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

Optimal. Leaf size=74 \[ \frac{1}{315} \left (63 d^2 x^5+90 d e x^7+35 e^2 x^9\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{25} b d^2 n x^5-\frac{2}{49} b d e n x^7-\frac{1}{81} b e^2 n x^9 \]

[Out]

-(b*d^2*n*x^5)/25 - (2*b*d*e*n*x^7)/49 - (b*e^2*n*x^9)/81 + ((63*d^2*x^5 + 90*d*e*x^7 + 35*e^2*x^9)*(a + b*Log
[c*x^n]))/315

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Rubi [A]  time = 0.0723918, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {270, 2334} \[ \frac{1}{315} \left (63 d^2 x^5+90 d e x^7+35 e^2 x^9\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{25} b d^2 n x^5-\frac{2}{49} b d e n x^7-\frac{1}{81} b e^2 n x^9 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*d^2*n*x^5)/25 - (2*b*d*e*n*x^7)/49 - (b*e^2*n*x^9)/81 + ((63*d^2*x^5 + 90*d*e*x^7 + 35*e^2*x^9)*(a + b*Log
[c*x^n]))/315

Rule 270

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 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 )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{1}{315} \left (63 d^2 x^5+90 d e x^7+35 e^2 x^9\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (\frac{d^2 x^4}{5}+\frac{2}{7} d e x^6+\frac{e^2 x^8}{9}\right ) \, dx\\ &=-\frac{1}{25} b d^2 n x^5-\frac{2}{49} b d e n x^7-\frac{1}{81} b e^2 n x^9+\frac{1}{315} \left (63 d^2 x^5+90 d e x^7+35 e^2 x^9\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.0352158, size = 95, normalized size = 1.28 \[ \frac{1}{5} d^2 x^5 \left (a+b \log \left (c x^n\right )\right )+\frac{2}{7} d e x^7 \left (a+b \log \left (c x^n\right )\right )+\frac{1}{9} e^2 x^9 \left (a+b \log \left (c x^n\right )\right )-\frac{1}{25} b d^2 n x^5-\frac{2}{49} b d e n x^7-\frac{1}{81} b e^2 n x^9 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*d^2*n*x^5)/25 - (2*b*d*e*n*x^7)/49 - (b*e^2*n*x^9)/81 + (d^2*x^5*(a + b*Log[c*x^n]))/5 + (2*d*e*x^7*(a + b
*Log[c*x^n]))/7 + (e^2*x^9*(a + b*Log[c*x^n]))/9

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Maple [C]  time = 0.195, size = 434, normalized size = 5.9 \begin{align*}{\frac{b{x}^{5} \left ( 35\,{e}^{2}{x}^{4}+90\,de{x}^{2}+63\,{d}^{2} \right ) \ln \left ({x}^{n} \right ) }{315}}+{\frac{i}{10}}\pi \,b{d}^{2}{x}^{5} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -{\frac{i}{18}}\pi \,b{e}^{2}{x}^{9}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) +{\frac{i}{18}}\pi \,b{e}^{2}{x}^{9} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -{\frac{i}{7}}\pi \,bde{x}^{7} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+{\frac{\ln \left ( c \right ) b{e}^{2}{x}^{9}}{9}}-{\frac{b{e}^{2}n{x}^{9}}{81}}+{\frac{a{e}^{2}{x}^{9}}{9}}+{\frac{i}{7}}\pi \,bde{x}^{7} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -{\frac{i}{7}}\pi \,bde{x}^{7}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) +{\frac{i}{7}}\pi \,bde{x}^{7}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+{\frac{i}{10}}\pi \,b{d}^{2}{x}^{5}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+{\frac{2\,\ln \left ( c \right ) bde{x}^{7}}{7}}-{\frac{2\,bden{x}^{7}}{49}}+{\frac{2\,ade{x}^{7}}{7}}-{\frac{i}{18}}\pi \,b{e}^{2}{x}^{9} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+{\frac{i}{18}}\pi \,b{e}^{2}{x}^{9}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-{\frac{i}{10}}\pi \,b{d}^{2}{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 \,b{d}^{2}{x}^{5} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+{\frac{\ln \left ( c \right ) b{d}^{2}{x}^{5}}{5}}-{\frac{b{d}^{2}n{x}^{5}}{25}}+{\frac{a{d}^{2}{x}^{5}}{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/315*b*x^5*(35*e^2*x^4+90*d*e*x^2+63*d^2)*ln(x^n)+1/10*I*Pi*b*d^2*x^5*csgn(I*c*x^n)^2*csgn(I*c)-1/18*I*Pi*b*e
^2*x^9*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+1/18*I*Pi*b*e^2*x^9*csgn(I*c*x^n)^2*csgn(I*c)-1/7*I*Pi*b*d*e*x^7*cs
gn(I*c*x^n)^3+1/9*ln(c)*b*e^2*x^9-1/81*b*e^2*n*x^9+1/9*a*e^2*x^9+1/7*I*Pi*b*d*e*x^7*csgn(I*c*x^n)^2*csgn(I*c)-
1/7*I*Pi*b*d*e*x^7*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+1/7*I*Pi*b*d*e*x^7*csgn(I*x^n)*csgn(I*c*x^n)^2+1/10*I*P
i*b*d^2*x^5*csgn(I*x^n)*csgn(I*c*x^n)^2+2/7*ln(c)*b*d*e*x^7-2/49*b*d*e*n*x^7+2/7*a*d*e*x^7-1/18*I*Pi*b*e^2*x^9
*csgn(I*c*x^n)^3+1/18*I*Pi*b*e^2*x^9*csgn(I*x^n)*csgn(I*c*x^n)^2-1/10*I*Pi*b*d^2*x^5*csgn(I*x^n)*csgn(I*c*x^n)
*csgn(I*c)-1/10*I*Pi*b*d^2*x^5*csgn(I*c*x^n)^3+1/5*ln(c)*b*d^2*x^5-1/25*b*d^2*n*x^5+1/5*a*d^2*x^5

