3.8.31 \(\int \frac {1}{x^3 (a+b x^2)^{4/3}} \, dx\) [731]

Optimal. Leaf size=123 \[ -\frac {2 b}{a^2 \sqrt [3]{a+b x^2}}-\frac {1}{2 a x^2 \sqrt [3]{a+b x^2}}-\frac {2 b \tan ^{-1}\left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x^2}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{7/3}}+\frac {2 b \log (x)}{3 a^{7/3}}-\frac {b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{a^{7/3}} \]

[Out]

-2*b/a^2/(b*x^2+a)^(1/3)-1/2/a/x^2/(b*x^2+a)^(1/3)+2/3*b*ln(x)/a^(7/3)-b*ln(a^(1/3)-(b*x^2+a)^(1/3))/a^(7/3)-2
/3*b*arctan(1/3*(a^(1/3)+2*(b*x^2+a)^(1/3))/a^(1/3)*3^(1/2))/a^(7/3)*3^(1/2)

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Rubi [A]
time = 0.06, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {272, 44, 53, 57, 631, 210, 31} \begin {gather*} -\frac {2 b \text {ArcTan}\left (\frac {2 \sqrt [3]{a+b x^2}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{7/3}}-\frac {b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{a^{7/3}}+\frac {2 b \log (x)}{3 a^{7/3}}-\frac {2 b}{a^2 \sqrt [3]{a+b x^2}}-\frac {1}{2 a x^2 \sqrt [3]{a+b x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(a + b*x^2)^(4/3)),x]

[Out]

(-2*b)/(a^2*(a + b*x^2)^(1/3)) - 1/(2*a*x^2*(a + b*x^2)^(1/3)) - (2*b*ArcTan[(a^(1/3) + 2*(a + b*x^2)^(1/3))/(
Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(7/3)) + (2*b*Log[x])/(3*a^(7/3)) - (b*Log[a^(1/3) - (a + b*x^2)^(1/3)])/a^(7/3)

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 44

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 53

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 57

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, Simp[-L
og[RemoveContent[a + b*x, x]]/(2*b*q), x] + (Dist[3/(2*b), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/
3)], x] - Dist[3/(2*b*q), Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && PosQ
[(b*c - a*d)/b]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 272

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

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rubi steps

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

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Mathematica [A]
time = 0.16, size = 135, normalized size = 1.10 \begin {gather*} \frac {-\frac {3 \sqrt [3]{a} \left (a+4 b x^2\right )}{x^2 \sqrt [3]{a+b x^2}}-4 \sqrt {3} b \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b x^2}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-4 b \log \left (-\sqrt [3]{a}+\sqrt [3]{a+b x^2}\right )+2 b \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}\right )}{6 a^{7/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(a + b*x^2)^(4/3)),x]

[Out]

((-3*a^(1/3)*(a + 4*b*x^2))/(x^2*(a + b*x^2)^(1/3)) - 4*Sqrt[3]*b*ArcTan[(1 + (2*(a + b*x^2)^(1/3))/a^(1/3))/S
qrt[3]] - 4*b*Log[-a^(1/3) + (a + b*x^2)^(1/3)] + 2*b*Log[a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2
/3)])/(6*a^(7/3))

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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{3} \left (b \,x^{2}+a \right )^{\frac {4}{3}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(b*x^2+a)^(4/3),x)

[Out]

int(1/x^3/(b*x^2+a)^(4/3),x)

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Maxima [A]
time = 0.56, size = 136, normalized size = 1.11 \begin {gather*} -\frac {2 \, \sqrt {3} b \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{3 \, a^{\frac {7}{3}}} - \frac {4 \, {\left (b x^{2} + a\right )} b - 3 \, a b}{2 \, {\left ({\left (b x^{2} + a\right )}^{\frac {4}{3}} a^{2} - {\left (b x^{2} + a\right )}^{\frac {1}{3}} a^{3}\right )}} + \frac {b \log \left ({\left (b x^{2} + a\right )}^{\frac {2}{3}} + {\left (b x^{2} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{3 \, a^{\frac {7}{3}}} - \frac {2 \, b \log \left ({\left (b x^{2} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{3 \, a^{\frac {7}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(b*x^2+a)^(4/3),x, algorithm="maxima")

