Integrand size = 22, antiderivative size = 166 \[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \, dx=\frac {b d^7 n \sqrt {x}}{4 e^7}-\frac {b d^6 n x}{8 e^6}+\frac {b d^5 n x^{3/2}}{12 e^5}-\frac {b d^4 n x^2}{16 e^4}+\frac {b d^3 n x^{5/2}}{20 e^3}-\frac {b d^2 n x^3}{24 e^2}+\frac {b d n x^{7/2}}{28 e}-\frac {1}{32} b n x^4-\frac {b d^8 n \log \left (d+e \sqrt {x}\right )}{4 e^8}+\frac {1}{4} x^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \]
-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
Time = 0.09 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.95 \[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \, dx=\frac {a x^4}{4}-\frac {1}{8} b e n \left (-\frac {2 d^7 \sqrt {x}}{e^8}+\frac {d^6 x}{e^7}-\frac {2 d^5 x^{3/2}}{3 e^6}+\frac {d^4 x^2}{2 e^5}-\frac {2 d^3 x^{5/2}}{5 e^4}+\frac {d^2 x^3}{3 e^3}-\frac {2 d x^{7/2}}{7 e^2}+\frac {x^4}{4 e}+\frac {2 d^8 \log \left (d+e \sqrt {x}\right )}{e^9}\right )+\frac {1}{4} b x^4 \log \left (c \left (d+e \sqrt {x}\right )^n\right ) \]
(a*x^4)/4 - (b*e*n*((-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) + (2*d^8*Log[d + e*Sqrt[x]])/e^9))/8 + (b*x^4*Log[c*(d + e*Sqrt[x])^n])/4
Time = 0.32 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.94, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2904, 2842, 49, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \, dx\) |
\(\Big \downarrow \) 2904 |
\(\displaystyle 2 \int x^{7/2} \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )d\sqrt {x}\) |
\(\Big \downarrow \) 2842 |
\(\displaystyle 2 \left (\frac {1}{8} x^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )-\frac {1}{8} b e n \int \frac {x^4}{d+e \sqrt {x}}d\sqrt {x}\right )\) |
\(\Big \downarrow \) 49 |
\(\displaystyle 2 \left (\frac {1}{8} x^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )-\frac {1}{8} b e n \int \left (\frac {d^8}{e^8 \left (d+e \sqrt {x}\right )}-\frac {d^7}{e^8}+\frac {\sqrt {x} d^6}{e^7}-\frac {x d^5}{e^6}+\frac {x^{3/2} d^4}{e^5}-\frac {x^2 d^3}{e^4}+\frac {x^{5/2} d^2}{e^3}-\frac {x^3 d}{e^2}+\frac {x^{7/2}}{e}\right )d\sqrt {x}\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (\frac {1}{8} x^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )-\frac {1}{8} b e n \left (\frac {d^8 \log \left (d+e \sqrt {x}\right )}{e^9}-\frac {d^7 \sqrt {x}}{e^8}+\frac {d^6 x}{2 e^7}-\frac {d^5 x^{3/2}}{3 e^6}+\frac {d^4 x^2}{4 e^5}-\frac {d^3 x^{5/2}}{5 e^4}+\frac {d^2 x^3}{6 e^3}-\frac {d x^{7/2}}{7 e^2}+\frac {x^4}{8 e}\right )\right )\) |
2*(-1/8*(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)) + (x^4*(a + b*Log[c*(d + e*Sqrt[x])^n]))/8)
3.4.100.3.1 Defintions of rubi rules used
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}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
