3.9.72 \(\int \frac {x^6}{(b+a x^4)^{3/4} (b^2+a^2 x^8)} \, dx\)

Optimal. Leaf size=66 \[ \frac {\text {RootSum}\left [\text {$\#$1}^8-2 \text {$\#$1}^4 a+2 a^2\& ,\frac {\log \left (\sqrt [4]{a x^4+b}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}^3 a-\text {$\#$1}^7}\& \right ]}{8 b} \]

________________________________________________________________________________________

Rubi [B]  time = 0.72, antiderivative size = 325, normalized size of antiderivative = 4.92, number of steps used = 10, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1529, 494, 298, 205} \begin {gather*} \frac {\left (-a^2\right )^{7/8} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{\sqrt {-a^2}-a} x}{\sqrt [8]{-a^2} \sqrt [4]{a x^4+b}}\right )}{4 a^{11/4} \left (\sqrt {-a^2}-a\right )^{3/4} b}-\frac {\left (-a^2\right )^{3/8} \tan ^{-1}\left (\frac {\left (-a^2\right )^{3/8} \sqrt [4]{\sqrt {-a^2}-a} x}{a^{3/4} \sqrt [4]{a x^4+b}}\right )}{4 a^{7/4} \left (\sqrt {-a^2}-a\right )^{3/4} b}+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{\sqrt {-a^2}+a} x}{\sqrt [8]{-a^2} \sqrt [4]{a x^4+b}}\right )}{4 a^{3/4} \sqrt [8]{-a^2} \left (\sqrt {-a^2}+a\right )^{3/4} b}-\frac {\sqrt [4]{a} \tan ^{-1}\left (\frac {\left (-a^2\right )^{3/8} \sqrt [4]{\sqrt {-a^2}+a} x}{a^{3/4} \sqrt [4]{a x^4+b}}\right )}{4 \left (-a^2\right )^{5/8} \left (\sqrt {-a^2}+a\right )^{3/4} b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^6/((b + a*x^4)^(3/4)*(b^2 + a^2*x^8)),x]

[Out]

((-a^2)^(7/8)*ArcTan[(a^(1/4)*(-a + Sqrt[-a^2])^(1/4)*x)/((-a^2)^(1/8)*(b + a*x^4)^(1/4))])/(4*a^(11/4)*(-a +
Sqrt[-a^2])^(3/4)*b) - ((-a^2)^(3/8)*ArcTan[((-a^2)^(3/8)*(-a + Sqrt[-a^2])^(1/4)*x)/(a^(3/4)*(b + a*x^4)^(1/4
))])/(4*a^(7/4)*(-a + Sqrt[-a^2])^(3/4)*b) + ArcTan[(a^(1/4)*(a + Sqrt[-a^2])^(1/4)*x)/((-a^2)^(1/8)*(b + a*x^
4)^(1/4))]/(4*a^(3/4)*(-a^2)^(1/8)*(a + Sqrt[-a^2])^(3/4)*b) - (a^(1/4)*ArcTan[((-a^2)^(3/8)*(a + Sqrt[-a^2])^
(1/4)*x)/(a^(3/4)*(b + a*x^4)^(1/4))])/(4*(-a^2)^(5/8)*(a + Sqrt[-a^2])^(3/4)*b)

Rule 205

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

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 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]

Rule 1529

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))^(q_))/((a_) + (c_.)*(x_)^(n2_.)), x_Symbol] :> Int[ExpandInte
grand[(d + e*x^n)^q, (f*x)^m/(a + c*x^(2*n)), x], x] /; FreeQ[{a, c, d, e, f, q, n}, x] && EqQ[n2, 2*n] && IGt
Q[n, 0] &&  !IntegerQ[q] && IntegerQ[m]

