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

Optimal. Leaf size=86 \[ d^2 x \left (a+b \log \left (c x^n\right )\right )+\frac{2}{3} d e x^3 \left (a+b \log \left (c x^n\right )\right )+\frac{1}{5} e^2 x^5 \left (a+b \log \left (c x^n\right )\right )-b d^2 n x-\frac{2}{9} b d e n x^3-\frac{1}{25} b e^2 n x^5 \]

[Out]

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

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Rubi [A]  time = 0.0351193, antiderivative size = 68, normalized size of antiderivative = 0.79, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {194, 2313} \[ \frac{1}{15} \left (15 d^2 x+10 d e x^3+3 e^2 x^5\right ) \left (a+b \log \left (c x^n\right )\right )-b d^2 n x-\frac{2}{9} b d e n x^3-\frac{1}{25} b e^2 n x^5 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*d^2*n*x) - (2*b*d*e*n*x^3)/9 - (b*e^2*n*x^5)/25 + ((15*d^2*x + 10*d*e*x^3 + 3*e^2*x^5)*(a + b*Log[c*x^n]))
/15

Rule 194

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

Rule 2313

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = IntHide[(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]

Rubi steps

\begin{align*} \int \left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{1}{15} \left (15 d^2 x+10 d e x^3+3 e^2 x^5\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (d^2+\frac{2}{3} d e x^2+\frac{e^2 x^4}{5}\right ) \, dx\\ &=-b d^2 n x-\frac{2}{9} b d e n x^3-\frac{1}{25} b e^2 n x^5+\frac{1}{15} \left (15 d^2 x+10 d e x^3+3 e^2 x^5\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.033711, size = 89, normalized size = 1.03 \[ \frac{2}{3} d e x^3 \left (a+b \log \left (c x^n\right )\right )+\frac{1}{5} e^2 x^5 \left (a+b \log \left (c x^n\right )\right )+a d^2 x+b d^2 x \log \left (c x^n\right )-b d^2 n x-\frac{2}{9} b d e n x^3-\frac{1}{25} b e^2 n x^5 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.192, size = 416, normalized size = 4.8 \begin{align*}{\frac{bx \left ( 3\,{e}^{2}{x}^{4}+10\,de{x}^{2}+15\,{d}^{2} \right ) \ln \left ({x}^{n} \right ) }{15}}+{\frac{i}{3}}\pi \,bde{x}^{3}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+{\frac{i}{2}}\pi \,b{d}^{2}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}x-{\frac{i}{3}}\pi \,bde{x}^{3} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}-{\frac{i}{10}}\pi \,b{e}^{2}{x}^{5}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -{\frac{i}{2}}\pi \,b{d}^{2} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}x+{\frac{i}{2}}\pi \,b{d}^{2} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) x+{\frac{i}{3}}\pi \,bde{x}^{3} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +{\frac{i}{10}}\pi \,b{e}^{2}{x}^{5}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-{\frac{i}{10}}\pi \,b{e}^{2}{x}^{5} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+{\frac{i}{10}}\pi \,b{e}^{2}{x}^{5} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -{\frac{i}{2}}\pi \,b{d}^{2}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) x-{\frac{i}{3}}\pi \,bde{x}^{3}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) +{\frac{\ln \left ( c \right ) b{e}^{2}{x}^{5}}{5}}-{\frac{b{e}^{2}n{x}^{5}}{25}}+{\frac{a{e}^{2}{x}^{5}}{5}}+{\frac{2\,\ln \left ( c \right ) bde{x}^{3}}{3}}-{\frac{2\,bden{x}^{3}}{9}}+{\frac{2\,ade{x}^{3}}{3}}+\ln \left ( c \right ) b{d}^{2}x-b{d}^{2}nx+a{d}^{2}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*b*x*(3*e^2*x^4+10*d*e*x^2+15*d^2)*ln(x^n)+1/3*I*Pi*b*d*e*x^3*csgn(I*x^n)*csgn(I*c*x^n)^2+1/2*I*Pi*b*d^2*c
sgn(I*x^n)*csgn(I*c*x^n)^2*x-1/3*I*Pi*b*d*e*x^3*csgn(I*c*x^n)^3-1/10*I*Pi*b*e^2*x^5*csgn(I*x^n)*csgn(I*c*x^n)*
csgn(I*c)-1/2*I*Pi*b*d^2*csgn(I*c*x^n)^3*x+1/2*I*Pi*b*d^2*csgn(I*c*x^n)^2*csgn(I*c)*x+1/3*I*Pi*b*d*e*x^3*csgn(
I*c*x^n)^2*csgn(I*c)+1/10*I*Pi*b*e^2*x^5*csgn(I*x^n)*csgn(I*c*x^n)^2-1/10*I*Pi*b*e^2*x^5*csgn(I*c*x^n)^3+1/10*
I*Pi*b*e^2*x^5*csgn(I*c*x^n)^2*csgn(I*c)-1/2*I*Pi*b*d^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x-1/3*I*Pi*b*d*e*x
^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+1/5*ln(c)*b*e^2*x^5-1/25*b*e^2*n*x^5+1/5*a*e^2*x^5+2/3*ln(c)*b*d*e*x^3-
2/9*b*d*e*n*x^3+2/3*a*d*e*x^3+ln(c)*b*d^2*x-b*d^2*n*x+a*d^2*x

