3.4.100 \(\int x^3 (a+b \log (c (d+e \sqrt {x})^n)) \, dx\) [400]

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

[Out]

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

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Rubi [A]
time = 0.09, antiderivative size = 166, normalized size of antiderivative = 1.00, 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 = {2504, 2442, 45} \begin {gather*} \frac {1}{4} x^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )-\frac {b d^8 n \log \left (d+e \sqrt {x}\right )}{4 e^8}+\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 \end {gather*}

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 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 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \, dx &=2 \text {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) \text {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) \text {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*}

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Mathematica [A]
time = 0.10, size = 159, normalized size = 0.96 \begin {gather*} \frac {a x^4}{4}-\frac {1}{4} b e n \left (-\frac {d^7 \sqrt {x}}{e^8}+\frac {d^6 x}{2 e^7}-\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 x^{7/2}}{7 e^2}+\frac {x^4}{8 e}+\frac {d^8 \log \left (d+e \sqrt {x}\right )}{e^9}\right )+\frac {1}{4} b x^4 \log \left (c \left (d+e \sqrt {x}\right )^n\right ) \end {gather*}

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 = 0.07, size = 0, normalized size = 0.00 \[\int x^{3} \left (a +b \ln \left (c \left (d +e \sqrt {x}\right )^{n}\right )\right )\, dx\]

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 = 0.27, size = 124, normalized size = 0.75 \begin {gather*} \frac {1}{4} \, b x^{4} \log \left ({\left (\sqrt {x} e + d\right )}^{n} c\right ) + \frac {1}{4} \, a x^{4} - \frac {1}{3360} \, {\left (840 \, d^{8} e^{\left (-9\right )} \log \left (\sqrt {x} e + d\right ) + {\left (420 \, d^{6} x e - 840 \, d^{7} \sqrt {x} - 280 \, d^{5} x^{\frac {3}{2}} e^{2} + 210 \, d^{4} x^{2} e^{3} - 168 \, d^{3} x^{\frac {5}{2}} e^{4} + 140 \, d^{2} x^{3} e^{5} - 120 \, d x^{\frac {7}{2}} e^{6} + 105 \, x^{4} e^{7}\right )} e^{\left (-8\right )}\right )} b n e \end {gather*}

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((sqrt(x)*e + d)^n*c) + 1/4*a*x^4 - 1/3360*(840*d^8*e^(-9)*log(sqrt(x)*e + d) + (420*d^6*x*e - 84
0*d^7*sqrt(x) - 280*d^5*x^(3/2)*e^2 + 210*d^4*x^2*e^3 - 168*d^3*x^(5/2)*e^4 + 140*d^2*x^3*e^5 - 120*d*x^(7/2)*
e^6 + 105*x^4*e^7)*e^(-8))*b*n*e

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Fricas [A]
time = 0.46, size = 137, normalized size = 0.83 \begin {gather*} -\frac {1}{3360} \, {\left (420 \, b d^{6} n x e^{2} + 210 \, b d^{4} n x^{2} e^{4} + 140 \, b d^{2} n x^{3} e^{6} - 840 \, b x^{4} e^{8} \log \left (c\right ) + 105 \, {\left (b n - 8 \, a\right )} x^{4} e^{8} + 840 \, {\left (b d^{8} n - b n x^{4} e^{8}\right )} \log \left (\sqrt {x} e + d\right ) - 8 \, {\left (105 \, b d^{7} n e + 35 \, b d^{5} n x e^{3} + 21 \, b d^{3} n x^{2} e^{5} + 15 \, b d n x^{3} e^{7}\right )} \sqrt {x}\right )} e^{\left (-8\right )} \end {gather*}

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*(420*b*d^6*n*x*e^2 + 210*b*d^4*n*x^2*e^4 + 140*b*d^2*n*x^3*e^6 - 840*b*x^4*e^8*log(c) + 105*(b*n - 8*a
)*x^4*e^8 + 840*(b*d^8*n - b*n*x^4*e^8)*log(sqrt(x)*e + d) - 8*(105*b*d^7*n*e + 35*b*d^5*n*x*e^3 + 21*b*d^3*n*
x^2*e^5 + 15*b*d*n*x^3*e^7)*sqrt(x))*e^(-8)

