3.7.59 \(\int \frac {x^9}{(a+c x^4)^2} \, dx\) [659]

Optimal. Leaf size=59 \[ \frac {3 x^2}{4 c^2}-\frac {x^6}{4 c \left (a+c x^4\right )}-\frac {3 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{4 c^{5/2}} \]

[Out]

3/4*x^2/c^2-1/4*x^6/c/(c*x^4+a)-3/4*arctan(x^2*c^(1/2)/a^(1/2))*a^(1/2)/c^(5/2)

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Rubi [A]
time = 0.02, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {281, 294, 327, 211} \begin {gather*} -\frac {3 \sqrt {a} \text {ArcTan}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{4 c^{5/2}}-\frac {x^6}{4 c \left (a+c x^4\right )}+\frac {3 x^2}{4 c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^9/(a + c*x^4)^2,x]

[Out]

(3*x^2)/(4*c^2) - x^6/(4*c*(a + c*x^4)) - (3*Sqrt[a]*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(4*c^(5/2))

Rule 211

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

Rule 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 294

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 327

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*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rubi steps

\begin {align*} \int \frac {x^9}{\left (a+c x^4\right )^2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^4}{\left (a+c x^2\right )^2} \, dx,x,x^2\right )\\ &=-\frac {x^6}{4 c \left (a+c x^4\right )}+\frac {3 \text {Subst}\left (\int \frac {x^2}{a+c x^2} \, dx,x,x^2\right )}{4 c}\\ &=\frac {3 x^2}{4 c^2}-\frac {x^6}{4 c \left (a+c x^4\right )}-\frac {(3 a) \text {Subst}\left (\int \frac {1}{a+c x^2} \, dx,x,x^2\right )}{4 c^2}\\ &=\frac {3 x^2}{4 c^2}-\frac {x^6}{4 c \left (a+c x^4\right )}-\frac {3 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{4 c^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 60, normalized size = 1.02 \begin {gather*} \frac {x^2}{2 c^2}+\frac {a x^2}{4 c^2 \left (a+c x^4\right )}-\frac {3 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{4 c^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^9/(a + c*x^4)^2,x]

[Out]

x^2/(2*c^2) + (a*x^2)/(4*c^2*(a + c*x^4)) - (3*Sqrt[a]*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(4*c^(5/2))

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Maple [A]
time = 0.14, size = 49, normalized size = 0.83

method	result	size
default	\(\frac {x^{2}}{2 c^{2}}-\frac {a \left (-\frac {x^{2}}{2 \left (x^{4} c +a \right )}+\frac {3 \arctan \left (\frac {c \,x^{2}}{\sqrt {a c}}\right )}{2 \sqrt {a c}}\right )}{2 c^{2}}\)	\(49\)
risch	\(\frac {x^{2}}{2 c^{2}}+\frac {a \,x^{2}}{4 c^{2} \left (x^{4} c +a \right )}+\frac {3 \sqrt {-a c}\, \ln \left (c \,x^{2}-\sqrt {-a c}\right )}{8 c^{3}}-\frac {3 \sqrt {-a c}\, \ln \left (c \,x^{2}+\sqrt {-a c}\right )}{8 c^{3}}\)	\(78\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^9/(c*x^4+a)^2,x,method=_RETURNVERBOSE)

[Out]

1/2*x^2/c^2-1/2/c^2*a*(-1/2*x^2/(c*x^4+a)+3/2/(a*c)^(1/2)*arctan(c*x^2/(a*c)^(1/2)))

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Maxima [A]
time = 0.50, size = 52, normalized size = 0.88 \begin {gather*} \frac {a x^{2}}{4 \, {\left (c^{3} x^{4} + a c^{2}\right )}} + \frac {x^{2}}{2 \, c^{2}} - \frac {3 \, a \arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{4 \, \sqrt {a c} c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9/(c*x^4+a)^2,x, algorithm="maxima")

