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

Optimal. Leaf size=166 \[ a^2 b^2 c x^6+\frac{6}{7} a^2 b^2 d x^7+\frac{3}{4} a^2 b^2 e x^8+\frac{4}{3} a^3 b c x^3+a^3 b d x^4+\frac{4}{5} a^3 b e x^5+a^4 c \log (x)+a^4 d x+\frac{1}{2} a^4 e x^2+\frac{4}{9} a b^3 c x^9+\frac{2}{5} a b^3 d x^{10}+\frac{4}{11} a b^3 e x^{11}+\frac{1}{12} b^4 c x^{12}+\frac{1}{13} b^4 d x^{13}+\frac{1}{14} b^4 e x^{14} \]

[Out]

a^4*d*x + (a^4*e*x^2)/2 + (4*a^3*b*c*x^3)/3 + a^3*b*d*x^4 + (4*a^3*b*e*x^5)/5 + a^2*b^2*c*x^6 + (6*a^2*b^2*d*x
^7)/7 + (3*a^2*b^2*e*x^8)/4 + (4*a*b^3*c*x^9)/9 + (2*a*b^3*d*x^10)/5 + (4*a*b^3*e*x^11)/11 + (b^4*c*x^12)/12 +
(b^4*d*x^13)/13 + (b^4*e*x^14)/14 + a^4*c*Log[x]

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Rubi [A]  time = 0.108534, 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} \[ a^2 b^2 c x^6+\frac{6}{7} a^2 b^2 d x^7+\frac{3}{4} a^2 b^2 e x^8+\frac{4}{3} a^3 b c x^3+a^3 b d x^4+\frac{4}{5} a^3 b e x^5+a^4 c \log (x)+a^4 d x+\frac{1}{2} a^4 e x^2+\frac{4}{9} a b^3 c x^9+\frac{2}{5} a b^3 d x^{10}+\frac{4}{11} a b^3 e x^{11}+\frac{1}{12} b^4 c x^{12}+\frac{1}{13} b^4 d x^{13}+\frac{1}{14} b^4 e x^{14} \]

Antiderivative was successfully verified.

[In]

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

[Out]

a^4*d*x + (a^4*e*x^2)/2 + (4*a^3*b*c*x^3)/3 + a^3*b*d*x^4 + (4*a^3*b*e*x^5)/5 + a^2*b^2*c*x^6 + (6*a^2*b^2*d*x
^7)/7 + (3*a^2*b^2*e*x^8)/4 + (4*a*b^3*c*x^9)/9 + (2*a*b^3*d*x^10)/5 + (4*a*b^3*e*x^11)/11 + (b^4*c*x^12)/12 +
(b^4*d*x^13)/13 + (b^4*e*x^14)/14 + a^4*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 )^4}{x} \, dx &=\int \left (a^4 d+\frac{a^4 c}{x}+a^4 e x+4 a^3 b c x^2+4 a^3 b d x^3+4 a^3 b e x^4+6 a^2 b^2 c x^5+6 a^2 b^2 d x^6+6 a^2 b^2 e x^7+4 a b^3 c x^8+4 a b^3 d x^9+4 a b^3 e x^{10}+b^4 c x^{11}+b^4 d x^{12}+b^4 e x^{13}\right ) \, dx\\ &=a^4 d x+\frac{1}{2} a^4 e x^2+\frac{4}{3} a^3 b c x^3+a^3 b d x^4+\frac{4}{5} a^3 b e x^5+a^2 b^2 c x^6+\frac{6}{7} a^2 b^2 d x^7+\frac{3}{4} a^2 b^2 e x^8+\frac{4}{9} a b^3 c x^9+\frac{2}{5} a b^3 d x^{10}+\frac{4}{11} a b^3 e x^{11}+\frac{1}{12} b^4 c x^{12}+\frac{1}{13} b^4 d x^{13}+\frac{1}{14} b^4 e x^{14}+a^4 c \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0088974, size = 166, normalized size = 1. \[ a^2 b^2 c x^6+\frac{6}{7} a^2 b^2 d x^7+\frac{3}{4} a^2 b^2 e x^8+\frac{4}{3} a^3 b c x^3+a^3 b d x^4+\frac{4}{5} a^3 b e x^5+a^4 c \log (x)+a^4 d x+\frac{1}{2} a^4 e x^2+\frac{4}{9} a b^3 c x^9+\frac{2}{5} a b^3 d x^{10}+\frac{4}{11} a b^3 e x^{11}+\frac{1}{12} b^4 c x^{12}+\frac{1}{13} b^4 d x^{13}+\frac{1}{14} b^4 e x^{14} \]

Antiderivative was successfully verified.

