\(\int \frac {(b+a x^4)^{3/4} (2 b+a x^4)}{x^8 (4 b+a x^4)} \, dx\) [1931]

Optimal result

Integrand size = 35, antiderivative size = 134 \[ \int \frac {\left (b+a x^4\right )^{3/4} \left (2 b+a x^4\right )}{x^8 \left (4 b+a x^4\right )} \, dx=\frac {\left (-12 b-19 a x^4\right ) \left (b+a x^4\right )^{3/4}}{168 b x^7}+\frac {3^{3/4} a^{7/4} \arctan \left (\frac {\sqrt [4]{3} \sqrt [4]{a} x}{\sqrt {2} \sqrt [4]{b+a x^4}}\right )}{32 \sqrt {2} b}+\frac {3^{3/4} a^{7/4} \text {arctanh}\left (\frac {\sqrt [4]{3} \sqrt [4]{a} x}{\sqrt {2} \sqrt [4]{b+a x^4}}\right )}{32 \sqrt {2} b} \]

[Out]

1/168*(-19*a*x^4-12*b)*(a*x^4+b)^(3/4)/b/x^7+1/64*3^(3/4)*a^(7/4)*arctan(1/2*3^(1/4)*a^(1/4)*x*2^(1/2)/(a*x^4+
b)^(1/4))*2^(1/2)/b+1/64*3^(3/4)*a^(7/4)*arctanh(1/2*3^(1/4)*a^(1/4)*x*2^(1/2)/(a*x^4+b)^(1/4))*2^(1/2)/b

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {594, 597, 12, 385, 218, 212, 209} \[ \int \frac {\left (b+a x^4\right )^{3/4} \left (2 b+a x^4\right )}{x^8 \left (4 b+a x^4\right )} \, dx=\frac {3^{3/4} a^{7/4} \arctan \left (\frac {\sqrt [4]{3} \sqrt [4]{a} x}{\sqrt {2} \sqrt [4]{a x^4+b}}\right )}{32 \sqrt {2} b}+\frac {3^{3/4} a^{7/4} \text {arctanh}\left (\frac {\sqrt [4]{3} \sqrt [4]{a} x}{\sqrt {2} \sqrt [4]{a x^4+b}}\right )}{32 \sqrt {2} b}-\frac {\left (a x^4+b\right )^{3/4}}{14 x^7}-\frac {19 a \left (a x^4+b\right )^{3/4}}{168 b x^3} \]

[In]

Int[((b + a*x^4)^(3/4)*(2*b + a*x^4))/(x^8*(4*b + a*x^4)),x]

[Out]

-1/14*(b + a*x^4)^(3/4)/x^7 - (19*a*(b + a*x^4)^(3/4))/(168*b*x^3) + (3^(3/4)*a^(7/4)*ArcTan[(3^(1/4)*a^(1/4)*
x)/(Sqrt[2]*(b + a*x^4)^(1/4))])/(32*Sqrt[2]*b) + (3^(3/4)*a^(7/4)*ArcTanh[(3^(1/4)*a^(1/4)*x)/(Sqrt[2]*(b + a
*x^4)^(1/4))])/(32*Sqrt[2]*b)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 218

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]},
Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !Gt
Q[a/b, 0]

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 594

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/(a*g*(m + 1))), x] - Dist[1/(a*g^n*(m + 1
)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f)*(m + 1) + e*n*(b*c*(p + 1) + a*d*q)
+ d*((b*e - a*f)*(m + 1) + b*e*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && IGtQ[n
, 0] && GtQ[q, 0] && LtQ[m, -1] &&  !(EqQ[q, 1] && SimplerQ[e + f*x^n, c + d*x^n])

