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

Optimal. Leaf size=166 \[ \frac{1}{4} x^4 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )+\frac{b d^5 n x^{3/2}}{12 e^5}-\frac{b d^4 n x^2}{16 e^4}+\frac{b d^3 n x^{5/2}}{20 e^3}-\frac{b d^2 n x^3}{24 e^2}+\frac{b d^7 n \sqrt{x}}{4 e^7}-\frac{b d^6 n x}{8 e^6}-\frac{b d^8 n \log \left (d+e \sqrt{x}\right )}{4 e^8}+\frac{b d n x^{7/2}}{28 e}-\frac{1}{32} b n x^4 \]

[Out]

(b*d^7*n*Sqrt[x])/(4*e^7) - (b*d^6*n*x)/(8*e^6) + (b*d^5*n*x^(3/2))/(12*e^5) - (b*d^4*n*x^2)/(16*e^4) + (b*d^3
*n*x^(5/2))/(20*e^3) - (b*d^2*n*x^3)/(24*e^2) + (b*d*n*x^(7/2))/(28*e) - (b*n*x^4)/32 - (b*d^8*n*Log[d + e*Sqr
t[x]])/(4*e^8) + (x^4*(a + b*Log[c*(d + e*Sqrt[x])^n]))/4

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Rubi [A]  time = 0.136079, antiderivative size = 166, 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} \[ \frac{1}{4} x^4 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )+\frac{b d^5 n x^{3/2}}{12 e^5}-\frac{b d^4 n x^2}{16 e^4}+\frac{b d^3 n x^{5/2}}{20 e^3}-\frac{b d^2 n x^3}{24 e^2}+\frac{b d^7 n \sqrt{x}}{4 e^7}-\frac{b d^6 n x}{8 e^6}-\frac{b d^8 n \log \left (d+e \sqrt{x}\right )}{4 e^8}+\frac{b d n x^{7/2}}{28 e}-\frac{1}{32} b n x^4 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*d^7*n*Sqrt[x])/(4*e^7) - (b*d^6*n*x)/(8*e^6) + (b*d^5*n*x^(3/2))/(12*e^5) - (b*d^4*n*x^2)/(16*e^4) + (b*d^3
*n*x^(5/2))/(20*e^3) - (b*d^2*n*x^3)/(24*e^2) + (b*d*n*x^(7/2))/(28*e) - (b*n*x^4)/32 - (b*d^8*n*Log[d + e*Sqr
t[x]])/(4*e^8) + (x^4*(a + b*Log[c*(d + e*Sqrt[x])^n]))/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 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 x^3 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right ) \, dx &=2 \operatorname{Subst}\left (\int x^7 \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{1}{4} x^4 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )-\frac{1}{4} (b e n) \operatorname{Subst}\left (\int \frac{x^8}{d+e x} \, dx,x,\sqrt{x}\right )\\ &=\frac{1}{4} x^4 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )-\frac{1}{4} (b e n) \operatorname{Subst}\left (\int \left (-\frac{d^7}{e^8}+\frac{d^6 x}{e^7}-\frac{d^5 x^2}{e^6}+\frac{d^4 x^3}{e^5}-\frac{d^3 x^4}{e^4}+\frac{d^2 x^5}{e^3}-\frac{d x^6}{e^2}+\frac{x^7}{e}+\frac{d^8}{e^8 (d+e x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{b d^7 n \sqrt{x}}{4 e^7}-\frac{b d^6 n x}{8 e^6}+\frac{b d^5 n x^{3/2}}{12 e^5}-\frac{b d^4 n x^2}{16 e^4}+\frac{b d^3 n x^{5/2}}{20 e^3}-\frac{b d^2 n x^3}{24 e^2}+\frac{b d n x^{7/2}}{28 e}-\frac{1}{32} b n x^4-\frac{b d^8 n \log \left (d+e \sqrt{x}\right )}{4 e^8}+\frac{1}{4} x^4 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.130716, size = 159, normalized size = 0.96 \[ \frac{a x^4}{4}+\frac{1}{4} b x^4 \log \left (c \left (d+e \sqrt{x}\right )^n\right )-\frac{1}{4} b e n \left (-\frac{d^5 x^{3/2}}{3 e^6}+\frac{d^4 x^2}{4 e^5}-\frac{d^3 x^{5/2}}{5 e^4}+\frac{d^2 x^3}{6 e^3}-\frac{d^7 \sqrt{x}}{e^8}+\frac{d^6 x}{2 e^7}+\frac{d^8 \log \left (d+e \sqrt{x}\right )}{e^9}-\frac{d x^{7/2}}{7 e^2}+\frac{x^4}{8 e}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*x^4)/4 - (b*e*n*(-((d^7*Sqrt[x])/e^8) + (d^6*x)/(2*e^7) - (d^5*x^(3/2))/(3*e^6) + (d^4*x^2)/(4*e^5) - (d^3*
x^(5/2))/(5*e^4) + (d^2*x^3)/(6*e^3) - (d*x^(7/2))/(7*e^2) + x^4/(8*e) + (d^8*Log[d + e*Sqrt[x]])/e^9))/4 + (b
*x^4*Log[c*(d + e*Sqrt[x])^n])/4

