3.1.46 \(\int x^3 \log (c (a+b \sqrt {x})^p) \, dx\) [46]

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

[Out]

-1/8*a^6*p*x/b^6+1/12*a^5*p*x^(3/2)/b^5-1/16*a^4*p*x^2/b^4+1/20*a^3*p*x^(5/2)/b^3-1/24*a^2*p*x^3/b^2+1/28*a*p*
x^(7/2)/b-1/32*p*x^4-1/4*a^8*p*ln(a+b*x^(1/2))/b^8+1/4*x^4*ln(c*(a+b*x^(1/2))^p)+1/4*a^7*p*x^(1/2)/b^7

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Rubi [A]
time = 0.08, antiderivative size = 153, normalized size of antiderivative = 1.00, 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 = {2504, 2442, 45} \begin {gather*} -\frac {a^8 p \log \left (a+b \sqrt {x}\right )}{4 b^8}+\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 {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} \end {gather*}

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 45

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])

Rule 2442

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 2504

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])

Rubi steps

\begin {align*} \int x^3 \log \left (c \left (a+b \sqrt {x}\right )^p\right ) \, dx &=2 \text {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) \text {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) \text {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*}

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Mathematica [A]
time = 0.09, size = 134, normalized size = 0.88 \begin {gather*} \frac {1}{4} \left (-\frac {p \left (-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 \left (a+b \sqrt {x}\right )\right )}{840 b^8}+x^4 \log \left (c \left (a+b \sqrt {x}\right )^p\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-1/840*(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]]))/b^8 + x^4*Log[c*(a + b*Sqrt[
x])^p])/4

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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int x^{3} \ln \left (c \left (a +b \sqrt {x}\right )^{p}\right )\, dx\]

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 = 0.29, size = 120, normalized size = 0.78 \begin {gather*} \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 {gather*}

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 = 0.49, size = 129, normalized size = 0.84 \begin {gather*} -\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 {gather*}

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 [A]
time = 13.63, size = 146, normalized size = 0.95 \begin {gather*} - \frac {b p \left (\frac {2 a^{8} \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^{8}} - \frac {2 a^{7} \sqrt {x}}{b^{8}} + \frac {a^{6} x}{b^{7}} - \frac {2 a^{5} x^{\frac {3}{2}}}{3 b^{6}} + \frac {a^{4} x^{2}}{2 b^{5}} - \frac {2 a^{3} x^{\frac {5}{2}}}{5 b^{4}} + \frac {a^{2} x^{3}}{3 b^{3}} - \frac {2 a x^{\frac {7}{2}}}{7 b^{2}} + \frac {x^{4}}{4 b}\right )}{8} + \frac {x^{4} \log {\left (c \left (a + b \sqrt {x}\right )^{p} \right )}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b*p*(2*a**8*Piecewise((sqrt(x)/a, Eq(b, 0)), (log(a + b*sqrt(x))/b, True))/b**8 - 2*a**7*sqrt(x)/b**8 + a**6*
x/b**7 - 2*a**5*x**(3/2)/(3*b**6) + a**4*x**2/(2*b**5) - 2*a**3*x**(5/2)/(5*b**4) + a**2*x**3/(3*b**3) - 2*a*x
**(7/2)/(7*b**2) + x**4/(4*b))/8 + x**4*log(c*(a + b*sqrt(x))**p)/4