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Sympy [A]
time = 9.72, size = 155, normalized size = 0.93 \begin {gather*} \frac {a x^{4}}{4} + b \left (- \frac {e n \left (\frac {2 d^{8} \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^{8}} - \frac {2 d^{7} \sqrt {x}}{e^{8}} + \frac {d^{6} x}{e^{7}} - \frac {2 d^{5} x^{\frac {3}{2}}}{3 e^{6}} + \frac {d^{4} x^{2}}{2 e^{5}} - \frac {2 d^{3} x^{\frac {5}{2}}}{5 e^{4}} + \frac {d^{2} x^{3}}{3 e^{3}} - \frac {2 d x^{\frac {7}{2}}}{7 e^{2}} + \frac {x^{4}}{4 e}\right )}{8} + \frac {x^{4} \log {\left (c \left (d + e \sqrt {x}\right )^{n} \right )}}{4}\right ) \end {gather*}

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]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 357 vs. \(2 (128) = 256\).
time = 2.71, size = 357, normalized size = 2.15 \begin {gather*} \frac {1}{3360} \, {\left (840 \, b x^{4} e \log \left (c\right ) + 840 \, a x^{4} e + {\left (840 \, {\left (\sqrt {x} e + d\right )}^{8} e^{\left (-7\right )} \log \left (\sqrt {x} e + d\right ) - 6720 \, {\left (\sqrt {x} e + d\right )}^{7} d e^{\left (-7\right )} \log \left (\sqrt {x} e + d\right ) + 23520 \, {\left (\sqrt {x} e + d\right )}^{6} d^{2} e^{\left (-7\right )} \log \left (\sqrt {x} e + d\right ) - 47040 \, {\left (\sqrt {x} e + d\right )}^{5} d^{3} e^{\left (-7\right )} \log \left (\sqrt {x} e + d\right ) + 58800 \, {\left (\sqrt {x} e + d\right )}^{4} d^{4} e^{\left (-7\right )} \log \left (\sqrt {x} e + d\right ) - 47040 \, {\left (\sqrt {x} e + d\right )}^{3} d^{5} e^{\left (-7\right )} \log \left (\sqrt {x} e + d\right ) + 23520 \, {\left (\sqrt {x} e + d\right )}^{2} d^{6} e^{\left (-7\right )} \log \left (\sqrt {x} e + d\right ) - 6720 \, {\left (\sqrt {x} e + d\right )} d^{7} e^{\left (-7\right )} \log \left (\sqrt {x} e + d\right ) - 105 \, {\left (\sqrt {x} e + d\right )}^{8} e^{\left (-7\right )} + 960 \, {\left (\sqrt {x} e + d\right )}^{7} d e^{\left (-7\right )} - 3920 \, {\left (\sqrt {x} e + d\right )}^{6} d^{2} e^{\left (-7\right )} + 9408 \, {\left (\sqrt {x} e + d\right )}^{5} d^{3} e^{\left (-7\right )} - 14700 \, {\left (\sqrt {x} e + d\right )}^{4} d^{4} e^{\left (-7\right )} + 15680 \, {\left (\sqrt {x} e + d\right )}^{3} d^{5} e^{\left (-7\right )} - 11760 \, {\left (\sqrt {x} e + d\right )}^{2} d^{6} e^{\left (-7\right )} + 6720 \, {\left (\sqrt {x} e + d\right )} d^{7} e^{\left (-7\right )}\right )} b n\right )} e^{\left (-1\right )} \end {gather*}

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

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Mupad [B]
time = 0.51, size = 137, normalized size = 0.83 \begin {gather*} \frac {a\,x^4}{4}-\frac {b\,n\,x^4}{32}+\frac {b\,x^4\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{4}+\frac {b\,d\,n\,x^{7/2}}{28\,e}-\frac {b\,d^6\,n\,x}{8\,e^6}-\frac {b\,d^8\,n\,\ln \left (d+e\,\sqrt {x}\right )}{4\,e^8}-\frac {b\,d^2\,n\,x^3}{24\,e^2}-\frac {b\,d^4\,n\,x^2}{16\,e^4}+\frac {b\,d^3\,n\,x^{5/2}}{20\,e^3}+\frac {b\,d^5\,n\,x^{3/2}}{12\,e^5}+\frac {b\,d^7\,n\,\sqrt {x}}{4\,e^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + b*log(c*(d + e*x^(1/2))^n)),x)

[Out]

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

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