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

Optimal. Leaf size=107 \[ \frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )+\frac{b e^3 n \sqrt{x}}{2 d^3}-\frac{b e^2 n x}{4 d^2}-\frac{b e^4 n \log \left (d+\frac{e}{\sqrt{x}}\right )}{2 d^4}-\frac{b e^4 n \log (x)}{4 d^4}+\frac{b e n x^{3/2}}{6 d} \]

[Out]

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

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Rubi [A]  time = 0.0727471, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {2454, 2395, 44} \[ \frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )+\frac{b e^3 n \sqrt{x}}{2 d^3}-\frac{b e^2 n x}{4 d^2}-\frac{b e^4 n \log \left (d+\frac{e}{\sqrt{x}}\right )}{2 d^4}-\frac{b e^4 n \log (x)}{4 d^4}+\frac{b e n x^{3/2}}{6 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 44

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] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right ) \, dx &=-\left (2 \operatorname{Subst}\left (\int \frac{a+b \log \left (c (d+e x)^n\right )}{x^5} \, dx,x,\frac{1}{\sqrt{x}}\right )\right )\\ &=\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )-\frac{1}{2} (b e n) \operatorname{Subst}\left (\int \frac{1}{x^4 (d+e x)} \, dx,x,\frac{1}{\sqrt{x}}\right )\\ &=\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )-\frac{1}{2} (b e n) \operatorname{Subst}\left (\int \left (\frac{1}{d x^4}-\frac{e}{d^2 x^3}+\frac{e^2}{d^3 x^2}-\frac{e^3}{d^4 x}+\frac{e^4}{d^4 (d+e x)}\right ) \, dx,x,\frac{1}{\sqrt{x}}\right )\\ &=\frac{b e^3 n \sqrt{x}}{2 d^3}-\frac{b e^2 n x}{4 d^2}+\frac{b e n x^{3/2}}{6 d}-\frac{b e^4 n \log \left (d+\frac{e}{\sqrt{x}}\right )}{2 d^4}+\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )-\frac{b e^4 n \log (x)}{4 d^4}\\ \end{align*}

Mathematica [A]  time = 0.0280822, size = 102, normalized size = 0.95 \[ \frac{a x^2}{2}+\frac{1}{2} b x^2 \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )-\frac{1}{2} b e n \left (-\frac{e^2 \sqrt{x}}{d^3}+\frac{e^3 \log \left (d+\frac{e}{\sqrt{x}}\right )}{d^4}+\frac{e^3 \log (x)}{2 d^4}+\frac{e x}{2 d^2}-\frac{x^{3/2}}{3 d}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.335, size = 0, normalized size = 0. \begin{align*} \int x \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(x*(a+b*ln(c*(d+e/x^(1/2))^n)),x)

[Out]

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

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Maxima [A]  time = 1.05956, size = 100, normalized size = 0.93 \begin{align*} -\frac{1}{12} \, b e n{\left (\frac{6 \, e^{3} \log \left (d \sqrt{x} + e\right )}{d^{4}} - \frac{2 \, d^{2} x^{\frac{3}{2}} - 3 \, d e x + 6 \, e^{2} \sqrt{x}}{d^{3}}\right )} + \frac{1}{2} \, b x^{2} \log \left (c{\left (d + \frac{e}{\sqrt{x}}\right )}^{n}\right ) + \frac{1}{2} \, a x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.79551, size = 300, normalized size = 2.8 \begin{align*} \frac{6 \, b d^{4} x^{2} \log \left (c\right ) - 3 \, b d^{2} e^{2} n x + 6 \, a d^{4} x^{2} - 6 \, b d^{4} n \log \left (\sqrt{x}\right ) + 6 \,{\left (b d^{4} - b e^{4}\right )} n \log \left (d \sqrt{x} + e\right ) + 6 \,{\left (b d^{4} n x^{2} - b d^{4} n\right )} \log \left (\frac{d x + e \sqrt{x}}{x}\right ) + 2 \,{\left (b d^{3} e n x + 3 \, b d e^{3} n\right )} \sqrt{x}}{12 \, d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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Giac [A]  time = 1.35307, size = 109, normalized size = 1.02 \begin{align*} \frac{1}{2} \, b x^{2} \log \left (c\right ) + \frac{1}{12} \,{\left (6 \, x^{2} \log \left (d + \frac{e}{\sqrt{x}}\right ) +{\left (\frac{2 \, d^{2} x^{\frac{3}{2}} - 3 \, d x e + 6 \, \sqrt{x} e^{2}}{d^{3}} - \frac{6 \, e^{3} \log \left ({\left | d \sqrt{x} + e \right |}\right )}{d^{4}}\right )} e\right )} b n + \frac{1}{2} \, a x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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