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

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

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 94, normalized size of antiderivative = 0.78, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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*}

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Mathematica [A]  time = 0.05, size = 124, normalized size = 1.02 \[ d^2 e x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {3}{5} d e^2 x^5 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{7} e^3 x^7 \left (a+b \log \left (c x^n\right )\right )+a d^3 x+b d^3 x \log \left (c x^n\right )-b d^3 n x-\frac {1}{3} b d^2 e n x^3-\frac {3}{25} b d e^2 n x^5-\frac {1}{49} b e^3 n x^7 \]

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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fricas [A]  time = 0.72, size = 161, normalized size = 1.33 \[ -\frac {1}{49} \, {\left (b e^{3} n - 7 \, a e^{3}\right )} x^{7} - \frac {3}{25} \, {\left (b d e^{2} n - 5 \, a d e^{2}\right )} x^{5} - \frac {1}{3} \, {\left (b d^{2} e n - 3 \, a d^{2} e\right )} x^{3} - {\left (b d^{3} n - a d^{3}\right )} x + \frac {1}{35} \, {\left (5 \, b e^{3} x^{7} + 21 \, b d e^{2} x^{5} + 35 \, b d^{2} e x^{3} + 35 \, b d^{3} x\right )} \log \relax (c) + \frac {1}{35} \, {\left (5 \, b e^{3} n x^{7} + 21 \, b d e^{2} n x^{5} + 35 \, b d^{2} e n x^{3} + 35 \, b d^{3} n x\right )} \log \relax (x) \]

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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giac [A]  time = 0.32, size = 159, normalized size = 1.31 \[ \frac {1}{7} \, b n x^{7} e^{3} \log \relax (x) - \frac {1}{49} \, b n x^{7} e^{3} + \frac {1}{7} \, b x^{7} e^{3} \log \relax (c) + \frac {3}{5} \, b d n x^{5} e^{2} \log \relax (x) + \frac {1}{7} \, a x^{7} e^{3} - \frac {3}{25} \, b d n x^{5} e^{2} + \frac {3}{5} \, b d x^{5} e^{2} \log \relax (c) + b d^{2} n x^{3} e \log \relax (x) + \frac {3}{5} \, a d x^{5} e^{2} - \frac {1}{3} \, b d^{2} n x^{3} e + b d^{2} x^{3} e \log \relax (c) + a d^{2} x^{3} e + b d^{3} n x \log \relax (x) - b d^{3} n x + b d^{3} x \log \relax (c) + a d^{3} x \]

