3.468 \(\int \frac{a+b \log (c (d+e x^{2/3})^n)}{x^2} \, dx\)

Optimal. Leaf size=68 \[ -\frac{a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )}{x}-\frac{2 b e^{3/2} n \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{d^{3/2}}-\frac{2 b e n}{d \sqrt [3]{x}} \]

[Out]

(-2*b*e*n)/(d*x^(1/3)) - (2*b*e^(3/2)*n*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]])/d^(3/2) - (a + b*Log[c*(d + e*x^(2/
3))^n])/x

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Rubi [A]  time = 0.0401322, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2455, 341, 325, 205} \[ -\frac{a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )}{x}-\frac{2 b e^{3/2} n \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{d^{3/2}}-\frac{2 b e n}{d \sqrt [3]{x}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*(d + e*x^(2/3))^n])/x^2,x]

[Out]

(-2*b*e*n)/(d*x^(1/3)) - (2*b*e^(3/2)*n*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]])/d^(3/2) - (a + b*Log[c*(d + e*x^(2/
3))^n])/x

Rule 2455

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.)*(x_))^(m_.), x_Symbol] :> Simp[((f*x)^(m
+ 1)*(a + b*Log[c*(d + e*x^n)^p]))/(f*(m + 1)), x] - Dist[(b*e*n*p)/(f*(m + 1)), Int[(x^(n - 1)*(f*x)^(m + 1))
/(d + e*x^n), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && NeQ[m, -1]

Rule 341

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

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

Mathematica [C]  time = 0.0177444, size = 59, normalized size = 0.87 \[ -\frac{a}{x}-\frac{b \log \left (c \left (d+e x^{2/3}\right )^n\right )}{x}-\frac{2 b e n \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};-\frac{e x^{2/3}}{d}\right )}{d \sqrt [3]{x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*(d + e*x^(2/3))^n])/x^2,x]

[Out]

-(a/x) - (2*b*e*n*Hypergeometric2F1[-1/2, 1, 1/2, -((e*x^(2/3))/d)])/(d*x^(1/3)) - (b*Log[c*(d + e*x^(2/3))^n]
)/x

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Maple [F]  time = 0.342, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \left ( a+b\ln \left ( c \left ( d+e{x}^{{\frac{2}{3}}} \right ) ^{n} \right ) \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*(d+e*x^(2/3))^n))/x^2,x)

[Out]

int((a+b*ln(c*(d+e*x^(2/3))^n))/x^2,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((a+b*log(c*(d+e*x^(2/3))^n))/x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.85809, size = 481, normalized size = 7.07 \begin{align*} \left [\frac{b e n x \sqrt{-\frac{e}{d}} \log \left (\frac{e^{3} x^{2} + 2 \, d^{2} e x \sqrt{-\frac{e}{d}} - d^{3} - 2 \,{\left (d e^{2} x \sqrt{-\frac{e}{d}} - d^{2} e\right )} x^{\frac{2}{3}} - 2 \,{\left (d e^{2} x + d^{3} \sqrt{-\frac{e}{d}}\right )} x^{\frac{1}{3}}}{e^{3} x^{2} + d^{3}}\right ) - b d n \log \left (e x^{\frac{2}{3}} + d\right ) - 2 \, b e n x^{\frac{2}{3}} - b d \log \left (c\right ) - a d}{d x}, -\frac{2 \, b e n x \sqrt{\frac{e}{d}} \arctan \left (x^{\frac{1}{3}} \sqrt{\frac{e}{d}}\right ) + b d n \log \left (e x^{\frac{2}{3}} + d\right ) + 2 \, b e n x^{\frac{2}{3}} + b d \log \left (c\right ) + a d}{d x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d+e*x^(2/3))^n))/x^2,x, algorithm="fricas")

[Out]

[(b*e*n*x*sqrt(-e/d)*log((e^3*x^2 + 2*d^2*e*x*sqrt(-e/d) - d^3 - 2*(d*e^2*x*sqrt(-e/d) - d^2*e)*x^(2/3) - 2*(d
*e^2*x + d^3*sqrt(-e/d))*x^(1/3))/(e^3*x^2 + d^3)) - b*d*n*log(e*x^(2/3) + d) - 2*b*e*n*x^(2/3) - b*d*log(c) -
a*d)/(d*x), -(2*b*e*n*x*sqrt(e/d)*arctan(x^(1/3)*sqrt(e/d)) + b*d*n*log(e*x^(2/3) + d) + 2*b*e*n*x^(2/3) + b*
d*log(c) + a*d)/(d*x)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*(d+e*x**(2/3))**n))/x**2,x)

[Out]

Timed out

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Giac [A]  time = 1.24545, size = 82, normalized size = 1.21 \begin{align*} -{\left (2 \,{\left (\frac{\arctan \left (\frac{x^{\frac{1}{3}} e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\frac{1}{2}}}{d^{\frac{3}{2}}} + \frac{1}{d x^{\frac{1}{3}}}\right )} e + \frac{\log \left (x^{\frac{2}{3}} e + d\right )}{x}\right )} b n - \frac{b \log \left (c\right )}{x} - \frac{a}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d+e*x^(2/3))^n))/x^2,x, algorithm="giac")

[Out]

-(2*(arctan(x^(1/3)*e^(1/2)/sqrt(d))*e^(1/2)/d^(3/2) + 1/(d*x^(1/3)))*e + log(x^(2/3)*e + d)/x)*b*n - b*log(c)
/x - a/x