Integrand size = 24, antiderivative size = 85 \[ \int \frac {\left (-2 b+a x^4\right ) \left (b+a x^4\right )^{3/4}}{x^8} \, dx=\frac {\left (6 b-a x^4\right ) \left (b+a x^4\right )^{3/4}}{21 x^7}+\frac {1}{2} a^{7/4} \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )+\frac {1}{2} a^{7/4} \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right ) \]
1/21*(-a*x^4+6*b)*(a*x^4+b)^(3/4)/x^7+1/2*a^(7/4)*arctan(a^(1/4)*x/(a*x^4+ b)^(1/4))+1/2*a^(7/4)*arctanh(a^(1/4)*x/(a*x^4+b)^(1/4))
Time = 0.27 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-2 b+a x^4\right ) \left (b+a x^4\right )^{3/4}}{x^8} \, dx=\frac {\left (6 b-a x^4\right ) \left (b+a x^4\right )^{3/4}}{21 x^7}+\frac {1}{2} a^{7/4} \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )+\frac {1}{2} a^{7/4} \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right ) \]
((6*b - a*x^4)*(b + a*x^4)^(3/4))/(21*x^7) + (a^(7/4)*ArcTan[(a^(1/4)*x)/( b + a*x^4)^(1/4)])/2 + (a^(7/4)*ArcTanh[(a^(1/4)*x)/(b + a*x^4)^(1/4)])/2
Time = 0.23 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.16, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {953, 809, 770, 756, 216, 219}
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.
\(\displaystyle \int \frac {\left (a x^4-2 b\right ) \left (a x^4+b\right )^{3/4}}{x^8} \, dx\) |
\(\Big \downarrow \) 953 |
\(\displaystyle a \int \frac {\left (a x^4+b\right )^{3/4}}{x^4}dx+\frac {2 \left (a x^4+b\right )^{7/4}}{7 x^7}\) |
\(\Big \downarrow \) 809 |
\(\displaystyle a \left (a \int \frac {1}{\sqrt [4]{a x^4+b}}dx-\frac {\left (a x^4+b\right )^{3/4}}{3 x^3}\right )+\frac {2 \left (a x^4+b\right )^{7/4}}{7 x^7}\) |
\(\Big \downarrow \) 770 |
\(\displaystyle a \left (a \int \frac {1}{1-\frac {a x^4}{a x^4+b}}d\frac {x}{\sqrt [4]{a x^4+b}}-\frac {\left (a x^4+b\right )^{3/4}}{3 x^3}\right )+\frac {2 \left (a x^4+b\right )^{7/4}}{7 x^7}\) |
\(\Big \downarrow \) 756 |
\(\displaystyle a \left (a \left (\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {a} x^2}{\sqrt {a x^4+b}}}d\frac {x}{\sqrt [4]{a x^4+b}}+\frac {1}{2} \int \frac {1}{\frac {\sqrt {a} x^2}{\sqrt {a x^4+b}}+1}d\frac {x}{\sqrt [4]{a x^4+b}}\right )-\frac {\left (a x^4+b\right )^{3/4}}{3 x^3}\right )+\frac {2 \left (a x^4+b\right )^{7/4}}{7 x^7}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle a \left (a \left (\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {a} x^2}{\sqrt {a x^4+b}}}d\frac {x}{\sqrt [4]{a x^4+b}}+\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a}}\right )-\frac {\left (a x^4+b\right )^{3/4}}{3 x^3}\right )+\frac {2 \left (a x^4+b\right )^{7/4}}{7 x^7}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle a \left (a \left (\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a}}\right )-\frac {\left (a x^4+b\right )^{3/4}}{3 x^3}\right )+\frac {2 \left (a x^4+b\right )^{7/4}}{7 x^7}\) |
(2*(b + a*x^4)^(7/4))/(7*x^7) + a*(-1/3*(b + a*x^4)^(3/4)/x^3 + a*(ArcTan[ (a^(1/4)*x)/(b + a*x^4)^(1/4)]/(2*a^(1/4)) + ArcTanh[(a^(1/4)*x)/(b + a*x^ 4)^(1/4)]/(2*a^(1/4))))
3.12.40.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + 1/n) Subst[In t[1/(1 - b*x^n)^(p + 1/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p + 1 /n]
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[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _)), x_Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), x] + Simp[d/e^n Int[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n*(p + 1) + 1, 0] && (IntegerQ[n] || GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && G tQ[m + n, -1]))
Time = 1.20 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.11
method | result | size |
pseudoelliptic | \(\frac {21 x^{7} \left (\ln \left (\frac {-a^{\frac {1}{4}} x -\left (a \,x^{4}+b \right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x -\left (a \,x^{4}+b \right )^{\frac {1}{4}}}\right )-2 \arctan \left (\frac {\left (a \,x^{4}+b \right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )\right ) a^{\frac {7}{4}}-4 \left (a \,x^{4}+b \right )^{\frac {3}{4}} \left (a \,x^{4}-6 b \right )}{84 x^{7}}\) | \(94\) |
1/84*(21*x^7*(ln((-a^(1/4)*x-(a*x^4+b)^(1/4))/(a^(1/4)*x-(a*x^4+b)^(1/4))) -2*arctan(1/a^(1/4)/x*(a*x^4+b)^(1/4)))*a^(7/4)-4*(a*x^4+b)^(3/4)*(a*x^4-6 *b))/x^7
Timed out. \[ \int \frac {\left (-2 b+a x^4\right ) \left (b+a x^4\right )^{3/4}}{x^8} \, dx=\text {Timed out} \]
Result contains complex when optimal does not.