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

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maple [C]  time = 0.22, size = 582, normalized size = 4.81 \[ b \,d^{2} e \,x^{3} \ln \relax (c )+\frac {3 a d \,e^{2} x^{5}}{5}+a \,d^{2} e \,x^{3}+\frac {3 b d \,e^{2} x^{5} \ln \relax (c )}{5}+a \,d^{3} x +\frac {a \,e^{3} x^{7}}{7}+\frac {b \,e^{3} x^{7} \ln \relax (c )}{7}+b \,d^{3} x \ln \relax (c )+\frac {\left (5 e^{3} x^{6}+21 d \,e^{2} x^{4}+35 d^{2} e \,x^{2}+35 d^{3}\right ) b x \ln \left (x^{n}\right )}{35}-\frac {b \,e^{3} n \,x^{7}}{49}-\frac {i \pi b \,e^{3} x^{7} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{14}+\frac {3 i \pi b d \,e^{2} x^{5} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{10}+\frac {3 i \pi b d \,e^{2} x^{5} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{10}+\frac {i \pi b \,d^{2} e \,x^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i \pi b \,d^{2} e \,x^{3} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i \pi b \,d^{3} x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2}-\frac {3 i \pi b d \,e^{2} x^{5} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{10}-\frac {i \pi b \,d^{2} e \,x^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2}-\frac {i \pi b \,e^{3} x^{7} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{14}-\frac {i \pi b \,d^{3} x \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{2}+\frac {i \pi b \,e^{3} x^{7} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{14}-\frac {3 i \pi b d \,e^{2} x^{5} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{10}-\frac {i \pi b \,d^{2} e \,x^{3} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{2}+\frac {i \pi b \,d^{3} x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i \pi b \,d^{3} x \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i \pi b \,e^{3} x^{7} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{14}-b \,d^{3} n x -\frac {3 b d \,e^{2} n \,x^{5}}{25}-\frac {b \,d^{2} e n \,x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*I*Pi*b*d^2*e*x^3*csgn(I*x^n)*csgn(I*c*x^n)^2+b*d^2*e*x^3*ln(c)+3/5*a*d*e^2*x^5+a*d^2*e*x^3+3/5*b*d*e^2*x^5
*ln(c)+3/10*I*Pi*b*d*e^2*x^5*csgn(I*x^n)*csgn(I*c*x^n)^2+3/10*I*Pi*b*d*e^2*x^5*csgn(I*c*x^n)^2*csgn(I*c)+a*d^3
*x+1/7*a*e^3*x^7+1/7*b*e^3*x^7*ln(c)+ln(c)*b*d^3*x+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*x^n)*csgn(I*c*x^n)^2*x-1/2*I*Pi*b*d^2*e*x^3*csgn(I*c*x^n)^3-3/10*I*Pi*b*d*e^2*x^5*csg
n(I*c*x^n)^3-1/49*b*e^3*n*x^7+1/2*I*Pi*b*d^2*e*x^3*csgn(I*c*x^n)^2*csgn(I*c)-1/14*I*Pi*b*e^3*x^7*csgn(I*c*x^n)
^3-1/2*I*Pi*b*d^3*csgn(I*c*x^n)^3*x-3/10*I*Pi*b*d*e^2*x^5*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-1/2*I*Pi*b*d^2*e
*x^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-1/2*I*Pi*b*d^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x-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^3*csgn(I*c*x^n)^2*csgn(I*c)*x+1/14*I*Pi*b*e^3*x^7*csgn(I*
x^n)*csgn(I*c*x^n)^2+1/14*I*Pi*b*e^3*x^7*csgn(I*c*x^n)^2*csgn(I*c)-b*d^3*n*x-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 = 0.46, size = 133, normalized size = 1.10 \[ -\frac {1}{49} \, b e^{3} n x^{7} + \frac {1}{7} \, b e^{3} x^{7} \log \left (c x^{n}\right ) + \frac {1}{7} \, a e^{3} x^{7} - \frac {3}{25} \, b d e^{2} n x^{5} + \frac {3}{5} \, b d e^{2} x^{5} \log \left (c x^{n}\right ) + \frac {3}{5} \, a d e^{2} x^{5} - \frac {1}{3} \, b d^{2} e n x^{3} + b d^{2} e x^{3} \log \left (c x^{n}\right ) + a d^{2} e x^{3} - b d^{3} n x + b d^{3} x \log \left (c x^{n}\right ) + a d^{3} x \]

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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mupad [B]  time = 3.72, size = 104, normalized size = 0.86 \[ \ln \left (c\,x^n\right )\,\left (b\,d^3\,x+b\,d^2\,e\,x^3+\frac {3\,b\,d\,e^2\,x^5}{5}+\frac {b\,e^3\,x^7}{7}\right )+\frac {e^3\,x^7\,\left (7\,a-b\,n\right )}{49}+d^3\,x\,\left (a-b\,n\right )+\frac {d^2\,e\,x^3\,\left (3\,a-b\,n\right )}{3}+\frac {3\,d\,e^2\,x^5\,\left (5\,a-b\,n\right )}{25} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 6.60, size = 204, normalized size = 1.69 \[ a d^{3} x + a d^{2} e x^{3} + \frac {3 a d e^{2} x^{5}}{5} + \frac {a e^{3} x^{7}}{7} + b d^{3} n x \log {\relax (x )} - b d^{3} n x + b d^{3} x \log {\relax (c )} + b d^{2} e n x^{3} \log {\relax (x )} - \frac {b d^{2} e n x^{3}}{3} + b d^{2} e x^{3} \log {\relax (c )} + \frac {3 b d e^{2} n x^{5} \log {\relax (x )}}{5} - \frac {3 b d e^{2} n x^{5}}{25} + \frac {3 b d e^{2} x^{5} \log {\relax (c )}}{5} + \frac {b e^{3} n x^{7} \log {\relax (x )}}{7} - \frac {b e^{3} n x^{7}}{49} + \frac {b e^{3} x^{7} \log {\relax (c )}}{7} \]

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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