3.7.52 \(\int \frac {1}{x^7 (a+c x^4)} \, dx\) [652]

Optimal. Leaf size=51 \[ -\frac {1}{6 a x^6}+\frac {c}{2 a^2 x^2}+\frac {c^{3/2} \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 a^{5/2}} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 4, 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 {c^{3/2} \text {ArcTan}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 a^{5/2}}+\frac {c}{2 a^2 x^2}-\frac {1}{6 a x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^7*(a + c*x^4)),x]

[Out]

-1/6*1/(a*x^6) + c/(2*a^2*x^2) + (c^(3/2)*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(2*a^(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 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^7 \left (a+c x^4\right )} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^4 \left (a+c x^2\right )} \, dx,x,x^2\right )\\ &=-\frac {1}{6 a x^6}-\frac {c \text {Subst}\left (\int \frac {1}{x^2 \left (a+c x^2\right )} \, dx,x,x^2\right )}{2 a}\\ &=-\frac {1}{6 a x^6}+\frac {c}{2 a^2 x^2}+\frac {c^2 \text {Subst}\left (\int \frac {1}{a+c x^2} \, dx,x,x^2\right )}{2 a^2}\\ &=-\frac {1}{6 a x^6}+\frac {c}{2 a^2 x^2}+\frac {c^{3/2} \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 a^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 88, normalized size = 1.73 \begin {gather*} -\frac {\sqrt {a} \left (a-3 c x^4\right )+3 c^{3/2} x^6 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+3 c^{3/2} x^6 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{6 a^{5/2} x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^7*(a + c*x^4)),x]

[Out]

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

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Maple [A]
time = 0.14, size = 43, normalized size = 0.84

method	result	size
default	\(\frac {c^{2} \arctan \left (\frac {c \,x^{2}}{\sqrt {a c}}\right )}{2 a^{2} \sqrt {a c}}-\frac {1}{6 x^{6} a}+\frac {c}{2 a^{2} x^{2}}\)	\(43\)
risch	\(\frac {\frac {c \,x^{4}}{2 a^{2}}-\frac {1}{6 a}}{x^{6}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (a^{5} \textit {\_Z}^{2}+c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (5 a^{5} \textit {\_R}^{2}+4 c^{3}\right ) x^{2}-a^{3} c \textit {\_R} \right )\right )}{4}\)	\(67\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^7/(c*x^4+a),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.51, size = 43, normalized size = 0.84 \begin {gather*} \frac {c^{2} \arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{2 \, \sqrt {a c} a^{2}} + \frac {3 \, c x^{4} - a}{6 \, a^{2} x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(c*x^4+a),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.38, size = 112, normalized size = 2.20 \begin {gather*} \left [\frac {3 \, c x^{6} \sqrt {-\frac {c}{a}} \log \left (\frac {c x^{4} + 2 \, a x^{2} \sqrt {-\frac {c}{a}} - a}{c x^{4} + a}\right ) + 6 \, c x^{4} - 2 \, a}{12 \, a^{2} x^{6}}, -\frac {3 \, c x^{6} \sqrt {\frac {c}{a}} \arctan \left (\frac {a \sqrt {\frac {c}{a}}}{c x^{2}}\right ) - 3 \, c x^{4} + a}{6 \, a^{2} x^{6}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(c*x^4+a),x, algorithm="fricas")

[Out]

[1/12*(3*c*x^6*sqrt(-c/a)*log((c*x^4 + 2*a*x^2*sqrt(-c/a) - a)/(c*x^4 + a)) + 6*c*x^4 - 2*a)/(a^2*x^6), -1/6*(
3*c*x^6*sqrt(c/a)*arctan(a*sqrt(c/a)/(c*x^2)) - 3*c*x^4 + a)/(a^2*x^6)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (44) = 88\).
time = 0.13, size = 90, normalized size = 1.76 \begin {gather*} - \frac {\sqrt {- \frac {c^{3}}{a^{5}}} \log {\left (- \frac {a^{3} \sqrt {- \frac {c^{3}}{a^{5}}}}{c^{2}} + x^{2} \right )}}{4} + \frac {\sqrt {- \frac {c^{3}}{a^{5}}} \log {\left (\frac {a^{3} \sqrt {- \frac {c^{3}}{a^{5}}}}{c^{2}} + x^{2} \right )}}{4} + \frac {- a + 3 c x^{4}}{6 a^{2} x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**7/(c*x**4+a),x)

[Out]

-sqrt(-c**3/a**5)*log(-a**3*sqrt(-c**3/a**5)/c**2 + x**2)/4 + sqrt(-c**3/a**5)*log(a**3*sqrt(-c**3/a**5)/c**2
+ x**2)/4 + (-a + 3*c*x**4)/(6*a**2*x**6)

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Giac [A]
time = 0.51, size = 43, normalized size = 0.84 \begin {gather*} \frac {c^{2} \arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{2 \, \sqrt {a c} a^{2}} + \frac {3 \, c x^{4} - a}{6 \, a^{2} x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(c*x^4+a),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.06, size = 40, normalized size = 0.78 \begin {gather*} \frac {c^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {c}\,x^2}{\sqrt {a}}\right )}{2\,a^{5/2}}-\frac {\frac {1}{6\,a}-\frac {c\,x^4}{2\,a^2}}{x^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^7*(a + c*x^4)),x)

[Out]

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

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