\(\int x^4 (d+e x^2)^3 (a+b \log (c x^n)) \, dx\) [202]

Optimal result

Integrand size = 23, antiderivative size = 100 \[ \int x^4 \left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{25} b d^3 n x^5-\frac {3}{49} b d^2 e n x^7-\frac {1}{27} b d e^2 n x^9-\frac {1}{121} b e^3 n x^{11}+\frac {\left (231 d^3 x^5+495 d^2 e x^7+385 d e^2 x^9+105 e^3 x^{11}\right ) \left (a+b \log \left (c x^n\right )\right )}{1155} \]

[Out]

-1/25*b*d^3*n*x^5-3/49*b*d^2*e*n*x^7-1/27*b*d*e^2*n*x^9-1/121*b*e^3*n*x^11+1/1155*(105*e^3*x^11+385*d*e^2*x^9+
495*d^2*e*x^7+231*d^3*x^5)*(a+b*ln(c*x^n))

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {276, 2371} \[ \int x^4 \left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {\left (231 d^3 x^5+495 d^2 e x^7+385 d e^2 x^9+105 e^3 x^{11}\right ) \left (a+b \log \left (c x^n\right )\right )}{1155}-\frac {1}{25} b d^3 n x^5-\frac {3}{49} b d^2 e n x^7-\frac {1}{27} b d e^2 n x^9-\frac {1}{121} b e^3 n x^{11} \]

[In]

Int[x^4*(d + e*x^2)^3*(a + b*Log[c*x^n]),x]

[Out]

-1/25*(b*d^3*n*x^5) - (3*b*d^2*e*n*x^7)/49 - (b*d*e^2*n*x^9)/27 - (b*e^3*n*x^11)/121 + ((231*d^3*x^5 + 495*d^2
*e*x^7 + 385*d*e^2*x^9 + 105*e^3*x^11)*(a + b*Log[c*x^n]))/1155

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 2371

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = I
ntHide[x^m*(d + e*x^r)^q, x]}, Simp[u*(a + b*Log[c*x^n]), x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]
] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IGtQ[m, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (231 d^3 x^5+495 d^2 e x^7+385 d e^2 x^9+105 e^3 x^{11}\right ) \left (a+b \log \left (c x^n\right )\right )}{1155}-(b n) \int \left (\frac {d^3 x^4}{5}+\frac {3}{7} d^2 e x^6+\frac {1}{3} d e^2 x^8+\frac {e^3 x^{10}}{11}\right ) \, dx \\ & = -\frac {1}{25} b d^3 n x^5-\frac {3}{49} b d^2 e n x^7-\frac {1}{27} b d e^2 n x^9-\frac {1}{121} b e^3 n x^{11}+\frac {\left (231 d^3 x^5+495 d^2 e x^7+385 d e^2 x^9+105 e^3 x^{11}\right ) \left (a+b \log \left (c x^n\right )\right )}{1155} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.33 \[ \int x^4 \left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{25} b d^3 n x^5-\frac {3}{49} b d^2 e n x^7-\frac {1}{27} b d e^2 n x^9-\frac {1}{121} b e^3 n x^{11}+\frac {1}{5} d^3 x^5 \left (a+b \log \left (c x^n\right )\right )+\frac {3}{7} d^2 e x^7 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{3} d e^2 x^9 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{11} e^3 x^{11} \left (a+b \log \left (c x^n\right )\right ) \]

[In]

Integrate[x^4*(d + e*x^2)^3*(a + b*Log[c*x^n]),x]

[Out]

-1/25*(b*d^3*n*x^5) - (3*b*d^2*e*n*x^7)/49 - (b*d*e^2*n*x^9)/27 - (b*e^3*n*x^11)/121 + (d^3*x^5*(a + b*Log[c*x
^n]))/5 + (3*d^2*e*x^7*(a + b*Log[c*x^n]))/7 + (d*e^2*x^9*(a + b*Log[c*x^n]))/3 + (e^3*x^11*(a + b*Log[c*x^n])
)/11

Maple [A] (verified)

