3.7.46 \(\int \frac {x^5}{a+c x^4} \, dx\) [646]

Optimal. Leaf size=40 \[ \frac {x^2}{2 c}-\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 c^{3/2}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {281, 327, 211} \begin {gather*} \frac {x^2}{2 c}-\frac {\sqrt {a} \text {ArcTan}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 c^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5/(a + c*x^4),x]

[Out]

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

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

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Mathematica [A]
time = 0.01, size = 40, normalized size = 1.00 \begin {gather*} \frac {x^2}{2 c}-\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 c^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5/(a + c*x^4),x]

[Out]

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

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Maple [A]
time = 0.18, size = 32, normalized size = 0.80

method	result	size
default	\(\frac {x^{2}}{2 c}-\frac {a \arctan \left (\frac {c \,x^{2}}{\sqrt {a c}}\right )}{2 c \sqrt {a c}}\)	\(32\)
risch	\(\frac {x^{2}}{2 c}+\frac {\sqrt {-a c}\, \ln \left (c \,x^{2}-\sqrt {-a c}\right )}{4 c^{2}}-\frac {\sqrt {-a c}\, \ln \left (c \,x^{2}+\sqrt {-a c}\right )}{4 c^{2}}\)	\(60\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(c*x^4+a),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.50, size = 31, normalized size = 0.78 \begin {gather*} \frac {x^{2}}{2 \, c} - \frac {a \arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{2 \, \sqrt {a c} c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(c*x^4+a),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.37, size = 89, normalized size = 2.22 \begin {gather*} \left [\frac {2 \, x^{2} + \sqrt {-\frac {a}{c}} \log \left (\frac {c x^{4} - 2 \, c x^{2} \sqrt {-\frac {a}{c}} - a}{c x^{4} + a}\right )}{4 \, c}, \frac {x^{2} - \sqrt {\frac {a}{c}} \arctan \left (\frac {c x^{2} \sqrt {\frac {a}{c}}}{a}\right )}{2 \, c}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(c*x^4+a),x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.08, size = 63, normalized size = 1.58 \begin {gather*} \frac {\sqrt {- \frac {a}{c^{3}}} \log {\left (- c \sqrt {- \frac {a}{c^{3}}} + x^{2} \right )}}{4} - \frac {\sqrt {- \frac {a}{c^{3}}} \log {\left (c \sqrt {- \frac {a}{c^{3}}} + x^{2} \right )}}{4} + \frac {x^{2}}{2 c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(c*x**4+a),x)

[Out]

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

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Giac [A]
time = 0.67, size = 31, normalized size = 0.78 \begin {gather*} \frac {x^{2}}{2 \, c} - \frac {a \arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{2 \, \sqrt {a c} c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(c*x^4+a),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.04, size = 28, normalized size = 0.70 \begin {gather*} \frac {x^2}{2\,c}-\frac {\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {c}\,x^2}{\sqrt {a}}\right )}{2\,c^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(a + c*x^4),x)

[Out]

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

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