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Maple [F]  time = 0.415, size = 0, normalized size = 0. \begin{align*} \int{x}^{3} \left ( a+b\ln \left ( c \left ( d+e\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 = 1.04502, size = 173, normalized size = 1.04 \begin{align*} \frac{1}{4} \, b x^{4} \log \left ({\left (e \sqrt{x} + d\right )}^{n} c\right ) + \frac{1}{4} \, a x^{4} - \frac{1}{3360} \, b e n{\left (\frac{840 \, d^{8} \log \left (e \sqrt{x} + d\right )}{e^{9}} + \frac{105 \, e^{7} x^{4} - 120 \, d e^{6} x^{\frac{7}{2}} + 140 \, d^{2} e^{5} x^{3} - 168 \, d^{3} e^{4} x^{\frac{5}{2}} + 210 \, d^{4} e^{3} x^{2} - 280 \, d^{5} e^{2} x^{\frac{3}{2}} + 420 \, d^{6} e x - 840 \, d^{7} \sqrt{x}}{e^{8}}\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((e*sqrt(x) + d)^n*c) + 1/4*a*x^4 - 1/3360*b*e*n*(840*d^8*log(e*sqrt(x) + d)/e^9 + (105*e^7*x^4 -
120*d*e^6*x^(7/2) + 140*d^2*e^5*x^3 - 168*d^3*e^4*x^(5/2) + 210*d^4*e^3*x^2 - 280*d^5*e^2*x^(3/2) + 420*d^6*e
*x - 840*d^7*sqrt(x))/e^8)

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Fricas [A]  time = 1.82234, size = 358, normalized size = 2.16 \begin{align*} \frac{840 \, b e^{8} x^{4} \log \left (c\right ) - 140 \, b d^{2} e^{6} n x^{3} - 210 \, b d^{4} e^{4} n x^{2} - 420 \, b d^{6} e^{2} n x - 105 \,{\left (b e^{8} n - 8 \, a e^{8}\right )} x^{4} + 840 \,{\left (b e^{8} n x^{4} - b d^{8} n\right )} \log \left (e \sqrt{x} + d\right ) + 8 \,{\left (15 \, b d e^{7} n x^{3} + 21 \, b d^{3} e^{5} n x^{2} + 35 \, b d^{5} e^{3} n x + 105 \, b d^{7} e n\right )} \sqrt{x}}{3360 \, e^{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]

1/3360*(840*b*e^8*x^4*log(c) - 140*b*d^2*e^6*n*x^3 - 210*b*d^4*e^4*n*x^2 - 420*b*d^6*e^2*n*x - 105*(b*e^8*n -
8*a*e^8)*x^4 + 840*(b*e^8*n*x^4 - b*d^8*n)*log(e*sqrt(x) + d) + 8*(15*b*d*e^7*n*x^3 + 21*b*d^3*e^5*n*x^2 + 35*
b*d^5*e^3*n*x + 105*b*d^7*e*n)*sqrt(x))/e^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 [B]  time = 1.37637, size = 779, normalized size = 4.69 \begin{align*} \text{result too large to display} \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]

1/3360*((840*(sqrt(x)*e + d)^8*e^(-6)*log(sqrt(x)*e + d) - 6720*(sqrt(x)*e + d)^7*d*e^(-6)*log(sqrt(x)*e + d)
+ 23520*(sqrt(x)*e + d)^6*d^2*e^(-6)*log(sqrt(x)*e + d) - 47040*(sqrt(x)*e + d)^5*d^3*e^(-6)*log(sqrt(x)*e + d
) + 58800*(sqrt(x)*e + d)^4*d^4*e^(-6)*log(sqrt(x)*e + d) - 47040*(sqrt(x)*e + d)^3*d^5*e^(-6)*log(sqrt(x)*e +
d) + 23520*(sqrt(x)*e + d)^2*d^6*e^(-6)*log(sqrt(x)*e + d) - 6720*(sqrt(x)*e + d)*d^7*e^(-6)*log(sqrt(x)*e +
d) - 105*(sqrt(x)*e + d)^8*e^(-6) + 960*(sqrt(x)*e + d)^7*d*e^(-6) - 3920*(sqrt(x)*e + d)^6*d^2*e^(-6) + 9408*
(sqrt(x)*e + d)^5*d^3*e^(-6) - 14700*(sqrt(x)*e + d)^4*d^4*e^(-6) + 15680*(sqrt(x)*e + d)^3*d^5*e^(-6) - 11760
*(sqrt(x)*e + d)^2*d^6*e^(-6) + 6720*(sqrt(x)*e + d)*d^7*e^(-6))*b*n*e^(-1) + 840*((sqrt(x)*e + d)^8 - 8*(sqrt
(x)*e + d)^7*d + 28*(sqrt(x)*e + d)^6*d^2 - 56*(sqrt(x)*e + d)^5*d^3 + 70*(sqrt(x)*e + d)^4*d^4 - 56*(sqrt(x)*
e + d)^3*d^5 + 28*(sqrt(x)*e + d)^2*d^6 - 8*(sqrt(x)*e + d)*d^7)*b*e^(-7)*log(c) + 840*((sqrt(x)*e + d)^8 - 8*
(sqrt(x)*e + d)^7*d + 28*(sqrt(x)*e + d)^6*d^2 - 56*(sqrt(x)*e + d)^5*d^3 + 70*(sqrt(x)*e + d)^4*d^4 - 56*(sqr
t(x)*e + d)^3*d^5 + 28*(sqrt(x)*e + d)^2*d^6 - 8*(sqrt(x)*e + d)*d^7)*a*e^(-7))*e^(-1)