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

Optimal. Leaf size=171 \[ \frac{1}{4} x^4 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )+\frac{b e^5 n x^{3/2}}{12 d^5}-\frac{b e^4 n x^2}{16 d^4}+\frac{b e^3 n x^{5/2}}{20 d^3}-\frac{b e^2 n x^3}{24 d^2}+\frac{b e^7 n \sqrt{x}}{4 d^7}-\frac{b e^6 n x}{8 d^6}-\frac{b e^8 n \log \left (d+\frac{e}{\sqrt{x}}\right )}{4 d^8}-\frac{b e^8 n \log (x)}{8 d^8}+\frac{b e n x^{7/2}}{28 d} \]

[Out]

(b*e^7*n*Sqrt[x])/(4*d^7) - (b*e^6*n*x)/(8*d^6) + (b*e^5*n*x^(3/2))/(12*d^5) - (b*e^4*n*x^2)/(16*d^4) + (b*e^3
*n*x^(5/2))/(20*d^3) - (b*e^2*n*x^3)/(24*d^2) + (b*e*n*x^(7/2))/(28*d) - (b*e^8*n*Log[d + e/Sqrt[x]])/(4*d^8)
+ (x^4*(a + b*Log[c*(d + e/Sqrt[x])^n]))/4 - (b*e^8*n*Log[x])/(8*d^8)

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Rubi [A]  time = 0.131568, antiderivative size = 171, 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, 44} \[ \frac{1}{4} x^4 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )+\frac{b e^5 n x^{3/2}}{12 d^5}-\frac{b e^4 n x^2}{16 d^4}+\frac{b e^3 n x^{5/2}}{20 d^3}-\frac{b e^2 n x^3}{24 d^2}+\frac{b e^7 n \sqrt{x}}{4 d^7}-\frac{b e^6 n x}{8 d^6}-\frac{b e^8 n \log \left (d+\frac{e}{\sqrt{x}}\right )}{4 d^8}-\frac{b e^8 n \log (x)}{8 d^8}+\frac{b e n x^{7/2}}{28 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*e^7*n*Sqrt[x])/(4*d^7) - (b*e^6*n*x)/(8*d^6) + (b*e^5*n*x^(3/2))/(12*d^5) - (b*e^4*n*x^2)/(16*d^4) + (b*e^3
*n*x^(5/2))/(20*d^3) - (b*e^2*n*x^3)/(24*d^2) + (b*e*n*x^(7/2))/(28*d) - (b*e^8*n*Log[d + e/Sqrt[x]])/(4*d^8)
+ (x^4*(a + b*Log[c*(d + e/Sqrt[x])^n]))/4 - (b*e^8*n*Log[x])/(8*d^8)

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^3 \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^9} \, dx,x,\frac{1}{\sqrt{x}}\right )\right )\\ &=\frac{1}{4} x^4 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )-\frac{1}{4} (b e n) \operatorname{Subst}\left (\int \frac{1}{x^8 (d+e x)} \, dx,x,\frac{1}{\sqrt{x}}\right )\\ &=\frac{1}{4} x^4 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )-\frac{1}{4} (b e n) \operatorname{Subst}\left (\int \left (\frac{1}{d x^8}-\frac{e}{d^2 x^7}+\frac{e^2}{d^3 x^6}-\frac{e^3}{d^4 x^5}+\frac{e^4}{d^5 x^4}-\frac{e^5}{d^6 x^3}+\frac{e^6}{d^7 x^2}-\frac{e^7}{d^8 x}+\frac{e^8}{d^8 (d+e x)}\right ) \, dx,x,\frac{1}{\sqrt{x}}\right )\\ &=\frac{b e^7 n \sqrt{x}}{4 d^7}-\frac{b e^6 n x}{8 d^6}+\frac{b e^5 n x^{3/2}}{12 d^5}-\frac{b e^4 n x^2}{16 d^4}+\frac{b e^3 n x^{5/2}}{20 d^3}-\frac{b e^2 n x^3}{24 d^2}+\frac{b e n x^{7/2}}{28 d}-\frac{b e^8 n \log \left (d+\frac{e}{\sqrt{x}}\right )}{4 d^8}+\frac{1}{4} x^4 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )-\frac{b e^8 n \log (x)}{8 d^8}\\ \end{align*}

Mathematica [A]  time = 0.132673, size = 158, normalized size = 0.92 \[ \frac{a x^4}{4}+\frac{1}{4} b x^4 \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )-\frac{1}{4} b e n \left (-\frac{e^4 x^{3/2}}{3 d^5}+\frac{e^3 x^2}{4 d^4}-\frac{e^2 x^{5/2}}{5 d^3}-\frac{e^6 \sqrt{x}}{d^7}+\frac{e^5 x}{2 d^6}+\frac{e^7 \log \left (d+\frac{e}{\sqrt{x}}\right )}{d^8}+\frac{e^7 \log (x)}{2 d^8}+\frac{e x^3}{6 d^2}-\frac{x^{7/2}}{7 d}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.42, size = 0, normalized size = 0. \begin{align*} \int{x}^{3} \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^3*(a+b*ln(c*(d+e/x^(1/2))^n)),x)