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] - Simp[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]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m _.), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*L og[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])
\[\int x^{3} \left (a +b \ln \left (c \left (d +e \sqrt {x}\right )^{n}\right )\right )d x\]
Time = 0.32 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.89 \[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \, dx=\frac {840 \, b e^{8} x^{4} \log \left (c\right ) - 140 \, b d^{2} e^{6} n x^{3} - 210 \, b d^{4} e^{4} n x^{2} - 420 \, b d^{6} e^{2} n x - 105 \, {\left (b e^{8} n - 8 \, a e^{8}\right )} x^{4} + 840 \, {\left (b e^{8} n x^{4} - b d^{8} n\right )} \log \left (e \sqrt {x} + d\right ) + 8 \, {\left (15 \, b d e^{7} n x^{3} + 21 \, b d^{3} e^{5} n x^{2} + 35 \, b d^{5} e^{3} n x + 105 \, b d^{7} e n\right )} \sqrt {x}}{3360 \, e^{8}} \]
1/3360*(840*b*e^8*x^4*log(c) - 140*b*d^2*e^6*n*x^3 - 210*b*d^4*e^4*n*x^2 - 420*b*d^6*e^2*n*x - 105*(b*e^8*n - 8*a*e^8)*x^4 + 840*(b*e^8*n*x^4 - b*d^ 8*n)*log(e*sqrt(x) + d) + 8*(15*b*d*e^7*n*x^3 + 21*b*d^3*e^5*n*x^2 + 35*b* d^5*e^3*n*x + 105*b*d^7*e*n)*sqrt(x))/e^8
Time = 8.47 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.93 \[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \, dx=\frac {a x^{4}}{4} + b \left (- \frac {e n \left (\frac {2 d^{8} \left (\begin {cases} \frac {\sqrt {x}}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e \sqrt {x} \right )}}{e} & \text {otherwise} \end {cases}\right )}{e^{8}} - \frac {2 d^{7} \sqrt {x}}{e^{8}} + \frac {d^{6} x}{e^{7}} - \frac {2 d^{5} x^{\frac {3}{2}}}{3 e^{6}} + \frac {d^{4} x^{2}}{2 e^{5}} - \frac {2 d^{3} x^{\frac {5}{2}}}{5 e^{4}} + \frac {d^{2} x^{3}}{3 e^{3}} - \frac {2 d x^{\frac {7}{2}}}{7 e^{2}} + \frac {x^{4}}{4 e}\right )}{8} + \frac {x^{4} \log {\left (c \left (d + e \sqrt {x}\right )^{n} \right )}}{4}\right ) \]
a*x**4/4 + b*(-e*n*(2*d**8*Piecewise((sqrt(x)/d, Eq(e, 0)), (log(d + e*sqr t(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)
Time = 0.19 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.77 \[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \, dx=\frac {1}{4} \, b x^{4} \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + \frac {1}{4} \, a x^{4} - \frac {1}{3360} \, b e n {\left (\frac {840 \, d^{8} \log \left (e \sqrt {x} + d\right )}{e^{9}} + \frac {105 \, e^{7} x^{4} - 120 \, d e^{6} x^{\frac {7}{2}} + 140 \, d^{2} e^{5} x^{3} - 168 \, d^{3} e^{4} x^{\frac {5}{2}} + 210 \, d^{4} e^{3} x^{2} - 280 \, d^{5} e^{2} x^{\frac {3}{2}} + 420 \, d^{6} e x - 840 \, d^{7} \sqrt {x}}{e^{8}}\right )} \]
1/4*b*x^4*log((e*sqrt(x) + d)^n*c) + 1/4*a*x^4 - 1/3360*b*e*n*(840*d^8*log (e*sqrt(x) + d)/e^9 + (105*e^7*x^4 - 120*d*e^6*x^(7/2) + 140*d^2*e^5*x^3 - 168*d^3*e^4*x^(5/2) + 210*d^4*e^3*x^2 - 280*d^5*e^2*x^(3/2) + 420*d^6*e*x - 840*d^7*sqrt(x))/e^8)
Leaf count of result is larger than twice the leaf count of optimal. 348 vs. \(2 (134) = 268\).
