3.46 \(\int x^3 \log (c (a+b \sqrt{x})^p) \, dx\)

Optimal. Leaf size=153 \[ \frac{a^5 p x^{3/2}}{12 b^5}-\frac{a^4 p x^2}{16 b^4}+\frac{a^3 p x^{5/2}}{20 b^3}-\frac{a^2 p x^3}{24 b^2}+\frac{a^7 p \sqrt{x}}{4 b^7}-\frac{a^6 p x}{8 b^6}-\frac{a^8 p \log \left (a+b \sqrt{x}\right )}{4 b^8}+\frac{1}{4} x^4 \log \left (c \left (a+b \sqrt{x}\right )^p\right )+\frac{a p x^{7/2}}{28 b}-\frac{p x^4}{32} \]

[Out]

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

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Rubi [A]  time = 0.117554, antiderivative size = 153, 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} \[ \frac{a^5 p x^{3/2}}{12 b^5}-\frac{a^4 p x^2}{16 b^4}+\frac{a^3 p x^{5/2}}{20 b^3}-\frac{a^2 p x^3}{24 b^2}+\frac{a^7 p \sqrt{x}}{4 b^7}-\frac{a^6 p x}{8 b^6}-\frac{a^8 p \log \left (a+b \sqrt{x}\right )}{4 b^8}+\frac{1}{4} x^4 \log \left (c \left (a+b \sqrt{x}\right )^p\right )+\frac{a p x^{7/2}}{28 b}-\frac{p x^4}{32} \]

Antiderivative was successfully verified.

[In]

Int[x^3*Log[c*(a + b*Sqrt[x])^p],x]

[Out]

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

Mathematica [A]  time = 0.131131, size = 134, normalized size = 0.88 \[ \frac{1}{4} \left (x^4 \log \left (c \left (a+b \sqrt{x}\right )^p\right )-\frac{p \left (-280 a^5 b^3 x^{3/2}+210 a^4 b^4 x^2-168 a^3 b^5 x^{5/2}+140 a^2 b^6 x^3+420 a^6 b^2 x-840 a^7 b \sqrt{x}+840 a^8 \log \left (a+b \sqrt{x}\right )-120 a b^7 x^{7/2}+105 b^8 x^4\right )}{840 b^8}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(p*(-840*a^7*b*Sqrt[x] + 420*a^6*b^2*x - 280*a^5*b^3*x^(3/2) + 210*a^4*b^4*x^2 - 168*a^3*b^5*x^(5/2) + 140*a
^2*b^6*x^3 - 120*a*b^7*x^(7/2) + 105*b^8*x^4 + 840*a^8*Log[a + b*Sqrt[x]]))/(840*b^8) + x^4*Log[c*(a + b*Sqrt[
x])^p])/4

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Maple [F]  time = 0.299, size = 0, normalized size = 0. \begin{align*} \int{x}^{3}\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^3*ln(c*(a+b*x^(1/2))^p),x)

[Out]

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

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Maxima [A]  time = 1.10123, size = 162, normalized size = 1.06 \begin{align*} \frac{1}{4} \, x^{4} \log \left ({\left (b \sqrt{x} + a\right )}^{p} c\right ) - \frac{1}{3360} \, b p{\left (\frac{840 \, a^{8} \log \left (b \sqrt{x} + a\right )}{b^{9}} + \frac{105 \, b^{7} x^{4} - 120 \, a b^{6} x^{\frac{7}{2}} + 140 \, a^{2} b^{5} x^{3} - 168 \, a^{3} b^{4} x^{\frac{5}{2}} + 210 \, a^{4} b^{3} x^{2} - 280 \, a^{5} b^{2} x^{\frac{3}{2}} + 420 \, a^{6} b x - 840 \, a^{7} \sqrt{x}}{b^{8}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x^4*log((b*sqrt(x) + a)^p*c) - 1/3360*b*p*(840*a^8*log(b*sqrt(x) + a)/b^9 + (105*b^7*x^4 - 120*a*b^6*x^(7/
2) + 140*a^2*b^5*x^3 - 168*a^3*b^4*x^(5/2) + 210*a^4*b^3*x^2 - 280*a^5*b^2*x^(3/2) + 420*a^6*b*x - 840*a^7*sqr
t(x))/b^8)

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Fricas [A]  time = 2.46633, size = 313, normalized size = 2.05 \begin{align*} -\frac{105 \, b^{8} p x^{4} - 840 \, b^{8} x^{4} \log \left (c\right ) + 140 \, a^{2} b^{6} p x^{3} + 210 \, a^{4} b^{4} p x^{2} + 420 \, a^{6} b^{2} p x - 840 \,{\left (b^{8} p x^{4} - a^{8} p\right )} \log \left (b \sqrt{x} + a\right ) - 8 \,{\left (15 \, a b^{7} p x^{3} + 21 \, a^{3} b^{5} p x^{2} + 35 \, a^{5} b^{3} p x + 105 \, a^{7} b p\right )} \sqrt{x}}{3360 \, b^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3360*(105*b^8*p*x^4 - 840*b^8*x^4*log(c) + 140*a^2*b^6*p*x^3 + 210*a^4*b^4*p*x^2 + 420*a^6*b^2*p*x - 840*(b
^8*p*x^4 - a^8*p)*log(b*sqrt(x) + a) - 8*(15*a*b^7*p*x^3 + 21*a^3*b^5*p*x^2 + 35*a^5*b^3*p*x + 105*a^7*b*p)*sq
rt(x))/b^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*ln(c*(a+b*x**(1/2))**p),x)