[In]

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

[Out]

a^4*d*x + (a^4*e*x^2)/2 + (4*a^3*b*c*x^3)/3 + a^3*b*d*x^4 + (4*a^3*b*e*x^5)/5 + a^2*b^2*c*x^6 + (6*a^2*b^2*d*x
^7)/7 + (3*a^2*b^2*e*x^8)/4 + (4*a*b^3*c*x^9)/9 + (2*a*b^3*d*x^10)/5 + (4*a*b^3*e*x^11)/11 + (b^4*c*x^12)/12 +
(b^4*d*x^13)/13 + (b^4*e*x^14)/14 + a^4*c*Log[x]

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Maple [A]  time = 0.003, size = 145, normalized size = 0.9 \begin{align*}{a}^{4}dx+{\frac{{a}^{4}e{x}^{2}}{2}}+{\frac{4\,{a}^{3}bc{x}^{3}}{3}}+{a}^{3}bd{x}^{4}+{\frac{4\,{a}^{3}be{x}^{5}}{5}}+{a}^{2}{b}^{2}c{x}^{6}+{\frac{6\,{a}^{2}{b}^{2}d{x}^{7}}{7}}+{\frac{3\,{a}^{2}{b}^{2}e{x}^{8}}{4}}+{\frac{4\,a{b}^{3}c{x}^{9}}{9}}+{\frac{2\,a{b}^{3}d{x}^{10}}{5}}+{\frac{4\,a{b}^{3}e{x}^{11}}{11}}+{\frac{{b}^{4}c{x}^{12}}{12}}+{\frac{{b}^{4}d{x}^{13}}{13}}+{\frac{{b}^{4}e{x}^{14}}{14}}+{a}^{4}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)^4/x,x)

[Out]

a^4*d*x+1/2*a^4*e*x^2+4/3*a^3*b*c*x^3+a^3*b*d*x^4+4/5*a^3*b*e*x^5+a^2*b^2*c*x^6+6/7*a^2*b^2*d*x^7+3/4*a^2*b^2*
e*x^8+4/9*a*b^3*c*x^9+2/5*a*b^3*d*x^10+4/11*a*b^3*e*x^11+1/12*b^4*c*x^12+1/13*b^4*d*x^13+1/14*b^4*e*x^14+a^4*c
*ln(x)

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Maxima [A]  time = 0.936197, size = 194, normalized size = 1.17 \begin{align*} \frac{1}{14} \, b^{4} e x^{14} + \frac{1}{13} \, b^{4} d x^{13} + \frac{1}{12} \, b^{4} c x^{12} + \frac{4}{11} \, a b^{3} e x^{11} + \frac{2}{5} \, a b^{3} d x^{10} + \frac{4}{9} \, a b^{3} c x^{9} + \frac{3}{4} \, a^{2} b^{2} e x^{8} + \frac{6}{7} \, a^{2} b^{2} d x^{7} + a^{2} b^{2} c x^{6} + \frac{4}{5} \, a^{3} b e x^{5} + a^{3} b d x^{4} + \frac{4}{3} \, a^{3} b c x^{3} + \frac{1}{2} \, a^{4} e x^{2} + a^{4} d x + a^{4} 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)^4/x,x, algorithm="maxima")

[Out]

1/14*b^4*e*x^14 + 1/13*b^4*d*x^13 + 1/12*b^4*c*x^12 + 4/11*a*b^3*e*x^11 + 2/5*a*b^3*d*x^10 + 4/9*a*b^3*c*x^9 +
3/4*a^2*b^2*e*x^8 + 6/7*a^2*b^2*d*x^7 + a^2*b^2*c*x^6 + 4/5*a^3*b*e*x^5 + a^3*b*d*x^4 + 4/3*a^3*b*c*x^3 + 1/2
*a^4*e*x^2 + a^4*d*x + a^4*c*log(x)