Rubi steps

\begin {align*} \int \frac {x^6}{\left (b+a x^4\right )^{3/4} \left (b^2+a^2 x^8\right )} \, dx &=\int \left (\frac {x^2}{2 \left (b+a x^4\right )^{3/4} \left (-\sqrt {-a^2} b+a^2 x^4\right )}+\frac {x^2}{2 \left (b+a x^4\right )^{3/4} \left (\sqrt {-a^2} b+a^2 x^4\right )}\right ) \, dx\\ &=\frac {1}{2} \int \frac {x^2}{\left (b+a x^4\right )^{3/4} \left (-\sqrt {-a^2} b+a^2 x^4\right )} \, dx+\frac {1}{2} \int \frac {x^2}{\left (b+a x^4\right )^{3/4} \left (\sqrt {-a^2} b+a^2 x^4\right )} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^2}{-\sqrt {-a^2} b-\left (-a^2 b-a \sqrt {-a^2} b\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {-a^2} b-\left (-a^2 b+a \sqrt {-a^2} b\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-a^2}-\sqrt {a} \sqrt {-a+\sqrt {-a^2}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{4 \sqrt {a} \sqrt {-a+\sqrt {-a^2}} b}-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-a^2}+\sqrt {a} \sqrt {-a+\sqrt {-a^2}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{4 \sqrt {a} \sqrt {-a+\sqrt {-a^2}} b}-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-a^2}-\sqrt {a} \sqrt {a+\sqrt {-a^2}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{4 \sqrt {a} \sqrt {a+\sqrt {-a^2}} b}+\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-a^2}+\sqrt {a} \sqrt {a+\sqrt {-a^2}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{4 \sqrt {a} \sqrt {a+\sqrt {-a^2}} b}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{-a+\sqrt {-a^2}} x}{\sqrt [8]{-a^2} \sqrt [4]{b+a x^4}}\right )}{4 a^{3/4} \sqrt [8]{-a^2} \left (-a+\sqrt {-a^2}\right )^{3/4} b}+\frac {\sqrt [4]{a} \tan ^{-1}\left (\frac {\left (-a^2\right )^{3/8} \sqrt [4]{-a+\sqrt {-a^2}} x}{a^{3/4} \sqrt [4]{b+a x^4}}\right )}{4 \left (-a^2\right )^{5/8} \left (-a+\sqrt {-a^2}\right )^{3/4} b}+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{a+\sqrt {-a^2}} x}{\sqrt [8]{-a^2} \sqrt [4]{b+a x^4}}\right )}{4 a^{3/4} \sqrt [8]{-a^2} \left (a+\sqrt {-a^2}\right )^{3/4} b}-\frac {\sqrt [4]{a} \tan ^{-1}\left (\frac {\left (-a^2\right )^{3/8} \sqrt [4]{a+\sqrt {-a^2}} x}{a^{3/4} \sqrt [4]{b+a x^4}}\right )}{4 \left (-a^2\right )^{5/8} \left (a+\sqrt {-a^2}\right )^{3/4} b}\\ \end {align*}

________________________________________________________________________________________

Mathematica [F]  time = 0.13, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^6}{\left (b+a x^4\right )^{3/4} \left (b^2+a^2 x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[x^6/((b + a*x^4)^(3/4)*(b^2 + a^2*x^8)),x]

[Out]

Integrate[x^6/((b + a*x^4)^(3/4)*(b^2 + a^2*x^8)), x]

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.64, size = 65, normalized size = 0.98 \begin {gather*} \frac {\text {RootSum}\left [2 a^2-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {\log (x)-\log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right )}{-a \text {$\#$1}^3+\text {$\#$1}^7}\&\right ]}{8 b} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^6/((b + a*x^4)^(3/4)*(b^2 + a^2*x^8)),x]

[Out]

RootSum[2*a^2 - 2*a*#1^4 + #1^8 & , (Log[x] - Log[(b + a*x^4)^(1/4) - x*#1])/(-(a*#1^3) + #1^7) & ]/(8*b)

________________________________________________________________________________________

fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(a*x^4+b)^(3/4)/(a^2*x^8+b^2),x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{6}}{{\left (a^{2} x^{8} + b^{2}\right )} {\left (a x^{4} + b\right )}^{\frac {3}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(a*x^4+b)^(3/4)/(a^2*x^8+b^2),x, algorithm="giac")

[Out]

integrate(x^6/((a^2*x^8 + b^2)*(a*x^4 + b)^(3/4)), x)

________________________________________________________________________________________

maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {x^{6}}{\left (a \,x^{4}+b \right )^{\frac {3}{4}} \left (a^{2} x^{8}+b^{2}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(a*x^4+b)^(3/4)/(a^2*x^8+b^2),x)

[Out]

int(x^6/(a*x^4+b)^(3/4)/(a^2*x^8+b^2),x)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{6}}{{\left (a^{2} x^{8} + b^{2}\right )} {\left (a x^{4} + b\right )}^{\frac {3}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(a*x^4+b)^(3/4)/(a^2*x^8+b^2),x, algorithm="maxima")

[Out]

integrate(x^6/((a^2*x^8 + b^2)*(a*x^4 + b)^(3/4)), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^6}{\left (a^2\,x^8+b^2\right )\,{\left (a\,x^4+b\right )}^{3/4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/((b^2 + a^2*x^8)*(b + a*x^4)^(3/4)),x)

[Out]

int(x^6/((b^2 + a^2*x^8)*(b + a*x^4)^(3/4)), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{6}}{\left (a x^{4} + b\right )^{\frac {3}{4}} \left (a^{2} x^{8} + b^{2}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6/(a*x**4+b)**(3/4)/(a**2*x**8+b**2),x)

[Out]

Integral(x**6/((a*x**4 + b)**(3/4)*(a**2*x**8 + b**2)), x)

________________________________________________________________________________________