\(\int x^4 (a+b x^3)^3 (c+d x+e x^2+f x^3+g x^4+h x^5) \, dx\) [393]

Optimal result

Integrand size = 38, antiderivative size = 223 \[ \int x^4 \left (a+b x^3\right )^3 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {1}{5} a^3 c x^5+\frac {1}{6} a^3 d x^6+\frac {1}{7} a^3 e x^7+\frac {1}{8} a^2 (3 b c+a f) x^8+\frac {1}{9} a^2 (3 b d+a g) x^9+\frac {1}{10} a^2 (3 b e+a h) x^{10}+\frac {3}{11} a b (b c+a f) x^{11}+\frac {1}{4} a b (b d+a g) x^{12}+\frac {3}{13} a b (b e+a h) x^{13}+\frac {1}{14} b^2 (b c+3 a f) x^{14}+\frac {1}{15} b^2 (b d+3 a g) x^{15}+\frac {1}{16} b^2 (b e+3 a h) x^{16}+\frac {1}{17} b^3 f x^{17}+\frac {1}{18} b^3 g x^{18}+\frac {1}{19} b^3 h x^{19} \]

[Out]

1/5*a^3*c*x^5+1/6*a^3*d*x^6+1/7*a^3*e*x^7+1/8*a^2*(a*f+3*b*c)*x^8+1/9*a^2*(a*g+3*b*d)*x^9+1/10*a^2*(a*h+3*b*e)
*x^10+3/11*a*b*(a*f+b*c)*x^11+1/4*a*b*(a*g+b*d)*x^12+3/13*a*b*(a*h+b*e)*x^13+1/14*b^2*(3*a*f+b*c)*x^14+1/15*b^
2*(3*a*g+b*d)*x^15+1/16*b^2*(3*a*h+b*e)*x^16+1/17*b^3*f*x^17+1/18*b^3*g*x^18+1/19*b^3*h*x^19

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {1834} \[ \int x^4 \left (a+b x^3\right )^3 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {1}{5} a^3 c x^5+\frac {1}{6} a^3 d x^6+\frac {1}{7} a^3 e x^7+\frac {1}{8} a^2 x^8 (a f+3 b c)+\frac {1}{9} a^2 x^9 (a g+3 b d)+\frac {1}{10} a^2 x^{10} (a h+3 b e)+\frac {1}{14} b^2 x^{14} (3 a f+b c)+\frac {1}{15} b^2 x^{15} (3 a g+b d)+\frac {1}{16} b^2 x^{16} (3 a h+b e)+\frac {3}{11} a b x^{11} (a f+b c)+\frac {1}{4} a b x^{12} (a g+b d)+\frac {3}{13} a b x^{13} (a h+b e)+\frac {1}{17} b^3 f x^{17}+\frac {1}{18} b^3 g x^{18}+\frac {1}{19} b^3 h x^{19} \]

[In]

Int[x^4*(a + b*x^3)^3*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5),x]

[Out]

(a^3*c*x^5)/5 + (a^3*d*x^6)/6 + (a^3*e*x^7)/7 + (a^2*(3*b*c + a*f)*x^8)/8 + (a^2*(3*b*d + a*g)*x^9)/9 + (a^2*(
3*b*e + a*h)*x^10)/10 + (3*a*b*(b*c + a*f)*x^11)/11 + (a*b*(b*d + a*g)*x^12)/4 + (3*a*b*(b*e + a*h)*x^13)/13 +
(b^2*(b*c + 3*a*f)*x^14)/14 + (b^2*(b*d + 3*a*g)*x^15)/15 + (b^2*(b*e + 3*a*h)*x^16)/16 + (b^3*f*x^17)/17 + (
b^3*g*x^18)/18 + (b^3*h*x^19)/19

