3.20.12 \(\int \frac {x^6}{(-b+a x^4) (b+a x^4)^{3/4}} \, dx\)

Optimal. Leaf size=133 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 a^{7/4}}+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2\ 2^{3/4} a^{7/4}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 a^{7/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2\ 2^{3/4} a^{7/4}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.11, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {494, 481, 298, 203, 206} \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 a^{7/4}}+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2\ 2^{3/4} a^{7/4}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 a^{7/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2\ 2^{3/4} a^{7/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^6/((-b + a*x^4)*(b + a*x^4)^(3/4)),x]

[Out]

-1/2*ArcTan[(a^(1/4)*x)/(b + a*x^4)^(1/4)]/a^(7/4) + ArcTan[(2^(1/4)*a^(1/4)*x)/(b + a*x^4)^(1/4)]/(2*2^(3/4)*
a^(7/4)) + ArcTanh[(a^(1/4)*x)/(b + a*x^4)^(1/4)]/(2*a^(7/4)) - ArcTanh[(2^(1/4)*a^(1/4)*x)/(b + a*x^4)^(1/4)]
/(2*2^(3/4)*a^(7/4))

Rule 203

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

Rule 206

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

Rule 298

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

Rule 481

Int[((e_.)*(x_))^(m_.)/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> -Dist[(a*e^n)/(b*c -
a*d), Int[(e*x)^(m - n)/(a + b*x^n), x], x] + Dist[(c*e^n)/(b*c - a*d), Int[(e*x)^(m - n)/(c + d*x^n), x], x]
/; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LeQ[n, m, 2*n - 1]

Rule 494

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{k = Denominato
r[p]}, Dist[(k*a^(p + (m + 1)/n))/n, Subst[Int[(x^((k*(m + 1))/n - 1)*(c - (b*c - a*d)*x^k)^q)/(1 - b*x^k)^(p
+ q + (m + 1)/n + 1), x], x, x^(n/k)/(a + b*x^n)^(1/k)], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && Ration
alQ[m, p] && IntegersQ[p + (m + 1)/n, q] && LtQ[-1, p, 0]

Rubi steps

\begin {align*} \int \frac {x^6}{\left (-b+a x^4\right ) \left (b+a x^4\right )^{3/4}} \, dx &=b \operatorname {Subst}\left (\int \frac {x^6}{\left (1-a x^4\right ) \left (-b+2 a b x^4\right )} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{a}+\frac {b \operatorname {Subst}\left (\int \frac {x^2}{-b+2 a b x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{a}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{2 a^{3/2}}-\frac {\operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{2 a^{3/2}}-\frac {\operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} \sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{2 \sqrt {2} a^{3/2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} \sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{2 \sqrt {2} a^{3/2}}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{2 a^{7/4}}+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{2\ 2^{3/4} a^{7/4}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{2 a^{7/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{2\ 2^{3/4} a^{7/4}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.06, size = 63, normalized size = 0.47 \begin {gather*} -\frac {x^7 \left (\frac {a x^4}{b}+1\right )^{3/4} F_1\left (\frac {7}{4};\frac {3}{4},1;\frac {11}{4};-\frac {a x^4}{b},\frac {a x^4}{b}\right )}{7 b \left (a x^4+b\right )^{3/4}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[x^6/((-b + a*x^4)*(b + a*x^4)^(3/4)),x]

[Out]

-1/7*(x^7*(1 + (a*x^4)/b)^(3/4)*AppellF1[7/4, 3/4, 1, 11/4, -((a*x^4)/b), (a*x^4)/b])/(b*(b + a*x^4)^(3/4))

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.59, size = 133, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{2 a^{7/4}}+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{2\ 2^{3/4} a^{7/4}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{2 a^{7/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{2\ 2^{3/4} a^{7/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^6/((-b + a*x^4)*(b + a*x^4)^(3/4)),x]

[Out]

-1/2*ArcTan[(a^(1/4)*x)/(b + a*x^4)^(1/4)]/a^(7/4) + ArcTan[(2^(1/4)*a^(1/4)*x)/(b + a*x^4)^(1/4)]/(2*2^(3/4)*
a^(7/4)) + ArcTanh[(a^(1/4)*x)/(b + a*x^4)^(1/4)]/(2*a^(7/4)) - ArcTanh[(2^(1/4)*a^(1/4)*x)/(b + a*x^4)^(1/4)]
/(2*2^(3/4)*a^(7/4))

