3.202 \(\int x^4 (d+e x^2)^3 (a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=100 \[ \frac{\left (495 d^2 e x^7+231 d^3 x^5+385 d e^2 x^9+105 e^3 x^{11}\right ) \left (a+b \log \left (c x^n\right )\right )}{1155}-\frac{3}{49} b d^2 e n x^7-\frac{1}{25} b d^3 n x^5-\frac{1}{27} b d e^2 n x^9-\frac{1}{121} b e^3 n x^{11} \]

[Out]

-(b*d^3*n*x^5)/25 - (3*b*d^2*e*n*x^7)/49 - (b*d*e^2*n*x^9)/27 - (b*e^3*n*x^11)/121 + ((231*d^3*x^5 + 495*d^2*e
*x^7 + 385*d*e^2*x^9 + 105*e^3*x^11)*(a + b*Log[c*x^n]))/1155

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Rubi [A]  time = 0.0875338, antiderivative size = 100, 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} \[ \frac{\left (495 d^2 e x^7+231 d^3 x^5+385 d e^2 x^9+105 e^3 x^{11}\right ) \left (a+b \log \left (c x^n\right )\right )}{1155}-\frac{3}{49} b d^2 e n x^7-\frac{1}{25} b d^3 n x^5-\frac{1}{27} b d e^2 n x^9-\frac{1}{121} b e^3 n x^{11} \]

Antiderivative was successfully verified.

[In]

Int[x^4*(d + e*x^2)^3*(a + b*Log[c*x^n]),x]

[Out]

-(b*d^3*n*x^5)/25 - (3*b*d^2*e*n*x^7)/49 - (b*d*e^2*n*x^9)/27 - (b*e^3*n*x^11)/121 + ((231*d^3*x^5 + 495*d^2*e
*x^7 + 385*d*e^2*x^9 + 105*e^3*x^11)*(a + b*Log[c*x^n]))/1155

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^4 \left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{\left (231 d^3 x^5+495 d^2 e x^7+385 d e^2 x^9+105 e^3 x^{11}\right ) \left (a+b \log \left (c x^n\right )\right )}{1155}-(b n) \int \left (\frac{d^3 x^4}{5}+\frac{3}{7} d^2 e x^6+\frac{1}{3} d e^2 x^8+\frac{e^3 x^{10}}{11}\right ) \, dx\\ &=-\frac{1}{25} b d^3 n x^5-\frac{3}{49} b d^2 e n x^7-\frac{1}{27} b d e^2 n x^9-\frac{1}{121} b e^3 n x^{11}+\frac{\left (231 d^3 x^5+495 d^2 e x^7+385 d e^2 x^9+105 e^3 x^{11}\right ) \left (a+b \log \left (c x^n\right )\right )}{1155}\\ \end{align*}

Mathematica [A]  time = 0.0471077, size = 133, normalized size = 1.33 \[ \frac{3}{7} d^2 e x^7 \left (a+b \log \left (c x^n\right )\right )+\frac{1}{5} d^3 x^5 \left (a+b \log \left (c x^n\right )\right )+\frac{1}{3} d e^2 x^9 \left (a+b \log \left (c x^n\right )\right )+\frac{1}{11} e^3 x^{11} \left (a+b \log \left (c x^n\right )\right )-\frac{3}{49} b d^2 e n x^7-\frac{1}{25} b d^3 n x^5-\frac{1}{27} b d e^2 n x^9-\frac{1}{121} b e^3 n x^{11} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*(d + e*x^2)^3*(a + b*Log[c*x^n]),x]

[Out]

-(b*d^3*n*x^5)/25 - (3*b*d^2*e*n*x^7)/49 - (b*d*e^2*n*x^9)/27 - (b*e^3*n*x^11)/121 + (d^3*x^5*(a + b*Log[c*x^n
]))/5 + (3*d^2*e*x^7*(a + b*Log[c*x^n]))/7 + (d*e^2*x^9*(a + b*Log[c*x^n]))/3 + (e^3*x^11*(a + b*Log[c*x^n]))/
11

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Maple [C]  time = 0.207, size = 602, normalized size = 6. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(e*x^2+d)^3*(a+b*ln(c*x^n)),x)

[Out]

