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

Optimal. Leaf size=222 \[ -\frac{3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{7/4} c^{5/4}}+\frac{3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{7/4} c^{5/4}}-\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{7/4} c^{5/4}}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt{2} a^{7/4} c^{5/4}}+\frac{x}{32 a c \left (a+c x^4\right )}-\frac{x}{8 c \left (a+c x^4\right )^2} \]

[Out]

-x/(8*c*(a + c*x^4)^2) + x/(32*a*c*(a + c*x^4)) - (3*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(64*Sqrt[2]*a^(7
/4)*c^(5/4)) + (3*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(64*Sqrt[2]*a^(7/4)*c^(5/4)) - (3*Log[Sqrt[a] - Sqr
t[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(128*Sqrt[2]*a^(7/4)*c^(5/4)) + (3*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4
)*x + Sqrt[c]*x^2])/(128*Sqrt[2]*a^(7/4)*c^(5/4))

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Rubi [A]  time = 0.140841, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615, Rules used = {288, 199, 211, 1165, 628, 1162, 617, 204} \[ -\frac{3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{7/4} c^{5/4}}+\frac{3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{7/4} c^{5/4}}-\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{7/4} c^{5/4}}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt{2} a^{7/4} c^{5/4}}+\frac{x}{32 a c \left (a+c x^4\right )}-\frac{x}{8 c \left (a+c x^4\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[x^4/(a + c*x^4)^3,x]

[Out]

-x/(8*c*(a + c*x^4)^2) + x/(32*a*c*(a + c*x^4)) - (3*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(64*Sqrt[2]*a^(7
/4)*c^(5/4)) + (3*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(64*Sqrt[2]*a^(7/4)*c^(5/4)) - (3*Log[Sqrt[a] - Sqr
t[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(128*Sqrt[2]*a^(7/4)*c^(5/4)) + (3*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4
)*x + Sqrt[c]*x^2])/(128*Sqrt[2]*a^(7/4)*c^(5/4))

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 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

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^4}{\left (a+c x^4\right )^3} \, dx &=-\frac{x}{8 c \left (a+c x^4\right )^2}+\frac{\int \frac{1}{\left (a+c x^4\right )^2} \, dx}{8 c}\\ &=-\frac{x}{8 c \left (a+c x^4\right )^2}+\frac{x}{32 a c \left (a+c x^4\right )}+\frac{3 \int \frac{1}{a+c x^4} \, dx}{32 a c}\\ &=-\frac{x}{8 c \left (a+c x^4\right )^2}+\frac{x}{32 a c \left (a+c x^4\right )}+\frac{3 \int \frac{\sqrt{a}-\sqrt{c} x^2}{a+c x^4} \, dx}{64 a^{3/2} c}+\frac{3 \int \frac{\sqrt{a}+\sqrt{c} x^2}{a+c x^4} \, dx}{64 a^{3/2} c}\\ &=-\frac{x}{8 c \left (a+c x^4\right )^2}+\frac{x}{32 a c \left (a+c x^4\right )}+\frac{3 \int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{128 a^{3/2} c^{3/2}}+\frac{3 \int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{128 a^{3/2} c^{3/2}}-\frac{3 \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^{7/4} c^{5/4}}-\frac{3 \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^{7/4} c^{5/4}}\\ &=-\frac{x}{8 c \left (a+c x^4\right )^2}+\frac{x}{32 a c \left (a+c x^4\right )}-\frac{3 \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{7/4} c^{5/4}}+\frac{3 \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{7/4} c^{5/4}}+\frac{3 \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^{7/4} c^{5/4}}-\frac{3 \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^{7/4} c^{5/4}}\\ &=-\frac{x}{8 c \left (a+c x^4\right )^2}+\frac{x}{32 a c \left (a+c x^4\right )}-\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{7/4} c^{5/4}}+\frac{3 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{7/4} c^{5/4}}-\frac{3 \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{7/4} c^{5/4}}+\frac{3 \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{7/4} c^{5/4}}\\ \end{align*}

Mathematica [A]  time = 0.0886666, size = 203, normalized size = 0.91 \[ \frac{\frac{8 \sqrt [4]{c} x}{a^2+a c x^4}-\frac{3 \sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{a^{7/4}}+\frac{3 \sqrt{2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{a^{7/4}}-\frac{6 \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{7/4}}+\frac{6 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{a^{7/4}}-\frac{32 \sqrt [4]{c} x}{\left (a+c x^4\right )^2}}{256 c^{5/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/(a + c*x^4)^3,x]

[Out]

