3.336 \(\int \frac{(c+d x+e x^2) (a+b x^3)^4}{x^3} \, dx\)

Optimal. Leaf size=166 \[ \frac{3}{2} a^2 b^2 c x^4+\frac{6}{5} a^2 b^2 d x^5+a^2 b^2 e x^6+4 a^3 b c x+2 a^3 b d x^2+\frac{4}{3} a^3 b e x^3-\frac{a^4 c}{2 x^2}-\frac{a^4 d}{x}+a^4 e \log (x)+\frac{4}{7} a b^3 c x^7+\frac{1}{2} a b^3 d x^8+\frac{4}{9} a b^3 e x^9+\frac{1}{10} b^4 c x^{10}+\frac{1}{11} b^4 d x^{11}+\frac{1}{12} b^4 e x^{12} \]

[Out]

-(a^4*c)/(2*x^2) - (a^4*d)/x + 4*a^3*b*c*x + 2*a^3*b*d*x^2 + (4*a^3*b*e*x^3)/3 + (3*a^2*b^2*c*x^4)/2 + (6*a^2*
b^2*d*x^5)/5 + a^2*b^2*e*x^6 + (4*a*b^3*c*x^7)/7 + (a*b^3*d*x^8)/2 + (4*a*b^3*e*x^9)/9 + (b^4*c*x^10)/10 + (b^
4*d*x^11)/11 + (b^4*e*x^12)/12 + a^4*e*Log[x]

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Rubi [A]  time = 0.124775, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {1628} \[ \frac{3}{2} a^2 b^2 c x^4+\frac{6}{5} a^2 b^2 d x^5+a^2 b^2 e x^6+4 a^3 b c x+2 a^3 b d x^2+\frac{4}{3} a^3 b e x^3-\frac{a^4 c}{2 x^2}-\frac{a^4 d}{x}+a^4 e \log (x)+\frac{4}{7} a b^3 c x^7+\frac{1}{2} a b^3 d x^8+\frac{4}{9} a b^3 e x^9+\frac{1}{10} b^4 c x^{10}+\frac{1}{11} b^4 d x^{11}+\frac{1}{12} b^4 e x^{12} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^4*c)/(2*x^2) - (a^4*d)/x + 4*a^3*b*c*x + 2*a^3*b*d*x^2 + (4*a^3*b*e*x^3)/3 + (3*a^2*b^2*c*x^4)/2 + (6*a^2*
b^2*d*x^5)/5 + a^2*b^2*e*x^6 + (4*a*b^3*c*x^7)/7 + (a*b^3*d*x^8)/2 + (4*a*b^3*e*x^9)/9 + (b^4*c*x^10)/10 + (b^
4*d*x^11)/11 + (b^4*e*x^12)/12 + a^4*e*Log[x]

Rule 1628

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

\begin{align*} \int \frac{\left (c+d x+e x^2\right ) \left (a+b x^3\right )^4}{x^3} \, dx &=\int \left (4 a^3 b c+\frac{a^4 c}{x^3}+\frac{a^4 d}{x^2}+\frac{a^4 e}{x}+4 a^3 b d x+4 a^3 b e x^2+6 a^2 b^2 c x^3+6 a^2 b^2 d x^4+6 a^2 b^2 e x^5+4 a b^3 c x^6+4 a b^3 d x^7+4 a b^3 e x^8+b^4 c x^9+b^4 d x^{10}+b^4 e x^{11}\right ) \, dx\\ &=-\frac{a^4 c}{2 x^2}-\frac{a^4 d}{x}+4 a^3 b c x+2 a^3 b d x^2+\frac{4}{3} a^3 b e x^3+\frac{3}{2} a^2 b^2 c x^4+\frac{6}{5} a^2 b^2 d x^5+a^2 b^2 e x^6+\frac{4}{7} a b^3 c x^7+\frac{1}{2} a b^3 d x^8+\frac{4}{9} a b^3 e x^9+\frac{1}{10} b^4 c x^{10}+\frac{1}{11} b^4 d x^{11}+\frac{1}{12} b^4 e x^{12}+a^4 e \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0089017, size = 166, normalized size = 1. \[ \frac{3}{2} a^2 b^2 c x^4+\frac{6}{5} a^2 b^2 d x^5+a^2 b^2 e x^6+4 a^3 b c x+2 a^3 b d x^2+\frac{4}{3} a^3 b e x^3-\frac{a^4 c}{2 x^2}-\frac{a^4 d}{x}+a^4 e \log (x)+\frac{4}{7} a b^3 c x^7+\frac{1}{2} a b^3 d x^8+\frac{4}{9} a b^3 e x^9+\frac{1}{10} b^4 c x^{10}+\frac{1}{11} b^4 d x^{11}+\frac{1}{12} b^4 e x^{12} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^4*c)/(2*x^2) - (a^4*d)/x + 4*a^3*b*c*x + 2*a^3*b*d*x^2 + (4*a^3*b*e*x^3)/3 + (3*a^2*b^2*c*x^4)/2 + (6*a^2*
b^2*d*x^5)/5 + a^2*b^2*e*x^6 + (4*a*b^3*c*x^7)/7 + (a*b^3*d*x^8)/2 + (4*a*b^3*e*x^9)/9 + (b^4*c*x^10)/10 + (b^
4*d*x^11)/11 + (b^4*e*x^12)/12 + a^4*e*Log[x]

