3.5.49 \(\int \frac {a+b \log (c (d+e \sqrt [3]{x})^n)}{x^4} \, dx\) [449]

Optimal. Leaf size=192 \[ -\frac {b e n}{24 d x^{8/3}}+\frac {b e^2 n}{21 d^2 x^{7/3}}-\frac {b e^3 n}{18 d^3 x^2}+\frac {b e^4 n}{15 d^4 x^{5/3}}-\frac {b e^5 n}{12 d^5 x^{4/3}}+\frac {b e^6 n}{9 d^6 x}-\frac {b e^7 n}{6 d^7 x^{2/3}}+\frac {b e^8 n}{3 d^8 \sqrt [3]{x}}-\frac {b e^9 n \log \left (d+e \sqrt [3]{x}\right )}{3 d^9}-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{3 x^3}+\frac {b e^9 n \log (x)}{9 d^9} \]

[Out]

-1/24*b*e*n/d/x^(8/3)+1/21*b*e^2*n/d^2/x^(7/3)-1/18*b*e^3*n/d^3/x^2+1/15*b*e^4*n/d^4/x^(5/3)-1/12*b*e^5*n/d^5/
x^(4/3)+1/9*b*e^6*n/d^6/x-1/6*b*e^7*n/d^7/x^(2/3)+1/3*b*e^8*n/d^8/x^(1/3)-1/3*b*e^9*n*ln(d+e*x^(1/3))/d^9+1/3*
(-a-b*ln(c*(d+e*x^(1/3))^n))/x^3+1/9*b*e^9*n*ln(x)/d^9

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Rubi [A]
time = 0.09, antiderivative size = 192, 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 )}{3 x^3}-\frac {b e^9 n \log \left (d+e \sqrt [3]{x}\right )}{3 d^9}+\frac {b e^9 n \log (x)}{9 d^9}+\frac {b e^8 n}{3 d^8 \sqrt [3]{x}}-\frac {b e^7 n}{6 d^7 x^{2/3}}+\frac {b e^6 n}{9 d^6 x}-\frac {b e^5 n}{12 d^5 x^{4/3}}+\frac {b e^4 n}{15 d^4 x^{5/3}}-\frac {b e^3 n}{18 d^3 x^2}+\frac {b e^2 n}{21 d^2 x^{7/3}}-\frac {b e n}{24 d x^{8/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*(d + e*x^(1/3))^n])/x^4,x]

[Out]

-1/24*(b*e*n)/(d*x^(8/3)) + (b*e^2*n)/(21*d^2*x^(7/3)) - (b*e^3*n)/(18*d^3*x^2) + (b*e^4*n)/(15*d^4*x^(5/3)) -
(b*e^5*n)/(12*d^5*x^(4/3)) + (b*e^6*n)/(9*d^6*x) - (b*e^7*n)/(6*d^7*x^(2/3)) + (b*e^8*n)/(3*d^8*x^(1/3)) - (b
*e^9*n*Log[d + e*x^(1/3)])/(3*d^9) - (a + b*Log[c*(d + e*x^(1/3))^n])/(3*x^3) + (b*e^9*n*Log[x])/(9*d^9)

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^4} \, dx &=3 \text {Subst}\left (\int \frac {a+b \log \left (c (d+e x)^n\right )}{x^{10}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{3 x^3}+\frac {1}{3} (b e n) \text {Subst}\left (\int \frac {1}{x^9 (d+e x)} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{3 x^3}+\frac {1}{3} (b e n) \text {Subst}\left (\int \left (\frac {1}{d x^9}-\frac {e}{d^2 x^8}+\frac {e^2}{d^3 x^7}-\frac {e^3}{d^4 x^6}+\frac {e^4}{d^5 x^5}-\frac {e^5}{d^6 x^4}+\frac {e^6}{d^7 x^3}-\frac {e^7}{d^8 x^2}+\frac {e^8}{d^9 x}-\frac {e^9}{d^9 (d+e x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {b e n}{24 d x^{8/3}}+\frac {b e^2 n}{21 d^2 x^{7/3}}-\frac {b e^3 n}{18 d^3 x^2}+\frac {b e^4 n}{15 d^4 x^{5/3}}-\frac {b e^5 n}{12 d^5 x^{4/3}}+\frac {b e^6 n}{9 d^6 x}-\frac {b e^7 n}{6 d^7 x^{2/3}}+\frac {b e^8 n}{3 d^8 \sqrt [3]{x}}-\frac {b e^9 n \log \left (d+e \sqrt [3]{x}\right )}{3 d^9}-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{3 x^3}+\frac {b e^9 n \log (x)}{9 d^9}\\ \end {align*}

