3.5.27 \(\int \frac {a+b \log (c (d+\frac {e}{\sqrt {x}})^n)}{x^3} \, dx\) [427]

Optimal. Leaf size=104 \[ \frac {b n}{8 x^2}-\frac {b d n}{6 e x^{3/2}}+\frac {b d^2 n}{4 e^2 x}-\frac {b d^3 n}{2 e^3 \sqrt {x}}+\frac {b d^4 n \log \left (d+\frac {e}{\sqrt {x}}\right )}{2 e^4}-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{2 x^2} \]

[Out]

1/8*b*n/x^2-1/6*b*d*n/e/x^(3/2)+1/4*b*d^2*n/e^2/x+1/2*b*d^4*n*ln(d+e/x^(1/2))/e^4+1/2*(-a-b*ln(c*(d+e/x^(1/2))
^n))/x^2-1/2*b*d^3*n/e^3/x^(1/2)

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Rubi [A]
time = 0.05, antiderivative size = 104, 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 )}{2 x^2}+\frac {b d^4 n \log \left (d+\frac {e}{\sqrt {x}}\right )}{2 e^4}-\frac {b d^3 n}{2 e^3 \sqrt {x}}+\frac {b d^2 n}{4 e^2 x}-\frac {b d n}{6 e x^{3/2}}+\frac {b n}{8 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*n)/(8*x^2) - (b*d*n)/(6*e*x^(3/2)) + (b*d^2*n)/(4*e^2*x) - (b*d^3*n)/(2*e^3*Sqrt[x]) + (b*d^4*n*Log[d + e/S
qrt[x]])/(2*e^4) - (a + b*Log[c*(d + e/Sqrt[x])^n])/(2*x^2)

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^3} \, dx &=-\left (2 \text {Subst}\left (\int x^3 \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 )}{2 x^2}+\frac {1}{2} (b e n) \text {Subst}\left (\int \frac {x^4}{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 )}{2 x^2}+\frac {1}{2} (b e n) \text {Subst}\left (\int \left (-\frac {d^3}{e^4}+\frac {d^2 x}{e^3}-\frac {d x^2}{e^2}+\frac {x^3}{e}+\frac {d^4}{e^4 (d+e x)}\right ) \, dx,x,\frac {1}{\sqrt {x}}\right )\\ &=\frac {b n}{8 x^2}-\frac {b d n}{6 e x^{3/2}}+\frac {b d^2 n}{4 e^2 x}-\frac {b d^3 n}{2 e^3 \sqrt {x}}+\frac {b d^4 n \log \left (d+\frac {e}{\sqrt {x}}\right )}{2 e^4}-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{2 x^2}\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 109, normalized size = 1.05 \begin {gather*} -\frac {a}{2 x^2}+\frac {b n}{8 x^2}-\frac {b d n}{6 e x^{3/2}}+\frac {b d^2 n}{4 e^2 x}-\frac {b d^3 n}{2 e^3 \sqrt {x}}+\frac {b d^4 n \log \left (d+\frac {e}{\sqrt {x}}\right )}{2 e^4}-\frac {b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 94, normalized size = 0.90 \begin {gather*} \frac {1}{24} \, {\left (12 \, d^{4} e^{\left (-5\right )} \log \left (d \sqrt {x} + e\right ) - 6 \, d^{4} e^{\left (-5\right )} \log \left (x\right ) - \frac {{\left (12 \, d^{3} x^{\frac {3}{2}} - 6 \, d^{2} x e + 4 \, d \sqrt {x} e^{2} - 3 \, e^{3}\right )} e^{\left (-4\right )}}{x^{2}}\right )} b n e - \frac {b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right )}{2 \, x^{2}} - \frac {a}{2 \, x^{2}} \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^3,x, algorithm="maxima")

[Out]

1/24*(12*d^4*e^(-5)*log(d*sqrt(x) + e) - 6*d^4*e^(-5)*log(x) - (12*d^3*x^(3/2) - 6*d^2*x*e + 4*d*sqrt(x)*e^2 -
3*e^3)*e^(-4)/x^2)*b*n*e - 1/2*b*log(c*(d + e/sqrt(x))^n)/x^2 - 1/2*a/x^2

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Fricas [A]
time = 0.38, size = 91, normalized size = 0.88 \begin {gather*} \frac {{\left (6 \, b d^{2} n x e^{2} - 12 \, b e^{4} \log \left (c\right ) + 3 \, {\left (b n - 4 \, a\right )} e^{4} + 12 \, {\left (b d^{4} n x^{2} - b n e^{4}\right )} \log \left (\frac {d x + \sqrt {x} e}{x}\right ) - 4 \, {\left (3 \, b d^{3} n x e + b d n e^{3}\right )} \sqrt {x}\right )} e^{\left (-4\right )}}{24 \, x^{2}} \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^3,x, algorithm="fricas")

[Out]

