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

Optimal. Leaf size=127 \[ a^2 b c x^3+\frac{3}{4} a^2 b d x^4+\frac{3}{5} a^2 b e x^5+a^3 c \log (x)+a^3 d x+\frac{1}{2} a^3 e x^2+\frac{1}{2} a b^2 c x^6+\frac{3}{7} a b^2 d x^7+\frac{3}{8} a b^2 e x^8+\frac{1}{9} b^3 c x^9+\frac{1}{10} b^3 d x^{10}+\frac{1}{11} b^3 e x^{11} \]

[Out]

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

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Rubi [A]  time = 0.0736885, antiderivative size = 127, 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} \[ a^2 b c x^3+\frac{3}{4} a^2 b d x^4+\frac{3}{5} a^2 b e x^5+a^3 c \log (x)+a^3 d x+\frac{1}{2} a^3 e x^2+\frac{1}{2} a b^2 c x^6+\frac{3}{7} a b^2 d x^7+\frac{3}{8} a b^2 e x^8+\frac{1}{9} b^3 c x^9+\frac{1}{10} b^3 d x^{10}+\frac{1}{11} b^3 e x^{11} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0087516, size = 127, normalized size = 1. \[ a^2 b c x^3+\frac{3}{4} a^2 b d x^4+\frac{3}{5} a^2 b e x^5+a^3 c \log (x)+a^3 d x+\frac{1}{2} a^3 e x^2+\frac{1}{2} a b^2 c x^6+\frac{3}{7} a b^2 d x^7+\frac{3}{8} a b^2 e x^8+\frac{1}{9} b^3 c x^9+\frac{1}{10} b^3 d x^{10}+\frac{1}{11} b^3 e x^{11} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.001, size = 110, normalized size = 0.9 \begin{align*}{a}^{3}dx+{\frac{{a}^{3}e{x}^{2}}{2}}+{a}^{2}bc{x}^{3}+{\frac{3\,{a}^{2}bd{x}^{4}}{4}}+{\frac{3\,{a}^{2}be{x}^{5}}{5}}+{\frac{a{b}^{2}c{x}^{6}}{2}}+{\frac{3\,a{b}^{2}d{x}^{7}}{7}}+{\frac{3\,a{b}^{2}e{x}^{8}}{8}}+{\frac{{b}^{3}c{x}^{9}}{9}}+{\frac{{b}^{3}d{x}^{10}}{10}}+{\frac{{b}^{3}e{x}^{11}}{11}}+{a}^{3}c\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)^3/x,x)

[Out]

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

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Maxima [A]  time = 0.952839, size = 147, normalized size = 1.16 \begin{align*} \frac{1}{11} \, b^{3} e x^{11} + \frac{1}{10} \, b^{3} d x^{10} + \frac{1}{9} \, b^{3} c x^{9} + \frac{3}{8} \, a b^{2} e x^{8} + \frac{3}{7} \, a b^{2} d x^{7} + \frac{1}{2} \, a b^{2} c x^{6} + \frac{3}{5} \, a^{2} b e x^{5} + \frac{3}{4} \, a^{2} b d x^{4} + a^{2} b c x^{3} + \frac{1}{2} \, a^{3} e x^{2} + a^{3} d x + a^{3} c \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.49194, size = 265, normalized size = 2.09 \begin{align*} \frac{1}{11} \, b^{3} e x^{11} + \frac{1}{10} \, b^{3} d x^{10} + \frac{1}{9} \, b^{3} c x^{9} + \frac{3}{8} \, a b^{2} e x^{8} + \frac{3}{7} \, a b^{2} d x^{7} + \frac{1}{2} \, a b^{2} c x^{6} + \frac{3}{5} \, a^{2} b e x^{5} + \frac{3}{4} \, a^{2} b d x^{4} + a^{2} b c x^{3} + \frac{1}{2} \, a^{3} e x^{2} + a^{3} d x + a^{3} c \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.375604, size = 131, normalized size = 1.03 \begin{align*} a^{3} c \log{\left (x \right )} + a^{3} d x + \frac{a^{3} e x^{2}}{2} + a^{2} b c x^{3} + \frac{3 a^{2} b d x^{4}}{4} + \frac{3 a^{2} b e x^{5}}{5} + \frac{a b^{2} c x^{6}}{2} + \frac{3 a b^{2} d x^{7}}{7} + \frac{3 a b^{2} e x^{8}}{8} + \frac{b^{3} c x^{9}}{9} + \frac{b^{3} d x^{10}}{10} + \frac{b^{3} e x^{11}}{11} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.07317, size = 154, normalized size = 1.21 \begin{align*} \frac{1}{11} \, b^{3} x^{11} e + \frac{1}{10} \, b^{3} d x^{10} + \frac{1}{9} \, b^{3} c x^{9} + \frac{3}{8} \, a b^{2} x^{8} e + \frac{3}{7} \, a b^{2} d x^{7} + \frac{1}{2} \, a b^{2} c x^{6} + \frac{3}{5} \, a^{2} b x^{5} e + \frac{3}{4} \, a^{2} b d x^{4} + a^{2} b c x^{3} + \frac{1}{2} \, a^{3} x^{2} e + a^{3} d x + a^{3} c \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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