3.7.50 \(\int \frac {1}{x^3 (a+c x^4)} \, dx\) [650]

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

[Out]

-1/2/a/x^2-1/2*arctan(x^2*c^(1/2)/a^(1/2))*c^(1/2)/a^(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, 331, 211} \begin {gather*} -\frac {\sqrt {c} \text {ArcTan}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 a^{3/2}}-\frac {1}{2 a x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(a + c*x^4)),x]

[Out]

-1/2*1/(a*x^2) - (Sqrt[c]*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(2*a^(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 331

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

Rubi steps

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

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Mathematica [A]
time = 0.02, size = 79, normalized size = 1.98 \begin {gather*} \frac {-\sqrt {a}+\sqrt {c} x^2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+\sqrt {c} x^2 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 a^{3/2} x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(a + c*x^4)),x]

[Out]

(-Sqrt[a] + Sqrt[c]*x^2*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)] + Sqrt[c]*x^2*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a
^(1/4)])/(2*a^(3/2)*x^2)

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Maple [A]
time = 0.14, size = 32, normalized size = 0.80

method	result	size
default	\(-\frac {1}{2 a \,x^{2}}-\frac {c \arctan \left (\frac {c \,x^{2}}{\sqrt {a c}}\right )}{2 a \sqrt {a c}}\)	\(32\)
risch	\(-\frac {1}{2 a \,x^{2}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (a^{3} \textit {\_Z}^{2}+c \right )}{\sum }\textit {\_R} \ln \left (\left (-5 a^{3} \textit {\_R}^{2}-4 c \right ) x^{2}-a^{2} \textit {\_R} \right )\right )}{4}\)	\(51\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(c*x^4+a),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.51, size = 31, normalized size = 0.78 \begin {gather*} -\frac {c \arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{2 \, \sqrt {a c} a} - \frac {1}{2 \, a x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(c*x^4+a),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.38, size = 94, normalized size = 2.35 \begin {gather*} \left [\frac {x^{2} \sqrt {-\frac {c}{a}} \log \left (\frac {c x^{4} - 2 \, a x^{2} \sqrt {-\frac {c}{a}} - a}{c x^{4} + a}\right ) - 2}{4 \, a x^{2}}, \frac {x^{2} \sqrt {\frac {c}{a}} \arctan \left (\frac {a \sqrt {\frac {c}{a}}}{c x^{2}}\right ) - 1}{2 \, a x^{2}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(c*x^4+a),x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.10, size = 71, normalized size = 1.78 \begin {gather*} \frac {\sqrt {- \frac {c}{a^{3}}} \log {\left (- \frac {a^{2} \sqrt {- \frac {c}{a^{3}}}}{c} + x^{2} \right )}}{4} - \frac {\sqrt {- \frac {c}{a^{3}}} \log {\left (\frac {a^{2} \sqrt {- \frac {c}{a^{3}}}}{c} + x^{2} \right )}}{4} - \frac {1}{2 a x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/(c*x**4+a),x)

[Out]

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

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Giac [A]
time = 0.52, size = 31, normalized size = 0.78 \begin {gather*} -\frac {c \arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{2 \, \sqrt {a c} a} - \frac {1}{2 \, a x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(c*x^4+a),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.05, size = 28, normalized size = 0.70 \begin {gather*} -\frac {1}{2\,a\,x^2}-\frac {\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {c}\,x^2}{\sqrt {a}}\right )}{2\,a^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(a + c*x^4)),x)

[Out]

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

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