Rule 1834

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a +
b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rubi steps \begin{align*} \text {integral}& = \int \left (a^3 c x^4+a^3 d x^5+a^3 e x^6+a^2 (3 b c+a f) x^7+a^2 (3 b d+a g) x^8+a^2 (3 b e+a h) x^9+3 a b (b c+a f) x^{10}+3 a b (b d+a g) x^{11}+3 a b (b e+a h) x^{12}+b^2 (b c+3 a f) x^{13}+b^2 (b d+3 a g) x^{14}+b^2 (b e+3 a h) x^{15}+b^3 f x^{16}+b^3 g x^{17}+b^3 h x^{18}\right ) \, dx \\ & = \frac {1}{5} a^3 c x^5+\frac {1}{6} a^3 d x^6+\frac {1}{7} a^3 e x^7+\frac {1}{8} a^2 (3 b c+a f) x^8+\frac {1}{9} a^2 (3 b d+a g) x^9+\frac {1}{10} a^2 (3 b e+a h) x^{10}+\frac {3}{11} a b (b c+a f) x^{11}+\frac {1}{4} a b (b d+a g) x^{12}+\frac {3}{13} a b (b e+a h) x^{13}+\frac {1}{14} b^2 (b c+3 a f) x^{14}+\frac {1}{15} b^2 (b d+3 a g) x^{15}+\frac {1}{16} b^2 (b e+3 a h) x^{16}+\frac {1}{17} b^3 f x^{17}+\frac {1}{18} b^3 g x^{18}+\frac {1}{19} b^3 h x^{19} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.00 \[ \int x^4 \left (a+b x^3\right )^3 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {1}{5} a^3 c x^5+\frac {1}{6} a^3 d x^6+\frac {1}{7} a^3 e x^7+\frac {1}{8} a^2 (3 b c+a f) x^8+\frac {1}{9} a^2 (3 b d+a g) x^9+\frac {1}{10} a^2 (3 b e+a h) x^{10}+\frac {3}{11} a b (b c+a f) x^{11}+\frac {1}{4} a b (b d+a g) x^{12}+\frac {3}{13} a b (b e+a h) x^{13}+\frac {1}{14} b^2 (b c+3 a f) x^{14}+\frac {1}{15} b^2 (b d+3 a g) x^{15}+\frac {1}{16} b^2 (b e+3 a h) x^{16}+\frac {1}{17} b^3 f x^{17}+\frac {1}{18} b^3 g x^{18}+\frac {1}{19} b^3 h x^{19} \]

[In]

Integrate[x^4*(a + b*x^3)^3*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5),x]

[Out]

(a^3*c*x^5)/5 + (a^3*d*x^6)/6 + (a^3*e*x^7)/7 + (a^2*(3*b*c + a*f)*x^8)/8 + (a^2*(3*b*d + a*g)*x^9)/9 + (a^2*(
3*b*e + a*h)*x^10)/10 + (3*a*b*(b*c + a*f)*x^11)/11 + (a*b*(b*d + a*g)*x^12)/4 + (3*a*b*(b*e + a*h)*x^13)/13 +
(b^2*(b*c + 3*a*f)*x^14)/14 + (b^2*(b*d + 3*a*g)*x^15)/15 + (b^2*(b*e + 3*a*h)*x^16)/16 + (b^3*f*x^17)/17 + (
b^3*g*x^18)/18 + (b^3*h*x^19)/19

Maple [A] (verified)