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Mathematica [A]
time = 0.14, size = 177, normalized size = 0.92 \begin {gather*} -\frac {a}{3 x^3}-\frac {b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{3 x^3}+\frac {1}{3} b e n \left (-\frac {1}{8 d x^{8/3}}+\frac {e}{7 d^2 x^{7/3}}-\frac {e^2}{6 d^3 x^2}+\frac {e^3}{5 d^4 x^{5/3}}-\frac {e^4}{4 d^5 x^{4/3}}+\frac {e^5}{3 d^6 x}-\frac {e^6}{2 d^7 x^{2/3}}+\frac {e^7}{d^8 \sqrt [3]{x}}-\frac {e^8 \log \left (d+e \sqrt [3]{x}\right )}{d^9}+\frac {e^8 \log (x)}{3 d^9}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*(d + e*x^(1/3))^n])/x^4,x]

[Out]

-1/3*a/x^3 - (b*Log[c*(d + e*x^(1/3))^n])/(3*x^3) + (b*e*n*(-1/8*1/(d*x^(8/3)) + e/(7*d^2*x^(7/3)) - e^2/(6*d^
3*x^2) + e^3/(5*d^4*x^(5/3)) - e^4/(4*d^5*x^(4/3)) + e^5/(3*d^6*x) - e^6/(2*d^7*x^(2/3)) + e^7/(d^8*x^(1/3)) -
(e^8*Log[d + e*x^(1/3)])/d^9 + (e^8*Log[x])/(3*d^9)))/3

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {a +b \ln \left (c \left (d +e \,x^{\frac {1}{3}}\right )^{n}\right )}{x^{4}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*(d+e*x^(1/3))^n))/x^4,x)

[Out]

int((a+b*ln(c*(d+e*x^(1/3))^n))/x^4,x)

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Maxima [A]
time = 0.28, size = 135, normalized size = 0.70 \begin {gather*} -\frac {1}{2520} \, b n {\left (\frac {840 \, e^{8} \log \left (x^{\frac {1}{3}} e + d\right )}{d^{9}} - \frac {280 \, e^{8} \log \left (x\right )}{d^{9}} - \frac {120 \, d^{6} x^{\frac {1}{3}} e - 105 \, d^{7} - 140 \, d^{5} x^{\frac {2}{3}} e^{2} + 168 \, d^{4} x e^{3} - 210 \, d^{3} x^{\frac {4}{3}} e^{4} + 280 \, d^{2} x^{\frac {5}{3}} e^{5} - 420 \, d x^{2} e^{6} + 840 \, x^{\frac {7}{3}} e^{7}}{d^{8} x^{\frac {8}{3}}}\right )} e - \frac {b \log \left ({\left (x^{\frac {1}{3}} e + d\right )}^{n} c\right )}{3 \, x^{3}} - \frac {a}{3 \, x^{3}} \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^4,x, algorithm="maxima")

[Out]

-1/2520*b*n*(840*e^8*log(x^(1/3)*e + d)/d^9 - 280*e^8*log(x)/d^9 - (120*d^6*x^(1/3)*e - 105*d^7 - 140*d^5*x^(2
/3)*e^2 + 168*d^4*x*e^3 - 210*d^3*x^(4/3)*e^4 + 280*d^2*x^(5/3)*e^5 - 420*d*x^2*e^6 + 840*x^(7/3)*e^7)/(d^8*x^
(8/3)))*e - 1/3*b*log((x^(1/3)*e + d)^n*c)/x^3 - 1/3*a/x^3

