3.5.28 \(\int \frac {a+b \log (c (d+\frac {e}{\sqrt {x}})^n)}{x^4} \, dx\) [428]

Optimal. Leaf size=136 \[ \frac {b n}{18 x^3}-\frac {b d n}{15 e x^{5/2}}+\frac {b d^2 n}{12 e^2 x^2}-\frac {b d^3 n}{9 e^3 x^{3/2}}+\frac {b d^4 n}{6 e^4 x}-\frac {b d^5 n}{3 e^5 \sqrt {x}}+\frac {b d^6 n \log \left (d+\frac {e}{\sqrt {x}}\right )}{3 e^6}-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{3 x^3} \]

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 136, 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, 45} \begin {gather*} -\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{3 x^3}+\frac {b d^6 n \log \left (d+\frac {e}{\sqrt {x}}\right )}{3 e^6}-\frac {b d^5 n}{3 e^5 \sqrt {x}}+\frac {b d^4 n}{6 e^4 x}-\frac {b d^3 n}{9 e^3 x^{3/2}}+\frac {b d^2 n}{12 e^2 x^2}-\frac {b d n}{15 e x^{5/2}}+\frac {b n}{18 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 45

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[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+\frac {e}{\sqrt {x}}\right )^n\right )}{x^4} \, dx &=-\left (2 \text {Subst}\left (\int x^5 \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx,x,\frac {1}{\sqrt {x}}\right )\right )\\ &=-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{3 x^3}+\frac {1}{3} (b e n) \text {Subst}\left (\int \frac {x^6}{d+e x} \, dx,x,\frac {1}{\sqrt {x}}\right )\\ &=-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{3 x^3}+\frac {1}{3} (b e n) \text {Subst}\left (\int \left (-\frac {d^5}{e^6}+\frac {d^4 x}{e^5}-\frac {d^3 x^2}{e^4}+\frac {d^2 x^3}{e^3}-\frac {d x^4}{e^2}+\frac {x^5}{e}+\frac {d^6}{e^6 (d+e x)}\right ) \, dx,x,\frac {1}{\sqrt {x}}\right )\\ &=\frac {b n}{18 x^3}-\frac {b d n}{15 e x^{5/2}}+\frac {b d^2 n}{12 e^2 x^2}-\frac {b d^3 n}{9 e^3 x^{3/2}}+\frac {b d^4 n}{6 e^4 x}-\frac {b d^5 n}{3 e^5 \sqrt {x}}+\frac {b d^6 n \log \left (d+\frac {e}{\sqrt {x}}\right )}{3 e^6}-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{3 x^3}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 133, normalized size = 0.98 \begin {gather*} -\frac {a}{3 x^3}+\frac {1}{3} b e n \left (\frac {1}{6 e x^3}-\frac {d}{5 e^2 x^{5/2}}+\frac {d^2}{4 e^3 x^2}-\frac {d^3}{3 e^4 x^{3/2}}+\frac {d^4}{2 e^5 x}-\frac {d^5}{e^6 \sqrt {x}}+\frac {d^6 \log \left (d+\frac {e}{\sqrt {x}}\right )}{e^7}\right )-\frac {b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {a +b \ln \left (c \left (d +\frac {e}{\sqrt {x}}\right )^{n}\right )}{x^{4}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*(d+e/x^(1/2))^n))/x^4,x)

[Out]

int((a+b*ln(c*(d+e/x^(1/2))^n))/x^4,x)

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Maxima [A]
time = 0.29, size = 114, normalized size = 0.84 \begin {gather*} \frac {1}{180} \, {\left (60 \, d^{6} e^{\left (-7\right )} \log \left (d \sqrt {x} + e\right ) - 30 \, d^{6} e^{\left (-7\right )} \log \left (x\right ) - \frac {{\left (60 \, d^{5} x^{\frac {5}{2}} - 30 \, d^{4} x^{2} e + 20 \, d^{3} x^{\frac {3}{2}} e^{2} - 15 \, d^{2} x e^{3} + 12 \, d \sqrt {x} e^{4} - 10 \, e^{5}\right )} e^{\left (-6\right )}}{x^{3}}\right )} b n e - \frac {b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\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^(1/2))^n))/x^4,x, algorithm="maxima")