Time = 2.06 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.99

method	result	size
norman	\(\frac {a^{3} c \,x^{5}}{5}+\frac {a^{3} d \,x^{6}}{6}+\frac {a^{3} e \,x^{7}}{7}+\left (\frac {1}{8} f \,a^{3}+\frac {3}{8} a^{2} b c \right ) x^{8}+\left (\frac {1}{9} a^{3} g +\frac {1}{3} d \,a^{2} b \right ) x^{9}+\left (\frac {1}{10} a^{3} h +\frac {3}{10} a^{2} b e \right ) x^{10}+\left (\frac {3}{11} f \,a^{2} b +\frac {3}{11} a \,b^{2} c \right ) x^{11}+\left (\frac {1}{4} a^{2} b g +\frac {1}{4} a \,b^{2} d \right ) x^{12}+\left (\frac {3}{13} a^{2} b h +\frac {3}{13} a \,b^{2} e \right ) x^{13}+\left (\frac {3}{14} a \,b^{2} f +\frac {1}{14} b^{3} c \right ) x^{14}+\left (\frac {1}{5} a \,b^{2} g +\frac {1}{15} b^{3} d \right ) x^{15}+\left (\frac {3}{16} a \,b^{2} h +\frac {1}{16} b^{3} e \right ) x^{16}+\frac {b^{3} f \,x^{17}}{17}+\frac {b^{3} g \,x^{18}}{18}+\frac {b^{3} h \,x^{19}}{19}\)	\(221\)
default	\(\frac {b^{3} h \,x^{19}}{19}+\frac {b^{3} g \,x^{18}}{18}+\frac {b^{3} f \,x^{17}}{17}+\frac {\left (3 a \,b^{2} h +b^{3} e \right ) x^{16}}{16}+\frac {\left (3 a \,b^{2} g +b^{3} d \right ) x^{15}}{15}+\frac {\left (3 a \,b^{2} f +b^{3} c \right ) x^{14}}{14}+\frac {\left (3 a^{2} b h +3 a \,b^{2} e \right ) x^{13}}{13}+\frac {\left (3 a^{2} b g +3 a \,b^{2} d \right ) x^{12}}{12}+\frac {\left (3 f \,a^{2} b +3 a \,b^{2} c \right ) x^{11}}{11}+\frac {\left (a^{3} h +3 a^{2} b e \right ) x^{10}}{10}+\frac {\left (a^{3} g +3 d \,a^{2} b \right ) x^{9}}{9}+\frac {\left (f \,a^{3}+3 a^{2} b c \right ) x^{8}}{8}+\frac {a^{3} e \,x^{7}}{7}+\frac {a^{3} d \,x^{6}}{6}+\frac {a^{3} c \,x^{5}}{5}\)	\(224\)
gosper	\(\frac {1}{5} a^{3} c \,x^{5}+\frac {1}{6} a^{3} d \,x^{6}+\frac {1}{7} a^{3} e \,x^{7}+\frac {1}{8} x^{8} f \,a^{3}+\frac {3}{8} x^{8} a^{2} b c +\frac {1}{9} x^{9} a^{3} g +\frac {1}{3} a^{2} b d \,x^{9}+\frac {1}{10} x^{10} a^{3} h +\frac {3}{10} a^{2} b e \,x^{10}+\frac {3}{11} a^{2} b f \,x^{11}+\frac {3}{11} x^{11} a \,b^{2} c +\frac {1}{4} x^{12} a^{2} b g +\frac {1}{4} x^{12} a \,b^{2} d +\frac {3}{13} x^{13} a^{2} b h +\frac {3}{13} x^{13} a \,b^{2} e +\frac {3}{14} x^{14} a \,b^{2} f +\frac {1}{14} x^{14} b^{3} c +\frac {1}{5} x^{15} a \,b^{2} g +\frac {1}{15} x^{15} b^{3} d +\frac {3}{16} x^{16} a \,b^{2} h +\frac {1}{16} x^{16} b^{3} e +\frac {1}{17} b^{3} f \,x^{17}+\frac {1}{18} b^{3} g \,x^{18}+\frac {1}{19} b^{3} h \,x^{19}\)	\(230\)
risch	\(\frac {1}{5} a^{3} c \,x^{5}+\frac {1}{6} a^{3} d \,x^{6}+\frac {1}{7} a^{3} e \,x^{7}+\frac {1}{8} x^{8} f \,a^{3}+\frac {3}{8} x^{8} a^{2} b c +\frac {1}{9} x^{9} a^{3} g +\frac {1}{3} a^{2} b d \,x^{9}+\frac {1}{10} x^{10} a^{3} h +\frac {3}{10} a^{2} b e \,x^{10}+\frac {3}{11} a^{2} b f \,x^{11}+\frac {3}{11} x^{11} a \,b^{2} c +\frac {1}{4} x^{12} a^{2} b g +\frac {1}{4} x^{12} a \,b^{2} d +\frac {3}{13} x^{13} a^{2} b h +\frac {3}{13} x^{13} a \,b^{2} e +\frac {3}{14} x^{14} a \,b^{2} f +\frac {1}{14} x^{14} b^{3} c +\frac {1}{5} x^{15} a \,b^{2} g +\frac {1}{15} x^{15} b^{3} d +\frac {3}{16} x^{16} a \,b^{2} h +\frac {1}{16} x^{16} b^{3} e +\frac {1}{17} b^{3} f \,x^{17}+\frac {1}{18} b^{3} g \,x^{18}+\frac {1}{19} b^{3} h \,x^{19}\)	\(230\)
parallelrisch	\(\frac {1}{5} a^{3} c \,x^{5}+\frac {1}{6} a^{3} d \,x^{6}+\frac {1}{7} a^{3} e \,x^{7}+\frac {1}{8} x^{8} f \,a^{3}+\frac {3}{8} x^{8} a^{2} b c +\frac {1}{9} x^{9} a^{3} g +\frac {1}{3} a^{2} b d \,x^{9}+\frac {1}{10} x^{10} a^{3} h +\frac {3}{10} a^{2} b e \,x^{10}+\frac {3}{11} a^{2} b f \,x^{11}+\frac {3}{11} x^{11} a \,b^{2} c +\frac {1}{4} x^{12} a^{2} b g +\frac {1}{4} x^{12} a \,b^{2} d +\frac {3}{13} x^{13} a^{2} b h +\frac {3}{13} x^{13} a \,b^{2} e +\frac {3}{14} x^{14} a \,b^{2} f +\frac {1}{14} x^{14} b^{3} c +\frac {1}{5} x^{15} a \,b^{2} g +\frac {1}{15} x^{15} b^{3} d +\frac {3}{16} x^{16} a \,b^{2} h +\frac {1}{16} x^{16} b^{3} e +\frac {1}{17} b^{3} f \,x^{17}+\frac {1}{18} b^{3} g \,x^{18}+\frac {1}{19} b^{3} h \,x^{19}\)	\(230\)