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Fricas [A]
time = 0.39, size = 156, normalized size = 0.81 \begin {gather*} -\frac {840 \, b d^{9} \log \left (c\right ) + 840 \, a d^{9} + 140 \, b d^{6} n x e^{3} - 280 \, b d^{3} n x^{2} e^{6} - 840 \, b n x^{3} e^{9} \log \left (x^{\frac {1}{3}}\right ) + 840 \, {\left (b d^{9} n + b n x^{3} e^{9}\right )} \log \left (x^{\frac {1}{3}} e + d\right ) - 30 \, {\left (4 \, b d^{7} n e^{2} - 7 \, b d^{4} n x e^{5} + 28 \, b d n x^{2} e^{8}\right )} x^{\frac {2}{3}} + 21 \, {\left (5 \, b d^{8} n e - 8 \, b d^{5} n x e^{4} + 20 \, b d^{2} n x^{2} e^{7}\right )} x^{\frac {1}{3}}}{2520 \, d^{9} x^{3}} \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^4,x, algorithm="fricas")

[Out]

-1/2520*(840*b*d^9*log(c) + 840*a*d^9 + 140*b*d^6*n*x*e^3 - 280*b*d^3*n*x^2*e^6 - 840*b*n*x^3*e^9*log(x^(1/3))
+ 840*(b*d^9*n + b*n*x^3*e^9)*log(x^(1/3)*e + d) - 30*(4*b*d^7*n*e^2 - 7*b*d^4*n*x*e^5 + 28*b*d*n*x^2*e^8)*x^
(2/3) + 21*(5*b*d^8*n*e - 8*b*d^5*n*x*e^4 + 20*b*d^2*n*x^2*e^7)*x^(1/3))/(d^9*x^3)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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**4,x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3062 deep

