3.177 \(\int x^4 (d+e x^2) (a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=48 \[ \frac {1}{35} \left (7 d x^5+5 e x^7\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{25} b d n x^5-\frac {1}{49} b e n x^7 \]

[Out]

-1/25*b*d*n*x^5-1/49*b*e*n*x^7+1/35*(5*e*x^7+7*d*x^5)*(a+b*ln(c*x^n))

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Rubi [A]  time = 0.04, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {14, 2334} \[ \frac {1}{35} \left (7 d x^5+5 e x^7\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{25} b d n x^5-\frac {1}{49} b e n x^7 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*d*n*x^5)/25 - (b*e*n*x^7)/49 + ((7*d*x^5 + 5*e*x^7)*(a + b*Log[c*x^n]))/35

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2334

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] && IntegerQ[m] &&  !(EqQ[q, 1] && EqQ[m, -1])

Rubi steps

\begin {align*} \int x^4 \left (d+e x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {1}{35} \left (7 d x^5+5 e x^7\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (\frac {d x^4}{5}+\frac {e x^6}{7}\right ) \, dx\\ &=-\frac {1}{25} b d n x^5-\frac {1}{49} b e n x^7+\frac {1}{35} \left (7 d x^5+5 e x^7\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 69, normalized size = 1.44 \[ \frac {1}{5} a d x^5+\frac {1}{7} a e x^7+\frac {1}{5} b d x^5 \log \left (c x^n\right )+\frac {1}{7} b e x^7 \log \left (c x^n\right )-\frac {1}{25} b d n x^5-\frac {1}{49} b e n x^7 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*d*x^5)/5 - (b*d*n*x^5)/25 + (a*e*x^7)/7 - (b*e*n*x^7)/49 + (b*d*x^5*Log[c*x^n])/5 + (b*e*x^7*Log[c*x^n])/7

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fricas [A]  time = 0.53, size = 69, normalized size = 1.44 \[ -\frac {1}{49} \, {\left (b e n - 7 \, a e\right )} x^{7} - \frac {1}{25} \, {\left (b d n - 5 \, a d\right )} x^{5} + \frac {1}{35} \, {\left (5 \, b e x^{7} + 7 \, b d x^{5}\right )} \log \relax (c) + \frac {1}{35} \, {\left (5 \, b e n x^{7} + 7 \, b d n x^{5}\right )} \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/49*(b*e*n - 7*a*e)*x^7 - 1/25*(b*d*n - 5*a*d)*x^5 + 1/35*(5*b*e*x^7 + 7*b*d*x^5)*log(c) + 1/35*(5*b*e*n*x^7
+ 7*b*d*n*x^5)*log(x)

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giac [A]  time = 0.38, size = 73, normalized size = 1.52 \[ \frac {1}{7} \, b n x^{7} e \log \relax (x) - \frac {1}{49} \, b n x^{7} e + \frac {1}{7} \, b x^{7} e \log \relax (c) + \frac {1}{7} \, a x^{7} e + \frac {1}{5} \, b d n x^{5} \log \relax (x) - \frac {1}{25} \, b d n x^{5} + \frac {1}{5} \, b d x^{5} \log \relax (c) + \frac {1}{5} \, a d x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*b*n*x^7*e*log(x) - 1/49*b*n*x^7*e + 1/7*b*x^7*e*log(c) + 1/7*a*x^7*e + 1/5*b*d*n*x^5*log(x) - 1/25*b*d*n*x
^5 + 1/5*b*d*x^5*log(c) + 1/5*a*d*x^5