Rule 597

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (b+a x^4\right )^{3/4}}{14 x^7}+\frac {\int \frac {38 a b^2+20 a^2 b x^4}{x^4 \sqrt [4]{b+a x^4} \left (4 b+a x^4\right )} \, dx}{28 b} \\ & = -\frac {\left (b+a x^4\right )^{3/4}}{14 x^7}-\frac {19 a \left (b+a x^4\right )^{3/4}}{168 b x^3}-\frac {\int -\frac {126 a^2 b^3}{\sqrt [4]{b+a x^4} \left (4 b+a x^4\right )} \, dx}{336 b^3} \\ & = -\frac {\left (b+a x^4\right )^{3/4}}{14 x^7}-\frac {19 a \left (b+a x^4\right )^{3/4}}{168 b x^3}+\frac {1}{8} \left (3 a^2\right ) \int \frac {1}{\sqrt [4]{b+a x^4} \left (4 b+a x^4\right )} \, dx \\ & = -\frac {\left (b+a x^4\right )^{3/4}}{14 x^7}-\frac {19 a \left (b+a x^4\right )^{3/4}}{168 b x^3}+\frac {1}{8} \left (3 a^2\right ) \text {Subst}\left (\int \frac {1}{4 b-3 a b x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right ) \\ & = -\frac {\left (b+a x^4\right )^{3/4}}{14 x^7}-\frac {19 a \left (b+a x^4\right )^{3/4}}{168 b x^3}+\frac {\left (3 a^2\right ) \text {Subst}\left (\int \frac {1}{2-\sqrt {3} \sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{32 b}+\frac {\left (3 a^2\right ) \text {Subst}\left (\int \frac {1}{2+\sqrt {3} \sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{32 b} \\ & = -\frac {\left (b+a x^4\right )^{3/4}}{14 x^7}-\frac {19 a \left (b+a x^4\right )^{3/4}}{168 b x^3}+\frac {3^{3/4} a^{7/4} \arctan \left (\frac {\sqrt [4]{3} \sqrt [4]{a} x}{\sqrt {2} \sqrt [4]{b+a x^4}}\right )}{32 \sqrt {2} b}+\frac {3^{3/4} a^{7/4} \text {arctanh}\left (\frac {\sqrt [4]{3} \sqrt [4]{a} x}{\sqrt {2} \sqrt [4]{b+a x^4}}\right )}{32 \sqrt {2} b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.60 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.00 \[ \int \frac {\left (b+a x^4\right )^{3/4} \left (2 b+a x^4\right )}{x^8 \left (4 b+a x^4\right )} \, dx=\frac {\left (-12 b-19 a x^4\right ) \left (b+a x^4\right )^{3/4}}{168 b x^7}+\frac {3^{3/4} a^{7/4} \arctan \left (\frac {\sqrt [4]{3} \sqrt [4]{a} x}{\sqrt {2} \sqrt [4]{b+a x^4}}\right )}{32 \sqrt {2} b}+\frac {3^{3/4} a^{7/4} \text {arctanh}\left (\frac {\sqrt [4]{3} \sqrt [4]{a} x}{\sqrt {2} \sqrt [4]{b+a x^4}}\right )}{32 \sqrt {2} b} \]

[In]

Integrate[((b + a*x^4)^(3/4)*(2*b + a*x^4))/(x^8*(4*b + a*x^4)),x]

[Out]

((-12*b - 19*a*x^4)*(b + a*x^4)^(3/4))/(168*b*x^7) + (3^(3/4)*a^(7/4)*ArcTan[(3^(1/4)*a^(1/4)*x)/(Sqrt[2]*(b +
a*x^4)^(1/4))])/(32*Sqrt[2]*b) + (3^(3/4)*a^(7/4)*ArcTanh[(3^(1/4)*a^(1/4)*x)/(Sqrt[2]*(b + a*x^4)^(1/4))])/(
32*Sqrt[2]*b)