Time = 1.60 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.34 \[ \int \frac {\left (-2 b+a x^4\right ) \left (b+a x^4\right )^{3/4}}{x^8} \, dx=- \frac {a^{\frac {7}{4}} \left (1 + \frac {b}{a x^{4}}\right )^{\frac {3}{4}} \Gamma \left (- \frac {7}{4}\right )}{2 \Gamma \left (- \frac {3}{4}\right )} - \frac {a^{\frac {3}{4}} b \left (1 + \frac {b}{a x^{4}}\right )^{\frac {3}{4}} \Gamma \left (- \frac {7}{4}\right )}{2 x^{4} \Gamma \left (- \frac {3}{4}\right )} + \frac {a b^{\frac {3}{4}} \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {3}{4} \\ \frac {1}{4} \end {matrix}\middle | {\frac {a x^{4} e^{i \pi }}{b}} \right )}}{4 x^{3} \Gamma \left (\frac {1}{4}\right )} \]
-a**(7/4)*(1 + b/(a*x**4))**(3/4)*gamma(-7/4)/(2*gamma(-3/4)) - a**(3/4)*b *(1 + b/(a*x**4))**(3/4)*gamma(-7/4)/(2*x**4*gamma(-3/4)) + a*b**(3/4)*gam ma(-3/4)*hyper((-3/4, -3/4), (1/4,), a*x**4*exp_polar(I*pi)/b)/(4*x**3*gam ma(1/4))
Time = 0.30 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.21 \[ \int \frac {\left (-2 b+a x^4\right ) \left (b+a x^4\right )^{3/4}}{x^8} \, dx=-\frac {1}{12} \, {\left (3 \, a {\left (\frac {2 \, \arctan \left (\frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )}{a^{\frac {1}{4}}} + \frac {\log \left (-\frac {a^{\frac {1}{4}} - \frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}}{a^{\frac {1}{4}} + \frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}}\right )}{a^{\frac {1}{4}}}\right )} + \frac {4 \, {\left (a x^{4} + b\right )}^{\frac {3}{4}}}{x^{3}}\right )} a + \frac {2 \, {\left (a x^{4} + b\right )}^{\frac {7}{4}}}{7 \, x^{7}} \]
-1/12*(3*a*(2*arctan((a*x^4 + b)^(1/4)/(a^(1/4)*x))/a^(1/4) + log(-(a^(1/4 ) - (a*x^4 + b)^(1/4)/x)/(a^(1/4) + (a*x^4 + b)^(1/4)/x))/a^(1/4)) + 4*(a* x^4 + b)^(3/4)/x^3)*a + 2/7*(a*x^4 + b)^(7/4)/x^7
\[ \int \frac {\left (-2 b+a x^4\right ) \left (b+a x^4\right )^{3/4}}{x^8} \, dx=\int { \frac {{\left (a x^{4} + b\right )}^{\frac {3}{4}} {\left (a x^{4} - 2 \, b\right )}}{x^{8}} \,d x } \]
Timed out. \[ \int \frac {\left (-2 b+a x^4\right ) \left (b+a x^4\right )^{3/4}}{x^8} \, dx=-\int \frac {{\left (a\,x^4+b\right )}^{3/4}\,\left (2\,b-a\,x^4\right )}{x^8} \,d x \]