3/7*a*d^2*e*x^7+1/3*a*d*e^2*x^9-3/14*I*Pi*b*d^2*e*x^7*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+3/7*ln(c)*b*d^2*e*x^
7+1/3*ln(c)*b*d*e^2*x^9+1/6*I*Pi*b*d*e^2*x^9*csgn(I*x^n)*csgn(I*c*x^n)^2+3/14*I*Pi*b*d^2*e*x^7*csgn(I*c*x^n)^2
*csgn(I*c)-1/22*I*Pi*b*e^3*x^11*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+3/14*I*Pi*b*d^2*e*x^7*csgn(I*x^n)*csgn(I*c
*x^n)^2-1/10*I*Pi*b*d^3*x^5*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+1/6*I*Pi*b*d*e^2*x^9*csgn(I*c*x^n)^2*csgn(I*c)
+1/5*a*d^3*x^5+1/1155*b*x^5*(105*e^3*x^6+385*d*e^2*x^4+495*d^2*e*x^2+231*d^3)*ln(x^n)-1/22*I*Pi*b*e^3*x^11*csg
n(I*c*x^n)^3-1/10*I*Pi*b*d^3*x^5*csgn(I*c*x^n)^3+1/5*ln(c)*b*d^3*x^5+1/11*ln(c)*b*e^3*x^11+1/11*a*e^3*x^11+1/1
0*I*Pi*b*d^3*x^5*csgn(I*x^n)*csgn(I*c*x^n)^2-3/14*I*Pi*b*d^2*e*x^7*csgn(I*c*x^n)^3-3/49*b*d^2*e*n*x^7-1/27*b*d
*e^2*n*x^9-1/6*I*Pi*b*d*e^2*x^9*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+1/22*I*Pi*b*e^3*x^11*csgn(I*x^n)*csgn(I*c*
x^n)^2+1/22*I*Pi*b*e^3*x^11*csgn(I*c*x^n)^2*csgn(I*c)+1/10*I*Pi*b*d^3*x^5*csgn(I*c*x^n)^2*csgn(I*c)-1/6*I*Pi*b
*d*e^2*x^9*csgn(I*c*x^n)^3-1/25*b*d^3*n*x^5-1/121*b*e^3*n*x^11

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Maxima [A]  time = 1.14965, size = 193, normalized size = 1.93 \begin{align*} -\frac{1}{121} \, b e^{3} n x^{11} + \frac{1}{11} \, b e^{3} x^{11} \log \left (c x^{n}\right ) + \frac{1}{11} \, a e^{3} x^{11} - \frac{1}{27} \, b d e^{2} n x^{9} + \frac{1}{3} \, b d e^{2} x^{9} \log \left (c x^{n}\right ) + \frac{1}{3} \, a d e^{2} x^{9} - \frac{3}{49} \, b d^{2} e n x^{7} + \frac{3}{7} \, b d^{2} e x^{7} \log \left (c x^{n}\right ) + \frac{3}{7} \, a d^{2} e x^{7} - \frac{1}{25} \, b d^{3} n x^{5} + \frac{1}{5} \, b d^{3} x^{5} \log \left (c x^{n}\right ) + \frac{1}{5} \, a d^{3} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x^2+d)^3*(a+b*log(c*x^n)),x, algorithm="maxima")

[Out]

-1/121*b*e^3*n*x^11 + 1/11*b*e^3*x^11*log(c*x^n) + 1/11*a*e^3*x^11 - 1/27*b*d*e^2*n*x^9 + 1/3*b*d*e^2*x^9*log(
c*x^n) + 1/3*a*d*e^2*x^9 - 3/49*b*d^2*e*n*x^7 + 3/7*b*d^2*e*x^7*log(c*x^n) + 3/7*a*d^2*e*x^7 - 1/25*b*d^3*n*x^
5 + 1/5*b*d^3*x^5*log(c*x^n) + 1/5*a*d^3*x^5

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Fricas [A]  time = 1.4723, size = 423, normalized size = 4.23 \begin{align*} -\frac{1}{121} \,{\left (b e^{3} n - 11 \, a e^{3}\right )} x^{11} - \frac{1}{27} \,{\left (b d e^{2} n - 9 \, a d e^{2}\right )} x^{9} - \frac{3}{49} \,{\left (b d^{2} e n - 7 \, a d^{2} e\right )} x^{7} - \frac{1}{25} \,{\left (b d^{3} n - 5 \, a d^{3}\right )} x^{5} + \frac{1}{1155} \,{\left (105 \, b e^{3} x^{11} + 385 \, b d e^{2} x^{9} + 495 \, b d^{2} e x^{7} + 231 \, b d^{3} x^{5}\right )} \log \left (c\right ) + \frac{1}{1155} \,{\left (105 \, b e^{3} n x^{11} + 385 \, b d e^{2} n x^{9} + 495 \, b d^{2} e n x^{7} + 231 \, b d^{3} n x^{5}\right )} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x^2+d)^3*(a+b*log(c*x^n)),x, algorithm="fricas")

[Out]

