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

Optimal. Leaf size=74 \[ \frac{1}{105} \left (35 d^2 x^3+42 d e x^5+15 e^2 x^7\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{9} b d^2 n x^3-\frac{2}{25} b d e n x^5-\frac{1}{49} b e^2 n x^7 \]

[Out]

-(b*d^2*n*x^3)/9 - (2*b*d*e*n*x^5)/25 - (b*e^2*n*x^7)/49 + ((35*d^2*x^3 + 42*d*e*x^5 + 15*e^2*x^7)*(a + b*Log[
c*x^n]))/105

________________________________________________________________________________________

Rubi [A]  time = 0.0711839, 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} \[ \frac{1}{105} \left (35 d^2 x^3+42 d e x^5+15 e^2 x^7\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{9} b d^2 n x^3-\frac{2}{25} b d e n x^5-\frac{1}{49} b e^2 n x^7 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*d^2*n*x^3)/9 - (2*b*d*e*n*x^5)/25 - (b*e^2*n*x^7)/49 + ((35*d^2*x^3 + 42*d*e*x^5 + 15*e^2*x^7)*(a + b*Log[
c*x^n]))/105

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^2 \left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{1}{105} \left (35 d^2 x^3+42 d e x^5+15 e^2 x^7\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (\frac{d^2 x^2}{3}+\frac{2}{5} d e x^4+\frac{e^2 x^6}{7}\right ) \, dx\\ &=-\frac{1}{9} b d^2 n x^3-\frac{2}{25} b d e n x^5-\frac{1}{49} b e^2 n x^7+\frac{1}{105} \left (35 d^2 x^3+42 d e x^5+15 e^2 x^7\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.0339193, size = 95, normalized size = 1.28 \[ \frac{1}{3} d^2 x^3 \left (a+b \log \left (c x^n\right )\right )+\frac{2}{5} d e x^5 \left (a+b \log \left (c x^n\right )\right )+\frac{1}{7} e^2 x^7 \left (a+b \log \left (c x^n\right )\right )-\frac{1}{9} b d^2 n x^3-\frac{2}{25} b d e n x^5-\frac{1}{49} b e^2 n x^7 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [C]  time = 0.201, size = 434, normalized size = 5.9 \begin{align*}{\frac{b{x}^{3} \left ( 15\,{e}^{2}{x}^{4}+42\,de{x}^{2}+35\,{d}^{2} \right ) \ln \left ({x}^{n} \right ) }{105}}+{\frac{i}{6}}\pi \,b{d}^{2}{x}^{3} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +{\frac{i}{14}}\pi \,b{e}^{2}{x}^{7} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -{\frac{i}{5}}\pi \,bde{x}^{5} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}-{\frac{i}{14}}\pi \,b{e}^{2}{x}^{7} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+{\frac{\ln \left ( c \right ) b{e}^{2}{x}^{7}}{7}}-{\frac{b{e}^{2}n{x}^{7}}{49}}+{\frac{a{e}^{2}{x}^{7}}{7}}+{\frac{i}{6}}\pi \,b{d}^{2}{x}^{3}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+{\frac{i}{5}}\pi \,bde{x}^{5} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -{\frac{i}{6}}\pi \,b{d}^{2}{x}^{3}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -{\frac{i}{14}}\pi \,b{e}^{2}{x}^{7}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) +{\frac{2\,\ln \left ( c \right ) bde{x}^{5}}{5}}-{\frac{2\,bden{x}^{5}}{25}}+{\frac{2\,ade{x}^{5}}{5}}+{\frac{i}{5}}\pi \,bde{x}^{5}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+{\frac{i}{14}}\pi \,b{e}^{2}{x}^{7}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-{\frac{i}{5}}\pi \,bde{x}^{5}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -{\frac{i}{6}}\pi \,b{d}^{2}{x}^{3} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+{\frac{\ln \left ( c \right ) b{d}^{2}{x}^{3}}{3}}-{\frac{b{d}^{2}n{x}^{3}}{9}}+{\frac{a{d}^{2}{x}^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*b*x^3*(15*e^2*x^4+42*d*e*x^2+35*d^2)*ln(x^n)+1/6*I*Pi*b*d^2*x^3*csgn(I*c*x^n)^2*csgn(I*c)+1/14*I*Pi*b*e^
2*x^7*csgn(I*c*x^n)^2*csgn(I*c)-1/5*I*Pi*b*d*e*x^5*csgn(I*c*x^n)^3-1/14*I*Pi*b*e^2*x^7*csgn(I*c*x^n)^3+1/7*ln(
c)*b*e^2*x^7-1/49*b*e^2*n*x^7+1/7*a*e^2*x^7+1/6*I*Pi*b*d^2*x^3*csgn(I*x^n)*csgn(I*c*x^n)^2+1/5*I*Pi*b*d*e*x^5*
csgn(I*c*x^n)^2*csgn(I*c)-1/6*I*Pi*b*d^2*x^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-1/14*I*Pi*b*e^2*x^7*csgn(I*x^
n)*csgn(I*c*x^n)*csgn(I*c)+2/5*ln(c)*b*d*e*x^5-2/25*b*d*e*n*x^5+2/5*a*d*e*x^5+1/5*I*Pi*b*d*e*x^5*csgn(I*x^n)*c
sgn(I*c*x^n)^2+1/14*I*Pi*b*e^2*x^7*csgn(I*x^n)*csgn(I*c*x^n)^2-1/5*I*Pi*b*d*e*x^5*csgn(I*x^n)*csgn(I*c*x^n)*cs
gn(I*c)-1/6*I*Pi*b*d^2*x^3*csgn(I*c*x^n)^3+1/3*ln(c)*b*d^2*x^3-1/9*b*d^2*n*x^3+1/3*a*d^2*x^3

