3.47 \(\int x^2 \log (c (a+b \sqrt{x})^p) \, dx\)

Optimal. Leaf size=123 \[ \frac{a^3 p x^{3/2}}{9 b^3}-\frac{a^2 p x^2}{12 b^2}+\frac{a^5 p \sqrt{x}}{3 b^5}-\frac{a^4 p x}{6 b^4}-\frac{a^6 p \log \left (a+b \sqrt{x}\right )}{3 b^6}+\frac{1}{3} x^3 \log \left (c \left (a+b \sqrt{x}\right )^p\right )+\frac{a p x^{5/2}}{15 b}-\frac{p x^3}{18} \]

[Out]

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

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Rubi [A]  time = 0.0880257, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2454, 2395, 43} \[ \frac{a^3 p x^{3/2}}{9 b^3}-\frac{a^2 p x^2}{12 b^2}+\frac{a^5 p \sqrt{x}}{3 b^5}-\frac{a^4 p x}{6 b^4}-\frac{a^6 p \log \left (a+b \sqrt{x}\right )}{3 b^6}+\frac{1}{3} x^3 \log \left (c \left (a+b \sqrt{x}\right )^p\right )+\frac{a p x^{5/2}}{15 b}-\frac{p x^3}{18} \]

Antiderivative was successfully verified.

[In]

Int[x^2*Log[c*(a + b*Sqrt[x])^p],x]

[Out]

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

Mathematica [A]  time = 0.052671, size = 112, normalized size = 0.91 \[ \frac{b p \sqrt{x} \left (-15 a^2 b^3 x^{3/2}+20 a^3 b^2 x-30 a^4 b \sqrt{x}+60 a^5+12 a b^4 x^2-10 b^5 x^{5/2}\right )-60 a^6 p \log \left (a+b \sqrt{x}\right )+60 b^6 x^3 \log \left (c \left (a+b \sqrt{x}\right )^p\right )}{180 b^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*p*Sqrt[x]*(60*a^5 - 30*a^4*b*Sqrt[x] + 20*a^3*b^2*x - 15*a^2*b^3*x^(3/2) + 12*a*b^4*x^2 - 10*b^5*x^(5/2)) -
60*a^6*p*Log[a + b*Sqrt[x]] + 60*b^6*x^3*Log[c*(a + b*Sqrt[x])^p])/(180*b^6)

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Maple [F]  time = 0.062, size = 0, normalized size = 0. \begin{align*} \int{x}^{2}\ln \left ( c \left ( a+b\sqrt{x} \right ) ^{p} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*ln(c*(a+b*x^(1/2))^p),x)

[Out]

int(x^2*ln(c*(a+b*x^(1/2))^p),x)

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Maxima [A]  time = 1.07278, size = 132, normalized size = 1.07 \begin{align*} \frac{1}{3} \, x^{3} \log \left ({\left (b \sqrt{x} + a\right )}^{p} c\right ) - \frac{1}{180} \, b p{\left (\frac{60 \, a^{6} \log \left (b \sqrt{x} + a\right )}{b^{7}} + \frac{10 \, b^{5} x^{3} - 12 \, a b^{4} x^{\frac{5}{2}} + 15 \, a^{2} b^{3} x^{2} - 20 \, a^{3} b^{2} x^{\frac{3}{2}} + 30 \, a^{4} b x - 60 \, a^{5} \sqrt{x}}{b^{6}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.36353, size = 248, normalized size = 2.02 \begin{align*} -\frac{10 \, b^{6} p x^{3} - 60 \, b^{6} x^{3} \log \left (c\right ) + 15 \, a^{2} b^{4} p x^{2} + 30 \, a^{4} b^{2} p x - 60 \,{\left (b^{6} p x^{3} - a^{6} p\right )} \log \left (b \sqrt{x} + a\right ) - 4 \,{\left (3 \, a b^{5} p x^{2} + 5 \, a^{3} b^{3} p x + 15 \, a^{5} b p\right )} \sqrt{x}}{180 \, b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 162.442, size = 119, normalized size = 0.97 \begin{align*} - \frac{b p \left (\frac{2 a^{6} \left (\begin{cases} \frac{\sqrt{x}}{a} & \text{for}\: b = 0 \\\frac{\log{\left (a + b \sqrt{x} \right )}}{b} & \text{otherwise} \end{cases}\right )}{b^{6}} - \frac{2 a^{5} \sqrt{x}}{b^{6}} + \frac{a^{4} x}{b^{5}} - \frac{2 a^{3} x^{\frac{3}{2}}}{3 b^{4}} + \frac{a^{2} x^{2}}{2 b^{3}} - \frac{2 a x^{\frac{5}{2}}}{5 b^{2}} + \frac{x^{3}}{3 b}\right )}{6} + \frac{x^{3} \log{\left (c \left (a + b \sqrt{x}\right )^{p} \right )}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*ln(c*(a+b*x**(1/2))**p),x)

