3.684 \(\int \frac{x^2}{(a+c x^4)^3} \, dx\)

Optimal. Leaf size=223 \[ \frac{5 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{9/4} c^{3/4}}-\frac{5 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{9/4} c^{3/4}}-\frac{5 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{9/4} c^{3/4}}+\frac{5 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt{2} a^{9/4} c^{3/4}}+\frac{5 x^3}{32 a^2 \left (a+c x^4\right )}+\frac{x^3}{8 a \left (a+c x^4\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.144461, antiderivative size = 223, 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 = {290, 297, 1162, 617, 204, 1165, 628} \[ \frac{5 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{9/4} c^{3/4}}-\frac{5 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{9/4} c^{3/4}}-\frac{5 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{9/4} c^{3/4}}+\frac{5 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt{2} a^{9/4} c^{3/4}}+\frac{5 x^3}{32 a^2 \left (a+c x^4\right )}+\frac{x^3}{8 a \left (a+c x^4\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + c*x^4)^3,x]

[Out]

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

Rule 290

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

Rule 297

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

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])

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]

Rubi steps

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

Mathematica [A]  time = 0.0847371, size = 204, normalized size = 0.91 \[ \frac{\frac{32 a^{5/4} x^3}{\left (a+c x^4\right )^2}+\frac{5 \sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{c^{3/4}}-\frac{5 \sqrt{2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{c^{3/4}}-\frac{10 \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{c^{3/4}}+\frac{10 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{c^{3/4}}+\frac{40 \sqrt [4]{a} x^3}{a+c x^4}}{256 a^{9/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + c*x^4)^3,x]

[Out]

((32*a^(5/4)*x^3)/(a + c*x^4)^2 + (40*a^(1/4)*x^3)/(a + c*x^4) - (10*Sqrt[2]*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^
(1/4)])/c^(3/4) + (10*Sqrt[2]*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/c^(3/4) + (5*Sqrt[2]*Log[Sqrt[a] - Sqrt
[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/c^(3/4) - (5*Sqrt[2]*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x
^2])/c^(3/4))/(256*a^(9/4))

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Maple [A]  time = 0.002, size = 171, normalized size = 0.8 \begin{align*}{\frac{{x}^{3}}{8\,a \left ( c{x}^{4}+a \right ) ^{2}}}+{\frac{5\,{x}^{3}}{32\,{a}^{2} \left ( c{x}^{4}+a \right ) }}+{\frac{5\,\sqrt{2}}{256\,{a}^{2}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{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{5\,\sqrt{2}}{128\,{a}^{2}c}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{5\,\sqrt{2}}{128\,{a}^{2}c}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*x^3/a/(c*x^4+a)^2+5/32*x^3/a^2/(c*x^4+a)+5/256/a^2/c/(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)))+5/128/a^2/c/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*
x+1)+5/128/a^2/c/(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^2/(c*x^4+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.96105, size = 571, normalized size = 2.56 \begin{align*} \frac{20 \, c x^{7} + 36 \, a x^{3} - 20 \,{\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )} \left (-\frac{1}{a^{9} c^{3}}\right )^{\frac{1}{4}} \arctan \left (-a^{2} c x \left (-\frac{1}{a^{9} c^{3}}\right )^{\frac{1}{4}} + \sqrt{-a^{5} c \sqrt{-\frac{1}{a^{9} c^{3}}} + x^{2}} a^{2} c \left (-\frac{1}{a^{9} c^{3}}\right )^{\frac{1}{4}}\right ) + 5 \,{\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )} \left (-\frac{1}{a^{9} c^{3}}\right )^{\frac{1}{4}} \log \left (a^{7} c^{2} \left (-\frac{1}{a^{9} c^{3}}\right )^{\frac{3}{4}} + x\right ) - 5 \,{\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )} \left (-\frac{1}{a^{9} c^{3}}\right )^{\frac{1}{4}} \log \left (-a^{7} c^{2} \left (-\frac{1}{a^{9} c^{3}}\right )^{\frac{3}{4}} + x\right )}{128 \,{\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.34807, size = 71, normalized size = 0.32 \begin{align*} \frac{9 a x^{3} + 5 c x^{7}}{32 a^{4} + 64 a^{3} c x^{4} + 32 a^{2} c^{2} x^{8}} + \operatorname{RootSum}{\left (268435456 t^{4} a^{9} c^{3} + 625, \left ( t \mapsto t \log{\left (\frac{2097152 t^{3} a^{7} c^{2}}{125} + x \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(9*a*x**3 + 5*c*x**7)/(32*a**4 + 64*a**3*c*x**4 + 32*a**2*c**2*x**8) + RootSum(268435456*_t**4*a**9*c**3 + 625
, Lambda(_t, _t*log(2097152*_t**3*a**7*c**2/125 + x)))

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Giac [A]  time = 1.13042, size = 278, normalized size = 1.25 \begin{align*} \frac{5 \, c x^{7} + 9 \, a x^{3}}{32 \,{\left (c x^{4} + a\right )}^{2} a^{2}} + \frac{5 \, \sqrt{2} \left (a c^{3}\right )^{\frac{3}{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^{3} c^{3}} + \frac{5 \, \sqrt{2} \left (a c^{3}\right )^{\frac{3}{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^{3} c^{3}} - \frac{5 \, \sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}} \log \left (x^{2} + \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{256 \, a^{3} c^{3}} + \frac{5 \, \sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}} \log \left (x^{2} - \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{256 \, a^{3} c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*(5*c*x^7 + 9*a*x^3)/((c*x^4 + a)^2*a^2) + 5/128*sqrt(2)*(a*c^3)^(3/4)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(
a/c)^(1/4))/(a/c)^(1/4))/(a^3*c^3) + 5/128*sqrt(2)*(a*c^3)^(3/4)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/c)^(1/4)
)/(a/c)^(1/4))/(a^3*c^3) - 5/256*sqrt(2)*(a*c^3)^(3/4)*log(x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a^3*c^3)
+ 5/256*sqrt(2)*(a*c^3)^(3/4)*log(x^2 - sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a^3*c^3)