3.470 \(\int \frac{a+b \log (c (d+e x^{2/3})^n)}{x^4} \, dx\)

Optimal. Leaf size=123 \[ -\frac{a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )}{3 x^3}+\frac{2 b e^2 n}{15 d^2 x^{5/3}}+\frac{2 b e^4 n}{3 d^4 \sqrt [3]{x}}-\frac{2 b e^3 n}{9 d^3 x}+\frac{2 b e^{9/2} n \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{3 d^{9/2}}-\frac{2 b e n}{21 d x^{7/3}} \]

[Out]

(-2*b*e*n)/(21*d*x^(7/3)) + (2*b*e^2*n)/(15*d^2*x^(5/3)) - (2*b*e^3*n)/(9*d^3*x) + (2*b*e^4*n)/(3*d^4*x^(1/3))
+ (2*b*e^(9/2)*n*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]])/(3*d^(9/2)) - (a + b*Log[c*(d + e*x^(2/3))^n])/(3*x^3)

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Rubi [A]  time = 0.0768091, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2455, 341, 325, 205} \[ -\frac{a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )}{3 x^3}+\frac{2 b e^2 n}{15 d^2 x^{5/3}}+\frac{2 b e^4 n}{3 d^4 \sqrt [3]{x}}-\frac{2 b e^3 n}{9 d^3 x}+\frac{2 b e^{9/2} n \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{3 d^{9/2}}-\frac{2 b e n}{21 d x^{7/3}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*(d + e*x^(2/3))^n])/x^4,x]

[Out]

(-2*b*e*n)/(21*d*x^(7/3)) + (2*b*e^2*n)/(15*d^2*x^(5/3)) - (2*b*e^3*n)/(9*d^3*x) + (2*b*e^4*n)/(3*d^4*x^(1/3))
+ (2*b*e^(9/2)*n*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]])/(3*d^(9/2)) - (a + b*Log[c*(d + e*x^(2/3))^n])/(3*x^3)

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^4} \, dx &=-\frac{a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )}{3 x^3}+\frac{1}{9} (2 b e n) \int \frac{1}{\left (d+e x^{2/3}\right ) x^{10/3}} \, dx\\ &=-\frac{a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )}{3 x^3}+\frac{1}{3} (2 b e n) \operatorname{Subst}\left (\int \frac{1}{x^8 \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{2 b e n}{21 d x^{7/3}}-\frac{a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )}{3 x^3}-\frac{\left (2 b e^2 n\right ) \operatorname{Subst}\left (\int \frac{1}{x^6 \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{3 d}\\ &=-\frac{2 b e n}{21 d x^{7/3}}+\frac{2 b e^2 n}{15 d^2 x^{5/3}}-\frac{a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )}{3 x^3}+\frac{\left (2 b e^3 n\right ) \operatorname{Subst}\left (\int \frac{1}{x^4 \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{3 d^2}\\ &=-\frac{2 b e n}{21 d x^{7/3}}+\frac{2 b e^2 n}{15 d^2 x^{5/3}}-\frac{2 b e^3 n}{9 d^3 x}-\frac{a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )}{3 x^3}-\frac{\left (2 b e^4 n\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{3 d^3}\\ &=-\frac{2 b e n}{21 d x^{7/3}}+\frac{2 b e^2 n}{15 d^2 x^{5/3}}-\frac{2 b e^3 n}{9 d^3 x}+\frac{2 b e^4 n}{3 d^4 \sqrt [3]{x}}-\frac{a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )}{3 x^3}+\frac{\left (2 b e^5 n\right ) \operatorname{Subst}\left (\int \frac{1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{3 d^4}\\ &=-\frac{2 b e n}{21 d x^{7/3}}+\frac{2 b e^2 n}{15 d^2 x^{5/3}}-\frac{2 b e^3 n}{9 d^3 x}+\frac{2 b e^4 n}{3 d^4 \sqrt [3]{x}}+\frac{2 b e^{9/2} n \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{3 d^{9/2}}-\frac{a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )}{3 x^3}\\ \end{align*}

Mathematica [C]  time = 0.0136037, size = 65, normalized size = 0.53 \[ -\frac{a}{3 x^3}-\frac{b \log \left (c \left (d+e x^{2/3}\right )^n\right )}{3 x^3}-\frac{2 b e n \, _2F_1\left (-\frac{7}{2},1;-\frac{5}{2};-\frac{e x^{2/3}}{d}\right )}{21 d x^{7/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*(d + e*x^(2/3))^n])/x^4,x]