[In]

int(x^4*(b*x^3+a)^3*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c),x,method=_RETURNVERBOSE)

[Out]

1/5*a^3*c*x^5+1/6*a^3*d*x^6+1/7*a^3*e*x^7+(1/8*f*a^3+3/8*a^2*b*c)*x^8+(1/9*a^3*g+1/3*d*a^2*b)*x^9+(1/10*a^3*h+
3/10*a^2*b*e)*x^10+(3/11*f*a^2*b+3/11*a*b^2*c)*x^11+(1/4*a^2*b*g+1/4*a*b^2*d)*x^12+(3/13*a^2*b*h+3/13*a*b^2*e)
*x^13+(3/14*a*b^2*f+1/14*b^3*c)*x^14+(1/5*a*b^2*g+1/15*b^3*d)*x^15+(3/16*a*b^2*h+1/16*b^3*e)*x^16+1/17*b^3*f*x
^17+1/18*b^3*g*x^18+1/19*b^3*h*x^19

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.97 \[ \int x^4 \left (a+b x^3\right )^3 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {1}{19} \, b^{3} h x^{19} + \frac {1}{18} \, b^{3} g x^{18} + \frac {1}{17} \, b^{3} f x^{17} + \frac {1}{16} \, {\left (b^{3} e + 3 \, a b^{2} h\right )} x^{16} + \frac {1}{15} \, {\left (b^{3} d + 3 \, a b^{2} g\right )} x^{15} + \frac {1}{14} \, {\left (b^{3} c + 3 \, a b^{2} f\right )} x^{14} + \frac {3}{13} \, {\left (a b^{2} e + a^{2} b h\right )} x^{13} + \frac {1}{4} \, {\left (a b^{2} d + a^{2} b g\right )} x^{12} + \frac {3}{11} \, {\left (a b^{2} c + a^{2} b f\right )} x^{11} + \frac {1}{7} \, a^{3} e x^{7} + \frac {1}{10} \, {\left (3 \, a^{2} b e + a^{3} h\right )} x^{10} + \frac {1}{6} \, a^{3} d x^{6} + \frac {1}{9} \, {\left (3 \, a^{2} b d + a^{3} g\right )} x^{9} + \frac {1}{5} \, a^{3} c x^{5} + \frac {1}{8} \, {\left (3 \, a^{2} b c + a^{3} f\right )} x^{8} \]

[In]

integrate(x^4*(b*x^3+a)^3*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c),x, algorithm="fricas")

[Out]

1/19*b^3*h*x^19 + 1/18*b^3*g*x^18 + 1/17*b^3*f*x^17 + 1/16*(b^3*e + 3*a*b^2*h)*x^16 + 1/15*(b^3*d + 3*a*b^2*g)
*x^15 + 1/14*(b^3*c + 3*a*b^2*f)*x^14 + 3/13*(a*b^2*e + a^2*b*h)*x^13 + 1/4*(a*b^2*d + a^2*b*g)*x^12 + 3/11*(a
*b^2*c + a^2*b*f)*x^11 + 1/7*a^3*e*x^7 + 1/10*(3*a^2*b*e + a^3*h)*x^10 + 1/6*a^3*d*x^6 + 1/9*(3*a^2*b*d + a^3*
g)*x^9 + 1/5*a^3*c*x^5 + 1/8*(3*a^2*b*c + a^3*f)*x^8