[Out]

-2/3*sqrt(3)*b*arctan(1/3*sqrt(3)*(2*(b*x^2 + a)^(1/3) + a^(1/3))/a^(1/3))/a^(7/3) - 1/2*(4*(b*x^2 + a)*b - 3*
a*b)/((b*x^2 + a)^(4/3)*a^2 - (b*x^2 + a)^(1/3)*a^3) + 1/3*b*log((b*x^2 + a)^(2/3) + (b*x^2 + a)^(1/3)*a^(1/3)
+ a^(2/3))/a^(7/3) - 2/3*b*log((b*x^2 + a)^(1/3) - a^(1/3))/a^(7/3)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 203 vs. \(2 (96) = 192\).
time = 0.78, size = 453, normalized size = 3.68 \begin {gather*} \left [\frac {6 \, \sqrt {\frac {1}{3}} {\left (a b^{2} x^{4} + a^{2} b x^{2}\right )} \sqrt {\frac {\left (-a\right )^{\frac {1}{3}}}{a}} \log \left (\frac {2 \, b x^{2} - 3 \, \sqrt {\frac {1}{3}} {\left (2 \, {\left (b x^{2} + a\right )}^{\frac {2}{3}} \left (-a\right )^{\frac {2}{3}} - {\left (b x^{2} + a\right )}^{\frac {1}{3}} a + \left (-a\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (-a\right )^{\frac {1}{3}}}{a}} - 3 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {2}{3}} + 3 \, a}{x^{2}}\right ) + 2 \, {\left (b^{2} x^{4} + a b x^{2}\right )} \left (-a\right )^{\frac {2}{3}} \log \left ({\left (b x^{2} + a\right )}^{\frac {2}{3}} - {\left (b x^{2} + a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} + \left (-a\right )^{\frac {2}{3}}\right ) - 4 \, {\left (b^{2} x^{4} + a b x^{2}\right )} \left (-a\right )^{\frac {2}{3}} \log \left ({\left (b x^{2} + a\right )}^{\frac {1}{3}} + \left (-a\right )^{\frac {1}{3}}\right ) - 3 \, {\left (4 \, a b x^{2} + a^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {2}{3}}}{6 \, {\left (a^{3} b x^{4} + a^{4} x^{2}\right )}}, -\frac {12 \, \sqrt {\frac {1}{3}} {\left (a b^{2} x^{4} + a^{2} b x^{2}\right )} \sqrt {-\frac {\left (-a\right )^{\frac {1}{3}}}{a}} \arctan \left (\sqrt {\frac {1}{3}} {\left (2 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} - \left (-a\right )^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-a\right )^{\frac {1}{3}}}{a}}\right ) - 2 \, {\left (b^{2} x^{4} + a b x^{2}\right )} \left (-a\right )^{\frac {2}{3}} \log \left ({\left (b x^{2} + a\right )}^{\frac {2}{3}} - {\left (b x^{2} + a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} + \left (-a\right )^{\frac {2}{3}}\right ) + 4 \, {\left (b^{2} x^{4} + a b x^{2}\right )} \left (-a\right )^{\frac {2}{3}} \log \left ({\left (b x^{2} + a\right )}^{\frac {1}{3}} + \left (-a\right )^{\frac {1}{3}}\right ) + 3 \, {\left (4 \, a b x^{2} + a^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {2}{3}}}{6 \, {\left (a^{3} b x^{4} + a^{4} x^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(b*x^2+a)^(4/3),x, algorithm="fricas")

[Out]

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

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Sympy [C] Result contains complex when optimal does not.
time = 0.87, size = 41, normalized size = 0.33 \begin {gather*} - \frac {\Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {4}{3}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{2}}} \right )}}{2 b^{\frac {4}{3}} x^{\frac {14}{3}} \Gamma \left (\frac {10}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/(b*x**2+a)**(4/3),x)

