3.401 \(\int x^2 (a+b \log (c (d+e \sqrt{x})^n)) \, dx\)

Optimal. Leaf size=134 \[ \frac{1}{3} x^3 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )+\frac{b d^3 n x^{3/2}}{9 e^3}-\frac{b d^2 n x^2}{12 e^2}+\frac{b d^5 n \sqrt{x}}{3 e^5}-\frac{b d^4 n x}{6 e^4}-\frac{b d^6 n \log \left (d+e \sqrt{x}\right )}{3 e^6}+\frac{b d n x^{5/2}}{15 e}-\frac{1}{18} b n x^3 \]

[Out]

(b*d^5*n*Sqrt[x])/(3*e^5) - (b*d^4*n*x)/(6*e^4) + (b*d^3*n*x^(3/2))/(9*e^3) - (b*d^2*n*x^2)/(12*e^2) + (b*d*n*
x^(5/2))/(15*e) - (b*n*x^3)/18 - (b*d^6*n*Log[d + e*Sqrt[x]])/(3*e^6) + (x^3*(a + b*Log[c*(d + e*Sqrt[x])^n]))
/3

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Rubi [A]  time = 0.0993512, antiderivative size = 134, 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} \[ \frac{1}{3} x^3 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )+\frac{b d^3 n x^{3/2}}{9 e^3}-\frac{b d^2 n x^2}{12 e^2}+\frac{b d^5 n \sqrt{x}}{3 e^5}-\frac{b d^4 n x}{6 e^4}-\frac{b d^6 n \log \left (d+e \sqrt{x}\right )}{3 e^6}+\frac{b d n x^{5/2}}{15 e}-\frac{1}{18} b n x^3 \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*Log[c*(d + e*Sqrt[x])^n]),x]

[Out]

(b*d^5*n*Sqrt[x])/(3*e^5) - (b*d^4*n*x)/(6*e^4) + (b*d^3*n*x^(3/2))/(9*e^3) - (b*d^2*n*x^2)/(12*e^2) + (b*d*n*
x^(5/2))/(15*e) - (b*n*x^3)/18 - (b*d^6*n*Log[d + e*Sqrt[x]])/(3*e^6) + (x^3*(a + b*Log[c*(d + e*Sqrt[x])^n]))
/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 x^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right ) \, dx &=2 \operatorname{Subst}\left (\int x^5 \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )-\frac{1}{3} (b e n) \operatorname{Subst}\left (\int \frac{x^6}{d+e x} \, dx,x,\sqrt{x}\right )\\ &=\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )-\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,\sqrt{x}\right )\\ &=\frac{b d^5 n \sqrt{x}}{3 e^5}-\frac{b d^4 n x}{6 e^4}+\frac{b d^3 n x^{3/2}}{9 e^3}-\frac{b d^2 n x^2}{12 e^2}+\frac{b d n x^{5/2}}{15 e}-\frac{1}{18} b n x^3-\frac{b d^6 n \log \left (d+e \sqrt{x}\right )}{3 e^6}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.0903966, size = 131, normalized size = 0.98 \[ \frac{a x^3}{3}+\frac{1}{3} b x^3 \log \left (c \left (d+e \sqrt{x}\right )^n\right )-\frac{1}{3} b e n \left (-\frac{d^3 x^{3/2}}{3 e^4}+\frac{d^2 x^2}{4 e^3}-\frac{d^5 \sqrt{x}}{e^6}+\frac{d^4 x}{2 e^5}+\frac{d^6 \log \left (d+e \sqrt{x}\right )}{e^7}-\frac{d x^{5/2}}{5 e^2}+\frac{x^3}{6 e}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*Log[c*(d + e*Sqrt[x])^n]),x]

[Out]

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

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Maple [F]  time = 0.098, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \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^2*(a+b*ln(c*(d+e*x^(1/2))^n)),x)

[Out]

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

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Maxima [A]  time = 1.06794, size = 143, normalized size = 1.07 \begin{align*} \frac{1}{3} \, b x^{3} \log \left ({\left (e \sqrt{x} + d\right )}^{n} c\right ) + \frac{1}{3} \, a x^{3} - \frac{1}{180} \, b e n{\left (\frac{60 \, d^{6} \log \left (e \sqrt{x} + d\right )}{e^{7}} + \frac{10 \, e^{5} x^{3} - 12 \, d e^{4} x^{\frac{5}{2}} + 15 \, d^{2} e^{3} x^{2} - 20 \, d^{3} e^{2} x^{\frac{3}{2}} + 30 \, d^{4} e x - 60 \, d^{5} \sqrt{x}}{e^{6}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.69733, size = 288, normalized size = 2.15 \begin{align*} \frac{60 \, b e^{6} x^{3} \log \left (c\right ) - 15 \, b d^{2} e^{4} n x^{2} - 30 \, b d^{4} e^{2} n x - 10 \,{\left (b e^{6} n - 6 \, a e^{6}\right )} x^{3} + 60 \,{\left (b e^{6} n x^{3} - b d^{6} n\right )} \log \left (e \sqrt{x} + d\right ) + 4 \,{\left (3 \, b d e^{5} n x^{2} + 5 \, b d^{3} e^{3} n x + 15 \, b d^{5} e n\right )} \sqrt{x}}{180 \, e^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 29.4452, size = 128, normalized size = 0.96 \begin{align*} \frac{a x^{3}}{3} + b \left (- \frac{e n \left (\frac{2 d^{6} \left (\begin{cases} \frac{\sqrt{x}}{d} & \text{for}\: e = 0 \\\frac{\log{\left (d + e \sqrt{x} \right )}}{e} & \text{otherwise} \end{cases}\right )}{e^{6}} - \frac{2 d^{5} \sqrt{x}}{e^{6}} + \frac{d^{4} x}{e^{5}} - \frac{2 d^{3} x^{\frac{3}{2}}}{3 e^{4}} + \frac{d^{2} x^{2}}{2 e^{3}} - \frac{2 d x^{\frac{5}{2}}}{5 e^{2}} + \frac{x^{3}}{3 e}\right )}{6} + \frac{x^{3} \log{\left (c \left (d + e \sqrt{x}\right )^{n} \right )}}{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x**3/3 + b*(-e*n*(2*d**6*Piecewise((sqrt(x)/d, Eq(e, 0)), (log(d + e*sqrt(x))/e, True))/e**6 - 2*d**5*sqrt(x
)/e**6 + d**4*x/e**5 - 2*d**3*x**(3/2)/(3*e**4) + d**2*x**2/(2*e**3) - 2*d*x**(5/2)/(5*e**2) + x**3/(3*e))/6 +
x**3*log(c*(d + e*sqrt(x))**n)/3)