[Out]

1/180*(60*d^6*e^(-7)*log(d*sqrt(x) + e) - 30*d^6*e^(-7)*log(x) - (60*d^5*x^(5/2) - 30*d^4*x^2*e + 20*d^3*x^(3/
2)*e^2 - 15*d^2*x*e^3 + 12*d*sqrt(x)*e^4 - 10*e^5)*e^(-6)/x^3)*b*n*e - 1/3*b*log(c*(d + e/sqrt(x))^n)/x^3 - 1/
3*a/x^3

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Fricas [A]
time = 0.43, size = 116, normalized size = 0.85 \begin {gather*} \frac {{\left (30 \, b d^{4} n x^{2} e^{2} + 15 \, b d^{2} n x e^{4} - 60 \, b e^{6} \log \left (c\right ) + 10 \, {\left (b n - 6 \, a\right )} e^{6} + 60 \, {\left (b d^{6} n x^{3} - b n e^{6}\right )} \log \left (\frac {d x + \sqrt {x} e}{x}\right ) - 4 \, {\left (15 \, b d^{5} n x^{2} e + 5 \, b d^{3} n x e^{3} + 3 \, b d n e^{5}\right )} \sqrt {x}\right )} e^{\left (-6\right )}}{180 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d+e/x^(1/2))^n))/x^4,x, algorithm="fricas")

[Out]

1/180*(30*b*d^4*n*x^2*e^2 + 15*b*d^2*n*x*e^4 - 60*b*e^6*log(c) + 10*(b*n - 6*a)*e^6 + 60*(b*d^6*n*x^3 - b*n*e^
6)*log((d*x + sqrt(x)*e)/x) - 4*(15*b*d^5*n*x^2*e + 5*b*d^3*n*x*e^3 + 3*b*d*n*e^5)*sqrt(x))*e^(-6)/x^3

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*(d+e/x**(1/2))**n))/x**4,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 535 vs. \(2 (106) = 212\).
time = 4.29, size = 535, normalized size = 3.93 \begin {gather*} \frac {1}{180} \, {\left (\frac {360 \, {\left (d \sqrt {x} + e\right )} b d^{5} n \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )}{\sqrt {x}} - \frac {900 \, {\left (d \sqrt {x} + e\right )}^{2} b d^{4} n \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )}{x} - \frac {360 \, {\left (d \sqrt {x} + e\right )} b d^{5} n}{\sqrt {x}} + \frac {360 \, {\left (d \sqrt {x} + e\right )} b d^{5} \log \left (c\right )}{\sqrt {x}} + \frac {1200 \, {\left (d \sqrt {x} + e\right )}^{3} b d^{3} n \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )}{x^{\frac {3}{2}}} + \frac {450 \, {\left (d \sqrt {x} + e\right )}^{2} b d^{4} n}{x} - \frac {900 \, {\left (d \sqrt {x} + e\right )}^{2} b d^{4} \log \left (c\right )}{x} - \frac {900 \, {\left (d \sqrt {x} + e\right )}^{4} b d^{2} n \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )}{x^{2}} - \frac {400 \, {\left (d \sqrt {x} + e\right )}^{3} b d^{3} n}{x^{\frac {3}{2}}} + \frac {360 \, {\left (d \sqrt {x} + e\right )} a d^{5}}{\sqrt {x}} + \frac {1200 \, {\left (d \sqrt {x} + e\right )}^{3} b d^{3} \log \left (c\right )}{x^{\frac {3}{2}}} + \frac {360 \, {\left (d \sqrt {x} + e\right )}^{5} b d n \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )}{x^{\frac {5}{2}}} + \frac {225 \, {\left (d \sqrt {x} + e\right )}^{4} b d^{2} n}{x^{2}} - \frac {900 \, {\left (d \sqrt {x} + e\right )}^{2} a d^{4}}{x} - \frac {900 \, {\left (d \sqrt {x} + e\right )}^{4} b d^{2} \log \left (c\right )}{x^{2}} - \frac {60 \, {\left (d \sqrt {x} + e\right )}^{6} b n \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )}{x^{3}} - \frac {72 \, {\left (d \sqrt {x} + e\right )}^{5} b d n}{x^{\frac {5}{2}}} + \frac {1200 \, {\left (d \sqrt {x} + e\right )}^{3} a d^{3}}{x^{\frac {3}{2}}} + \frac {360 \, {\left (d \sqrt {x} + e\right )}^{5} b d \log \left (c\right )}{x^{\frac {5}{2}}} + \frac {10 \, {\left (d \sqrt {x} + e\right )}^{6} b n}{x^{3}} - \frac {900 \, {\left (d \sqrt {x} + e\right )}^{4} a d^{2}}{x^{2}} - \frac {60 \, {\left (d \sqrt {x} + e\right )}^{6} b \log \left (c\right )}{x^{3}} + \frac {360 \, {\left (d \sqrt {x} + e\right )}^{5} a d}{x^{\frac {5}{2}}} - \frac {60 \, {\left (d \sqrt {x} + e\right )}^{6} a}{x^{3}}\right )} e^{\left (-6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d+e/x^(1/2))^n))/x^4,x, algorithm="giac")

