3.683 \(\int \frac{(a+b x^2)^{2/3}}{x^5} \, dx\)

Optimal. Leaf size=135 \[ -\frac{b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{12 a^{4/3}}-\frac{b^2 \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^2}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{6 \sqrt{3} a^{4/3}}+\frac{b^2 \log (x)}{18 a^{4/3}}-\frac{b \left (a+b x^2\right )^{2/3}}{6 a x^2}-\frac{\left (a+b x^2\right )^{2/3}}{4 x^4} \]

[Out]

-(a + b*x^2)^(2/3)/(4*x^4) - (b*(a + b*x^2)^(2/3))/(6*a*x^2) - (b^2*ArcTan[(a^(1/3) + 2*(a + b*x^2)^(1/3))/(Sq
rt[3]*a^(1/3))])/(6*Sqrt[3]*a^(4/3)) + (b^2*Log[x])/(18*a^(4/3)) - (b^2*Log[a^(1/3) - (a + b*x^2)^(1/3)])/(12*
a^(4/3))

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Rubi [A]  time = 0.0877556, antiderivative size = 135, normalized size of antiderivative = 1., 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 = {266, 47, 51, 55, 617, 204, 31} \[ -\frac{b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{12 a^{4/3}}-\frac{b^2 \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^2}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{6 \sqrt{3} a^{4/3}}+\frac{b^2 \log (x)}{18 a^{4/3}}-\frac{b \left (a+b x^2\right )^{2/3}}{6 a x^2}-\frac{\left (a+b x^2\right )^{2/3}}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(2/3)/x^5,x]

[Out]

-(a + b*x^2)^(2/3)/(4*x^4) - (b*(a + b*x^2)^(2/3))/(6*a*x^2) - (b^2*ArcTan[(a^(1/3) + 2*(a + b*x^2)^(1/3))/(Sq
rt[3]*a^(1/3))])/(6*Sqrt[3]*a^(4/3)) + (b^2*Log[x])/(18*a^(4/3)) - (b^2*Log[a^(1/3) - (a + b*x^2)^(1/3)])/(12*
a^(4/3))

Rule 266

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 47

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

Rule 51

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 55

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 617

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]

Rule 204

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

Rule 31

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

Rubi steps

\begin{align*} \int \frac{\left (a+b x^2\right )^{2/3}}{x^5} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^{2/3}}{x^3} \, dx,x,x^2\right )\\ &=-\frac{\left (a+b x^2\right )^{2/3}}{4 x^4}+\frac{1}{6} b \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt [3]{a+b x}} \, dx,x,x^2\right )\\ &=-\frac{\left (a+b x^2\right )^{2/3}}{4 x^4}-\frac{b \left (a+b x^2\right )^{2/3}}{6 a x^2}-\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{x \sqrt [3]{a+b x}} \, dx,x,x^2\right )}{18 a}\\ &=-\frac{\left (a+b x^2\right )^{2/3}}{4 x^4}-\frac{b \left (a+b x^2\right )^{2/3}}{6 a x^2}+\frac{b^2 \log (x)}{18 a^{4/3}}+\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x^2}\right )}{12 a^{4/3}}-\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x^2}\right )}{12 a}\\ &=-\frac{\left (a+b x^2\right )^{2/3}}{4 x^4}-\frac{b \left (a+b x^2\right )^{2/3}}{6 a x^2}+\frac{b^2 \log (x)}{18 a^{4/3}}-\frac{b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{12 a^{4/3}}+\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{a+b x^2}}{\sqrt [3]{a}}\right )}{6 a^{4/3}}\\ &=-\frac{\left (a+b x^2\right )^{2/3}}{4 x^4}-\frac{b \left (a+b x^2\right )^{2/3}}{6 a x^2}-\frac{b^2 \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{a+b x^2}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{6 \sqrt{3} a^{4/3}}+\frac{b^2 \log (x)}{18 a^{4/3}}-\frac{b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{12 a^{4/3}}\\ \end{align*}

Mathematica [C]  time = 0.008453, size = 39, normalized size = 0.29 \[ -\frac{3 b^2 \left (a+b x^2\right )^{5/3} \, _2F_1\left (\frac{5}{3},3;\frac{8}{3};\frac{b x^2}{a}+1\right )}{10 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(2/3)/x^5,x]

[Out]

(-3*b^2*(a + b*x^2)^(5/3)*Hypergeometric2F1[5/3, 3, 8/3, 1 + (b*x^2)/a])/(10*a^3)

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Maple [F]  time = 0.03, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{5}} \left ( b{x}^{2}+a \right ) ^{{\frac{2}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^(2/3)/x^5,x)

[Out]

int((b*x^2+a)^(2/3)/x^5,x)