[Out]

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

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Maxima [A]  time = 1.05057, size = 159, normalized size = 0.93 \begin{align*} \frac{1}{4} \, b x^{4} \log \left (c{\left (d + \frac{e}{\sqrt{x}}\right )}^{n}\right ) + \frac{1}{4} \, a x^{4} - \frac{1}{1680} \, b e n{\left (\frac{420 \, e^{7} \log \left (d \sqrt{x} + e\right )}{d^{8}} - \frac{60 \, d^{6} x^{\frac{7}{2}} - 70 \, d^{5} e x^{3} + 84 \, d^{4} e^{2} x^{\frac{5}{2}} - 105 \, d^{3} e^{3} x^{2} + 140 \, d^{2} e^{4} x^{\frac{3}{2}} - 210 \, d e^{5} x + 420 \, e^{6} \sqrt{x}}{d^{7}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*log(c*(d+e/x^(1/2))^n)),x, algorithm="maxima")

[Out]

1/4*b*x^4*log(c*(d + e/sqrt(x))^n) + 1/4*a*x^4 - 1/1680*b*e*n*(420*e^7*log(d*sqrt(x) + e)/d^8 - (60*d^6*x^(7/2
) - 70*d^5*e*x^3 + 84*d^4*e^2*x^(5/2) - 105*d^3*e^3*x^2 + 140*d^2*e^4*x^(3/2) - 210*d*e^5*x + 420*e^6*sqrt(x))
/d^7)

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Fricas [A]  time = 1.78585, size = 440, normalized size = 2.57 \begin{align*} \frac{420 \, b d^{8} x^{4} \log \left (c\right ) - 70 \, b d^{6} e^{2} n x^{3} + 420 \, a d^{8} x^{4} - 105 \, b d^{4} e^{4} n x^{2} - 210 \, b d^{2} e^{6} n x - 420 \, b d^{8} n \log \left (\sqrt{x}\right ) + 420 \,{\left (b d^{8} - b e^{8}\right )} n \log \left (d \sqrt{x} + e\right ) + 420 \,{\left (b d^{8} n x^{4} - b d^{8} n\right )} \log \left (\frac{d x + e \sqrt{x}}{x}\right ) + 4 \,{\left (15 \, b d^{7} e n x^{3} + 21 \, b d^{5} e^{3} n x^{2} + 35 \, b d^{3} e^{5} n x + 105 \, b d e^{7} n\right )} \sqrt{x}}{1680 \, d^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*log(c*(d+e/x^(1/2))^n)),x, algorithm="fricas")

[Out]

1/1680*(420*b*d^8*x^4*log(c) - 70*b*d^6*e^2*n*x^3 + 420*a*d^8*x^4 - 105*b*d^4*e^4*n*x^2 - 210*b*d^2*e^6*n*x -
420*b*d^8*n*log(sqrt(x)) + 420*(b*d^8 - b*e^8)*n*log(d*sqrt(x) + e) + 420*(b*d^8*n*x^4 - b*d^8*n)*log((d*x + e
*sqrt(x))/x) + 4*(15*b*d^7*e*n*x^3 + 21*b*d^5*e^3*n*x^2 + 35*b*d^3*e^5*n*x + 105*b*d*e^7*n)*sqrt(x))/d^8

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

[Out]

Timed out

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Giac [A]  time = 1.31369, size = 163, normalized size = 0.95 \begin{align*} \frac{1}{4} \, b x^{4} \log \left (c\right ) + \frac{1}{4} \, a x^{4} + \frac{1}{1680} \,{\left (420 \, x^{4} \log \left (d + \frac{e}{\sqrt{x}}\right ) +{\left (\frac{60 \, d^{6} x^{\frac{7}{2}} - 70 \, d^{5} x^{3} e + 84 \, d^{4} x^{\frac{5}{2}} e^{2} - 105 \, d^{3} x^{2} e^{3} + 140 \, d^{2} x^{\frac{3}{2}} e^{4} - 210 \, d x e^{5} + 420 \, \sqrt{x} e^{6}}{d^{7}} - \frac{420 \, e^{7} \log \left ({\left | d \sqrt{x} + e \right |}\right )}{d^{8}}\right )} e\right )} b n \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*log(c*(d+e/x^(1/2))^n)),x, algorithm="giac")

[Out]

1/4*b*x^4*log(c) + 1/4*a*x^4 + 1/1680*(420*x^4*log(d + e/sqrt(x)) + ((60*d^6*x^(7/2) - 70*d^5*x^3*e + 84*d^4*x
^(5/2)*e^2 - 105*d^3*x^2*e^3 + 140*d^2*x^(3/2)*e^4 - 210*d*x*e^5 + 420*sqrt(x)*e^6)/d^7 - 420*e^7*log(abs(d*sq
rt(x) + e))/d^8)*e)*b*n