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Giac [B]  time = 1.29028, size = 582, normalized size = 4.34 \begin{align*} \frac{1}{180} \,{\left ({\left (60 \,{\left (\sqrt{x} e + d\right )}^{6} e^{\left (-4\right )} \log \left (\sqrt{x} e + d\right ) - 360 \,{\left (\sqrt{x} e + d\right )}^{5} d e^{\left (-4\right )} \log \left (\sqrt{x} e + d\right ) + 900 \,{\left (\sqrt{x} e + d\right )}^{4} d^{2} e^{\left (-4\right )} \log \left (\sqrt{x} e + d\right ) - 1200 \,{\left (\sqrt{x} e + d\right )}^{3} d^{3} e^{\left (-4\right )} \log \left (\sqrt{x} e + d\right ) + 900 \,{\left (\sqrt{x} e + d\right )}^{2} d^{4} e^{\left (-4\right )} \log \left (\sqrt{x} e + d\right ) - 360 \,{\left (\sqrt{x} e + d\right )} d^{5} e^{\left (-4\right )} \log \left (\sqrt{x} e + d\right ) - 10 \,{\left (\sqrt{x} e + d\right )}^{6} e^{\left (-4\right )} + 72 \,{\left (\sqrt{x} e + d\right )}^{5} d e^{\left (-4\right )} - 225 \,{\left (\sqrt{x} e + d\right )}^{4} d^{2} e^{\left (-4\right )} + 400 \,{\left (\sqrt{x} e + d\right )}^{3} d^{3} e^{\left (-4\right )} - 450 \,{\left (\sqrt{x} e + d\right )}^{2} d^{4} e^{\left (-4\right )} + 360 \,{\left (\sqrt{x} e + d\right )} d^{5} e^{\left (-4\right )}\right )} b n e^{\left (-1\right )} + 60 \,{\left ({\left (\sqrt{x} e + d\right )}^{6} - 6 \,{\left (\sqrt{x} e + d\right )}^{5} d + 15 \,{\left (\sqrt{x} e + d\right )}^{4} d^{2} - 20 \,{\left (\sqrt{x} e + d\right )}^{3} d^{3} + 15 \,{\left (\sqrt{x} e + d\right )}^{2} d^{4} - 6 \,{\left (\sqrt{x} e + d\right )} d^{5}\right )} b e^{\left (-5\right )} \log \left (c\right ) + 60 \,{\left ({\left (\sqrt{x} e + d\right )}^{6} - 6 \,{\left (\sqrt{x} e + d\right )}^{5} d + 15 \,{\left (\sqrt{x} e + d\right )}^{4} d^{2} - 20 \,{\left (\sqrt{x} e + d\right )}^{3} d^{3} + 15 \,{\left (\sqrt{x} e + d\right )}^{2} d^{4} - 6 \,{\left (\sqrt{x} e + d\right )} d^{5}\right )} a e^{\left (-5\right )}\right )} e^{\left (-1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/180*((60*(sqrt(x)*e + d)^6*e^(-4)*log(sqrt(x)*e + d) - 360*(sqrt(x)*e + d)^5*d*e^(-4)*log(sqrt(x)*e + d) + 9
00*(sqrt(x)*e + d)^4*d^2*e^(-4)*log(sqrt(x)*e + d) - 1200*(sqrt(x)*e + d)^3*d^3*e^(-4)*log(sqrt(x)*e + d) + 90
0*(sqrt(x)*e + d)^2*d^4*e^(-4)*log(sqrt(x)*e + d) - 360*(sqrt(x)*e + d)*d^5*e^(-4)*log(sqrt(x)*e + d) - 10*(sq
rt(x)*e + d)^6*e^(-4) + 72*(sqrt(x)*e + d)^5*d*e^(-4) - 225*(sqrt(x)*e + d)^4*d^2*e^(-4) + 400*(sqrt(x)*e + d)
^3*d^3*e^(-4) - 450*(sqrt(x)*e + d)^2*d^4*e^(-4) + 360*(sqrt(x)*e + d)*d^5*e^(-4))*b*n*e^(-1) + 60*((sqrt(x)*e
+ d)^6 - 6*(sqrt(x)*e + d)^5*d + 15*(sqrt(x)*e + d)^4*d^2 - 20*(sqrt(x)*e + d)^3*d^3 + 15*(sqrt(x)*e + d)^2*d
^4 - 6*(sqrt(x)*e + d)*d^5)*b*e^(-5)*log(c) + 60*((sqrt(x)*e + d)^6 - 6*(sqrt(x)*e + d)^5*d + 15*(sqrt(x)*e +
d)^4*d^2 - 20*(sqrt(x)*e + d)^3*d^3 + 15*(sqrt(x)*e + d)^2*d^4 - 6*(sqrt(x)*e + d)*d^5)*a*e^(-5))*e^(-1)