3.5.47 \(\int \frac {a+b \log (c (d+e \sqrt [3]{x})^n)}{x^2} \, dx\) [447]

Optimal. Leaf size=87 \[ -\frac {b e n}{2 d x^{2/3}}+\frac {b e^2 n}{d^2 \sqrt [3]{x}}-\frac {b e^3 n \log \left (d+e \sqrt [3]{x}\right )}{d^3}-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{x}+\frac {b e^3 n \log (x)}{3 d^3} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2504, 2442, 46} \begin {gather*} -\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{x}-\frac {b e^3 n \log \left (d+e \sqrt [3]{x}\right )}{d^3}+\frac {b e^3 n \log (x)}{3 d^3}+\frac {b e^2 n}{d^2 \sqrt [3]{x}}-\frac {b e n}{2 d x^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 46

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

Rule 2442

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[(f + g*
x)^(q + 1)*((a + b*Log[c*(d + e*x)^n])/(g*(q + 1))), x] - Dist[b*e*(n/(g*(q + 1))), Int[(f + g*x)^(q + 1)/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]

Rule 2504

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rubi steps

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

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Mathematica [A]
time = 0.03, size = 84, normalized size = 0.97 \begin {gather*} -\frac {a}{x}-\frac {b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{x}+b e n \left (-\frac {1}{2 d x^{2/3}}+\frac {e}{d^2 \sqrt [3]{x}}-\frac {e^2 \log \left (d+e \sqrt [3]{x}\right )}{d^3}+\frac {e^2 \log (x)}{3 d^3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {a +b \ln \left (c \left (d +e \,x^{\frac {1}{3}}\right )^{n}\right )}{x^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*(d+e*x^(1/3))^n))/x^2,x)

[Out]

int((a+b*ln(c*(d+e*x^(1/3))^n))/x^2,x)

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Maxima [A]
time = 0.27, size = 77, normalized size = 0.89 \begin {gather*} -\frac {1}{6} \, b n {\left (\frac {6 \, e^{2} \log \left (x^{\frac {1}{3}} e + d\right )}{d^{3}} - \frac {2 \, e^{2} \log \left (x\right )}{d^{3}} - \frac {3 \, {\left (2 \, x^{\frac {1}{3}} e - d\right )}}{d^{2} x^{\frac {2}{3}}}\right )} e - \frac {b \log \left ({\left (x^{\frac {1}{3}} e + d\right )}^{n} c\right )}{x} - \frac {a}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d+e*x^(1/3))^n))/x^2,x, algorithm="maxima")

[Out]

-1/6*b*n*(6*e^2*log(x^(1/3)*e + d)/d^3 - 2*e^2*log(x)/d^3 - 3*(2*x^(1/3)*e - d)/(d^2*x^(2/3)))*e - b*log((x^(1
/3)*e + d)^n*c)/x - a/x

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Fricas [A]
time = 0.50, size = 79, normalized size = 0.91 \begin {gather*} -\frac {b d^{2} n x^{\frac {1}{3}} e + 2 \, b d^{3} \log \left (c\right ) - 2 \, b n x e^{3} \log \left (x^{\frac {1}{3}}\right ) - 2 \, b d n x^{\frac {2}{3}} e^{2} + 2 \, a d^{3} + 2 \, {\left (b d^{3} n + b n x e^{3}\right )} \log \left (x^{\frac {1}{3}} e + d\right )}{2 \, d^{3} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d+e*x^(1/3))^n))/x^2,x, algorithm="fricas")

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 450 vs. \(2 (83) = 166\).
time = 106.56, size = 450, normalized size = 5.17 \begin {gather*} \begin {cases} - \frac {6 a d^{4} x^{\frac {2}{3}}}{6 d^{4} x^{\frac {5}{3}} + 6 d^{3} e x^{2}} - \frac {6 a d^{3} e x}{6 d^{4} x^{\frac {5}{3}} + 6 d^{3} e x^{2}} - \frac {6 b d^{4} x^{\frac {2}{3}} \log {\left (c \left (d + e \sqrt [3]{x}\right )^{n} \right )}}{6 d^{4} x^{\frac {5}{3}} + 6 d^{3} e x^{2}} - \frac {3 b d^{3} e n x}{6 d^{4} x^{\frac {5}{3}} + 6 d^{3} e x^{2}} - \frac {6 b d^{3} e x \log {\left (c \left (d + e \sqrt [3]{x}\right )^{n} \right )}}{6 d^{4} x^{\frac {5}{3}} + 6 d^{3} e x^{2}} + \frac {3 b d^{2} e^{2} n x^{\frac {4}{3}}}{6 d^{4} x^{\frac {5}{3}} + 6 d^{3} e x^{2}} + \frac {2 b d e^{3} n x^{\frac {5}{3}} \log {\left (x \right )}}{6 d^{4} x^{\frac {5}{3}} + 6 d^{3} e x^{2}} + \frac {6 b d e^{3} n x^{\frac {5}{3}}}{6 d^{4} x^{\frac {5}{3}} + 6 d^{3} e x^{2}} - \frac {6 b d e^{3} x^{\frac {5}{3}} \log {\left (c \left (d + e \sqrt [3]{x}\right )^{n} \right )}}{6 d^{4} x^{\frac {5}{3}} + 6 d^{3} e x^{2}} + \frac {2 b e^{4} n x^{2} \log {\left (x \right )}}{6 d^{4} x^{\frac {5}{3}} + 6 d^{3} e x^{2}} - \frac {6 b e^{4} x^{2} \log {\left (c \left (d + e \sqrt [3]{x}\right )^{n} \right )}}{6 d^{4} x^{\frac {5}{3}} + 6 d^{3} e x^{2}} & \text {for}\: d \neq 0 \\- \frac {a}{x} - \frac {b n}{3 x} - \frac {b \log {\left (c \left (e \sqrt [3]{x}\right )^{n} \right )}}{x} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*(d+e*x**(1/3))**n))/x**2,x)

