3.204 \(\int (d+e x^2)^3 (a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=121 \[ d^2 e x^3 \left (a+b \log \left (c x^n\right )\right )+d^3 x \left (a+b \log \left (c x^n\right )\right )+\frac{3}{5} d e^2 x^5 \left (a+b \log \left (c x^n\right )\right )+\frac{1}{7} e^3 x^7 \left (a+b \log \left (c x^n\right )\right )-\frac{1}{3} b d^2 e n x^3-b d^3 n x-\frac{3}{25} b d e^2 n x^5-\frac{1}{49} b e^3 n x^7 \]

[Out]

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

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Rubi [A]  time = 0.0479855, antiderivative size = 94, normalized size of antiderivative = 0.78, 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} \[ \frac{1}{35} \left (35 d^2 e x^3+35 d^3 x+21 d e^2 x^5+5 e^3 x^7\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{3} b d^2 e n x^3-b d^3 n x-\frac{3}{25} b d e^2 n x^5-\frac{1}{49} b e^3 n x^7 \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x^2)^3*(a + b*Log[c*x^n]),x]

[Out]

-(b*d^3*n*x) - (b*d^2*e*n*x^3)/3 - (3*b*d*e^2*n*x^5)/25 - (b*e^3*n*x^7)/49 + ((35*d^3*x + 35*d^2*e*x^3 + 21*d*
e^2*x^5 + 5*e^3*x^7)*(a + b*Log[c*x^n]))/35

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 )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{1}{35} \left (35 d^3 x+35 d^2 e x^3+21 d e^2 x^5+5 e^3 x^7\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (d^3+d^2 e x^2+\frac{3}{5} d e^2 x^4+\frac{e^3 x^6}{7}\right ) \, dx\\ &=-b d^3 n x-\frac{1}{3} b d^2 e n x^3-\frac{3}{25} b d e^2 n x^5-\frac{1}{49} b e^3 n x^7+\frac{1}{35} \left (35 d^3 x+35 d^2 e x^3+21 d e^2 x^5+5 e^3 x^7\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.044881, size = 124, normalized size = 1.02 \[ d^2 e x^3 \left (a+b \log \left (c x^n\right )\right )+\frac{3}{5} d e^2 x^5 \left (a+b \log \left (c x^n\right )\right )+\frac{1}{7} e^3 x^7 \left (a+b \log \left (c x^n\right )\right )+a d^3 x+b d^3 x \log \left (c x^n\right )-\frac{1}{3} b d^2 e n x^3-b d^3 n x-\frac{3}{25} b d e^2 n x^5-\frac{1}{49} b e^3 n x^7 \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x^2)^3*(a + b*Log[c*x^n]),x]

[Out]

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

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Maple [C]  time = 0.201, size = 582, normalized size = 4.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)^3*(a+b*ln(c*x^n)),x)

[Out]

ln(c)*b*d^2*e*x^3-1/2*I*Pi*b*d^2*e*x^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-3/10*I*Pi*b*d*e^2*x^5*csgn(I*x^n)*c
sgn(I*c*x^n)*csgn(I*c)+3/5*a*d*e^2*x^5+a*d^2*e*x^3+3/5*ln(c)*b*d*e^2*x^5+1/7*a*e^3*x^7+a*d^3*x+3/10*I*Pi*b*d*e
^2*x^5*csgn(I*x^n)*csgn(I*c*x^n)^2-1/14*I*Pi*b*e^3*x^7*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+1/2*I*Pi*b*d^2*e*x^
3*csgn(I*c*x^n)^2*csgn(I*c)+1/7*ln(c)*b*e^3*x^7+3/10*I*Pi*b*d*e^2*x^5*csgn(I*c*x^n)^2*csgn(I*c)+1/35*b*x*(5*e^
3*x^6+21*d*e^2*x^4+35*d^2*e*x^2+35*d^3)*ln(x^n)-1/2*I*Pi*b*d^3*csgn(I*c*x^n)^3*x-1/2*I*Pi*b*d^3*csgn(I*x^n)*cs
gn(I*c*x^n)*csgn(I*c)*x+ln(c)*b*d^3*x+1/2*I*Pi*b*d^2*e*x^3*csgn(I*x^n)*csgn(I*c*x^n)^2-1/14*I*Pi*b*e^3*x^7*csg
n(I*c*x^n)^3+1/2*I*Pi*b*d^3*csgn(I*c*x^n)^2*csgn(I*c)*x+1/2*I*Pi*b*d^3*csgn(I*x^n)*csgn(I*c*x^n)^2*x-b*d^3*n*x
-1/2*I*Pi*b*d^2*e*x^3*csgn(I*c*x^n)^3+1/14*I*Pi*b*e^3*x^7*csgn(I*c*x^n)^2*csgn(I*c)-3/10*I*Pi*b*d*e^2*x^5*csgn
(I*c*x^n)^3+1/14*I*Pi*b*e^3*x^7*csgn(I*x^n)*csgn(I*c*x^n)^2-1/49*b*e^3*n*x^7-3/25*b*d*e^2*n*x^5-1/3*b*d^2*e*n*
x^3