((-32*c^(1/4)*x)/(a + c*x^4)^2 + (8*c^(1/4)*x)/(a^2 + a*c*x^4) - (6*Sqrt[2]*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(
1/4)])/a^(7/4) + (6*Sqrt[2]*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/a^(7/4) - (3*Sqrt[2]*Log[Sqrt[a] - Sqrt[2
]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/a^(7/4) + (3*Sqrt[2]*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2
])/a^(7/4))/(256*c^(5/4))

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Maple [A]  time = 0.01, size = 162, normalized size = 0.7 \begin{align*}{\frac{1}{ \left ( c{x}^{4}+a \right ) ^{2}} \left ({\frac{{x}^{5}}{32\,a}}-{\frac{3\,x}{32\,c}} \right ) }+{\frac{3\,\sqrt{2}}{256\,{a}^{2}c}\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{3\,\sqrt{2}}{128\,{a}^{2}c}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }+{\frac{3\,\sqrt{2}}{128\,{a}^{2}c}\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^4/(c*x^4+a)^3,x)

[Out]

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

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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^4/(c*x^4+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.79208, size = 579, normalized size = 2.61 \begin{align*} \frac{4 \, c x^{5} + 12 \,{\left (a c^{3} x^{8} + 2 \, a^{2} c^{2} x^{4} + a^{3} c\right )} \left (-\frac{1}{a^{7} c^{5}}\right )^{\frac{1}{4}} \arctan \left (-a^{5} c^{4} x \left (-\frac{1}{a^{7} c^{5}}\right )^{\frac{3}{4}} + \sqrt{a^{4} c^{2} \sqrt{-\frac{1}{a^{7} c^{5}}} + x^{2}} a^{5} c^{4} \left (-\frac{1}{a^{7} c^{5}}\right )^{\frac{3}{4}}\right ) + 3 \,{\left (a c^{3} x^{8} + 2 \, a^{2} c^{2} x^{4} + a^{3} c\right )} \left (-\frac{1}{a^{7} c^{5}}\right )^{\frac{1}{4}} \log \left (a^{2} c \left (-\frac{1}{a^{7} c^{5}}\right )^{\frac{1}{4}} + x\right ) - 3 \,{\left (a c^{3} x^{8} + 2 \, a^{2} c^{2} x^{4} + a^{3} c\right )} \left (-\frac{1}{a^{7} c^{5}}\right )^{\frac{1}{4}} \log \left (-a^{2} c \left (-\frac{1}{a^{7} c^{5}}\right )^{\frac{1}{4}} + x\right ) - 12 \, a x}{128 \,{\left (a c^{3} x^{8} + 2 \, a^{2} c^{2} x^{4} + a^{3} c\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/128*(4*c*x^5 + 12*(a*c^3*x^8 + 2*a^2*c^2*x^4 + a^3*c)*(-1/(a^7*c^5))^(1/4)*arctan(-a^5*c^4*x*(-1/(a^7*c^5))^
(3/4) + sqrt(a^4*c^2*sqrt(-1/(a^7*c^5)) + x^2)*a^5*c^4*(-1/(a^7*c^5))^(3/4)) + 3*(a*c^3*x^8 + 2*a^2*c^2*x^4 +
a^3*c)*(-1/(a^7*c^5))^(1/4)*log(a^2*c*(-1/(a^7*c^5))^(1/4) + x) - 3*(a*c^3*x^8 + 2*a^2*c^2*x^4 + a^3*c)*(-1/(a
^7*c^5))^(1/4)*log(-a^2*c*(-1/(a^7*c^5))^(1/4) + x) - 12*a*x)/(a*c^3*x^8 + 2*a^2*c^2*x^4 + a^3*c)

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Sympy [A]  time = 1.58854, size = 66, normalized size = 0.3 \begin{align*} \frac{- 3 a x + c x^{5}}{32 a^{3} c + 64 a^{2} c^{2} x^{4} + 32 a c^{3} x^{8}} + \operatorname{RootSum}{\left (268435456 t^{4} a^{7} c^{5} + 81, \left ( t \mapsto t \log{\left (\frac{128 t a^{2} c}{3} + x \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-3*a*x + c*x**5)/(32*a**3*c + 64*a**2*c**2*x**4 + 32*a*c**3*x**8) + RootSum(268435456*_t**4*a**7*c**5 + 81, L
ambda(_t, _t*log(128*_t*a**2*c/3 + x)))

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Giac [A]  time = 1.14634, size = 278, normalized size = 1.25 \begin{align*} \frac{3 \, \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^{2} c^{2}} + \frac{3 \, \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^{2} c^{2}} + \frac{3 \, \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^{2} c^{2}} - \frac{3 \, \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^{2} c^{2}} + \frac{c x^{5} - 3 \, a x}{32 \,{\left (c x^{4} + a\right )}^{2} a c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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