Time = 1.37 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.44

method	result	size
parallelrisch	\(\frac {x^{11} b \ln \left (c \,x^{n}\right ) e^{3}}{11}-\frac {b \,e^{3} n \,x^{11}}{121}+\frac {x^{11} a \,e^{3}}{11}+\frac {x^{9} b \ln \left (c \,x^{n}\right ) d \,e^{2}}{3}-\frac {b d \,e^{2} n \,x^{9}}{27}+\frac {x^{9} a d \,e^{2}}{3}+\frac {3 x^{7} b \ln \left (c \,x^{n}\right ) d^{2} e}{7}-\frac {3 b \,d^{2} e n \,x^{7}}{49}+\frac {3 x^{7} a \,d^{2} e}{7}+\frac {x^{5} b \ln \left (c \,x^{n}\right ) d^{3}}{5}-\frac {b \,d^{3} n \,x^{5}}{25}+\frac {x^{5} a \,d^{3}}{5}\)	\(144\)
risch	\(\frac {x^{11} a \,e^{3}}{11}+\frac {x^{5} a \,d^{3}}{5}+\frac {3 \ln \left (c \right ) b \,d^{2} e \,x^{7}}{7}+\frac {\ln \left (c \right ) b d \,e^{2} x^{9}}{3}-\frac {3 i \pi b \,d^{2} e \,x^{7} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{14}+\frac {i \pi b \,e^{3} x^{11} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{22}+\frac {x^{9} a d \,e^{2}}{3}+\frac {3 x^{7} a \,d^{2} e}{7}-\frac {i \pi b \,d^{3} x^{5} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{10}-\frac {i \pi b \,e^{3} x^{11} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{22}+\frac {i \pi b \,d^{3} x^{5} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{10}+\frac {3 i \pi b \,d^{2} e \,x^{7} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{14}+\frac {3 i \pi b \,d^{2} e \,x^{7} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{14}-\frac {i \pi b \,d^{3} x^{5} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{10}-\frac {3 i \pi b \,d^{2} e \,x^{7} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{14}-\frac {i \pi b d \,e^{2} x^{9} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{6}+\frac {b \,x^{5} \left (105 e^{3} x^{6}+385 e^{2} d \,x^{4}+495 d^{2} e \,x^{2}+231 d^{3}\right ) \ln \left (x^{n}\right )}{1155}+\frac {i \pi b d \,e^{2} x^{9} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{6}+\frac {i \pi b d \,e^{2} x^{9} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{6}-\frac {i \pi b \,e^{3} x^{11} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{22}+\frac {\ln \left (c \right ) b \,d^{3} x^{5}}{5}+\frac {\ln \left (c \right ) b \,e^{3} x^{11}}{11}+\frac {i \pi b \,e^{3} x^{11} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{22}-\frac {i \pi b d \,e^{2} x^{9} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{6}+\frac {i \pi b \,d^{3} x^{5} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{10}-\frac {3 b \,d^{2} e n \,x^{7}}{49}-\frac {b d \,e^{2} n \,x^{9}}{27}-\frac {b \,d^{3} n \,x^{5}}{25}-\frac {b \,e^{3} n \,x^{11}}{121}\)	\(602\)

[In]

int(x^4*(e*x^2+d)^3*(a+b*ln(c*x^n)),x,method=_RETURNVERBOSE)

[Out]

1/11*x^11*b*ln(c*x^n)*e^3-1/121*b*e^3*n*x^11+1/11*x^11*a*e^3+1/3*x^9*b*ln(c*x^n)*d*e^2-1/27*b*d*e^2*n*x^9+1/3*
x^9*a*d*e^2+3/7*x^7*b*ln(c*x^n)*d^2*e-3/49*b*d^2*e*n*x^7+3/7*x^7*a*d^2*e+1/5*x^5*b*ln(c*x^n)*d^3-1/25*b*d^3*n*
x^5+1/5*x^5*a*d^3