________________________________________________________________________________________

fricas [B]  time = 0.61, size = 302, normalized size = 2.27 \begin {gather*} \left (\frac {1}{8}\right )^{\frac {1}{4}} \frac {1}{a^{7}}^{\frac {1}{4}} \arctan \left (\frac {4 \, {\left (\left (\frac {1}{8}\right )^{\frac {3}{4}} a^{5} x \sqrt {\frac {2 \, \sqrt {\frac {1}{2}} a^{4} \sqrt {\frac {1}{a^{7}}} x^{2} + \sqrt {a x^{4} + b}}{x^{2}}} \frac {1}{a^{7}}^{\frac {3}{4}} - \left (\frac {1}{8}\right )^{\frac {3}{4}} {\left (a x^{4} + b\right )}^{\frac {1}{4}} a^{5} \frac {1}{a^{7}}^{\frac {3}{4}}\right )}}{x}\right ) - \frac {1}{4} \, \left (\frac {1}{8}\right )^{\frac {1}{4}} \frac {1}{a^{7}}^{\frac {1}{4}} \log \left (\frac {2 \, \left (\frac {1}{8}\right )^{\frac {1}{4}} a^{2} \frac {1}{a^{7}}^{\frac {1}{4}} x + {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{4} \, \left (\frac {1}{8}\right )^{\frac {1}{4}} \frac {1}{a^{7}}^{\frac {1}{4}} \log \left (-\frac {2 \, \left (\frac {1}{8}\right )^{\frac {1}{4}} a^{2} \frac {1}{a^{7}}^{\frac {1}{4}} x - {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{a^{7}}^{\frac {1}{4}} \arctan \left (\frac {a^{5} x \sqrt {\frac {a^{4} \sqrt {\frac {1}{a^{7}}} x^{2} + \sqrt {a x^{4} + b}}{x^{2}}} \frac {1}{a^{7}}^{\frac {3}{4}} - {\left (a x^{4} + b\right )}^{\frac {1}{4}} a^{5} \frac {1}{a^{7}}^{\frac {3}{4}}}{x}\right ) + \frac {1}{4} \, \frac {1}{a^{7}}^{\frac {1}{4}} \log \left (\frac {a^{2} \frac {1}{a^{7}}^{\frac {1}{4}} x + {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{4} \, \frac {1}{a^{7}}^{\frac {1}{4}} \log \left (-\frac {a^{2} \frac {1}{a^{7}}^{\frac {1}{4}} x - {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(a*x^4-b)/(a*x^4+b)^(3/4),x, algorithm="fricas")

[Out]

(1/8)^(1/4)*(a^(-7))^(1/4)*arctan(4*((1/8)^(3/4)*a^5*x*sqrt((2*sqrt(1/2)*a^4*sqrt(a^(-7))*x^2 + sqrt(a*x^4 + b
))/x^2)*(a^(-7))^(3/4) - (1/8)^(3/4)*(a*x^4 + b)^(1/4)*a^5*(a^(-7))^(3/4))/x) - 1/4*(1/8)^(1/4)*(a^(-7))^(1/4)
*log((2*(1/8)^(1/4)*a^2*(a^(-7))^(1/4)*x + (a*x^4 + b)^(1/4))/x) + 1/4*(1/8)^(1/4)*(a^(-7))^(1/4)*log(-(2*(1/8
)^(1/4)*a^2*(a^(-7))^(1/4)*x - (a*x^4 + b)^(1/4))/x) - (a^(-7))^(1/4)*arctan((a^5*x*sqrt((a^4*sqrt(a^(-7))*x^2
+ sqrt(a*x^4 + b))/x^2)*(a^(-7))^(3/4) - (a*x^4 + b)^(1/4)*a^5*(a^(-7))^(3/4))/x) + 1/4*(a^(-7))^(1/4)*log((a
^2*(a^(-7))^(1/4)*x + (a*x^4 + b)^(1/4))/x) - 1/4*(a^(-7))^(1/4)*log(-(a^2*(a^(-7))^(1/4)*x - (a*x^4 + b)^(1/4
))/x)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{6}}{{\left (a x^{4} + b\right )}^{\frac {3}{4}} {\left (a x^{4} - b\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(a*x^4-b)/(a*x^4+b)^(3/4),x, algorithm="giac")

[Out]

integrate(x^6/((a*x^4 + b)^(3/4)*(a*x^4 - b)), x)

________________________________________________________________________________________

maple [F]  time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {x^{6}}{\left (a \,x^{4}-b \right ) \left (a \,x^{4}+b \right )^{\frac {3}{4}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(a*x^4-b)/(a*x^4+b)^(3/4),x)

[Out]

int(x^6/(a*x^4-b)/(a*x^4+b)^(3/4),x)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{6}}{{\left (a x^{4} + b\right )}^{\frac {3}{4}} {\left (a x^{4} - b\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(a*x^4-b)/(a*x^4+b)^(3/4),x, algorithm="maxima")

[Out]

integrate(x^6/((a*x^4 + b)^(3/4)*(a*x^4 - b)), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {x^6}{{\left (a\,x^4+b\right )}^{3/4}\,\left (b-a\,x^4\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-x^6/((b + a*x^4)^(3/4)*(b - a*x^4)),x)

[Out]

-int(x^6/((b + a*x^4)^(3/4)*(b - a*x^4)), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{6}}{\left (a x^{4} - b\right ) \left (a x^{4} + b\right )^{\frac {3}{4}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6/(a*x**4-b)/(a*x**4+b)**(3/4),x)

[Out]

Integral(x**6/((a*x**4 - b)*(a*x**4 + b)**(3/4)), x)

________________________________________________________________________________________