3.652 \(\int \frac{1}{x^7 (a+c x^4)} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0263047, antiderivative size = 51, normalized size of antiderivative = 1., 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 = {275, 325, 205} \[ \frac{c^{3/2} \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 a^{5/2}}+\frac{c}{2 a^2 x^2}-\frac{1}{6 a x^6} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^7*(a + c*x^4)),x]

[Out]

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

Rule 275

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 325

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]

Rule 205

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

Rubi steps

\begin{align*} \int \frac{1}{x^7 \left (a+c x^4\right )} \, dx &=\frac{1}{2} \operatorname{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 \operatorname{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 \operatorname{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*}

Mathematica [A]  time = 0.0361623, size = 88, normalized size = 1.73 \[ -\frac{3 c^{3/2} x^6 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+3 c^{3/2} x^6 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )+\sqrt{a} \left (a-3 c x^4\right )}{6 a^{5/2} x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^7*(a + c*x^4)),x]

[Out]

-(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 + (Sq
rt[2]*c^(1/4)*x)/a^(1/4)])/(6*a^(5/2)*x^6)

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Maple [A]  time = 0.006, size = 43, normalized size = 0.8 \begin{align*}{\frac{{c}^{2}}{2\,{a}^{2}}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}-{\frac{1}{6\,{x}^{6}a}}+{\frac{c}{2\,{a}^{2}{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.69261, size = 250, normalized size = 4.9 \begin{align*} \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{align*}

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]  time = 0.857069, size = 90, normalized size = 1.76 \begin{align*} - \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{align*}

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 = 1.10819, size = 58, normalized size = 1.14 \begin{align*} \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{align*}

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)