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 808 vs. \(2 (148) = 296\).
time = 6.25, size = 808, normalized size = 4.21 \begin {gather*} -\frac {{\left (840 \, {\left (x^{\frac {1}{3}} e + d\right )}^{9} b n e^{10} \log \left (x^{\frac {1}{3}} e + d\right ) - 7560 \, {\left (x^{\frac {1}{3}} e + d\right )}^{8} b d n e^{10} \log \left (x^{\frac {1}{3}} e + d\right ) + 30240 \, {\left (x^{\frac {1}{3}} e + d\right )}^{7} b d^{2} n e^{10} \log \left (x^{\frac {1}{3}} e + d\right ) - 70560 \, {\left (x^{\frac {1}{3}} e + d\right )}^{6} b d^{3} n e^{10} \log \left (x^{\frac {1}{3}} e + d\right ) + 105840 \, {\left (x^{\frac {1}{3}} e + d\right )}^{5} b d^{4} n e^{10} \log \left (x^{\frac {1}{3}} e + d\right ) - 105840 \, {\left (x^{\frac {1}{3}} e + d\right )}^{4} b d^{5} n e^{10} \log \left (x^{\frac {1}{3}} e + d\right ) + 70560 \, {\left (x^{\frac {1}{3}} e + d\right )}^{3} b d^{6} n e^{10} \log \left (x^{\frac {1}{3}} e + d\right ) - 30240 \, {\left (x^{\frac {1}{3}} e + d\right )}^{2} b d^{7} n e^{10} \log \left (x^{\frac {1}{3}} e + d\right ) + 7560 \, {\left (x^{\frac {1}{3}} e + d\right )} b d^{8} n e^{10} \log \left (x^{\frac {1}{3}} e + d\right ) - 840 \, {\left (x^{\frac {1}{3}} e + d\right )}^{9} b n e^{10} \log \left (x^{\frac {1}{3}} e\right ) + 7560 \, {\left (x^{\frac {1}{3}} e + d\right )}^{8} b d n e^{10} \log \left (x^{\frac {1}{3}} e\right ) - 30240 \, {\left (x^{\frac {1}{3}} e + d\right )}^{7} b d^{2} n e^{10} \log \left (x^{\frac {1}{3}} e\right ) + 70560 \, {\left (x^{\frac {1}{3}} e + d\right )}^{6} b d^{3} n e^{10} \log \left (x^{\frac {1}{3}} e\right ) - 105840 \, {\left (x^{\frac {1}{3}} e + d\right )}^{5} b d^{4} n e^{10} \log \left (x^{\frac {1}{3}} e\right ) + 105840 \, {\left (x^{\frac {1}{3}} e + d\right )}^{4} b d^{5} n e^{10} \log \left (x^{\frac {1}{3}} e\right ) - 70560 \, {\left (x^{\frac {1}{3}} e + d\right )}^{3} b d^{6} n e^{10} \log \left (x^{\frac {1}{3}} e\right ) + 30240 \, {\left (x^{\frac {1}{3}} e + d\right )}^{2} b d^{7} n e^{10} \log \left (x^{\frac {1}{3}} e\right ) - 7560 \, {\left (x^{\frac {1}{3}} e + d\right )} b d^{8} n e^{10} \log \left (x^{\frac {1}{3}} e\right ) + 840 \, b d^{9} n e^{10} \log \left (x^{\frac {1}{3}} e\right ) - 840 \, {\left (x^{\frac {1}{3}} e + d\right )}^{8} b d n e^{10} + 7140 \, {\left (x^{\frac {1}{3}} e + d\right )}^{7} b d^{2} n e^{10} - 26740 \, {\left (x^{\frac {1}{3}} e + d\right )}^{6} b d^{3} n e^{10} + 57750 \, {\left (x^{\frac {1}{3}} e + d\right )}^{5} b d^{4} n e^{10} - 78918 \, {\left (x^{\frac {1}{3}} e + d\right )}^{4} b d^{5} n e^{10} + 70252 \, {\left (x^{\frac {1}{3}} e + d\right )}^{3} b d^{6} n e^{10} - 40188 \, {\left (x^{\frac {1}{3}} e + d\right )}^{2} b d^{7} n e^{10} + 13827 \, {\left (x^{\frac {1}{3}} e + d\right )} b d^{8} n e^{10} - 2283 \, b d^{9} n e^{10} + 840 \, b d^{9} e^{10} \log \left (c\right ) + 840 \, a d^{9} e^{10}\right )} e^{\left (-1\right )}}{2520 \, {\left ({\left (x^{\frac {1}{3}} e + d\right )}^{9} d^{9} - 9 \, {\left (x^{\frac {1}{3}} e + d\right )}^{8} d^{10} + 36 \, {\left (x^{\frac {1}{3}} e + d\right )}^{7} d^{11} - 84 \, {\left (x^{\frac {1}{3}} e + d\right )}^{6} d^{12} + 126 \, {\left (x^{\frac {1}{3}} e + d\right )}^{5} d^{13} - 126 \, {\left (x^{\frac {1}{3}} e + d\right )}^{4} d^{14} + 84 \, {\left (x^{\frac {1}{3}} e + d\right )}^{3} d^{15} - 36 \, {\left (x^{\frac {1}{3}} e + d\right )}^{2} d^{16} + 9 \, {\left (x^{\frac {1}{3}} e + d\right )} d^{17} - d^{18}\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^4,x, algorithm="giac")

[Out]