Fricas [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.67 \[ \int x^4 \left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{121} \, {\left (b e^{3} n - 11 \, a e^{3}\right )} x^{11} - \frac {1}{27} \, {\left (b d e^{2} n - 9 \, a d e^{2}\right )} x^{9} - \frac {3}{49} \, {\left (b d^{2} e n - 7 \, a d^{2} e\right )} x^{7} - \frac {1}{25} \, {\left (b d^{3} n - 5 \, a d^{3}\right )} x^{5} + \frac {1}{1155} \, {\left (105 \, b e^{3} x^{11} + 385 \, b d e^{2} x^{9} + 495 \, b d^{2} e x^{7} + 231 \, b d^{3} x^{5}\right )} \log \left (c\right ) + \frac {1}{1155} \, {\left (105 \, b e^{3} n x^{11} + 385 \, b d e^{2} n x^{9} + 495 \, b d^{2} e n x^{7} + 231 \, b d^{3} n x^{5}\right )} \log \left (x\right ) \]

[In]

integrate(x^4*(e*x^2+d)^3*(a+b*log(c*x^n)),x, algorithm="fricas")

[Out]

-1/121*(b*e^3*n - 11*a*e^3)*x^11 - 1/27*(b*d*e^2*n - 9*a*d*e^2)*x^9 - 3/49*(b*d^2*e*n - 7*a*d^2*e)*x^7 - 1/25*
(b*d^3*n - 5*a*d^3)*x^5 + 1/1155*(105*b*e^3*x^11 + 385*b*d*e^2*x^9 + 495*b*d^2*e*x^7 + 231*b*d^3*x^5)*log(c) +
1/1155*(105*b*e^3*n*x^11 + 385*b*d*e^2*n*x^9 + 495*b*d^2*e*n*x^7 + 231*b*d^3*n*x^5)*log(x)

Sympy [A] (verification not implemented)

Time = 2.28 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.70 \[ \int x^4 \left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {a d^{3} x^{5}}{5} + \frac {3 a d^{2} e x^{7}}{7} + \frac {a d e^{2} x^{9}}{3} + \frac {a e^{3} x^{11}}{11} - \frac {b d^{3} n x^{5}}{25} + \frac {b d^{3} x^{5} \log {\left (c x^{n} \right )}}{5} - \frac {3 b d^{2} e n x^{7}}{49} + \frac {3 b d^{2} e x^{7} \log {\left (c x^{n} \right )}}{7} - \frac {b d e^{2} n x^{9}}{27} + \frac {b d e^{2} x^{9} \log {\left (c x^{n} \right )}}{3} - \frac {b e^{3} n x^{11}}{121} + \frac {b e^{3} x^{11} \log {\left (c x^{n} \right )}}{11} \]

[In]

integrate(x**4*(e*x**2+d)**3*(a+b*ln(c*x**n)),x)

[Out]

a*d**3*x**5/5 + 3*a*d**2*e*x**7/7 + a*d*e**2*x**9/3 + a*e**3*x**11/11 - b*d**3*n*x**5/25 + b*d**3*x**5*log(c*x
**n)/5 - 3*b*d**2*e*n*x**7/49 + 3*b*d**2*e*x**7*log(c*x**n)/7 - b*d*e**2*n*x**9/27 + b*d*e**2*x**9*log(c*x**n)
/3 - b*e**3*n*x**11/121 + b*e**3*x**11*log(c*x**n)/11

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.43 \[ \int x^4 \left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{121} \, b e^{3} n x^{11} + \frac {1}{11} \, b e^{3} x^{11} \log \left (c x^{n}\right ) + \frac {1}{11} \, a e^{3} x^{11} - \frac {1}{27} \, b d e^{2} n x^{9} + \frac {1}{3} \, b d e^{2} x^{9} \log \left (c x^{n}\right ) + \frac {1}{3} \, a d e^{2} x^{9} - \frac {3}{49} \, b d^{2} e n x^{7} + \frac {3}{7} \, b d^{2} e x^{7} \log \left (c x^{n}\right ) + \frac {3}{7} \, a d^{2} e x^{7} - \frac {1}{25} \, b d^{3} n x^{5} + \frac {1}{5} \, b d^{3} x^{5} \log \left (c x^{n}\right ) + \frac {1}{5} \, a d^{3} x^{5} \]