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Maxima [A]  time = 1.05893, size = 124, normalized size = 1.44 \begin{align*} -\frac{1}{25} \, b e^{2} n x^{5} + \frac{1}{5} \, b e^{2} x^{5} \log \left (c x^{n}\right ) + \frac{1}{5} \, a e^{2} x^{5} - \frac{2}{9} \, b d e n x^{3} + \frac{2}{3} \, b d e x^{3} \log \left (c x^{n}\right ) + \frac{2}{3} \, a d e x^{3} - b d^{2} n x + b d^{2} x \log \left (c x^{n}\right ) + a d^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.27679, size = 271, normalized size = 3.15 \begin{align*} -\frac{1}{25} \,{\left (b e^{2} n - 5 \, a e^{2}\right )} x^{5} - \frac{2}{9} \,{\left (b d e n - 3 \, a d e\right )} x^{3} -{\left (b d^{2} n - a d^{2}\right )} x + \frac{1}{15} \,{\left (3 \, b e^{2} x^{5} + 10 \, b d e x^{3} + 15 \, b d^{2} x\right )} \log \left (c\right ) + \frac{1}{15} \,{\left (3 \, b e^{2} n x^{5} + 10 \, b d e n x^{3} + 15 \, b d^{2} n x\right )} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/25*(b*e^2*n - 5*a*e^2)*x^5 - 2/9*(b*d*e*n - 3*a*d*e)*x^3 - (b*d^2*n - a*d^2)*x + 1/15*(3*b*e^2*x^5 + 10*b*d
*e*x^3 + 15*b*d^2*x)*log(c) + 1/15*(3*b*e^2*n*x^5 + 10*b*d*e*n*x^3 + 15*b*d^2*n*x)*log(x)

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Sympy [A]  time = 3.30075, size = 144, normalized size = 1.67 \begin{align*} a d^{2} x + \frac{2 a d e x^{3}}{3} + \frac{a e^{2} x^{5}}{5} + b d^{2} n x \log{\left (x \right )} - b d^{2} n x + b d^{2} x \log{\left (c \right )} + \frac{2 b d e n x^{3} \log{\left (x \right )}}{3} - \frac{2 b d e n x^{3}}{9} + \frac{2 b d e x^{3} \log{\left (c \right )}}{3} + \frac{b e^{2} n x^{5} \log{\left (x \right )}}{5} - \frac{b e^{2} n x^{5}}{25} + \frac{b e^{2} x^{5} \log{\left (c \right )}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.31353, size = 151, normalized size = 1.76 \begin{align*} \frac{1}{5} \, b n x^{5} e^{2} \log \left (x\right ) - \frac{1}{25} \, b n x^{5} e^{2} + \frac{1}{5} \, b x^{5} e^{2} \log \left (c\right ) + \frac{2}{3} \, b d n x^{3} e \log \left (x\right ) + \frac{1}{5} \, a x^{5} e^{2} - \frac{2}{9} \, b d n x^{3} e + \frac{2}{3} \, b d x^{3} e \log \left (c\right ) + \frac{2}{3} \, a d x^{3} e + b d^{2} n x \log \left (x\right ) - b d^{2} n x + b d^{2} x \log \left (c\right ) + a d^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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