[Out]

-a/(3*x^3) - (2*b*e*n*Hypergeometric2F1[-7/2, 1, -5/2, -((e*x^(2/3))/d)])/(21*d*x^(7/3)) - (b*Log[c*(d + e*x^(
2/3))^n])/(3*x^3)

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Maple [F]  time = 0.332, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4}} \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^4,x)

[Out]

int((a+b*ln(c*(d+e*x^(2/3))^n))/x^4,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^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.93159, size = 755, normalized size = 6.14 \begin{align*} \left [\frac{105 \, b e^{4} n x^{3} \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 ) - 70 \, b d e^{3} n x^{2} + 42 \, b d^{2} e^{2} n x^{\frac{4}{3}} - 105 \, b d^{4} n \log \left (e x^{\frac{2}{3}} + d\right ) - 105 \, b d^{4} \log \left (c\right ) - 105 \, a d^{4} + 30 \,{\left (7 \, b e^{4} n x^{2} - b d^{3} e n\right )} x^{\frac{2}{3}}}{315 \, d^{4} x^{3}}, \frac{210 \, b e^{4} n x^{3} \sqrt{\frac{e}{d}} \arctan \left (x^{\frac{1}{3}} \sqrt{\frac{e}{d}}\right ) - 70 \, b d e^{3} n x^{2} + 42 \, b d^{2} e^{2} n x^{\frac{4}{3}} - 105 \, b d^{4} n \log \left (e x^{\frac{2}{3}} + d\right ) - 105 \, b d^{4} \log \left (c\right ) - 105 \, a d^{4} + 30 \,{\left (7 \, b e^{4} n x^{2} - b d^{3} e n\right )} x^{\frac{2}{3}}}{315 \, d^{4} x^{3}}\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^4,x, algorithm="fricas")

[Out]

[1/315*(105*b*e^4*n*x^3*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)) - 70*b*d*e^3*n*x^2 + 42*b*d^2*e^2*n*x^(4/3) -
105*b*d^4*n*log(e*x^(2/3) + d) - 105*b*d^4*log(c) - 105*a*d^4 + 30*(7*b*e^4*n*x^2 - b*d^3*e*n)*x^(2/3))/(d^4*
x^3), 1/315*(210*b*e^4*n*x^3*sqrt(e/d)*arctan(x^(1/3)*sqrt(e/d)) - 70*b*d*e^3*n*x^2 + 42*b*d^2*e^2*n*x^(4/3) -
105*b*d^4*n*log(e*x^(2/3) + d) - 105*b*d^4*log(c) - 105*a*d^4 + 30*(7*b*e^4*n*x^2 - b*d^3*e*n)*x^(2/3))/(d^4*
x^3)]

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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**4,x)

[Out]

Timed out

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Giac [A]  time = 1.36582, size = 127, normalized size = 1.03 \begin{align*} \frac{1}{315} \,{\left (2 \,{\left (\frac{105 \, \arctan \left (\frac{x^{\frac{1}{3}} e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\frac{7}{2}}}{d^{\frac{9}{2}}} + \frac{21 \, d^{2} x^{\frac{2}{3}} e - 35 \, d x^{\frac{4}{3}} e^{2} - 15 \, d^{3} + 105 \, x^{2} e^{3}}{d^{4} x^{\frac{7}{3}}}\right )} e - \frac{105 \, \log \left (x^{\frac{2}{3}} e + d\right )}{x^{3}}\right )} b n - \frac{b \log \left (c\right )}{3 \, x^{3}} - \frac{a}{3 \, x^{3}} \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^4,x, algorithm="giac")

[Out]

1/315*(2*(105*arctan(x^(1/3)*e^(1/2)/sqrt(d))*e^(7/2)/d^(9/2) + (21*d^2*x^(2/3)*e - 35*d*x^(4/3)*e^2 - 15*d^3
+ 105*x^2*e^3)/(d^4*x^(7/3)))*e - 105*log(x^(2/3)*e + d)/x^3)*b*n - 1/3*b*log(c)/x^3 - 1/3*a/x^3