[Out]

-gamma(7/3)*hyper((4/3, 7/3), (10/3,), a*exp_polar(I*pi)/(b*x**2))/(2*b**(4/3)*x**(14/3)*gamma(10/3))

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Giac [A]
time = 1.25, size = 134, normalized size = 1.09 \begin {gather*} -\frac {2 \, \sqrt {3} b \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{3 \, a^{\frac {7}{3}}} + \frac {b \log \left ({\left (b x^{2} + a\right )}^{\frac {2}{3}} + {\left (b x^{2} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{3 \, a^{\frac {7}{3}}} - \frac {2 \, b \log \left ({\left | {\left (b x^{2} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{3 \, a^{\frac {7}{3}}} - \frac {4 \, {\left (b x^{2} + a\right )} b - 3 \, a b}{2 \, {\left ({\left (b x^{2} + a\right )}^{\frac {4}{3}} - {\left (b x^{2} + a\right )}^{\frac {1}{3}} a\right )} a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(b*x^2+a)^(4/3),x, algorithm="giac")

[Out]

-2/3*sqrt(3)*b*arctan(1/3*sqrt(3)*(2*(b*x^2 + a)^(1/3) + a^(1/3))/a^(1/3))/a^(7/3) + 1/3*b*log((b*x^2 + a)^(2/
3) + (b*x^2 + a)^(1/3)*a^(1/3) + a^(2/3))/a^(7/3) - 2/3*b*log(abs((b*x^2 + a)^(1/3) - a^(1/3)))/a^(7/3) - 1/2*
(4*(b*x^2 + a)*b - 3*a*b)/(((b*x^2 + a)^(4/3) - (b*x^2 + a)^(1/3)*a)*a^2)

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Mupad [B]
time = 5.65, size = 178, normalized size = 1.45 \begin {gather*} -\frac {\frac {3\,b}{a}-\frac {4\,b\,\left (b\,x^2+a\right )}{a^2}}{2\,a\,{\left (b\,x^2+a\right )}^{1/3}-2\,{\left (b\,x^2+a\right )}^{4/3}}-\frac {2\,b\,\ln \left (4\,a^{7/3}\,b^2-4\,a^2\,b^2\,{\left (b\,x^2+a\right )}^{1/3}\right )}{3\,a^{7/3}}+\frac {\ln \left (a^{7/3}\,{\left (b-\sqrt {3}\,b\,1{}\mathrm {i}\right )}^2-4\,a^2\,b^2\,{\left (b\,x^2+a\right )}^{1/3}\right )\,\left (b-\sqrt {3}\,b\,1{}\mathrm {i}\right )}{3\,a^{7/3}}+\frac {\ln \left (a^{7/3}\,{\left (b+\sqrt {3}\,b\,1{}\mathrm {i}\right )}^2-4\,a^2\,b^2\,{\left (b\,x^2+a\right )}^{1/3}\right )\,\left (b+\sqrt {3}\,b\,1{}\mathrm {i}\right )}{3\,a^{7/3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(a + b*x^2)^(4/3)),x)

[Out]

(log(a^(7/3)*(b - 3^(1/2)*b*1i)^2 - 4*a^2*b^2*(a + b*x^2)^(1/3))*(b - 3^(1/2)*b*1i))/(3*a^(7/3)) - (2*b*log(4*
a^(7/3)*b^2 - 4*a^2*b^2*(a + b*x^2)^(1/3)))/(3*a^(7/3)) - ((3*b)/a - (4*b*(a + b*x^2))/a^2)/(2*a*(a + b*x^2)^(
1/3) - 2*(a + b*x^2)^(4/3)) + (log(a^(7/3)*(b + 3^(1/2)*b*1i)^2 - 4*a^2*b^2*(a + b*x^2)^(1/3))*(b + 3^(1/2)*b*
1i))/(3*a^(7/3))

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