3.681 \(\int \frac{x^8}{(a+c x^4)^3} \, dx\)

Optimal. Leaf size=221 \[ -\frac{5 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{3/4} c^{9/4}}+\frac{5 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{3/4} c^{9/4}}-\frac{5 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{3/4} c^{9/4}}+\frac{5 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt{2} a^{3/4} c^{9/4}}-\frac{5 x}{32 c^2 \left (a+c x^4\right )}-\frac{x^5}{8 c \left (a+c x^4\right )^2} \]

[Out]

-x^5/(8*c*(a + c*x^4)^2) - (5*x)/(32*c^2*(a + c*x^4)) - (5*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(64*Sqrt[2
]*a^(3/4)*c^(9/4)) + (5*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(64*Sqrt[2]*a^(3/4)*c^(9/4)) - (5*Log[Sqrt[a]
- Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(128*Sqrt[2]*a^(3/4)*c^(9/4)) + (5*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*
c^(1/4)*x + Sqrt[c]*x^2])/(128*Sqrt[2]*a^(3/4)*c^(9/4))

________________________________________________________________________________________

Rubi [A]  time = 0.137943, antiderivative size = 221, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538, Rules used = {288, 211, 1165, 628, 1162, 617, 204} \[ -\frac{5 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{3/4} c^{9/4}}+\frac{5 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{3/4} c^{9/4}}-\frac{5 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{3/4} c^{9/4}}+\frac{5 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt{2} a^{3/4} c^{9/4}}-\frac{5 x}{32 c^2 \left (a+c x^4\right )}-\frac{x^5}{8 c \left (a+c x^4\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[x^8/(a + c*x^4)^3,x]

[Out]

-x^5/(8*c*(a + c*x^4)^2) - (5*x)/(32*c^2*(a + c*x^4)) - (5*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(64*Sqrt[2
]*a^(3/4)*c^(9/4)) + (5*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(64*Sqrt[2]*a^(3/4)*c^(9/4)) - (5*Log[Sqrt[a]
- Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(128*Sqrt[2]*a^(3/4)*c^(9/4)) + (5*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*
c^(1/4)*x + Sqrt[c]*x^2])/(128*Sqrt[2]*a^(3/4)*c^(9/4))

Rule 288

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 211

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

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

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

Rubi steps

\begin{align*} \int \frac{x^8}{\left (a+c x^4\right )^3} \, dx &=-\frac{x^5}{8 c \left (a+c x^4\right )^2}+\frac{5 \int \frac{x^4}{\left (a+c x^4\right )^2} \, dx}{8 c}\\ &=-\frac{x^5}{8 c \left (a+c x^4\right )^2}-\frac{5 x}{32 c^2 \left (a+c x^4\right )}+\frac{5 \int \frac{1}{a+c x^4} \, dx}{32 c^2}\\ &=-\frac{x^5}{8 c \left (a+c x^4\right )^2}-\frac{5 x}{32 c^2 \left (a+c x^4\right )}+\frac{5 \int \frac{\sqrt{a}-\sqrt{c} x^2}{a+c x^4} \, dx}{64 \sqrt{a} c^2}+\frac{5 \int \frac{\sqrt{a}+\sqrt{c} x^2}{a+c x^4} \, dx}{64 \sqrt{a} c^2}\\ &=-\frac{x^5}{8 c \left (a+c x^4\right )^2}-\frac{5 x}{32 c^2 \left (a+c x^4\right )}+\frac{5 \int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{128 \sqrt{a} c^{5/2}}+\frac{5 \int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{128 \sqrt{a} c^{5/2}}-\frac{5 \int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{c}}+2 x}{-\frac{\sqrt{a}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{128 \sqrt{2} a^{3/4} c^{9/4}}-\frac{5 \int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{c}}-2 x}{-\frac{\sqrt{a}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{128 \sqrt{2} a^{3/4} c^{9/4}}\\ &=-\frac{x^5}{8 c \left (a+c x^4\right )^2}-\frac{5 x}{32 c^2 \left (a+c x^4\right )}-\frac{5 \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{3/4} c^{9/4}}+\frac{5 \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{3/4} c^{9/4}}+\frac{5 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{3/4} c^{9/4}}-\frac{5 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{3/4} c^{9/4}}\\ &=-\frac{x^5}{8 c \left (a+c x^4\right )^2}-\frac{5 x}{32 c^2 \left (a+c x^4\right )}-\frac{5 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{3/4} c^{9/4}}+\frac{5 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{3/4} c^{9/4}}-\frac{5 \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{3/4} c^{9/4}}+\frac{5 \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{3/4} c^{9/4}}\\ \end{align*}

Mathematica [A]  time = 0.0888749, size = 201, normalized size = 0.91 \[ \frac{-\frac{5 \sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{a^{3/4}}+\frac{5 \sqrt{2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{a^{3/4}}-\frac{10 \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{3/4}}+\frac{10 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{a^{3/4}}-\frac{72 \sqrt [4]{c} x}{a+c x^4}+\frac{32 a \sqrt [4]{c} x}{\left (a+c x^4\right )^2}}{256 c^{9/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^8/(a + c*x^4)^3,x]

[Out]

((32*a*c^(1/4)*x)/(a + c*x^4)^2 - (72*c^(1/4)*x)/(a + c*x^4) - (10*Sqrt[2]*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1
/4)])/a^(3/4) + (10*Sqrt[2]*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/a^(3/4) - (5*Sqrt[2]*Log[Sqrt[a] - Sqrt[2
]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/a^(3/4) + (5*Sqrt[2]*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2
])/a^(3/4))/(256*c^(9/4))

