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]

-(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

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Rubi [A]  time = 0.0431169, antiderivative size = 48, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0022063, 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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Maple [C]  time = 0.187, size = 266, normalized size = 5.5 \begin{align*}{\frac{b{x}^{5} \left ( 5\,e{x}^{2}+7\,d \right ) \ln \left ({x}^{n} \right ) }{35}}+{\frac{i}{14}}\pi \,be{x}^{7}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-{\frac{i}{14}}\pi \,be{x}^{7}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -{\frac{i}{14}}\pi \,be{x}^{7} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+{\frac{i}{14}}\pi \,be{x}^{7} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +{\frac{\ln \left ( c \right ) be{x}^{7}}{7}}-{\frac{ben{x}^{7}}{49}}+{\frac{ae{x}^{7}}{7}}+{\frac{i}{10}}\pi \,bd{x}^{5}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-{\frac{i}{10}}\pi \,bd{x}^{5}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -{\frac{i}{10}}\pi \,bd{x}^{5} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+{\frac{i}{10}}\pi \,bd{x}^{5} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +{\frac{\ln \left ( c \right ) bd{x}^{5}}{5}}-{\frac{bdn{x}^{5}}{25}}+{\frac{ad{x}^{5}}{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(e*x^2+d)*(a+b*ln(c*x^n)),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 = 1.08784, size = 77, normalized size = 1.6 \begin{align*} -\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} \end{align*}

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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Fricas [A]  time = 1.22252, size = 181, normalized size = 3.77 \begin{align*} -\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 \left (c\right ) + \frac{1}{35} \,{\left (5 \, b e n x^{7} + 7 \, b d n x^{5}\right )} \log \left (x\right ) \end{align*}

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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Sympy [B]  time = 9.31182, size = 87, normalized size = 1.81 \begin{align*} \frac{a d x^{5}}{5} + \frac{a e x^{7}}{7} + \frac{b d n x^{5} \log{\left (x \right )}}{5} - \frac{b d n x^{5}}{25} + \frac{b d x^{5} \log{\left (c \right )}}{5} + \frac{b e n x^{7} \log{\left (x \right )}}{7} - \frac{b e n x^{7}}{49} + \frac{b e x^{7} \log{\left (c \right )}}{7} \end{align*}

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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Giac [A]  time = 1.28399, size = 99, normalized size = 2.06 \begin{align*} \frac{1}{7} \, b n x^{7} e \log \left (x\right ) - \frac{1}{49} \, b n x^{7} e + \frac{1}{7} \, b x^{7} e \log \left (c\right ) + \frac{1}{7} \, a x^{7} e + \frac{1}{5} \, b d n x^{5} \log \left (x\right ) - \frac{1}{25} \, b d n x^{5} + \frac{1}{5} \, b d x^{5} \log \left (c\right ) + \frac{1}{5} \, a d x^{5} \end{align*}

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