\(\int \frac {(c+d x+e x^2) (a+b x^3)^3}{x^3} \, dx\) [330]

Optimal result

Integrand size = 23, antiderivative size = 126 \[ \int \frac {\left (c+d x+e x^2\right ) \left (a+b x^3\right )^3}{x^3} \, dx=-\frac {a^3 c}{2 x^2}-\frac {a^3 d}{x}+3 a^2 b c x+\frac {3}{2} a^2 b d x^2+a^2 b e x^3+\frac {3}{4} a b^2 c x^4+\frac {3}{5} a b^2 d x^5+\frac {1}{2} a b^2 e x^6+\frac {1}{7} b^3 c x^7+\frac {1}{8} b^3 d x^8+\frac {1}{9} b^3 e x^9+a^3 e \log (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {1642} \[ \int \frac {\left (c+d x+e x^2\right ) \left (a+b x^3\right )^3}{x^3} \, dx=-\frac {a^3 c}{2 x^2}-\frac {a^3 d}{x}+a^3 e \log (x)+3 a^2 b c x+\frac {3}{2} a^2 b d x^2+a^2 b e x^3+\frac {3}{4} a b^2 c x^4+\frac {3}{5} a b^2 d x^5+\frac {1}{2} a b^2 e x^6+\frac {1}{7} b^3 c x^7+\frac {1}{8} b^3 d x^8+\frac {1}{9} b^3 e x^9 \]

[In]

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

[Out]

-1/2*(a^3*c)/x^2 - (a^3*d)/x + 3*a^2*b*c*x + (3*a^2*b*d*x^2)/2 + a^2*b*e*x^3 + (3*a*b^2*c*x^4)/4 + (3*a*b^2*d*
x^5)/5 + (a*b^2*e*x^6)/2 + (b^3*c*x^7)/7 + (b^3*d*x^8)/8 + (b^3*e*x^9)/9 + a^3*e*Log[x]

Rule 1642

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00 \[ \int \frac {\left (c+d x+e x^2\right ) \left (a+b x^3\right )^3}{x^3} \, dx=-\frac {a^3 c}{2 x^2}-\frac {a^3 d}{x}+3 a^2 b c x+\frac {3}{2} a^2 b d x^2+a^2 b e x^3+\frac {3}{4} a b^2 c x^4+\frac {3}{5} a b^2 d x^5+\frac {1}{2} a b^2 e x^6+\frac {1}{7} b^3 c x^7+\frac {1}{8} b^3 d x^8+\frac {1}{9} b^3 e x^9+a^3 e \log (x) \]

[In]

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

[Out]

-1/2*(a^3*c)/x^2 - (a^3*d)/x + 3*a^2*b*c*x + (3*a^2*b*d*x^2)/2 + a^2*b*e*x^3 + (3*a*b^2*c*x^4)/4 + (3*a*b^2*d*
x^5)/5 + (a*b^2*e*x^6)/2 + (b^3*c*x^7)/7 + (b^3*d*x^8)/8 + (b^3*e*x^9)/9 + a^3*e*Log[x]

Maple [A] (verified)

Time = 1.54 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.88

method	result	size
default	\(-\frac {a^{3} c}{2 x^{2}}-\frac {a^{3} d}{x}+3 a^{2} b c x +\frac {3 a^{2} b d \,x^{2}}{2}+a^{2} b e \,x^{3}+\frac {3 a \,b^{2} c \,x^{4}}{4}+\frac {3 a \,b^{2} d \,x^{5}}{5}+\frac {a \,b^{2} e \,x^{6}}{2}+\frac {b^{3} c \,x^{7}}{7}+\frac {b^{3} d \,x^{8}}{8}+\frac {b^{3} e \,x^{9}}{9}+a^{3} e \ln \left (x \right )\)	\(111\)
risch	\(\frac {b^{3} e \,x^{9}}{9}+\frac {b^{3} d \,x^{8}}{8}+\frac {b^{3} c \,x^{7}}{7}+\frac {a \,b^{2} e \,x^{6}}{2}+\frac {3 a \,b^{2} d \,x^{5}}{5}+\frac {3 a \,b^{2} c \,x^{4}}{4}+a^{2} b e \,x^{3}+\frac {3 a^{2} b d \,x^{2}}{2}+3 a^{2} b c x +\frac {-a^{3} d x -\frac {1}{2} c \,a^{3}}{x^{2}}+a^{3} e \ln \left (x \right )\)	\(111\)
norman	\(\frac {a^{2} b e \,x^{5}-\frac {1}{2} c \,a^{3}-a^{3} d x +\frac {1}{7} b^{3} c \,x^{9}+\frac {1}{8} b^{3} d \,x^{10}+\frac {1}{9} b^{3} e \,x^{11}+\frac {3}{4} a \,b^{2} c \,x^{6}+\frac {3}{5} a \,b^{2} d \,x^{7}+\frac {1}{2} a \,b^{2} e \,x^{8}+\frac {3}{2} a^{2} b d \,x^{4}+3 a^{2} x^{3} b c}{x^{2}}+a^{3} e \ln \left (x \right )\)	\(113\)
parallelrisch	\(\frac {280 b^{3} e \,x^{11}+315 b^{3} d \,x^{10}+360 b^{3} c \,x^{9}+1260 a \,b^{2} e \,x^{8}+1512 a \,b^{2} d \,x^{7}+1890 a \,b^{2} c \,x^{6}+2520 a^{2} b e \,x^{5}+3780 a^{2} b d \,x^{4}+2520 e \,a^{3} \ln \left (x \right ) x^{2}+7560 a^{2} x^{3} b c -2520 a^{3} d x -1260 c \,a^{3}}{2520 x^{2}}\)	\(118\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.93 \[ \int \frac {\left (c+d x+e x^2\right ) \left (a+b x^3\right )^3}{x^3} \, dx=\frac {280 \, b^{3} e x^{11} + 315 \, b^{3} d x^{10} + 360 \, b^{3} c x^{9} + 1260 \, a b^{2} e x^{8} + 1512 \, a b^{2} d x^{7} + 1890 \, a b^{2} c x^{6} + 2520 \, a^{2} b e x^{5} + 3780 \, a^{2} b d x^{4} + 7560 \, a^{2} b c x^{3} + 2520 \, a^{3} e x^{2} \log \left (x\right ) - 2520 \, a^{3} d x - 1260 \, a^{3} c}{2520 \, x^{2}} \]