[Out]

Piecewise((-6*a*d**4*x**(2/3)/(6*d**4*x**(5/3) + 6*d**3*e*x**2) - 6*a*d**3*e*x/(6*d**4*x**(5/3) + 6*d**3*e*x**
2) - 6*b*d**4*x**(2/3)*log(c*(d + e*x**(1/3))**n)/(6*d**4*x**(5/3) + 6*d**3*e*x**2) - 3*b*d**3*e*n*x/(6*d**4*x
**(5/3) + 6*d**3*e*x**2) - 6*b*d**3*e*x*log(c*(d + e*x**(1/3))**n)/(6*d**4*x**(5/3) + 6*d**3*e*x**2) + 3*b*d**
2*e**2*n*x**(4/3)/(6*d**4*x**(5/3) + 6*d**3*e*x**2) + 2*b*d*e**3*n*x**(5/3)*log(x)/(6*d**4*x**(5/3) + 6*d**3*e
*x**2) + 6*b*d*e**3*n*x**(5/3)/(6*d**4*x**(5/3) + 6*d**3*e*x**2) - 6*b*d*e**3*x**(5/3)*log(c*(d + e*x**(1/3))*
*n)/(6*d**4*x**(5/3) + 6*d**3*e*x**2) + 2*b*e**4*n*x**2*log(x)/(6*d**4*x**(5/3) + 6*d**3*e*x**2) - 6*b*e**4*x*
*2*log(c*(d + e*x**(1/3))**n)/(6*d**4*x**(5/3) + 6*d**3*e*x**2), Ne(d, 0)), (-a/x - b*n/(3*x) - b*log(c*(e*x**
(1/3))**n)/x, True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 280 vs. \(2 (75) = 150\).
time = 4.20, size = 280, normalized size = 3.22 \begin {gather*} -\frac {{\left (2 \, {\left (x^{\frac {1}{3}} e + d\right )}^{3} b n e^{4} \log \left (x^{\frac {1}{3}} e + d\right ) - 6 \, {\left (x^{\frac {1}{3}} e + d\right )}^{2} b d n e^{4} \log \left (x^{\frac {1}{3}} e + d\right ) + 6 \, {\left (x^{\frac {1}{3}} e + d\right )} b d^{2} n e^{4} \log \left (x^{\frac {1}{3}} e + d\right ) - 2 \, {\left (x^{\frac {1}{3}} e + d\right )}^{3} b n e^{4} \log \left (x^{\frac {1}{3}} e\right ) + 6 \, {\left (x^{\frac {1}{3}} e + d\right )}^{2} b d n e^{4} \log \left (x^{\frac {1}{3}} e\right ) - 6 \, {\left (x^{\frac {1}{3}} e + d\right )} b d^{2} n e^{4} \log \left (x^{\frac {1}{3}} e\right ) + 2 \, b d^{3} n e^{4} \log \left (x^{\frac {1}{3}} e\right ) - 2 \, {\left (x^{\frac {1}{3}} e + d\right )}^{2} b d n e^{4} + 5 \, {\left (x^{\frac {1}{3}} e + d\right )} b d^{2} n e^{4} - 3 \, b d^{3} n e^{4} + 2 \, b d^{3} e^{4} \log \left (c\right ) + 2 \, a d^{3} e^{4}\right )} e^{\left (-1\right )}}{2 \, {\left ({\left (x^{\frac {1}{3}} e + d\right )}^{3} d^{3} - 3 \, {\left (x^{\frac {1}{3}} e + d\right )}^{2} d^{4} + 3 \, {\left (x^{\frac {1}{3}} e + d\right )} d^{5} - d^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d+e*x^(1/3))^n))/x^2,x, algorithm="giac")

[Out]

-1/2*(2*(x^(1/3)*e + d)^3*b*n*e^4*log(x^(1/3)*e + d) - 6*(x^(1/3)*e + d)^2*b*d*n*e^4*log(x^(1/3)*e + d) + 6*(x
^(1/3)*e + d)*b*d^2*n*e^4*log(x^(1/3)*e + d) - 2*(x^(1/3)*e + d)^3*b*n*e^4*log(x^(1/3)*e) + 6*(x^(1/3)*e + d)^
2*b*d*n*e^4*log(x^(1/3)*e) - 6*(x^(1/3)*e + d)*b*d^2*n*e^4*log(x^(1/3)*e) + 2*b*d^3*n*e^4*log(x^(1/3)*e) - 2*(
x^(1/3)*e + d)^2*b*d*n*e^4 + 5*(x^(1/3)*e + d)*b*d^2*n*e^4 - 3*b*d^3*n*e^4 + 2*b*d^3*e^4*log(c) + 2*a*d^3*e^4)
*e^(-1)/((x^(1/3)*e + d)^3*d^3 - 3*(x^(1/3)*e + d)^2*d^4 + 3*(x^(1/3)*e + d)*d^5 - d^6)

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Mupad [B]
time = 0.59, size = 74, normalized size = 0.85 \begin {gather*} -\frac {\frac {b\,e\,n}{2\,d}-\frac {b\,e^2\,n\,x^{1/3}}{d^2}}{x^{2/3}}-\frac {a}{x}-\frac {b\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{x}-\frac {2\,b\,e^3\,n\,\mathrm {atanh}\left (\frac {2\,e\,x^{1/3}}{d}+1\right )}{d^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*log(c*(d + e*x^(1/3))^n))/x^2,x)

[Out]

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

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