*x**4+a)**(3/4)/x**6,x)
Output:
-b**(3/4)*hyper((-3/4, 1/2), (3/2,), a*exp_polar(I*pi)/(b*x**4))/(2*x**2)
\[ \int \frac {\left (a+b x^4\right )^{3/4}}{x^6} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {3}{4}}}{x^{6}} \,d x } \] Input:
integrate((b*x^4+a)^(3/4)/x^6,x, algorithm="maxima")
Output:
integrate((b*x^4 + a)^(3/4)/x^6, x)
\[ \int \frac {\left (a+b x^4\right )^{3/4}}{x^6} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {3}{4}}}{x^{6}} \,d x } \] Input:
integrate((b*x^4+a)^(3/4)/x^6,x, algorithm="giac")
Output:
integrate((b*x^4 + a)^(3/4)/x^6, x)
Timed out. \[ \int \frac {\left (a+b x^4\right )^{3/4}}{x^6} \, dx=\int \frac {{\left (b\,x^4+a\right )}^{3/4}}{x^6} \,d x \] Input:
int((a + b*x^4)^(3/4)/x^6,x)
Output:
int((a + b*x^4)^(3/4)/x^6, x)
\[ \int \frac {\left (a+b x^4\right )^{3/4}}{x^6} \, dx=\frac {-\left (b \,x^{4}+a \right )^{\frac {3}{4}}-3 \left (\int \frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}}}{b \,x^{10}+a \,x^{6}}d x \right ) a \,x^{5}}{2 x^{5}} \] Input:
int((b*x^4+a)^(3/4)/x^6,x)
Output:
( - (a + b*x**4)**(3/4) - 3*int((a + b*x**4)**(3/4)/(a*x**6 + b*x**10),x)* a*x**5)/(2*x**5)