Time = 0.32 (sec) , antiderivative size = 348, normalized size of antiderivative = 2.10 \[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \, dx=\frac {840 \, b e x^{4} \log \left (c\right ) + 840 \, a e x^{4} + {\left (\frac {840 \, {\left (e \sqrt {x} + d\right )}^{8} \log \left (e \sqrt {x} + d\right )}{e^{7}} - \frac {6720 \, {\left (e \sqrt {x} + d\right )}^{7} d \log \left (e \sqrt {x} + d\right )}{e^{7}} + \frac {23520 \, {\left (e \sqrt {x} + d\right )}^{6} d^{2} \log \left (e \sqrt {x} + d\right )}{e^{7}} - \frac {47040 \, {\left (e \sqrt {x} + d\right )}^{5} d^{3} \log \left (e \sqrt {x} + d\right )}{e^{7}} + \frac {58800 \, {\left (e \sqrt {x} + d\right )}^{4} d^{4} \log \left (e \sqrt {x} + d\right )}{e^{7}} - \frac {47040 \, {\left (e \sqrt {x} + d\right )}^{3} d^{5} \log \left (e \sqrt {x} + d\right )}{e^{7}} + \frac {23520 \, {\left (e \sqrt {x} + d\right )}^{2} d^{6} \log \left (e \sqrt {x} + d\right )}{e^{7}} - \frac {6720 \, {\left (e \sqrt {x} + d\right )} d^{7} \log \left (e \sqrt {x} + d\right )}{e^{7}} - \frac {105 \, {\left (e \sqrt {x} + d\right )}^{8}}{e^{7}} + \frac {960 \, {\left (e \sqrt {x} + d\right )}^{7} d}{e^{7}} - \frac {3920 \, {\left (e \sqrt {x} + d\right )}^{6} d^{2}}{e^{7}} + \frac {9408 \, {\left (e \sqrt {x} + d\right )}^{5} d^{3}}{e^{7}} - \frac {14700 \, {\left (e \sqrt {x} + d\right )}^{4} d^{4}}{e^{7}} + \frac {15680 \, {\left (e \sqrt {x} + d\right )}^{3} d^{5}}{e^{7}} - \frac {11760 \, {\left (e \sqrt {x} + d\right )}^{2} d^{6}}{e^{7}} + \frac {6720 \, {\left (e \sqrt {x} + d\right )} d^{7}}{e^{7}}\right )} b n}{3360 \, e} \]
1/3360*(840*b*e*x^4*log(c) + 840*a*e*x^4 + (840*(e*sqrt(x) + d)^8*log(e*sq rt(x) + d)/e^7 - 6720*(e*sqrt(x) + d)^7*d*log(e*sqrt(x) + d)/e^7 + 23520*( e*sqrt(x) + d)^6*d^2*log(e*sqrt(x) + d)/e^7 - 47040*(e*sqrt(x) + d)^5*d^3* log(e*sqrt(x) + d)/e^7 + 58800*(e*sqrt(x) + d)^4*d^4*log(e*sqrt(x) + d)/e^ 7 - 47040*(e*sqrt(x) + d)^3*d^5*log(e*sqrt(x) + d)/e^7 + 23520*(e*sqrt(x) + d)^2*d^6*log(e*sqrt(x) + d)/e^7 - 6720*(e*sqrt(x) + d)*d^7*log(e*sqrt(x) + d)/e^7 - 105*(e*sqrt(x) + d)^8/e^7 + 960*(e*sqrt(x) + d)^7*d/e^7 - 3920 *(e*sqrt(x) + d)^6*d^2/e^7 + 9408*(e*sqrt(x) + d)^5*d^3/e^7 - 14700*(e*sqr t(x) + d)^4*d^4/e^7 + 15680*(e*sqrt(x) + d)^3*d^5/e^7 - 11760*(e*sqrt(x) + d)^2*d^6/e^7 + 6720*(e*sqrt(x) + d)*d^7/e^7)*b*n)/e
Time = 1.80 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.83 \[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \, dx=\frac {a\,x^4}{4}-\frac {b\,n\,x^4}{32}+\frac {b\,x^4\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{4}+\frac {b\,d\,n\,x^{7/2}}{28\,e}-\frac {b\,d^6\,n\,x}{8\,e^6}-\frac {b\,d^8\,n\,\ln \left (d+e\,\sqrt {x}\right )}{4\,e^8}-\frac {b\,d^2\,n\,x^3}{24\,e^2}-\frac {b\,d^4\,n\,x^2}{16\,e^4}+\frac {b\,d^3\,n\,x^{5/2}}{20\,e^3}+\frac {b\,d^5\,n\,x^{3/2}}{12\,e^5}+\frac {b\,d^7\,n\,\sqrt {x}}{4\,e^7} \]