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Fricas [A]  time = 1.46817, size = 347, normalized size = 2.09 \begin{align*} \frac{1}{14} \, b^{4} e x^{14} + \frac{1}{13} \, b^{4} d x^{13} + \frac{1}{12} \, b^{4} c x^{12} + \frac{4}{11} \, a b^{3} e x^{11} + \frac{2}{5} \, a b^{3} d x^{10} + \frac{4}{9} \, a b^{3} c x^{9} + \frac{3}{4} \, a^{2} b^{2} e x^{8} + \frac{6}{7} \, a^{2} b^{2} d x^{7} + a^{2} b^{2} c x^{6} + \frac{4}{5} \, a^{3} b e x^{5} + a^{3} b d x^{4} + \frac{4}{3} \, a^{3} b c x^{3} + \frac{1}{2} \, a^{4} e x^{2} + a^{4} d x + a^{4} 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)^4/x,x, algorithm="fricas")

[Out]

1/14*b^4*e*x^14 + 1/13*b^4*d*x^13 + 1/12*b^4*c*x^12 + 4/11*a*b^3*e*x^11 + 2/5*a*b^3*d*x^10 + 4/9*a*b^3*c*x^9 +
3/4*a^2*b^2*e*x^8 + 6/7*a^2*b^2*d*x^7 + a^2*b^2*c*x^6 + 4/5*a^3*b*e*x^5 + a^3*b*d*x^4 + 4/3*a^3*b*c*x^3 + 1/2
*a^4*e*x^2 + a^4*d*x + a^4*c*log(x)

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Sympy [A]  time = 0.456358, size = 175, normalized size = 1.05 \begin{align*} a^{4} c \log{\left (x \right )} + a^{4} d x + \frac{a^{4} e x^{2}}{2} + \frac{4 a^{3} b c x^{3}}{3} + a^{3} b d x^{4} + \frac{4 a^{3} b e x^{5}}{5} + a^{2} b^{2} c x^{6} + \frac{6 a^{2} b^{2} d x^{7}}{7} + \frac{3 a^{2} b^{2} e x^{8}}{4} + \frac{4 a b^{3} c x^{9}}{9} + \frac{2 a b^{3} d x^{10}}{5} + \frac{4 a b^{3} e x^{11}}{11} + \frac{b^{4} c x^{12}}{12} + \frac{b^{4} d x^{13}}{13} + \frac{b^{4} e x^{14}}{14} \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,x)

[Out]

a**4*c*log(x) + a**4*d*x + a**4*e*x**2/2 + 4*a**3*b*c*x**3/3 + a**3*b*d*x**4 + 4*a**3*b*e*x**5/5 + a**2*b**2*c
*x**6 + 6*a**2*b**2*d*x**7/7 + 3*a**2*b**2*e*x**8/4 + 4*a*b**3*c*x**9/9 + 2*a*b**3*d*x**10/5 + 4*a*b**3*e*x**1
1/11 + b**4*c*x**12/12 + b**4*d*x**13/13 + b**4*e*x**14/14

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Giac [A]  time = 1.06076, size = 203, normalized size = 1.22 \begin{align*} \frac{1}{14} \, b^{4} x^{14} e + \frac{1}{13} \, b^{4} d x^{13} + \frac{1}{12} \, b^{4} c x^{12} + \frac{4}{11} \, a b^{3} x^{11} e + \frac{2}{5} \, a b^{3} d x^{10} + \frac{4}{9} \, a b^{3} c x^{9} + \frac{3}{4} \, a^{2} b^{2} x^{8} e + \frac{6}{7} \, a^{2} b^{2} d x^{7} + a^{2} b^{2} c x^{6} + \frac{4}{5} \, a^{3} b x^{5} e + a^{3} b d x^{4} + \frac{4}{3} \, a^{3} b c x^{3} + \frac{1}{2} \, a^{4} x^{2} e + a^{4} d x + a^{4} 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)^4/x,x, algorithm="giac")

[Out]

1/14*b^4*x^14*e + 1/13*b^4*d*x^13 + 1/12*b^4*c*x^12 + 4/11*a*b^3*x^11*e + 2/5*a*b^3*d*x^10 + 4/9*a*b^3*c*x^9 +
3/4*a^2*b^2*x^8*e + 6/7*a^2*b^2*d*x^7 + a^2*b^2*c*x^6 + 4/5*a^3*b*x^5*e + a^3*b*d*x^4 + 4/3*a^3*b*c*x^3 + 1/2
*a^4*x^2*e + a^4*d*x + a^4*c*log(abs(x))