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.10 \[ \int x^4 \left (a+b x^3\right )^3 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {a^{3} c x^{5}}{5} + \frac {a^{3} d x^{6}}{6} + \frac {a^{3} e x^{7}}{7} + \frac {b^{3} f x^{17}}{17} + \frac {b^{3} g x^{18}}{18} + \frac {b^{3} h x^{19}}{19} + x^{16} \cdot \left (\frac {3 a b^{2} h}{16} + \frac {b^{3} e}{16}\right ) + x^{15} \left (\frac {a b^{2} g}{5} + \frac {b^{3} d}{15}\right ) + x^{14} \cdot \left (\frac {3 a b^{2} f}{14} + \frac {b^{3} c}{14}\right ) + x^{13} \cdot \left (\frac {3 a^{2} b h}{13} + \frac {3 a b^{2} e}{13}\right ) + x^{12} \left (\frac {a^{2} b g}{4} + \frac {a b^{2} d}{4}\right ) + x^{11} \cdot \left (\frac {3 a^{2} b f}{11} + \frac {3 a b^{2} c}{11}\right ) + x^{10} \left (\frac {a^{3} h}{10} + \frac {3 a^{2} b e}{10}\right ) + x^{9} \left (\frac {a^{3} g}{9} + \frac {a^{2} b d}{3}\right ) + x^{8} \left (\frac {a^{3} f}{8} + \frac {3 a^{2} b c}{8}\right ) \]

[In]

integrate(x**4*(b*x**3+a)**3*(h*x**5+g*x**4+f*x**3+e*x**2+d*x+c),x)

[Out]

a**3*c*x**5/5 + a**3*d*x**6/6 + a**3*e*x**7/7 + b**3*f*x**17/17 + b**3*g*x**18/18 + b**3*h*x**19/19 + x**16*(3
*a*b**2*h/16 + b**3*e/16) + x**15*(a*b**2*g/5 + b**3*d/15) + x**14*(3*a*b**2*f/14 + b**3*c/14) + x**13*(3*a**2
*b*h/13 + 3*a*b**2*e/13) + x**12*(a**2*b*g/4 + a*b**2*d/4) + x**11*(3*a**2*b*f/11 + 3*a*b**2*c/11) + x**10*(a*
*3*h/10 + 3*a**2*b*e/10) + x**9*(a**3*g/9 + a**2*b*d/3) + x**8*(a**3*f/8 + 3*a**2*b*c/8)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.97 \[ \int x^4 \left (a+b x^3\right )^3 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {1}{19} \, b^{3} h x^{19} + \frac {1}{18} \, b^{3} g x^{18} + \frac {1}{17} \, b^{3} f x^{17} + \frac {1}{16} \, {\left (b^{3} e + 3 \, a b^{2} h\right )} x^{16} + \frac {1}{15} \, {\left (b^{3} d + 3 \, a b^{2} g\right )} x^{15} + \frac {1}{14} \, {\left (b^{3} c + 3 \, a b^{2} f\right )} x^{14} + \frac {3}{13} \, {\left (a b^{2} e + a^{2} b h\right )} x^{13} + \frac {1}{4} \, {\left (a b^{2} d + a^{2} b g\right )} x^{12} + \frac {3}{11} \, {\left (a b^{2} c + a^{2} b f\right )} x^{11} + \frac {1}{7} \, a^{3} e x^{7} + \frac {1}{10} \, {\left (3 \, a^{2} b e + a^{3} h\right )} x^{10} + \frac {1}{6} \, a^{3} d x^{6} + \frac {1}{9} \, {\left (3 \, a^{2} b d + a^{3} g\right )} x^{9} + \frac {1}{5} \, a^{3} c x^{5} + \frac {1}{8} \, {\left (3 \, a^{2} b c + a^{3} f\right )} x^{8} \]

[In]

integrate(x^4*(b*x^3+a)^3*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c),x, algorithm="maxima")

[Out]

1/19*b^3*h*x^19 + 1/18*b^3*g*x^18 + 1/17*b^3*f*x^17 + 1/16*(b^3*e + 3*a*b^2*h)*x^16 + 1/15*(b^3*d + 3*a*b^2*g)
*x^15 + 1/14*(b^3*c + 3*a*b^2*f)*x^14 + 3/13*(a*b^2*e + a^2*b*h)*x^13 + 1/4*(a*b^2*d + a^2*b*g)*x^12 + 3/11*(a
*b^2*c + a^2*b*f)*x^11 + 1/7*a^3*e*x^7 + 1/10*(3*a^2*b*e + a^3*h)*x^10 + 1/6*a^3*d*x^6 + 1/9*(3*a^2*b*d + a^3*
g)*x^9 + 1/5*a^3*c*x^5 + 1/8*(3*a^2*b*c + a^3*f)*x^8

