3.3.4 \(\int (d+e x^2)^3 (a+b \log (c x^n)) \, dx\) [204]

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

[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 = 0.04, antiderivative size = 121, normalized size of antiderivative = 1.00, 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 = {200, 2350} \begin {gather*} 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 \end {gather*}

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 + 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

Rule 200

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 2350

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = IntHide[(d +
e*x^r)^q, x]}, Dist[a + b*Log[c*x^n], u, 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 = 0.03, size = 124, normalized size = 1.02 \begin {gather*} a d^3 x-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+b d^3 x \log \left (c x^n\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 ) \end {gather*}

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] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.14, size = 582, normalized size = 4.81

method	result	size
risch	\(x a \,d^{3}+\frac {x^{7} a \,e^{3}}{7}+\frac {3 \ln \left (c \right ) b d \,e^{2} x^{5}}{5}+x^{3} a \,d^{2} e +\frac {3 x^{5} a d \,e^{2}}{5}+\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}-b \,d^{3} n x +\ln \left (c \right ) b \,d^{3} x +\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 \,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 {i \pi b \,d^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) x}{2}+\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 {b x \left (5 e^{3} x^{6}+21 d \,e^{2} x^{4}+35 d^{2} e \,x^{2}+35 d^{3}\right ) \ln \left (x^{n}\right )}{35}+\ln \left (c \right ) b \,d^{2} e \,x^{3}-\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} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x}{2}+\frac {i \pi b \,d^{3} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x}{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}-\frac {3 i \pi b d \,e^{2} x^{5} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{10}+\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 {b \,e^{3} n \,x^{7}}{49}-\frac {3 b d \,e^{2} n \,x^{5}}{25}-\frac {b \,d^{2} e n \,x^{3}}{3}-\frac {i \pi b \,e^{3} x^{7} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{14}-\frac {i \pi b \,d^{3} \mathrm {csgn}\left (i c \,x^{n}\right )^{3} x}{2}-\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 {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 {\ln \left (c \right ) b \,e^{3} x^{7}}{7}\)	\(582\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.26, size = 130, normalized size = 1.07 \begin {gather*} -\frac {1}{49} \, b n x^{7} e^{3} + \frac {1}{7} \, b x^{7} e^{3} \log \left (c x^{n}\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 x^{n}\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 x^{n}\right ) + a d^{2} x^{3} e - b d^{3} n x + b d^{3} x \log \left (c x^{n}\right ) + a d^{3} x \end {gather*}

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

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Fricas [A]
time = 0.38, size = 151, normalized size = 1.25 \begin {gather*} -\frac {1}{49} \, {\left (b n - 7 \, a\right )} x^{7} e^{3} - \frac {3}{25} \, {\left (b d n - 5 \, a d\right )} x^{5} e^{2} - \frac {1}{3} \, {\left (b d^{2} n - 3 \, a d^{2}\right )} x^{3} e - {\left (b d^{3} n - a d^{3}\right )} x + \frac {1}{35} \, {\left (5 \, b x^{7} e^{3} + 21 \, b d x^{5} e^{2} + 35 \, b d^{2} x^{3} e + 35 \, b d^{3} x\right )} \log \left (c\right ) + \frac {1}{35} \, {\left (5 \, b n x^{7} e^{3} + 21 \, b d n x^{5} e^{2} + 35 \, b d^{2} n x^{3} e + 35 \, b d^{3} n x\right )} \log \left (x\right ) \end {gather*}

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

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Sympy [A]
time = 0.97, size = 156, normalized size = 1.29 \begin {gather*} 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 + b d^{3} x \log {\left (c x^{n} \right )} - \frac {b d^{2} e n x^{3}}{3} + b d^{2} e x^{3} \log {\left (c x^{n} \right )} - \frac {3 b d e^{2} n x^{5}}{25} + \frac {3 b d e^{2} x^{5} \log {\left (c x^{n} \right )}}{5} - \frac {b e^{3} n x^{7}}{49} + \frac {b e^{3} x^{7} \log {\left (c x^{n} \right )}}{7} \end {gather*}

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

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Giac [A]
time = 3.92, size = 159, normalized size = 1.31 \begin {gather*} \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 {gather*}

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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Mupad [B]
time = 3.72, size = 104, normalized size = 0.86 \begin {gather*} \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} \end {gather*}

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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