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Maple [A]  time = 0.006, size = 147, normalized size = 0.9 \begin{align*} -{\frac{{a}^{4}c}{2\,{x}^{2}}}-{\frac{{a}^{4}d}{x}}+4\,{a}^{3}bcx+2\,{a}^{3}bd{x}^{2}+{\frac{4\,{a}^{3}be{x}^{3}}{3}}+{\frac{3\,{a}^{2}{b}^{2}c{x}^{4}}{2}}+{\frac{6\,{a}^{2}{b}^{2}d{x}^{5}}{5}}+{a}^{2}{b}^{2}e{x}^{6}+{\frac{4\,a{b}^{3}c{x}^{7}}{7}}+{\frac{a{b}^{3}d{x}^{8}}{2}}+{\frac{4\,a{b}^{3}e{x}^{9}}{9}}+{\frac{{b}^{4}c{x}^{10}}{10}}+{\frac{{b}^{4}d{x}^{11}}{11}}+{\frac{{b}^{4}e{x}^{12}}{12}}+{a}^{4}e\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*a^4*c/x^2-a^4*d/x+4*a^3*b*c*x+2*a^3*b*d*x^2+4/3*a^3*b*e*x^3+3/2*a^2*b^2*c*x^4+6/5*a^2*b^2*d*x^5+a^2*b^2*e
*x^6+4/7*a*b^3*c*x^7+1/2*a*b^3*d*x^8+4/9*a*b^3*e*x^9+1/10*b^4*c*x^10+1/11*b^4*d*x^11+1/12*b^4*e*x^12+a^4*e*ln(
x)

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Maxima [A]  time = 0.949211, size = 197, normalized size = 1.19 \begin{align*} \frac{1}{12} \, b^{4} e x^{12} + \frac{1}{11} \, b^{4} d x^{11} + \frac{1}{10} \, b^{4} c x^{10} + \frac{4}{9} \, a b^{3} e x^{9} + \frac{1}{2} \, a b^{3} d x^{8} + \frac{4}{7} \, a b^{3} c x^{7} + a^{2} b^{2} e x^{6} + \frac{6}{5} \, a^{2} b^{2} d x^{5} + \frac{3}{2} \, a^{2} b^{2} c x^{4} + \frac{4}{3} \, a^{3} b e x^{3} + 2 \, a^{3} b d x^{2} + 4 \, a^{3} b c x + a^{4} e \log \left (x\right ) - \frac{2 \, a^{4} d x + a^{4} c}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*b^4*e*x^12 + 1/11*b^4*d*x^11 + 1/10*b^4*c*x^10 + 4/9*a*b^3*e*x^9 + 1/2*a*b^3*d*x^8 + 4/7*a*b^3*c*x^7 + a^
2*b^2*e*x^6 + 6/5*a^2*b^2*d*x^5 + 3/2*a^2*b^2*c*x^4 + 4/3*a^3*b*e*x^3 + 2*a^3*b*d*x^2 + 4*a^3*b*c*x + a^4*e*lo
g(x) - 1/2*(2*a^4*d*x + a^4*c)/x^2