[Out]

Timed out

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Giac [B]  time = 1.30382, size = 601, normalized size = 3.93 \begin{align*} \frac{\frac{{\left (\frac{840 \,{\left (b \sqrt{x} + a\right )}^{8} \log \left (b \sqrt{x} + a\right )}{b^{6}} - \frac{6720 \,{\left (b \sqrt{x} + a\right )}^{7} a \log \left (b \sqrt{x} + a\right )}{b^{6}} + \frac{23520 \,{\left (b \sqrt{x} + a\right )}^{6} a^{2} \log \left (b \sqrt{x} + a\right )}{b^{6}} - \frac{47040 \,{\left (b \sqrt{x} + a\right )}^{5} a^{3} \log \left (b \sqrt{x} + a\right )}{b^{6}} + \frac{58800 \,{\left (b \sqrt{x} + a\right )}^{4} a^{4} \log \left (b \sqrt{x} + a\right )}{b^{6}} - \frac{47040 \,{\left (b \sqrt{x} + a\right )}^{3} a^{5} \log \left (b \sqrt{x} + a\right )}{b^{6}} + \frac{23520 \,{\left (b \sqrt{x} + a\right )}^{2} a^{6} \log \left (b \sqrt{x} + a\right )}{b^{6}} - \frac{6720 \,{\left (b \sqrt{x} + a\right )} a^{7} \log \left (b \sqrt{x} + a\right )}{b^{6}} - \frac{105 \,{\left (b \sqrt{x} + a\right )}^{8}}{b^{6}} + \frac{960 \,{\left (b \sqrt{x} + a\right )}^{7} a}{b^{6}} - \frac{3920 \,{\left (b \sqrt{x} + a\right )}^{6} a^{2}}{b^{6}} + \frac{9408 \,{\left (b \sqrt{x} + a\right )}^{5} a^{3}}{b^{6}} - \frac{14700 \,{\left (b \sqrt{x} + a\right )}^{4} a^{4}}{b^{6}} + \frac{15680 \,{\left (b \sqrt{x} + a\right )}^{3} a^{5}}{b^{6}} - \frac{11760 \,{\left (b \sqrt{x} + a\right )}^{2} a^{6}}{b^{6}} + \frac{6720 \,{\left (b \sqrt{x} + a\right )} a^{7}}{b^{6}}\right )} p}{b} + \frac{840 \,{\left ({\left (b \sqrt{x} + a\right )}^{8} - 8 \,{\left (b \sqrt{x} + a\right )}^{7} a + 28 \,{\left (b \sqrt{x} + a\right )}^{6} a^{2} - 56 \,{\left (b \sqrt{x} + a\right )}^{5} a^{3} + 70 \,{\left (b \sqrt{x} + a\right )}^{4} a^{4} - 56 \,{\left (b \sqrt{x} + a\right )}^{3} a^{5} + 28 \,{\left (b \sqrt{x} + a\right )}^{2} a^{6} - 8 \,{\left (b \sqrt{x} + a\right )} a^{7}\right )} \log \left (c\right )}{b^{7}}}{3360 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3360*((840*(b*sqrt(x) + a)^8*log(b*sqrt(x) + a)/b^6 - 6720*(b*sqrt(x) + a)^7*a*log(b*sqrt(x) + a)/b^6 + 2352
0*(b*sqrt(x) + a)^6*a^2*log(b*sqrt(x) + a)/b^6 - 47040*(b*sqrt(x) + a)^5*a^3*log(b*sqrt(x) + a)/b^6 + 58800*(b
*sqrt(x) + a)^4*a^4*log(b*sqrt(x) + a)/b^6 - 47040*(b*sqrt(x) + a)^3*a^5*log(b*sqrt(x) + a)/b^6 + 23520*(b*sqr
t(x) + a)^2*a^6*log(b*sqrt(x) + a)/b^6 - 6720*(b*sqrt(x) + a)*a^7*log(b*sqrt(x) + a)/b^6 - 105*(b*sqrt(x) + a)
^8/b^6 + 960*(b*sqrt(x) + a)^7*a/b^6 - 3920*(b*sqrt(x) + a)^6*a^2/b^6 + 9408*(b*sqrt(x) + a)^5*a^3/b^6 - 14700
*(b*sqrt(x) + a)^4*a^4/b^6 + 15680*(b*sqrt(x) + a)^3*a^5/b^6 - 11760*(b*sqrt(x) + a)^2*a^6/b^6 + 6720*(b*sqrt(
x) + a)*a^7/b^6)*p/b + 840*((b*sqrt(x) + a)^8 - 8*(b*sqrt(x) + a)^7*a + 28*(b*sqrt(x) + a)^6*a^2 - 56*(b*sqrt(
x) + a)^5*a^3 + 70*(b*sqrt(x) + a)^4*a^4 - 56*(b*sqrt(x) + a)^3*a^5 + 28*(b*sqrt(x) + a)^2*a^6 - 8*(b*sqrt(x)
+ a)*a^7)*log(c)/b^7)/b