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.13

method	result	size
pseudoelliptic	\(-\frac {\left (126 \arctan \left (\frac {3^{\frac {3}{4}} \sqrt {2}\, \left (a \,x^{4}+b \right )^{\frac {1}{4}}}{3 a^{\frac {1}{4}} x}\right ) a^{2} \sqrt {2}\, x^{7}-63 \ln \left (\frac {\sqrt {2}\, 3^{\frac {1}{4}} a^{\frac {1}{4}} x +2 \left (a \,x^{4}+b \right )^{\frac {1}{4}}}{-\sqrt {2}\, 3^{\frac {1}{4}} a^{\frac {1}{4}} x +2 \left (a \,x^{4}+b \right )^{\frac {1}{4}}}\right ) a^{2} \sqrt {2}\, x^{7}+304 a^{\frac {5}{4}} \left (a \,x^{4}+b \right )^{\frac {3}{4}} x^{4} 3^{\frac {1}{4}}+192 b \left (a \,x^{4}+b \right )^{\frac {3}{4}} 3^{\frac {1}{4}} a^{\frac {1}{4}}\right ) 3^{\frac {3}{4}}}{8064 x^{7} a^{\frac {1}{4}} b}\)	\(151\)

[In]

int((a*x^4+b)^(3/4)*(a*x^4+2*b)/x^8/(a*x^4+4*b),x,method=_RETURNVERBOSE)

[Out]

-1/8064*(126*arctan(1/3*3^(3/4)/a^(1/4)/x*2^(1/2)*(a*x^4+b)^(1/4))*a^2*2^(1/2)*x^7-63*ln((2^(1/2)*3^(1/4)*a^(1
/4)*x+2*(a*x^4+b)^(1/4))/(-2^(1/2)*3^(1/4)*a^(1/4)*x+2*(a*x^4+b)^(1/4)))*a^2*2^(1/2)*x^7+304*a^(5/4)*(a*x^4+b)
^(3/4)*x^4*3^(1/4)+192*b*(a*x^4+b)^(3/4)*3^(1/4)*a^(1/4))/x^7*3^(3/4)/a^(1/4)/b