[Out]

1/180*(360*(d*sqrt(x) + e)*b*d^5*n*log((d*sqrt(x) + e)/sqrt(x))/sqrt(x) - 900*(d*sqrt(x) + e)^2*b*d^4*n*log((d
*sqrt(x) + e)/sqrt(x))/x - 360*(d*sqrt(x) + e)*b*d^5*n/sqrt(x) + 360*(d*sqrt(x) + e)*b*d^5*log(c)/sqrt(x) + 12
00*(d*sqrt(x) + e)^3*b*d^3*n*log((d*sqrt(x) + e)/sqrt(x))/x^(3/2) + 450*(d*sqrt(x) + e)^2*b*d^4*n/x - 900*(d*s
qrt(x) + e)^2*b*d^4*log(c)/x - 900*(d*sqrt(x) + e)^4*b*d^2*n*log((d*sqrt(x) + e)/sqrt(x))/x^2 - 400*(d*sqrt(x)
+ e)^3*b*d^3*n/x^(3/2) + 360*(d*sqrt(x) + e)*a*d^5/sqrt(x) + 1200*(d*sqrt(x) + e)^3*b*d^3*log(c)/x^(3/2) + 36
0*(d*sqrt(x) + e)^5*b*d*n*log((d*sqrt(x) + e)/sqrt(x))/x^(5/2) + 225*(d*sqrt(x) + e)^4*b*d^2*n/x^2 - 900*(d*sq
rt(x) + e)^2*a*d^4/x - 900*(d*sqrt(x) + e)^4*b*d^2*log(c)/x^2 - 60*(d*sqrt(x) + e)^6*b*n*log((d*sqrt(x) + e)/s
qrt(x))/x^3 - 72*(d*sqrt(x) + e)^5*b*d*n/x^(5/2) + 1200*(d*sqrt(x) + e)^3*a*d^3/x^(3/2) + 360*(d*sqrt(x) + e)^
5*b*d*log(c)/x^(5/2) + 10*(d*sqrt(x) + e)^6*b*n/x^3 - 900*(d*sqrt(x) + e)^4*a*d^2/x^2 - 60*(d*sqrt(x) + e)^6*b
*log(c)/x^3 + 360*(d*sqrt(x) + e)^5*a*d/x^(5/2) - 60*(d*sqrt(x) + e)^6*a/x^3)*e^(-6)

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Mupad [B]
time = 0.44, size = 113, normalized size = 0.83 \begin {gather*} \frac {b\,n}{18\,x^3}-\frac {a}{3\,x^3}-\frac {b\,\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )}{3\,x^3}-\frac {b\,d\,n}{15\,e\,x^{5/2}}+\frac {b\,d^6\,n\,\ln \left (d+\frac {e}{\sqrt {x}}\right )}{3\,e^6}+\frac {b\,d^2\,n}{12\,e^2\,x^2}+\frac {b\,d^4\,n}{6\,e^4\,x}-\frac {b\,d^3\,n}{9\,e^3\,x^{3/2}}-\frac {b\,d^5\,n}{3\,e^5\,\sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*log(c*(d + e/x^(1/2))^n))/x^4,x)

[Out]

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

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