[Out]

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

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Giac [B]  time = 1.23736, size = 450, normalized size = 3.66 \begin{align*} \frac{\frac{{\left (\frac{60 \,{\left (b \sqrt{x} + a\right )}^{6} \log \left (b \sqrt{x} + a\right )}{b^{4}} - \frac{360 \,{\left (b \sqrt{x} + a\right )}^{5} a \log \left (b \sqrt{x} + a\right )}{b^{4}} + \frac{900 \,{\left (b \sqrt{x} + a\right )}^{4} a^{2} \log \left (b \sqrt{x} + a\right )}{b^{4}} - \frac{1200 \,{\left (b \sqrt{x} + a\right )}^{3} a^{3} \log \left (b \sqrt{x} + a\right )}{b^{4}} + \frac{900 \,{\left (b \sqrt{x} + a\right )}^{2} a^{4} \log \left (b \sqrt{x} + a\right )}{b^{4}} - \frac{360 \,{\left (b \sqrt{x} + a\right )} a^{5} \log \left (b \sqrt{x} + a\right )}{b^{4}} - \frac{10 \,{\left (b \sqrt{x} + a\right )}^{6}}{b^{4}} + \frac{72 \,{\left (b \sqrt{x} + a\right )}^{5} a}{b^{4}} - \frac{225 \,{\left (b \sqrt{x} + a\right )}^{4} a^{2}}{b^{4}} + \frac{400 \,{\left (b \sqrt{x} + a\right )}^{3} a^{3}}{b^{4}} - \frac{450 \,{\left (b \sqrt{x} + a\right )}^{2} a^{4}}{b^{4}} + \frac{360 \,{\left (b \sqrt{x} + a\right )} a^{5}}{b^{4}}\right )} p}{b} + \frac{60 \,{\left ({\left (b \sqrt{x} + a\right )}^{6} - 6 \,{\left (b \sqrt{x} + a\right )}^{5} a + 15 \,{\left (b \sqrt{x} + a\right )}^{4} a^{2} - 20 \,{\left (b \sqrt{x} + a\right )}^{3} a^{3} + 15 \,{\left (b \sqrt{x} + a\right )}^{2} a^{4} - 6 \,{\left (b \sqrt{x} + a\right )} a^{5}\right )} \log \left (c\right )}{b^{5}}}{180 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/180*((60*(b*sqrt(x) + a)^6*log(b*sqrt(x) + a)/b^4 - 360*(b*sqrt(x) + a)^5*a*log(b*sqrt(x) + a)/b^4 + 900*(b*
sqrt(x) + a)^4*a^2*log(b*sqrt(x) + a)/b^4 - 1200*(b*sqrt(x) + a)^3*a^3*log(b*sqrt(x) + a)/b^4 + 900*(b*sqrt(x)
+ a)^2*a^4*log(b*sqrt(x) + a)/b^4 - 360*(b*sqrt(x) + a)*a^5*log(b*sqrt(x) + a)/b^4 - 10*(b*sqrt(x) + a)^6/b^4
+ 72*(b*sqrt(x) + a)^5*a/b^4 - 225*(b*sqrt(x) + a)^4*a^2/b^4 + 400*(b*sqrt(x) + a)^3*a^3/b^4 - 450*(b*sqrt(x)
+ a)^2*a^4/b^4 + 360*(b*sqrt(x) + a)*a^5/b^4)*p/b + 60*((b*sqrt(x) + a)^6 - 6*(b*sqrt(x) + a)^5*a + 15*(b*sqr
t(x) + a)^4*a^2 - 20*(b*sqrt(x) + a)^3*a^3 + 15*(b*sqrt(x) + a)^2*a^4 - 6*(b*sqrt(x) + a)*a^5)*log(c)/b^5)/b