[Out]

1/4*a*x^2/(c^3*x^4 + a*c^2) + 1/2*x^2/c^2 - 3/4*a*arctan(c*x^2/sqrt(a*c))/(sqrt(a*c)*c^2)

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Fricas [A]
time = 0.36, size = 144, normalized size = 2.44 \begin {gather*} \left [\frac {4 \, c x^{6} + 6 \, a x^{2} + 3 \, {\left (c x^{4} + a\right )} \sqrt {-\frac {a}{c}} \log \left (\frac {c x^{4} - 2 \, c x^{2} \sqrt {-\frac {a}{c}} - a}{c x^{4} + a}\right )}{8 \, {\left (c^{3} x^{4} + a c^{2}\right )}}, \frac {2 \, c x^{6} + 3 \, a x^{2} - 3 \, {\left (c x^{4} + a\right )} \sqrt {\frac {a}{c}} \arctan \left (\frac {c x^{2} \sqrt {\frac {a}{c}}}{a}\right )}{4 \, {\left (c^{3} x^{4} + a c^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9/(c*x^4+a)^2,x, algorithm="fricas")

[Out]

[1/8*(4*c*x^6 + 6*a*x^2 + 3*(c*x^4 + a)*sqrt(-a/c)*log((c*x^4 - 2*c*x^2*sqrt(-a/c) - a)/(c*x^4 + a)))/(c^3*x^4
+ a*c^2), 1/4*(2*c*x^6 + 3*a*x^2 - 3*(c*x^4 + a)*sqrt(a/c)*arctan(c*x^2*sqrt(a/c)/a))/(c^3*x^4 + a*c^2)]

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Sympy [A]
time = 0.17, size = 92, normalized size = 1.56 \begin {gather*} \frac {a x^{2}}{4 a c^{2} + 4 c^{3} x^{4}} + \frac {3 \sqrt {- \frac {a}{c^{5}}} \log {\left (- c^{2} \sqrt {- \frac {a}{c^{5}}} + x^{2} \right )}}{8} - \frac {3 \sqrt {- \frac {a}{c^{5}}} \log {\left (c^{2} \sqrt {- \frac {a}{c^{5}}} + x^{2} \right )}}{8} + \frac {x^{2}}{2 c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**9/(c*x**4+a)**2,x)

[Out]

a*x**2/(4*a*c**2 + 4*c**3*x**4) + 3*sqrt(-a/c**5)*log(-c**2*sqrt(-a/c**5) + x**2)/8 - 3*sqrt(-a/c**5)*log(c**2
*sqrt(-a/c**5) + x**2)/8 + x**2/(2*c**2)

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Giac [A]
time = 0.50, size = 49, normalized size = 0.83 \begin {gather*} \frac {a x^{2}}{4 \, {\left (c x^{4} + a\right )} c^{2}} + \frac {x^{2}}{2 \, c^{2}} - \frac {3 \, a \arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{4 \, \sqrt {a c} c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9/(c*x^4+a)^2,x, algorithm="giac")

[Out]

1/4*a*x^2/((c*x^4 + a)*c^2) + 1/2*x^2/c^2 - 3/4*a*arctan(c*x^2/sqrt(a*c))/(sqrt(a*c)*c^2)

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Mupad [B]
time = 1.07, size = 50, normalized size = 0.85 \begin {gather*} \frac {x^2}{2\,c^2}+\frac {a\,x^2}{4\,\left (c^3\,x^4+a\,c^2\right )}-\frac {3\,\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {c}\,x^2}{\sqrt {a}}\right )}{4\,c^{5/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^9/(a + c*x^4)^2,x)

[Out]

x^2/(2*c^2) + (a*x^2)/(4*(a*c^2 + c^3*x^4)) - (3*a^(1/2)*atan((c^(1/2)*x^2)/a^(1/2)))/(4*c^(5/2))

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