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 339 vs. \(2 (121) = 242\).
time = 4.64, size = 339, normalized size = 2.22 \begin {gather*} \frac {840 \, b x^{4} \log \left (c\right ) + {\left (\frac {840 \, {\left (b \sqrt {x} + a\right )}^{8} \log \left (b \sqrt {x} + a\right )}{b^{7}} - \frac {6720 \, {\left (b \sqrt {x} + a\right )}^{7} a \log \left (b \sqrt {x} + a\right )}{b^{7}} + \frac {23520 \, {\left (b \sqrt {x} + a\right )}^{6} a^{2} \log \left (b \sqrt {x} + a\right )}{b^{7}} - \frac {47040 \, {\left (b \sqrt {x} + a\right )}^{5} a^{3} \log \left (b \sqrt {x} + a\right )}{b^{7}} + \frac {58800 \, {\left (b \sqrt {x} + a\right )}^{4} a^{4} \log \left (b \sqrt {x} + a\right )}{b^{7}} - \frac {47040 \, {\left (b \sqrt {x} + a\right )}^{3} a^{5} \log \left (b \sqrt {x} + a\right )}{b^{7}} + \frac {23520 \, {\left (b \sqrt {x} + a\right )}^{2} a^{6} \log \left (b \sqrt {x} + a\right )}{b^{7}} - \frac {6720 \, {\left (b \sqrt {x} + a\right )} a^{7} \log \left (b \sqrt {x} + a\right )}{b^{7}} - \frac {105 \, {\left (b \sqrt {x} + a\right )}^{8}}{b^{7}} + \frac {960 \, {\left (b \sqrt {x} + a\right )}^{7} a}{b^{7}} - \frac {3920 \, {\left (b \sqrt {x} + a\right )}^{6} a^{2}}{b^{7}} + \frac {9408 \, {\left (b \sqrt {x} + a\right )}^{5} a^{3}}{b^{7}} - \frac {14700 \, {\left (b \sqrt {x} + a\right )}^{4} a^{4}}{b^{7}} + \frac {15680 \, {\left (b \sqrt {x} + a\right )}^{3} a^{5}}{b^{7}} - \frac {11760 \, {\left (b \sqrt {x} + a\right )}^{2} a^{6}}{b^{7}} + \frac {6720 \, {\left (b \sqrt {x} + a\right )} a^{7}}{b^{7}}\right )} p}{3360 \, b} \end {gather*}

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*x^4*log(c) + (840*(b*sqrt(x) + a)^8*log(b*sqrt(x) + a)/b^7 - 6720*(b*sqrt(x) + a)^7*a*log(b*sqrt
(x) + a)/b^7 + 23520*(b*sqrt(x) + a)^6*a^2*log(b*sqrt(x) + a)/b^7 - 47040*(b*sqrt(x) + a)^5*a^3*log(b*sqrt(x)
+ a)/b^7 + 58800*(b*sqrt(x) + a)^4*a^4*log(b*sqrt(x) + a)/b^7 - 47040*(b*sqrt(x) + a)^3*a^5*log(b*sqrt(x) + a)
/b^7 + 23520*(b*sqrt(x) + a)^2*a^6*log(b*sqrt(x) + a)/b^7 - 6720*(b*sqrt(x) + a)*a^7*log(b*sqrt(x) + a)/b^7 -
105*(b*sqrt(x) + a)^8/b^7 + 960*(b*sqrt(x) + a)^7*a/b^7 - 3920*(b*sqrt(x) + a)^6*a^2/b^7 + 9408*(b*sqrt(x) + a
)^5*a^3/b^7 - 14700*(b*sqrt(x) + a)^4*a^4/b^7 + 15680*(b*sqrt(x) + a)^3*a^5/b^7 - 11760*(b*sqrt(x) + a)^2*a^6/
b^7 + 6720*(b*sqrt(x) + a)*a^7/b^7)*p)/b

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Mupad [B]
time = 0.33, size = 121, normalized size = 0.79 \begin {gather*} \frac {x^4\,\ln \left (c\,{\left (a+b\,\sqrt {x}\right )}^p\right )}{4}-\frac {p\,x^4}{32}-\frac {a^8\,p\,\ln \left (a+b\,\sqrt {x}\right )}{4\,b^8}-\frac {a^2\,p\,x^3}{24\,b^2}-\frac {a^4\,p\,x^2}{16\,b^4}+\frac {a^3\,p\,x^{5/2}}{20\,b^3}+\frac {a^5\,p\,x^{3/2}}{12\,b^5}+\frac {a^7\,p\,\sqrt {x}}{4\,b^7}+\frac {a\,p\,x^{7/2}}{28\,b}-\frac {a^6\,p\,x}{8\,b^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*log(c*(a + b*x^(1/2))^p),x)

[Out]

(x^4*log(c*(a + b*x^(1/2))^p))/4 - (p*x^4)/32 - (a^8*p*log(a + b*x^(1/2)))/(4*b^8) - (a^2*p*x^3)/(24*b^2) - (a
^4*p*x^2)/(16*b^4) + (a^3*p*x^(5/2))/(20*b^3) + (a^5*p*x^(3/2))/(12*b^5) + (a^7*p*x^(1/2))/(4*b^7) + (a*p*x^(7
/2))/(28*b) - (a^6*p*x)/(8*b^6)

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