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

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

[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)

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Rubi [A]  time = 0.0970842, antiderivative size = 136, normalized size of antiderivative = 1., 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 = {2454, 2395, 43} \[ -\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )}{3 x^3}-\frac{b d^3 n}{9 e^3 x^{3/2}}+\frac{b d^2 n}{12 e^2 x^2}-\frac{b d^5 n}{3 e^5 \sqrt{x}}+\frac{b d^4 n}{6 e^4 x}+\frac{b d^6 n \log \left (d+\frac{e}{\sqrt{x}}\right )}{3 e^6}-\frac{b d n}{15 e x^{5/2}}+\frac{b n}{18 x^3} \]

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 2454

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])

Rule 2395

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 43

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])

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 \operatorname{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) \operatorname{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) \operatorname{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*}

Mathematica [A]  time = 0.0857543, size = 133, normalized size = 0.98 \[ -\frac{a}{3 x^3}-\frac{b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )}{3 x^3}+\frac{1}{3} b e n \left (-\frac{d^3}{3 e^4 x^{3/2}}+\frac{d^2}{4 e^3 x^2}-\frac{d^5}{e^6 \sqrt{x}}+\frac{d^4}{2 e^5 x}+\frac{d^6 \log \left (d+\frac{e}{\sqrt{x}}\right )}{e^7}-\frac{d}{5 e^2 x^{5/2}}+\frac{1}{6 e x^3}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-a/(3*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 = 0.328, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4}} \left ( a+b\ln \left ( c \left ( d+{e{\frac{1}{\sqrt{x}}}} \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^(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 = 1.0244, size = 158, normalized size = 1.16 \begin{align*} \frac{1}{180} \, b e n{\left (\frac{60 \, d^{6} \log \left (d \sqrt{x} + e\right )}{e^{7}} - \frac{30 \, d^{6} \log \left (x\right )}{e^{7}} - \frac{60 \, d^{5} x^{\frac{5}{2}} - 30 \, d^{4} e x^{2} + 20 \, d^{3} e^{2} x^{\frac{3}{2}} - 15 \, d^{2} e^{3} x + 12 \, d e^{4} \sqrt{x} - 10 \, e^{5}}{e^{6} x^{3}}\right )} - \frac{b \log \left (c{\left (d + \frac{e}{\sqrt{x}}\right )}^{n}\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^(1/2))^n))/x^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.88629, size = 292, normalized size = 2.15 \begin{align*} \frac{30 \, b d^{4} e^{2} n x^{2} + 15 \, b d^{2} e^{4} n x + 10 \, b e^{6} n - 60 \, b e^{6} \log \left (c\right ) - 60 \, a e^{6} + 60 \,{\left (b d^{6} n x^{3} - b e^{6} n\right )} \log \left (\frac{d x + e \sqrt{x}}{x}\right ) - 4 \,{\left (15 \, b d^{5} e n x^{2} + 5 \, b d^{3} e^{3} n x + 3 \, b d e^{5} n\right )} \sqrt{x}}{180 \, e^{6} x^{3}} \end{align*}

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

[Out]

Timed out

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Giac [A]  time = 1.21075, size = 189, normalized size = 1.39 \begin{align*} \frac{{\left (60 \, b d^{6} n x^{3} \log \left (d \sqrt{x} + e\right ) - 60 \, b d^{6} n x^{3} \log \left (\sqrt{x}\right ) - 60 \, b d^{5} n x^{\frac{5}{2}} e + 30 \, b d^{4} n x^{2} e^{2} - 20 \, b d^{3} n x^{\frac{3}{2}} e^{3} + 15 \, b d^{2} n x e^{4} - 12 \, b d n \sqrt{x} e^{5} - 60 \, b n e^{6} \log \left (d \sqrt{x} + e\right ) + 60 \, b n e^{6} \log \left (\sqrt{x}\right ) + 10 \, b n e^{6} - 60 \, b e^{6} \log \left (c\right ) - 60 \, a e^{6}\right )} e^{\left (-6\right )}}{180 \, x^{3}} \end{align*}

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