1/24*(6*b*d^2*n*x*e^2 - 12*b*e^4*log(c) + 3*(b*n - 4*a)*e^4 + 12*(b*d^4*n*x^2 - b*n*e^4)*log((d*x + sqrt(x)*e)
/x) - 4*(3*b*d^3*n*x*e + b*d*n*e^3)*sqrt(x))*e^(-4)/x^2

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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**3,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 349 vs. \(2 (82) = 164\).
time = 3.32, size = 349, normalized size = 3.36 \begin {gather*} \frac {1}{24} \, {\left (\frac {48 \, {\left (d \sqrt {x} + e\right )} b d^{3} n \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )}{\sqrt {x}} - \frac {72 \, {\left (d \sqrt {x} + e\right )}^{2} b d^{2} n \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )}{x} - \frac {48 \, {\left (d \sqrt {x} + e\right )} b d^{3} n}{\sqrt {x}} + \frac {48 \, {\left (d \sqrt {x} + e\right )} b d^{3} \log \left (c\right )}{\sqrt {x}} + \frac {48 \, {\left (d \sqrt {x} + e\right )}^{3} b d n \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )}{x^{\frac {3}{2}}} + \frac {36 \, {\left (d \sqrt {x} + e\right )}^{2} b d^{2} n}{x} - \frac {72 \, {\left (d \sqrt {x} + e\right )}^{2} b d^{2} \log \left (c\right )}{x} - \frac {12 \, {\left (d \sqrt {x} + e\right )}^{4} b n \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )}{x^{2}} - \frac {16 \, {\left (d \sqrt {x} + e\right )}^{3} b d n}{x^{\frac {3}{2}}} + \frac {48 \, {\left (d \sqrt {x} + e\right )} a d^{3}}{\sqrt {x}} + \frac {48 \, {\left (d \sqrt {x} + e\right )}^{3} b d \log \left (c\right )}{x^{\frac {3}{2}}} + \frac {3 \, {\left (d \sqrt {x} + e\right )}^{4} b n}{x^{2}} - \frac {72 \, {\left (d \sqrt {x} + e\right )}^{2} a d^{2}}{x} - \frac {12 \, {\left (d \sqrt {x} + e\right )}^{4} b \log \left (c\right )}{x^{2}} + \frac {48 \, {\left (d \sqrt {x} + e\right )}^{3} a d}{x^{\frac {3}{2}}} - \frac {12 \, {\left (d \sqrt {x} + e\right )}^{4} a}{x^{2}}\right )} e^{\left (-4\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^3,x, algorithm="giac")

[Out]

1/24*(48*(d*sqrt(x) + e)*b*d^3*n*log((d*sqrt(x) + e)/sqrt(x))/sqrt(x) - 72*(d*sqrt(x) + e)^2*b*d^2*n*log((d*sq
rt(x) + e)/sqrt(x))/x - 48*(d*sqrt(x) + e)*b*d^3*n/sqrt(x) + 48*(d*sqrt(x) + e)*b*d^3*log(c)/sqrt(x) + 48*(d*s
qrt(x) + e)^3*b*d*n*log((d*sqrt(x) + e)/sqrt(x))/x^(3/2) + 36*(d*sqrt(x) + e)^2*b*d^2*n/x - 72*(d*sqrt(x) + e)
^2*b*d^2*log(c)/x - 12*(d*sqrt(x) + e)^4*b*n*log((d*sqrt(x) + e)/sqrt(x))/x^2 - 16*(d*sqrt(x) + e)^3*b*d*n/x^(
3/2) + 48*(d*sqrt(x) + e)*a*d^3/sqrt(x) + 48*(d*sqrt(x) + e)^3*b*d*log(c)/x^(3/2) + 3*(d*sqrt(x) + e)^4*b*n/x^
2 - 72*(d*sqrt(x) + e)^2*a*d^2/x - 12*(d*sqrt(x) + e)^4*b*log(c)/x^2 + 48*(d*sqrt(x) + e)^3*a*d/x^(3/2) - 12*(
d*sqrt(x) + e)^4*a/x^2)*e^(-4)

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Mupad [B]
time = 0.42, size = 87, normalized size = 0.84 \begin {gather*} \frac {b\,n}{8\,x^2}-\frac {a}{2\,x^2}-\frac {b\,\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )}{2\,x^2}-\frac {b\,d\,n}{6\,e\,x^{3/2}}+\frac {b\,d^4\,n\,\ln \left (d+\frac {e}{\sqrt {x}}\right )}{2\,e^4}+\frac {b\,d^2\,n}{4\,e^2\,x}-\frac {b\,d^3\,n}{2\,e^3\,\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^3,x)

[Out]

(b*n)/(8*x^2) - a/(2*x^2) - (b*log(c*(d + e/x^(1/2))^n))/(2*x^2) - (b*d*n)/(6*e*x^(3/2)) + (b*d^4*n*log(d + e/
x^(1/2)))/(2*e^4) + (b*d^2*n)/(4*e^2*x) - (b*d^3*n)/(2*e^3*x^(1/2))

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