-1/121*(b*e^3*n - 11*a*e^3)*x^11 - 1/27*(b*d*e^2*n - 9*a*d*e^2)*x^9 - 3/49*(b*d^2*e*n - 7*a*d^2*e)*x^7 - 1/25*
(b*d^3*n - 5*a*d^3)*x^5 + 1/1155*(105*b*e^3*x^11 + 385*b*d*e^2*x^9 + 495*b*d^2*e*x^7 + 231*b*d^3*x^5)*log(c) +
1/1155*(105*b*e^3*n*x^11 + 385*b*d*e^2*n*x^9 + 495*b*d^2*e*n*x^7 + 231*b*d^3*n*x^5)*log(x)

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Sympy [B]  time = 42.5899, size = 223, normalized size = 2.23 \begin{align*} \frac{a d^{3} x^{5}}{5} + \frac{3 a d^{2} e x^{7}}{7} + \frac{a d e^{2} x^{9}}{3} + \frac{a e^{3} x^{11}}{11} + \frac{b d^{3} n x^{5} \log{\left (x \right )}}{5} - \frac{b d^{3} n x^{5}}{25} + \frac{b d^{3} x^{5} \log{\left (c \right )}}{5} + \frac{3 b d^{2} e n x^{7} \log{\left (x \right )}}{7} - \frac{3 b d^{2} e n x^{7}}{49} + \frac{3 b d^{2} e x^{7} \log{\left (c \right )}}{7} + \frac{b d e^{2} n x^{9} \log{\left (x \right )}}{3} - \frac{b d e^{2} n x^{9}}{27} + \frac{b d e^{2} x^{9} \log{\left (c \right )}}{3} + \frac{b e^{3} n x^{11} \log{\left (x \right )}}{11} - \frac{b e^{3} n x^{11}}{121} + \frac{b e^{3} x^{11} \log{\left (c \right )}}{11} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(e*x**2+d)**3*(a+b*ln(c*x**n)),x)

[Out]

a*d**3*x**5/5 + 3*a*d**2*e*x**7/7 + a*d*e**2*x**9/3 + a*e**3*x**11/11 + b*d**3*n*x**5*log(x)/5 - b*d**3*n*x**5
/25 + b*d**3*x**5*log(c)/5 + 3*b*d**2*e*n*x**7*log(x)/7 - 3*b*d**2*e*n*x**7/49 + 3*b*d**2*e*x**7*log(c)/7 + b*
d*e**2*n*x**9*log(x)/3 - b*d*e**2*n*x**9/27 + b*d*e**2*x**9*log(c)/3 + b*e**3*n*x**11*log(x)/11 - b*e**3*n*x**
11/121 + b*e**3*x**11*log(c)/11

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Giac [A]  time = 1.28929, size = 234, normalized size = 2.34 \begin{align*} \frac{1}{11} \, b n x^{11} e^{3} \log \left (x\right ) - \frac{1}{121} \, b n x^{11} e^{3} + \frac{1}{11} \, b x^{11} e^{3} \log \left (c\right ) + \frac{1}{3} \, b d n x^{9} e^{2} \log \left (x\right ) + \frac{1}{11} \, a x^{11} e^{3} - \frac{1}{27} \, b d n x^{9} e^{2} + \frac{1}{3} \, b d x^{9} e^{2} \log \left (c\right ) + \frac{3}{7} \, b d^{2} n x^{7} e \log \left (x\right ) + \frac{1}{3} \, a d x^{9} e^{2} - \frac{3}{49} \, b d^{2} n x^{7} e + \frac{3}{7} \, b d^{2} x^{7} e \log \left (c\right ) + \frac{3}{7} \, a d^{2} x^{7} e + \frac{1}{5} \, b d^{3} n x^{5} \log \left (x\right ) - \frac{1}{25} \, b d^{3} n x^{5} + \frac{1}{5} \, b d^{3} x^{5} \log \left (c\right ) + \frac{1}{5} \, a d^{3} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x^2+d)^3*(a+b*log(c*x^n)),x, algorithm="giac")

[Out]

1/11*b*n*x^11*e^3*log(x) - 1/121*b*n*x^11*e^3 + 1/11*b*x^11*e^3*log(c) + 1/3*b*d*n*x^9*e^2*log(x) + 1/11*a*x^1
1*e^3 - 1/27*b*d*n*x^9*e^2 + 1/3*b*d*x^9*e^2*log(c) + 3/7*b*d^2*n*x^7*e*log(x) + 1/3*a*d*x^9*e^2 - 3/49*b*d^2*
n*x^7*e + 3/7*b*d^2*x^7*e*log(c) + 3/7*a*d^2*x^7*e + 1/5*b*d^3*n*x^5*log(x) - 1/25*b*d^3*n*x^5 + 1/5*b*d^3*x^5
*log(c) + 1/5*a*d^3*x^5