[In]

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

[Out]

1/2520*(280*b^3*e*x^11 + 315*b^3*d*x^10 + 360*b^3*c*x^9 + 1260*a*b^2*e*x^8 + 1512*a*b^2*d*x^7 + 1890*a*b^2*c*x
^6 + 2520*a^2*b*e*x^5 + 3780*a^2*b*d*x^4 + 7560*a^2*b*c*x^3 + 2520*a^3*e*x^2*log(x) - 2520*a^3*d*x - 1260*a^3*
c)/x^2

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.04 \[ \int \frac {\left (c+d x+e x^2\right ) \left (a+b x^3\right )^3}{x^3} \, dx=a^{3} e \log {\left (x \right )} + 3 a^{2} b c x + \frac {3 a^{2} b d x^{2}}{2} + a^{2} b e x^{3} + \frac {3 a b^{2} c x^{4}}{4} + \frac {3 a b^{2} d x^{5}}{5} + \frac {a b^{2} e x^{6}}{2} + \frac {b^{3} c x^{7}}{7} + \frac {b^{3} d x^{8}}{8} + \frac {b^{3} e x^{9}}{9} + \frac {- a^{3} c - 2 a^{3} d x}{2 x^{2}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.87 \[ \int \frac {\left (c+d x+e x^2\right ) \left (a+b x^3\right )^3}{x^3} \, dx=\frac {1}{9} \, b^{3} e x^{9} + \frac {1}{8} \, b^{3} d x^{8} + \frac {1}{7} \, b^{3} c x^{7} + \frac {1}{2} \, a b^{2} e x^{6} + \frac {3}{5} \, a b^{2} d x^{5} + \frac {3}{4} \, a b^{2} c x^{4} + a^{2} b e x^{3} + \frac {3}{2} \, a^{2} b d x^{2} + 3 \, a^{2} b c x + a^{3} e \log \left (x\right ) - \frac {2 \, a^{3} d x + a^{3} c}{2 \, x^{2}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.88 \[ \int \frac {\left (c+d x+e x^2\right ) \left (a+b x^3\right )^3}{x^3} \, dx=\frac {1}{9} \, b^{3} e x^{9} + \frac {1}{8} \, b^{3} d x^{8} + \frac {1}{7} \, b^{3} c x^{7} + \frac {1}{2} \, a b^{2} e x^{6} + \frac {3}{5} \, a b^{2} d x^{5} + \frac {3}{4} \, a b^{2} c x^{4} + a^{2} b e x^{3} + \frac {3}{2} \, a^{2} b d x^{2} + 3 \, a^{2} b c x + a^{3} e \log \left ({\left | x \right |}\right ) - \frac {2 \, a^{3} d x + a^{3} c}{2 \, x^{2}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 8.98 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.87 \[ \int \frac {\left (c+d x+e x^2\right ) \left (a+b x^3\right )^3}{x^3} \, dx=\frac {b^3\,c\,x^7}{7}-\frac {\frac {a^3\,c}{2}+a^3\,d\,x}{x^2}+\frac {b^3\,d\,x^8}{8}+\frac {b^3\,e\,x^9}{9}+a^3\,e\,\ln \left (x\right )+3\,a^2\,b\,c\,x+\frac {3\,a\,b^2\,c\,x^4}{4}+\frac {3\,a^2\,b\,d\,x^2}{2}+\frac {3\,a\,b^2\,d\,x^5}{5}+a^2\,b\,e\,x^3+\frac {a\,b^2\,e\,x^6}{2} \]

[In]

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

[Out]

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