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

Optimal. Leaf size=162 \[ \frac{6}{5} a^2 b^2 c x^5+a^2 b^2 d x^6+\frac{6}{7} a^2 b^2 e x^7+2 a^3 b c x^2+\frac{4}{3} a^3 b d x^3+a^3 b e x^4-\frac{a^4 c}{x}+a^4 d \log (x)+a^4 e x+\frac{1}{2} a b^3 c x^8+\frac{4}{9} a b^3 d x^9+\frac{2}{5} a b^3 e x^{10}+\frac{1}{11} b^4 c x^{11}+\frac{1}{12} b^4 d x^{12}+\frac{1}{13} b^4 e x^{13} \]

[Out]

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

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Rubi [A]  time = 0.13332, antiderivative size = 162, 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} \[ \frac{6}{5} a^2 b^2 c x^5+a^2 b^2 d x^6+\frac{6}{7} a^2 b^2 e x^7+2 a^3 b c x^2+\frac{4}{3} a^3 b d x^3+a^3 b e x^4-\frac{a^4 c}{x}+a^4 d \log (x)+a^4 e x+\frac{1}{2} a b^3 c x^8+\frac{4}{9} a b^3 d x^9+\frac{2}{5} a b^3 e x^{10}+\frac{1}{11} b^4 c x^{11}+\frac{1}{12} b^4 d x^{12}+\frac{1}{13} b^4 e x^{13} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0087761, size = 162, normalized size = 1. \[ \frac{6}{5} a^2 b^2 c x^5+a^2 b^2 d x^6+\frac{6}{7} a^2 b^2 e x^7+2 a^3 b c x^2+\frac{4}{3} a^3 b d x^3+a^3 b e x^4-\frac{a^4 c}{x}+a^4 d \log (x)+a^4 e x+\frac{1}{2} a b^3 c x^8+\frac{4}{9} a b^3 d x^9+\frac{2}{5} a b^3 e x^{10}+\frac{1}{11} b^4 c x^{11}+\frac{1}{12} b^4 d x^{12}+\frac{1}{13} b^4 e x^{13} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.006, size = 145, normalized size = 0.9 \begin{align*} -{\frac{{a}^{4}c}{x}}+{a}^{4}ex+2\,{a}^{3}bc{x}^{2}+{\frac{4\,{a}^{3}bd{x}^{3}}{3}}+{a}^{3}be{x}^{4}+{\frac{6\,{a}^{2}{b}^{2}c{x}^{5}}{5}}+{a}^{2}{b}^{2}d{x}^{6}+{\frac{6\,{a}^{2}{b}^{2}e{x}^{7}}{7}}+{\frac{a{b}^{3}c{x}^{8}}{2}}+{\frac{4\,a{b}^{3}d{x}^{9}}{9}}+{\frac{2\,a{b}^{3}e{x}^{10}}{5}}+{\frac{{b}^{4}c{x}^{11}}{11}}+{\frac{{b}^{4}d{x}^{12}}{12}}+{\frac{{b}^{4}e{x}^{13}}{13}}+{a}^{4}d\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^2,x)

[Out]

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

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Maxima [A]  time = 0.941723, size = 194, normalized size = 1.2 \begin{align*} \frac{1}{13} \, b^{4} e x^{13} + \frac{1}{12} \, b^{4} d x^{12} + \frac{1}{11} \, b^{4} c x^{11} + \frac{2}{5} \, a b^{3} e x^{10} + \frac{4}{9} \, a b^{3} d x^{9} + \frac{1}{2} \, a b^{3} c x^{8} + \frac{6}{7} \, a^{2} b^{2} e x^{7} + a^{2} b^{2} d x^{6} + \frac{6}{5} \, a^{2} b^{2} c x^{5} + a^{3} b e x^{4} + \frac{4}{3} \, a^{3} b d x^{3} + 2 \, a^{3} b c x^{2} + a^{4} e x + a^{4} d \log \left (x\right ) - \frac{a^{4} c}{x} \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^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.4897, size = 433, normalized size = 2.67 \begin{align*} \frac{13860 \, b^{4} e x^{14} + 15015 \, b^{4} d x^{13} + 16380 \, b^{4} c x^{12} + 72072 \, a b^{3} e x^{11} + 80080 \, a b^{3} d x^{10} + 90090 \, a b^{3} c x^{9} + 154440 \, a^{2} b^{2} e x^{8} + 180180 \, a^{2} b^{2} d x^{7} + 216216 \, a^{2} b^{2} c x^{6} + 180180 \, a^{3} b e x^{5} + 240240 \, a^{3} b d x^{4} + 360360 \, a^{3} b c x^{3} + 180180 \, a^{4} e x^{2} + 180180 \, a^{4} d x \log \left (x\right ) - 180180 \, a^{4} c}{180180 \, x} \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^2,x, algorithm="fricas")

[Out]

1/180180*(13860*b^4*e*x^14 + 15015*b^4*d*x^13 + 16380*b^4*c*x^12 + 72072*a*b^3*e*x^11 + 80080*a*b^3*d*x^10 + 9
0090*a*b^3*c*x^9 + 154440*a^2*b^2*e*x^8 + 180180*a^2*b^2*d*x^7 + 216216*a^2*b^2*c*x^6 + 180180*a^3*b*e*x^5 + 2
40240*a^3*b*d*x^4 + 360360*a^3*b*c*x^3 + 180180*a^4*e*x^2 + 180180*a^4*d*x*log(x) - 180180*a^4*c)/x

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Sympy [A]  time = 0.47288, size = 168, normalized size = 1.04 \begin{align*} - \frac{a^{4} c}{x} + a^{4} d \log{\left (x \right )} + a^{4} e x + 2 a^{3} b c x^{2} + \frac{4 a^{3} b d x^{3}}{3} + a^{3} b e x^{4} + \frac{6 a^{2} b^{2} c x^{5}}{5} + a^{2} b^{2} d x^{6} + \frac{6 a^{2} b^{2} e x^{7}}{7} + \frac{a b^{3} c x^{8}}{2} + \frac{4 a b^{3} d x^{9}}{9} + \frac{2 a b^{3} e x^{10}}{5} + \frac{b^{4} c x^{11}}{11} + \frac{b^{4} d x^{12}}{12} + \frac{b^{4} e x^{13}}{13} \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**2,x)

[Out]

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

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Giac [A]  time = 1.0459, size = 203, normalized size = 1.25 \begin{align*} \frac{1}{13} \, b^{4} x^{13} e + \frac{1}{12} \, b^{4} d x^{12} + \frac{1}{11} \, b^{4} c x^{11} + \frac{2}{5} \, a b^{3} x^{10} e + \frac{4}{9} \, a b^{3} d x^{9} + \frac{1}{2} \, a b^{3} c x^{8} + \frac{6}{7} \, a^{2} b^{2} x^{7} e + a^{2} b^{2} d x^{6} + \frac{6}{5} \, a^{2} b^{2} c x^{5} + a^{3} b x^{4} e + \frac{4}{3} \, a^{3} b d x^{3} + 2 \, a^{3} b c x^{2} + a^{4} x e + a^{4} d \log \left ({\left | x \right |}\right ) - \frac{a^{4} c}{x} \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^2,x, algorithm="giac")

[Out]

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