________________________________________________________________________________________

Maple [A]  time = 0.011, size = 163, normalized size = 0.7 \begin{align*}{\frac{1}{ \left ( c{x}^{4}+a \right ) ^{2}} \left ( -{\frac{9\,{x}^{5}}{32\,c}}-{\frac{5\,ax}{32\,{c}^{2}}} \right ) }+{\frac{5\,\sqrt{2}}{256\,{c}^{2}a}\sqrt [4]{{\frac{a}{c}}}\ln \left ({ \left ({x}^{2}+\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ) }+{\frac{5\,\sqrt{2}}{128\,{c}^{2}a}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }+{\frac{5\,\sqrt{2}}{128\,{c}^{2}a}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-9/32/c*x^5-5/32*a/c^2*x)/(c*x^4+a)^2+5/256/c^2*(a/c)^(1/4)/a*2^(1/2)*ln((x^2+(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/
2))/(x^2-(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2)))+5/128/c^2*(a/c)^(1/4)/a*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x+1)+5
/128/c^2*(a/c)^(1/4)/a*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x-1)

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.77623, size = 571, normalized size = 2.58 \begin{align*} -\frac{36 \, c x^{5} - 20 \,{\left (c^{4} x^{8} + 2 \, a c^{3} x^{4} + a^{2} c^{2}\right )} \left (-\frac{1}{a^{3} c^{9}}\right )^{\frac{1}{4}} \arctan \left (-a^{2} c^{7} x \left (-\frac{1}{a^{3} c^{9}}\right )^{\frac{3}{4}} + \sqrt{a^{2} c^{4} \sqrt{-\frac{1}{a^{3} c^{9}}} + x^{2}} a^{2} c^{7} \left (-\frac{1}{a^{3} c^{9}}\right )^{\frac{3}{4}}\right ) - 5 \,{\left (c^{4} x^{8} + 2 \, a c^{3} x^{4} + a^{2} c^{2}\right )} \left (-\frac{1}{a^{3} c^{9}}\right )^{\frac{1}{4}} \log \left (a c^{2} \left (-\frac{1}{a^{3} c^{9}}\right )^{\frac{1}{4}} + x\right ) + 5 \,{\left (c^{4} x^{8} + 2 \, a c^{3} x^{4} + a^{2} c^{2}\right )} \left (-\frac{1}{a^{3} c^{9}}\right )^{\frac{1}{4}} \log \left (-a c^{2} \left (-\frac{1}{a^{3} c^{9}}\right )^{\frac{1}{4}} + x\right ) + 20 \, a x}{128 \,{\left (c^{4} x^{8} + 2 \, a c^{3} x^{4} + a^{2} c^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/128*(36*c*x^5 - 20*(c^4*x^8 + 2*a*c^3*x^4 + a^2*c^2)*(-1/(a^3*c^9))^(1/4)*arctan(-a^2*c^7*x*(-1/(a^3*c^9))^
(3/4) + sqrt(a^2*c^4*sqrt(-1/(a^3*c^9)) + x^2)*a^2*c^7*(-1/(a^3*c^9))^(3/4)) - 5*(c^4*x^8 + 2*a*c^3*x^4 + a^2*
c^2)*(-1/(a^3*c^9))^(1/4)*log(a*c^2*(-1/(a^3*c^9))^(1/4) + x) + 5*(c^4*x^8 + 2*a*c^3*x^4 + a^2*c^2)*(-1/(a^3*c
^9))^(1/4)*log(-a*c^2*(-1/(a^3*c^9))^(1/4) + x) + 20*a*x)/(c^4*x^8 + 2*a*c^3*x^4 + a^2*c^2)

________________________________________________________________________________________

Sympy [A]  time = 1.59375, size = 66, normalized size = 0.3 \begin{align*} - \frac{5 a x + 9 c x^{5}}{32 a^{2} c^{2} + 64 a c^{3} x^{4} + 32 c^{4} x^{8}} + \operatorname{RootSum}{\left (268435456 t^{4} a^{3} c^{9} + 625, \left ( t \mapsto t \log{\left (\frac{128 t a c^{2}}{5} + x \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(5*a*x + 9*c*x**5)/(32*a**2*c**2 + 64*a*c**3*x**4 + 32*c**4*x**8) + RootSum(268435456*_t**4*a**3*c**9 + 625,
Lambda(_t, _t*log(128*_t*a*c**2/5 + x)))

________________________________________________________________________________________

Giac [A]  time = 1.13381, size = 275, normalized size = 1.24 \begin{align*} \frac{5 \, \sqrt{2} \left (a c^{3}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{128 \, a c^{3}} + \frac{5 \, \sqrt{2} \left (a c^{3}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{128 \, a c^{3}} + \frac{5 \, \sqrt{2} \left (a c^{3}\right )^{\frac{1}{4}} \log \left (x^{2} + \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{256 \, a c^{3}} - \frac{5 \, \sqrt{2} \left (a c^{3}\right )^{\frac{1}{4}} \log \left (x^{2} - \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{256 \, a c^{3}} - \frac{9 \, c x^{5} + 5 \, a x}{32 \,{\left (c x^{4} + a\right )}^{2} c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/128*sqrt(2)*(a*c^3)^(1/4)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(a*c^3) + 5/128*sqrt(2
)*(a*c^3)^(1/4)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(a*c^3) + 5/256*sqrt(2)*(a*c^3)^(1
/4)*log(x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a*c^3) - 5/256*sqrt(2)*(a*c^3)^(1/4)*log(x^2 - sqrt(2)*x*(a/
c)^(1/4) + sqrt(a/c))/(a*c^3) - 1/32*(9*c*x^5 + 5*a*x)/((c*x^4 + a)^2*c^2)