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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((b*x^2+a)^(2/3)/x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.85463, size = 994, normalized size = 7.36 \begin{align*} \left [\frac{3 \, \sqrt{\frac{1}{3}} a b^{2} x^{4} \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 ) + \left (-a\right )^{\frac{2}{3}} b^{2} x^{4} \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 ) - 2 \, \left (-a\right )^{\frac{2}{3}} b^{2} x^{4} \log \left ({\left (b x^{2} + a\right )}^{\frac{1}{3}} + \left (-a\right )^{\frac{1}{3}}\right ) - 3 \,{\left (2 \, a b x^{2} + 3 \, a^{2}\right )}{\left (b x^{2} + a\right )}^{\frac{2}{3}}}{36 \, a^{2} x^{4}}, -\frac{6 \, \sqrt{\frac{1}{3}} a b^{2} x^{4} \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 ) - \left (-a\right )^{\frac{2}{3}} b^{2} x^{4} \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 ) + 2 \, \left (-a\right )^{\frac{2}{3}} b^{2} x^{4} \log \left ({\left (b x^{2} + a\right )}^{\frac{1}{3}} + \left (-a\right )^{\frac{1}{3}}\right ) + 3 \,{\left (2 \, a b x^{2} + 3 \, a^{2}\right )}{\left (b x^{2} + a\right )}^{\frac{2}{3}}}{36 \, a^{2} x^{4}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(2/3)/x^5,x, algorithm="fricas")

[Out]

[1/36*(3*sqrt(1/3)*a*b^2*x^4*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) + (-a)^(2/3
)*b^2*x^4*log((b*x^2 + a)^(2/3) - (b*x^2 + a)^(1/3)*(-a)^(1/3) + (-a)^(2/3)) - 2*(-a)^(2/3)*b^2*x^4*log((b*x^2
+ a)^(1/3) + (-a)^(1/3)) - 3*(2*a*b*x^2 + 3*a^2)*(b*x^2 + a)^(2/3))/(a^2*x^4), -1/36*(6*sqrt(1/3)*a*b^2*x^4*s
qrt(-(-a)^(1/3)/a)*arctan(sqrt(1/3)*(2*(b*x^2 + a)^(1/3) - (-a)^(1/3))*sqrt(-(-a)^(1/3)/a)) - (-a)^(2/3)*b^2*x
^4*log((b*x^2 + a)^(2/3) - (b*x^2 + a)^(1/3)*(-a)^(1/3) + (-a)^(2/3)) + 2*(-a)^(2/3)*b^2*x^4*log((b*x^2 + a)^(
1/3) + (-a)^(1/3)) + 3*(2*a*b*x^2 + 3*a^2)*(b*x^2 + a)^(2/3))/(a^2*x^4)]

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Sympy [C]  time = 1.64532, size = 42, normalized size = 0.31 \begin{align*} - \frac{b^{\frac{2}{3}} \Gamma \left (\frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, \frac{4}{3} \\ \frac{7}{3} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{2}}} \right )}}{2 x^{\frac{8}{3}} \Gamma \left (\frac{7}{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**(2/3)/x**5,x)

[Out]

-b**(2/3)*gamma(4/3)*hyper((-2/3, 4/3), (7/3,), a*exp_polar(I*pi)/(b*x**2))/(2*x**(8/3)*gamma(7/3))

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Giac [A]  time = 4.25879, size = 170, normalized size = 1.26 \begin{align*} -\frac{1}{36} \, b^{2}{\left (\frac{2 \, \sqrt{3} \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 )}{a^{\frac{4}{3}}} - \frac{\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 )}{a^{\frac{4}{3}}} + \frac{2 \, \log \left ({\left |{\left (b x^{2} + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}} \right |}\right )}{a^{\frac{4}{3}}} + \frac{3 \,{\left (2 \,{\left (b x^{2} + a\right )}^{\frac{5}{3}} +{\left (b x^{2} + a\right )}^{\frac{2}{3}} a\right )}}{a b^{2} x^{4}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(2/3)/x^5,x, algorithm="giac")

[Out]

-1/36*b^2*(2*sqrt(3)*arctan(1/3*sqrt(3)*(2*(b*x^2 + a)^(1/3) + a^(1/3))/a^(1/3))/a^(4/3) - log((b*x^2 + a)^(2/
3) + (b*x^2 + a)^(1/3)*a^(1/3) + a^(2/3))/a^(4/3) + 2*log(abs((b*x^2 + a)^(1/3) - a^(1/3)))/a^(4/3) + 3*(2*(b*
x^2 + a)^(5/3) + (b*x^2 + a)^(2/3)*a)/(a*b^2*x^4))