________________________________________________________________________________________

Maxima [A]  time = 1.22336, size = 135, normalized size = 1.82 \begin{align*} -\frac{1}{49} \, b e^{2} n x^{7} + \frac{1}{7} \, b e^{2} x^{7} \log \left (c x^{n}\right ) + \frac{1}{7} \, a e^{2} x^{7} - \frac{2}{25} \, b d e n x^{5} + \frac{2}{5} \, b d e x^{5} \log \left (c x^{n}\right ) + \frac{2}{5} \, a d e x^{5} - \frac{1}{9} \, b d^{2} n x^{3} + \frac{1}{3} \, b d^{2} x^{3} \log \left (c x^{n}\right ) + \frac{1}{3} \, a d^{2} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.28705, size = 294, normalized size = 3.97 \begin{align*} -\frac{1}{49} \,{\left (b e^{2} n - 7 \, a e^{2}\right )} x^{7} - \frac{2}{25} \,{\left (b d e n - 5 \, a d e\right )} x^{5} - \frac{1}{9} \,{\left (b d^{2} n - 3 \, a d^{2}\right )} x^{3} + \frac{1}{105} \,{\left (15 \, b e^{2} x^{7} + 42 \, b d e x^{5} + 35 \, b d^{2} x^{3}\right )} \log \left (c\right ) + \frac{1}{105} \,{\left (15 \, b e^{2} n x^{7} + 42 \, b d e n x^{5} + 35 \, b d^{2} n x^{3}\right )} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/49*(b*e^2*n - 7*a*e^2)*x^7 - 2/25*(b*d*e*n - 5*a*d*e)*x^5 - 1/9*(b*d^2*n - 3*a*d^2)*x^3 + 1/105*(15*b*e^2*x
^7 + 42*b*d*e*x^5 + 35*b*d^2*x^3)*log(c) + 1/105*(15*b*e^2*n*x^7 + 42*b*d*e*n*x^5 + 35*b*d^2*n*x^3)*log(x)

________________________________________________________________________________________

Sympy [B]  time = 8.61375, size = 158, normalized size = 2.14 \begin{align*} \frac{a d^{2} x^{3}}{3} + \frac{2 a d e x^{5}}{5} + \frac{a e^{2} x^{7}}{7} + \frac{b d^{2} n x^{3} \log{\left (x \right )}}{3} - \frac{b d^{2} n x^{3}}{9} + \frac{b d^{2} x^{3} \log{\left (c \right )}}{3} + \frac{2 b d e n x^{5} \log{\left (x \right )}}{5} - \frac{2 b d e n x^{5}}{25} + \frac{2 b d e x^{5} \log{\left (c \right )}}{5} + \frac{b e^{2} n x^{7} \log{\left (x \right )}}{7} - \frac{b e^{2} n x^{7}}{49} + \frac{b e^{2} x^{7} \log{\left (c \right )}}{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.30035, size = 166, normalized size = 2.24 \begin{align*} \frac{1}{7} \, b n x^{7} e^{2} \log \left (x\right ) - \frac{1}{49} \, b n x^{7} e^{2} + \frac{1}{7} \, b x^{7} e^{2} \log \left (c\right ) + \frac{2}{5} \, b d n x^{5} e \log \left (x\right ) + \frac{1}{7} \, a x^{7} e^{2} - \frac{2}{25} \, b d n x^{5} e + \frac{2}{5} \, b d x^{5} e \log \left (c\right ) + \frac{2}{5} \, a d x^{5} e + \frac{1}{3} \, b d^{2} n x^{3} \log \left (x\right ) - \frac{1}{9} \, b d^{2} n x^{3} + \frac{1}{3} \, b d^{2} x^{3} \log \left (c\right ) + \frac{1}{3} \, a d^{2} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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