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Maxima [A]  time = 1.02537, size = 180, normalized size = 1.49 \begin{align*} -\frac{1}{49} \, b e^{3} n x^{7} + \frac{1}{7} \, b e^{3} x^{7} \log \left (c x^{n}\right ) + \frac{1}{7} \, a e^{3} x^{7} - \frac{3}{25} \, b d e^{2} n x^{5} + \frac{3}{5} \, b d e^{2} x^{5} \log \left (c x^{n}\right ) + \frac{3}{5} \, a d e^{2} x^{5} - \frac{1}{3} \, b d^{2} e n x^{3} + b d^{2} e x^{3} \log \left (c x^{n}\right ) + a d^{2} e x^{3} - b d^{3} n x + b d^{3} x \log \left (c x^{n}\right ) + a d^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^3*(a+b*log(c*x^n)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.32004, size = 378, normalized size = 3.12 \begin{align*} -\frac{1}{49} \,{\left (b e^{3} n - 7 \, a e^{3}\right )} x^{7} - \frac{3}{25} \,{\left (b d e^{2} n - 5 \, a d e^{2}\right )} x^{5} - \frac{1}{3} \,{\left (b d^{2} e n - 3 \, a d^{2} e\right )} x^{3} -{\left (b d^{3} n - a d^{3}\right )} x + \frac{1}{35} \,{\left (5 \, b e^{3} x^{7} + 21 \, b d e^{2} x^{5} + 35 \, b d^{2} e x^{3} + 35 \, b d^{3} x\right )} \log \left (c\right ) + \frac{1}{35} \,{\left (5 \, b e^{3} n x^{7} + 21 \, b d e^{2} n x^{5} + 35 \, b d^{2} e n x^{3} + 35 \, b d^{3} 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)^3*(a+b*log(c*x^n)),x, algorithm="fricas")

[Out]

-1/49*(b*e^3*n - 7*a*e^3)*x^7 - 3/25*(b*d*e^2*n - 5*a*d*e^2)*x^5 - 1/3*(b*d^2*e*n - 3*a*d^2*e)*x^3 - (b*d^3*n
- a*d^3)*x + 1/35*(5*b*e^3*x^7 + 21*b*d*e^2*x^5 + 35*b*d^2*e*x^3 + 35*b*d^3*x)*log(c) + 1/35*(5*b*e^3*n*x^7 +
21*b*d*e^2*n*x^5 + 35*b*d^2*e*n*x^3 + 35*b*d^3*n*x)*log(x)

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Sympy [A]  time = 8.74092, size = 204, normalized size = 1.69 \begin{align*} a d^{3} x + a d^{2} e x^{3} + \frac{3 a d e^{2} x^{5}}{5} + \frac{a e^{3} x^{7}}{7} + b d^{3} n x \log{\left (x \right )} - b d^{3} n x + b d^{3} x \log{\left (c \right )} + b d^{2} e n x^{3} \log{\left (x \right )} - \frac{b d^{2} e n x^{3}}{3} + b d^{2} e x^{3} \log{\left (c \right )} + \frac{3 b d e^{2} n x^{5} \log{\left (x \right )}}{5} - \frac{3 b d e^{2} n x^{5}}{25} + \frac{3 b d e^{2} x^{5} \log{\left (c \right )}}{5} + \frac{b e^{3} n x^{7} \log{\left (x \right )}}{7} - \frac{b e^{3} n x^{7}}{49} + \frac{b e^{3} x^{7} \log{\left (c \right )}}{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**2+d)**3*(a+b*ln(c*x**n)),x)

[Out]

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

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Giac [A]  time = 1.311, size = 215, normalized size = 1.78 \begin{align*} \frac{1}{7} \, b n x^{7} e^{3} \log \left (x\right ) - \frac{1}{49} \, b n x^{7} e^{3} + \frac{1}{7} \, b x^{7} e^{3} \log \left (c\right ) + \frac{3}{5} \, b d n x^{5} e^{2} \log \left (x\right ) + \frac{1}{7} \, a x^{7} e^{3} - \frac{3}{25} \, b d n x^{5} e^{2} + \frac{3}{5} \, b d x^{5} e^{2} \log \left (c\right ) + b d^{2} n x^{3} e \log \left (x\right ) + \frac{3}{5} \, a d x^{5} e^{2} - \frac{1}{3} \, b d^{2} n x^{3} e + b d^{2} x^{3} e \log \left (c\right ) + a d^{2} x^{3} e + b d^{3} n x \log \left (x\right ) - b d^{3} n x + b d^{3} x \log \left (c\right ) + a d^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^3*(a+b*log(c*x^n)),x, algorithm="giac")

[Out]

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