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Fricas [A]  time = 1.52702, size = 413, normalized size = 2.49 \begin{align*} \frac{1155 \, b^{4} e x^{14} + 1260 \, b^{4} d x^{13} + 1386 \, b^{4} c x^{12} + 6160 \, a b^{3} e x^{11} + 6930 \, a b^{3} d x^{10} + 7920 \, a b^{3} c x^{9} + 13860 \, a^{2} b^{2} e x^{8} + 16632 \, a^{2} b^{2} d x^{7} + 20790 \, a^{2} b^{2} c x^{6} + 18480 \, a^{3} b e x^{5} + 27720 \, a^{3} b d x^{4} + 55440 \, a^{3} b c x^{3} + 13860 \, a^{4} e x^{2} \log \left (x\right ) - 13860 \, a^{4} d x - 6930 \, a^{4} c}{13860 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13860*(1155*b^4*e*x^14 + 1260*b^4*d*x^13 + 1386*b^4*c*x^12 + 6160*a*b^3*e*x^11 + 6930*a*b^3*d*x^10 + 7920*a*
b^3*c*x^9 + 13860*a^2*b^2*e*x^8 + 16632*a^2*b^2*d*x^7 + 20790*a^2*b^2*c*x^6 + 18480*a^3*b*e*x^5 + 27720*a^3*b*
d*x^4 + 55440*a^3*b*c*x^3 + 13860*a^4*e*x^2*log(x) - 13860*a^4*d*x - 6930*a^4*c)/x^2

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Sympy [A]  time = 0.558303, size = 173, normalized size = 1.04 \begin{align*} a^{4} e \log{\left (x \right )} + 4 a^{3} b c x + 2 a^{3} b d x^{2} + \frac{4 a^{3} b e x^{3}}{3} + \frac{3 a^{2} b^{2} c x^{4}}{2} + \frac{6 a^{2} b^{2} d x^{5}}{5} + a^{2} b^{2} e x^{6} + \frac{4 a b^{3} c x^{7}}{7} + \frac{a b^{3} d x^{8}}{2} + \frac{4 a b^{3} e x^{9}}{9} + \frac{b^{4} c x^{10}}{10} + \frac{b^{4} d x^{11}}{11} + \frac{b^{4} e x^{12}}{12} - \frac{a^{4} c + 2 a^{4} d x}{2 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**4*e*log(x) + 4*a**3*b*c*x + 2*a**3*b*d*x**2 + 4*a**3*b*e*x**3/3 + 3*a**2*b**2*c*x**4/2 + 6*a**2*b**2*d*x**5
/5 + a**2*b**2*e*x**6 + 4*a*b**3*c*x**7/7 + a*b**3*d*x**8/2 + 4*a*b**3*e*x**9/9 + b**4*c*x**10/10 + b**4*d*x**
11/11 + b**4*e*x**12/12 - (a**4*c + 2*a**4*d*x)/(2*x**2)

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Giac [A]  time = 1.05512, size = 205, normalized size = 1.23 \begin{align*} \frac{1}{12} \, b^{4} x^{12} e + \frac{1}{11} \, b^{4} d x^{11} + \frac{1}{10} \, b^{4} c x^{10} + \frac{4}{9} \, a b^{3} x^{9} e + \frac{1}{2} \, a b^{3} d x^{8} + \frac{4}{7} \, a b^{3} c x^{7} + a^{2} b^{2} x^{6} e + \frac{6}{5} \, a^{2} b^{2} d x^{5} + \frac{3}{2} \, a^{2} b^{2} c x^{4} + \frac{4}{3} \, a^{3} b x^{3} e + 2 \, a^{3} b d x^{2} + 4 \, a^{3} b c x + a^{4} e \log \left ({\left | x \right |}\right ) - \frac{2 \, a^{4} d x + a^{4} c}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*b^4*x^12*e + 1/11*b^4*d*x^11 + 1/10*b^4*c*x^10 + 4/9*a*b^3*x^9*e + 1/2*a*b^3*d*x^8 + 4/7*a*b^3*c*x^7 + a^
2*b^2*x^6*e + 6/5*a^2*b^2*d*x^5 + 3/2*a^2*b^2*c*x^4 + 4/3*a^3*b*x^3*e + 2*a^3*b*d*x^2 + 4*a^3*b*c*x + a^4*e*lo
g(abs(x)) - 1/2*(2*a^4*d*x + a^4*c)/x^2