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Maxima [A]  time = 1.10956, size = 135, normalized size = 1.82 \begin{align*} -\frac{1}{81} \, b e^{2} n x^{9} + \frac{1}{9} \, b e^{2} x^{9} \log \left (c x^{n}\right ) + \frac{1}{9} \, a e^{2} x^{9} - \frac{2}{49} \, b d e n x^{7} + \frac{2}{7} \, b d e x^{7} \log \left (c x^{n}\right ) + \frac{2}{7} \, a d e x^{7} - \frac{1}{25} \, b d^{2} n x^{5} + \frac{1}{5} \, b d^{2} x^{5} \log \left (c x^{n}\right ) + \frac{1}{5} \, a d^{2} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/81*b*e^2*n*x^9 + 1/9*b*e^2*x^9*log(c*x^n) + 1/9*a*e^2*x^9 - 2/49*b*d*e*n*x^7 + 2/7*b*d*e*x^7*log(c*x^n) + 2
/7*a*d*e*x^7 - 1/25*b*d^2*n*x^5 + 1/5*b*d^2*x^5*log(c*x^n) + 1/5*a*d^2*x^5

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Fricas [A]  time = 1.34951, size = 296, normalized size = 4. \begin{align*} -\frac{1}{81} \,{\left (b e^{2} n - 9 \, a e^{2}\right )} x^{9} - \frac{2}{49} \,{\left (b d e n - 7 \, a d e\right )} x^{7} - \frac{1}{25} \,{\left (b d^{2} n - 5 \, a d^{2}\right )} x^{5} + \frac{1}{315} \,{\left (35 \, b e^{2} x^{9} + 90 \, b d e x^{7} + 63 \, b d^{2} x^{5}\right )} \log \left (c\right ) + \frac{1}{315} \,{\left (35 \, b e^{2} n x^{9} + 90 \, b d e n x^{7} + 63 \, b d^{2} 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)^2*(a+b*log(c*x^n)),x, algorithm="fricas")

[Out]

-1/81*(b*e^2*n - 9*a*e^2)*x^9 - 2/49*(b*d*e*n - 7*a*d*e)*x^7 - 1/25*(b*d^2*n - 5*a*d^2)*x^5 + 1/315*(35*b*e^2*
x^9 + 90*b*d*e*x^7 + 63*b*d^2*x^5)*log(c) + 1/315*(35*b*e^2*n*x^9 + 90*b*d*e*n*x^7 + 63*b*d^2*n*x^5)*log(x)

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Sympy [B]  time = 25.8695, size = 158, normalized size = 2.14 \begin{align*} \frac{a d^{2} x^{5}}{5} + \frac{2 a d e x^{7}}{7} + \frac{a e^{2} x^{9}}{9} + \frac{b d^{2} n x^{5} \log{\left (x \right )}}{5} - \frac{b d^{2} n x^{5}}{25} + \frac{b d^{2} x^{5} \log{\left (c \right )}}{5} + \frac{2 b d e n x^{7} \log{\left (x \right )}}{7} - \frac{2 b d e n x^{7}}{49} + \frac{2 b d e x^{7} \log{\left (c \right )}}{7} + \frac{b e^{2} n x^{9} \log{\left (x \right )}}{9} - \frac{b e^{2} n x^{9}}{81} + \frac{b e^{2} x^{9} \log{\left (c \right )}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*d**2*x**5/5 + 2*a*d*e*x**7/7 + a*e**2*x**9/9 + b*d**2*n*x**5*log(x)/5 - b*d**2*n*x**5/25 + b*d**2*x**5*log(c
)/5 + 2*b*d*e*n*x**7*log(x)/7 - 2*b*d*e*n*x**7/49 + 2*b*d*e*x**7*log(c)/7 + b*e**2*n*x**9*log(x)/9 - b*e**2*n*
x**9/81 + b*e**2*x**9*log(c)/9

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Giac [A]  time = 1.29064, size = 166, normalized size = 2.24 \begin{align*} \frac{1}{9} \, b n x^{9} e^{2} \log \left (x\right ) - \frac{1}{81} \, b n x^{9} e^{2} + \frac{1}{9} \, b x^{9} e^{2} \log \left (c\right ) + \frac{2}{7} \, b d n x^{7} e \log \left (x\right ) + \frac{1}{9} \, a x^{9} e^{2} - \frac{2}{49} \, b d n x^{7} e + \frac{2}{7} \, b d x^{7} e \log \left (c\right ) + \frac{2}{7} \, a d x^{7} e + \frac{1}{5} \, b d^{2} n x^{5} \log \left (x\right ) - \frac{1}{25} \, b d^{2} n x^{5} + \frac{1}{5} \, b d^{2} x^{5} \log \left (c\right ) + \frac{1}{5} \, a d^{2} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*b*n*x^9*e^2*log(x) - 1/81*b*n*x^9*e^2 + 1/9*b*x^9*e^2*log(c) + 2/7*b*d*n*x^7*e*log(x) + 1/9*a*x^9*e^2 - 2/
49*b*d*n*x^7*e + 2/7*b*d*x^7*e*log(c) + 2/7*a*d*x^7*e + 1/5*b*d^2*n*x^5*log(x) - 1/25*b*d^2*n*x^5 + 1/5*b*d^2*
x^5*log(c) + 1/5*a*d^2*x^5