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 52.67 (sec) , antiderivative size = 594, normalized size of antiderivative = 4.43 \[ \int \frac {\left (b+a x^4\right )^{3/4} \left (2 b+a x^4\right )}{x^8 \left (4 b+a x^4\right )} \, dx=\frac {21 \, \left (\frac {27}{4}\right )^{\frac {1}{4}} \left (\frac {a^{7}}{b^{4}}\right )^{\frac {1}{4}} b x^{7} \log \left (\frac {18 \, \sqrt {3} {\left (a x^{4} + b\right )}^{\frac {1}{4}} \sqrt {\frac {a^{7}}{b^{4}}} a^{2} b^{2} x^{3} + 8 \, \left (\frac {27}{4}\right )^{\frac {3}{4}} \sqrt {a x^{4} + b} \left (\frac {a^{7}}{b^{4}}\right )^{\frac {3}{4}} b^{3} x^{2} + 36 \, {\left (a x^{4} + b\right )}^{\frac {3}{4}} a^{5} x + 3 \, \left (\frac {27}{4}\right )^{\frac {1}{4}} {\left (7 \, a^{4} b x^{4} + 4 \, a^{3} b^{2}\right )} \left (\frac {a^{7}}{b^{4}}\right )^{\frac {1}{4}}}{4 \, {\left (a x^{4} + 4 \, b\right )}}\right ) + 21 i \, \left (\frac {27}{4}\right )^{\frac {1}{4}} \left (\frac {a^{7}}{b^{4}}\right )^{\frac {1}{4}} b x^{7} \log \left (-\frac {18 \, \sqrt {3} {\left (a x^{4} + b\right )}^{\frac {1}{4}} \sqrt {\frac {a^{7}}{b^{4}}} a^{2} b^{2} x^{3} + 8 i \, \left (\frac {27}{4}\right )^{\frac {3}{4}} \sqrt {a x^{4} + b} \left (\frac {a^{7}}{b^{4}}\right )^{\frac {3}{4}} b^{3} x^{2} - 36 \, {\left (a x^{4} + b\right )}^{\frac {3}{4}} a^{5} x + 3 \, \left (\frac {27}{4}\right )^{\frac {1}{4}} {\left (-7 i \, a^{4} b x^{4} - 4 i \, a^{3} b^{2}\right )} \left (\frac {a^{7}}{b^{4}}\right )^{\frac {1}{4}}}{4 \, {\left (a x^{4} + 4 \, b\right )}}\right ) - 21 i \, \left (\frac {27}{4}\right )^{\frac {1}{4}} \left (\frac {a^{7}}{b^{4}}\right )^{\frac {1}{4}} b x^{7} \log \left (-\frac {18 \, \sqrt {3} {\left (a x^{4} + b\right )}^{\frac {1}{4}} \sqrt {\frac {a^{7}}{b^{4}}} a^{2} b^{2} x^{3} - 8 i \, \left (\frac {27}{4}\right )^{\frac {3}{4}} \sqrt {a x^{4} + b} \left (\frac {a^{7}}{b^{4}}\right )^{\frac {3}{4}} b^{3} x^{2} - 36 \, {\left (a x^{4} + b\right )}^{\frac {3}{4}} a^{5} x + 3 \, \left (\frac {27}{4}\right )^{\frac {1}{4}} {\left (7 i \, a^{4} b x^{4} + 4 i \, a^{3} b^{2}\right )} \left (\frac {a^{7}}{b^{4}}\right )^{\frac {1}{4}}}{4 \, {\left (a x^{4} + 4 \, b\right )}}\right ) - 21 \, \left (\frac {27}{4}\right )^{\frac {1}{4}} \left (\frac {a^{7}}{b^{4}}\right )^{\frac {1}{4}} b x^{7} \log \left (\frac {18 \, \sqrt {3} {\left (a x^{4} + b\right )}^{\frac {1}{4}} \sqrt {\frac {a^{7}}{b^{4}}} a^{2} b^{2} x^{3} - 8 \, \left (\frac {27}{4}\right )^{\frac {3}{4}} \sqrt {a x^{4} + b} \left (\frac {a^{7}}{b^{4}}\right )^{\frac {3}{4}} b^{3} x^{2} + 36 \, {\left (a x^{4} + b\right )}^{\frac {3}{4}} a^{5} x - 3 \, \left (\frac {27}{4}\right )^{\frac {1}{4}} {\left (7 \, a^{4} b x^{4} + 4 \, a^{3} b^{2}\right )} \left (\frac {a^{7}}{b^{4}}\right )^{\frac {1}{4}}}{4 \, {\left (a x^{4} + 4 \, b\right )}}\right ) - 16 \, {\left (19 \, a x^{4} + 12 \, b\right )} {\left (a x^{4} + b\right )}^{\frac {3}{4}}}{2688 \, b x^{7}} \]

[In]

integrate((a*x^4+b)^(3/4)*(a*x^4+2*b)/x^8/(a*x^4+4*b),x, algorithm="fricas")

[Out]

1/2688*(21*(27/4)^(1/4)*(a^7/b^4)^(1/4)*b*x^7*log(1/4*(18*sqrt(3)*(a*x^4 + b)^(1/4)*sqrt(a^7/b^4)*a^2*b^2*x^3
+ 8*(27/4)^(3/4)*sqrt(a*x^4 + b)*(a^7/b^4)^(3/4)*b^3*x^2 + 36*(a*x^4 + b)^(3/4)*a^5*x + 3*(27/4)^(1/4)*(7*a^4*
b*x^4 + 4*a^3*b^2)*(a^7/b^4)^(1/4))/(a*x^4 + 4*b)) + 21*I*(27/4)^(1/4)*(a^7/b^4)^(1/4)*b*x^7*log(-1/4*(18*sqrt
(3)*(a*x^4 + b)^(1/4)*sqrt(a^7/b^4)*a^2*b^2*x^3 + 8*I*(27/4)^(3/4)*sqrt(a*x^4 + b)*(a^7/b^4)^(3/4)*b^3*x^2 - 3
6*(a*x^4 + b)^(3/4)*a^5*x + 3*(27/4)^(1/4)*(-7*I*a^4*b*x^4 - 4*I*a^3*b^2)*(a^7/b^4)^(1/4))/(a*x^4 + 4*b)) - 21
*I*(27/4)^(1/4)*(a^7/b^4)^(1/4)*b*x^7*log(-1/4*(18*sqrt(3)*(a*x^4 + b)^(1/4)*sqrt(a^7/b^4)*a^2*b^2*x^3 - 8*I*(
27/4)^(3/4)*sqrt(a*x^4 + b)*(a^7/b^4)^(3/4)*b^3*x^2 - 36*(a*x^4 + b)^(3/4)*a^5*x + 3*(27/4)^(1/4)*(7*I*a^4*b*x
^4 + 4*I*a^3*b^2)*(a^7/b^4)^(1/4))/(a*x^4 + 4*b)) - 21*(27/4)^(1/4)*(a^7/b^4)^(1/4)*b*x^7*log(1/4*(18*sqrt(3)*
(a*x^4 + b)^(1/4)*sqrt(a^7/b^4)*a^2*b^2*x^3 - 8*(27/4)^(3/4)*sqrt(a*x^4 + b)*(a^7/b^4)^(3/4)*b^3*x^2 + 36*(a*x
^4 + b)^(3/4)*a^5*x - 3*(27/4)^(1/4)*(7*a^4*b*x^4 + 4*a^3*b^2)*(a^7/b^4)^(1/4))/(a*x^4 + 4*b)) - 16*(19*a*x^4
+ 12*b)*(a*x^4 + b)^(3/4))/(b*x^7)