-1/2520*(840*(x^(1/3)*e + d)^9*b*n*e^10*log(x^(1/3)*e + d) - 7560*(x^(1/3)*e + d)^8*b*d*n*e^10*log(x^(1/3)*e +
d) + 30240*(x^(1/3)*e + d)^7*b*d^2*n*e^10*log(x^(1/3)*e + d) - 70560*(x^(1/3)*e + d)^6*b*d^3*n*e^10*log(x^(1/
3)*e + d) + 105840*(x^(1/3)*e + d)^5*b*d^4*n*e^10*log(x^(1/3)*e + d) - 105840*(x^(1/3)*e + d)^4*b*d^5*n*e^10*l
og(x^(1/3)*e + d) + 70560*(x^(1/3)*e + d)^3*b*d^6*n*e^10*log(x^(1/3)*e + d) - 30240*(x^(1/3)*e + d)^2*b*d^7*n*
e^10*log(x^(1/3)*e + d) + 7560*(x^(1/3)*e + d)*b*d^8*n*e^10*log(x^(1/3)*e + d) - 840*(x^(1/3)*e + d)^9*b*n*e^1
0*log(x^(1/3)*e) + 7560*(x^(1/3)*e + d)^8*b*d*n*e^10*log(x^(1/3)*e) - 30240*(x^(1/3)*e + d)^7*b*d^2*n*e^10*log
(x^(1/3)*e) + 70560*(x^(1/3)*e + d)^6*b*d^3*n*e^10*log(x^(1/3)*e) - 105840*(x^(1/3)*e + d)^5*b*d^4*n*e^10*log(
x^(1/3)*e) + 105840*(x^(1/3)*e + d)^4*b*d^5*n*e^10*log(x^(1/3)*e) - 70560*(x^(1/3)*e + d)^3*b*d^6*n*e^10*log(x
^(1/3)*e) + 30240*(x^(1/3)*e + d)^2*b*d^7*n*e^10*log(x^(1/3)*e) - 7560*(x^(1/3)*e + d)*b*d^8*n*e^10*log(x^(1/3
)*e) + 840*b*d^9*n*e^10*log(x^(1/3)*e) - 840*(x^(1/3)*e + d)^8*b*d*n*e^10 + 7140*(x^(1/3)*e + d)^7*b*d^2*n*e^1
0 - 26740*(x^(1/3)*e + d)^6*b*d^3*n*e^10 + 57750*(x^(1/3)*e + d)^5*b*d^4*n*e^10 - 78918*(x^(1/3)*e + d)^4*b*d^
5*n*e^10 + 70252*(x^(1/3)*e + d)^3*b*d^6*n*e^10 - 40188*(x^(1/3)*e + d)^2*b*d^7*n*e^10 + 13827*(x^(1/3)*e + d)
*b*d^8*n*e^10 - 2283*b*d^9*n*e^10 + 840*b*d^9*e^10*log(c) + 840*a*d^9*e^10)*e^(-1)/((x^(1/3)*e + d)^9*d^9 - 9*
(x^(1/3)*e + d)^8*d^10 + 36*(x^(1/3)*e + d)^7*d^11 - 84*(x^(1/3)*e + d)^6*d^12 + 126*(x^(1/3)*e + d)^5*d^13 -
126*(x^(1/3)*e + d)^4*d^14 + 84*(x^(1/3)*e + d)^3*d^15 - 36*(x^(1/3)*e + d)^2*d^16 + 9*(x^(1/3)*e + d)*d^17 -
d^18)

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Mupad [B]
time = 0.61, size = 154, normalized size = 0.80 \begin {gather*} -\frac {\frac {a\,d^9}{3}+\frac {b\,d^9\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{3}+\frac {b\,d^6\,e^3\,n\,x}{18}+\frac {b\,d^8\,e\,n\,x^{1/3}}{24}-\frac {b\,d\,e^8\,n\,x^{8/3}}{3}-\frac {b\,d^3\,e^6\,n\,x^2}{9}-\frac {b\,d^7\,e^2\,n\,x^{2/3}}{21}-\frac {b\,d^5\,e^4\,n\,x^{4/3}}{15}+\frac {b\,d^4\,e^5\,n\,x^{5/3}}{12}+\frac {b\,d^2\,e^7\,n\,x^{7/3}}{6}}{d^9\,x^3}-\frac {2\,b\,e^9\,n\,\mathrm {atanh}\left (\frac {2\,e\,x^{1/3}}{d}+1\right )}{3\,d^9} \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^4,x)

[Out]

- ((a*d^9)/3 + (b*d^9*log(c*(d + e*x^(1/3))^n))/3 + (b*d^6*e^3*n*x)/18 + (b*d^8*e*n*x^(1/3))/24 - (b*d*e^8*n*x
^(8/3))/3 - (b*d^3*e^6*n*x^2)/9 - (b*d^7*e^2*n*x^(2/3))/21 - (b*d^5*e^4*n*x^(4/3))/15 + (b*d^4*e^5*n*x^(5/3))/
12 + (b*d^2*e^7*n*x^(7/3))/6)/(d^9*x^3) - (2*b*e^9*n*atanh((2*e*x^(1/3))/d + 1))/(3*d^9)

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