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maple [C]  time = 0.20, size = 266, normalized size = 5.54 \[ -\frac {i \pi b e \,x^{7} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{14}+\frac {i \pi b e \,x^{7} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{14}+\frac {i \pi b e \,x^{7} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{14}-\frac {i \pi b e \,x^{7} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{14}-\frac {i \pi b d \,x^{5} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{10}+\frac {i \pi b d \,x^{5} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{10}+\frac {i \pi b d \,x^{5} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{10}-\frac {i \pi b d \,x^{5} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{10}-\frac {b e n \,x^{7}}{49}+\frac {b e \,x^{7} \ln \relax (c )}{7}+\frac {a e \,x^{7}}{7}-\frac {b d n \,x^{5}}{25}+\frac {b d \,x^{5} \ln \relax (c )}{5}+\frac {a d \,x^{5}}{5}+\frac {\left (5 e \,x^{2}+7 d \right ) b \,x^{5} \ln \left (x^{n}\right )}{35} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/35*b*x^5*(5*e*x^2+7*d)*ln(x^n)+1/14*I*Pi*b*e*x^7*csgn(I*x^n)*csgn(I*c*x^n)^2-1/14*I*Pi*b*e*x^7*csgn(I*x^n)*c
sgn(I*c*x^n)*csgn(I*c)-1/14*I*Pi*b*e*x^7*csgn(I*c*x^n)^3+1/14*I*Pi*b*e*x^7*csgn(I*c*x^n)^2*csgn(I*c)+1/7*ln(c)
*b*e*x^7-1/49*b*e*n*x^7+1/7*a*e*x^7+1/10*I*Pi*b*d*x^5*csgn(I*x^n)*csgn(I*c*x^n)^2-1/10*I*Pi*b*d*x^5*csgn(I*x^n
)*csgn(I*c*x^n)*csgn(I*c)-1/10*I*Pi*b*d*x^5*csgn(I*c*x^n)^3+1/10*I*Pi*b*d*x^5*csgn(I*c*x^n)^2*csgn(I*c)+1/5*ln
(c)*b*d*x^5-1/25*b*d*n*x^5+1/5*a*d*x^5

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maxima [A]  time = 0.47, size = 57, normalized size = 1.19 \[ -\frac {1}{49} \, b e n x^{7} + \frac {1}{7} \, b e x^{7} \log \left (c x^{n}\right ) + \frac {1}{7} \, a e x^{7} - \frac {1}{25} \, b d n x^{5} + \frac {1}{5} \, b d x^{5} \log \left (c x^{n}\right ) + \frac {1}{5} \, a d x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/49*b*e*n*x^7 + 1/7*b*e*x^7*log(c*x^n) + 1/7*a*e*x^7 - 1/25*b*d*n*x^5 + 1/5*b*d*x^5*log(c*x^n) + 1/5*a*d*x^5

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mupad [B]  time = 3.35, size = 51, normalized size = 1.06 \[ \ln \left (c\,x^n\right )\,\left (\frac {b\,e\,x^7}{7}+\frac {b\,d\,x^5}{5}\right )+\frac {d\,x^5\,\left (5\,a-b\,n\right )}{25}+\frac {e\,x^7\,\left (7\,a-b\,n\right )}{49} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(c*x^n)*((b*d*x^5)/5 + (b*e*x^7)/7) + (d*x^5*(5*a - b*n))/25 + (e*x^7*(7*a - b*n))/49

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sympy [B]  time = 5.84, size = 87, normalized size = 1.81 \[ \frac {a d x^{5}}{5} + \frac {a e x^{7}}{7} + \frac {b d n x^{5} \log {\relax (x )}}{5} - \frac {b d n x^{5}}{25} + \frac {b d x^{5} \log {\relax (c )}}{5} + \frac {b e n x^{7} \log {\relax (x )}}{7} - \frac {b e n x^{7}}{49} + \frac {b e x^{7} \log {\relax (c )}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*d*x**5/5 + a*e*x**7/7 + b*d*n*x**5*log(x)/5 - b*d*n*x**5/25 + b*d*x**5*log(c)/5 + b*e*n*x**7*log(x)/7 - b*e*
n*x**7/49 + b*e*x**7*log(c)/7

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