Sympy [F]

\[ \int \frac {\left (b+a x^4\right )^{3/4} \left (2 b+a x^4\right )}{x^8 \left (4 b+a x^4\right )} \, dx=\int \frac {\left (a x^{4} + b\right )^{\frac {3}{4}} \left (a x^{4} + 2 b\right )}{x^{8} \left (a x^{4} + 4 b\right )}\, dx \]

[In]

integrate((a*x**4+b)**(3/4)*(a*x**4+2*b)/x**8/(a*x**4+4*b),x)

[Out]

Integral((a*x**4 + b)**(3/4)*(a*x**4 + 2*b)/(x**8*(a*x**4 + 4*b)), x)

Maxima [F]

\[ \int \frac {\left (b+a x^4\right )^{3/4} \left (2 b+a x^4\right )}{x^8 \left (4 b+a x^4\right )} \, dx=\int { \frac {{\left (a x^{4} + 2 \, b\right )} {\left (a x^{4} + b\right )}^{\frac {3}{4}}}{{\left (a x^{4} + 4 \, b\right )} x^{8}} \,d x } \]

[In]

integrate((a*x^4+b)^(3/4)*(a*x^4+2*b)/x^8/(a*x^4+4*b),x, algorithm="maxima")

[Out]

integrate((a*x^4 + 2*b)*(a*x^4 + b)^(3/4)/((a*x^4 + 4*b)*x^8), x)

Giac [F]

\[ \int \frac {\left (b+a x^4\right )^{3/4} \left (2 b+a x^4\right )}{x^8 \left (4 b+a x^4\right )} \, dx=\int { \frac {{\left (a x^{4} + 2 \, b\right )} {\left (a x^{4} + b\right )}^{\frac {3}{4}}}{{\left (a x^{4} + 4 \, b\right )} x^{8}} \,d x } \]

[In]

integrate((a*x^4+b)^(3/4)*(a*x^4+2*b)/x^8/(a*x^4+4*b),x, algorithm="giac")

[Out]

integrate((a*x^4 + 2*b)*(a*x^4 + b)^(3/4)/((a*x^4 + 4*b)*x^8), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (b+a x^4\right )^{3/4} \left (2 b+a x^4\right )}{x^8 \left (4 b+a x^4\right )} \, dx=\int \frac {{\left (a\,x^4+b\right )}^{3/4}\,\left (a\,x^4+2\,b\right )}{x^8\,\left (a\,x^4+4\,b\right )} \,d x \]

[In]

int(((b + a*x^4)^(3/4)*(2*b + a*x^4))/(x^8*(4*b + a*x^4)),x)

[Out]

int(((b + a*x^4)^(3/4)*(2*b + a*x^4))/(x^8*(4*b + a*x^4)), x)