[In]

integrate(x^4*(e*x^2+d)^3*(a+b*log(c*x^n)),x, algorithm="maxima")

[Out]

-1/121*b*e^3*n*x^11 + 1/11*b*e^3*x^11*log(c*x^n) + 1/11*a*e^3*x^11 - 1/27*b*d*e^2*n*x^9 + 1/3*b*d*e^2*x^9*log(
c*x^n) + 1/3*a*d*e^2*x^9 - 3/49*b*d^2*e*n*x^7 + 3/7*b*d^2*e*x^7*log(c*x^n) + 3/7*a*d^2*e*x^7 - 1/25*b*d^3*n*x^
5 + 1/5*b*d^3*x^5*log(c*x^n) + 1/5*a*d^3*x^5

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.77 \[ \int x^4 \left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {1}{11} \, b e^{3} n x^{11} \log \left (x\right ) - \frac {1}{121} \, b e^{3} n x^{11} + \frac {1}{11} \, b e^{3} x^{11} \log \left (c\right ) + \frac {1}{11} \, a e^{3} x^{11} + \frac {1}{3} \, b d e^{2} n x^{9} \log \left (x\right ) - \frac {1}{27} \, b d e^{2} n x^{9} + \frac {1}{3} \, b d e^{2} x^{9} \log \left (c\right ) + \frac {1}{3} \, a d e^{2} x^{9} + \frac {3}{7} \, b d^{2} e n x^{7} \log \left (x\right ) - \frac {3}{49} \, b d^{2} e n x^{7} + \frac {3}{7} \, b d^{2} e x^{7} \log \left (c\right ) + \frac {3}{7} \, a d^{2} e x^{7} + \frac {1}{5} \, b d^{3} n x^{5} \log \left (x\right ) - \frac {1}{25} \, b d^{3} n x^{5} + \frac {1}{5} \, b d^{3} x^{5} \log \left (c\right ) + \frac {1}{5} \, a d^{3} x^{5} \]

[In]

integrate(x^4*(e*x^2+d)^3*(a+b*log(c*x^n)),x, algorithm="giac")

[Out]

1/11*b*e^3*n*x^11*log(x) - 1/121*b*e^3*n*x^11 + 1/11*b*e^3*x^11*log(c) + 1/11*a*e^3*x^11 + 1/3*b*d*e^2*n*x^9*l
og(x) - 1/27*b*d*e^2*n*x^9 + 1/3*b*d*e^2*x^9*log(c) + 1/3*a*d*e^2*x^9 + 3/7*b*d^2*e*n*x^7*log(x) - 3/49*b*d^2*
e*n*x^7 + 3/7*b*d^2*e*x^7*log(c) + 3/7*a*d^2*e*x^7 + 1/5*b*d^3*n*x^5*log(x) - 1/25*b*d^3*n*x^5 + 1/5*b*d^3*x^5
*log(c) + 1/5*a*d^3*x^5

Mupad [B] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.13 \[ \int x^4 \left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\ln \left (c\,x^n\right )\,\left (\frac {b\,d^3\,x^5}{5}+\frac {3\,b\,d^2\,e\,x^7}{7}+\frac {b\,d\,e^2\,x^9}{3}+\frac {b\,e^3\,x^{11}}{11}\right )+\frac {d^3\,x^5\,\left (5\,a-b\,n\right )}{25}+\frac {e^3\,x^{11}\,\left (11\,a-b\,n\right )}{121}+\frac {3\,d^2\,e\,x^7\,\left (7\,a-b\,n\right )}{49}+\frac {d\,e^2\,x^9\,\left (9\,a-b\,n\right )}{27} \]

[In]

int(x^4*(d + e*x^2)^3*(a + b*log(c*x^n)),x)

[Out]

log(c*x^n)*((b*d^3*x^5)/5 + (b*e^3*x^11)/11 + (3*b*d^2*e*x^7)/7 + (b*d*e^2*x^9)/3) + (d^3*x^5*(5*a - b*n))/25
+ (e^3*x^11*(11*a - b*n))/121 + (3*d^2*e*x^7*(7*a - b*n))/49 + (d*e^2*x^9*(9*a - b*n))/27