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.03 \[ \int x^4 \left (a+b x^3\right )^3 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {1}{19} \, b^{3} h x^{19} + \frac {1}{18} \, b^{3} g x^{18} + \frac {1}{17} \, b^{3} f x^{17} + \frac {1}{16} \, b^{3} e x^{16} + \frac {3}{16} \, a b^{2} h x^{16} + \frac {1}{15} \, b^{3} d x^{15} + \frac {1}{5} \, a b^{2} g x^{15} + \frac {1}{14} \, b^{3} c x^{14} + \frac {3}{14} \, a b^{2} f x^{14} + \frac {3}{13} \, a b^{2} e x^{13} + \frac {3}{13} \, a^{2} b h x^{13} + \frac {1}{4} \, a b^{2} d x^{12} + \frac {1}{4} \, a^{2} b g x^{12} + \frac {3}{11} \, a b^{2} c x^{11} + \frac {3}{11} \, a^{2} b f x^{11} + \frac {3}{10} \, a^{2} b e x^{10} + \frac {1}{10} \, a^{3} h x^{10} + \frac {1}{3} \, a^{2} b d x^{9} + \frac {1}{9} \, a^{3} g x^{9} + \frac {3}{8} \, a^{2} b c x^{8} + \frac {1}{8} \, a^{3} f x^{8} + \frac {1}{7} \, a^{3} e x^{7} + \frac {1}{6} \, a^{3} d x^{6} + \frac {1}{5} \, a^{3} c x^{5} \]

[In]

integrate(x^4*(b*x^3+a)^3*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c),x, algorithm="giac")

[Out]

1/19*b^3*h*x^19 + 1/18*b^3*g*x^18 + 1/17*b^3*f*x^17 + 1/16*b^3*e*x^16 + 3/16*a*b^2*h*x^16 + 1/15*b^3*d*x^15 +
1/5*a*b^2*g*x^15 + 1/14*b^3*c*x^14 + 3/14*a*b^2*f*x^14 + 3/13*a*b^2*e*x^13 + 3/13*a^2*b*h*x^13 + 1/4*a*b^2*d*x
^12 + 1/4*a^2*b*g*x^12 + 3/11*a*b^2*c*x^11 + 3/11*a^2*b*f*x^11 + 3/10*a^2*b*e*x^10 + 1/10*a^3*h*x^10 + 1/3*a^2
*b*d*x^9 + 1/9*a^3*g*x^9 + 3/8*a^2*b*c*x^8 + 1/8*a^3*f*x^8 + 1/7*a^3*e*x^7 + 1/6*a^3*d*x^6 + 1/5*a^3*c*x^5

Mupad [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.92 \[ \int x^4 \left (a+b x^3\right )^3 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=x^8\,\left (\frac {f\,a^3}{8}+\frac {3\,b\,c\,a^2}{8}\right )+x^{14}\,\left (\frac {c\,b^3}{14}+\frac {3\,a\,f\,b^2}{14}\right )+x^9\,\left (\frac {g\,a^3}{9}+\frac {b\,d\,a^2}{3}\right )+x^{15}\,\left (\frac {d\,b^3}{15}+\frac {a\,g\,b^2}{5}\right )+x^{10}\,\left (\frac {h\,a^3}{10}+\frac {3\,b\,e\,a^2}{10}\right )+x^{16}\,\left (\frac {e\,b^3}{16}+\frac {3\,a\,h\,b^2}{16}\right )+\frac {a^3\,c\,x^5}{5}+\frac {a^3\,d\,x^6}{6}+\frac {a^3\,e\,x^7}{7}+\frac {b^3\,f\,x^{17}}{17}+\frac {b^3\,g\,x^{18}}{18}+\frac {b^3\,h\,x^{19}}{19}+\frac {3\,a\,b\,x^{11}\,\left (b\,c+a\,f\right )}{11}+\frac {a\,b\,x^{12}\,\left (b\,d+a\,g\right )}{4}+\frac {3\,a\,b\,x^{13}\,\left (b\,e+a\,h\right )}{13} \]

[In]

int(x^4*(a + b*x^3)^3*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5),x)

[Out]

x^8*((a^3*f)/8 + (3*a^2*b*c)/8) + x^14*((b^3*c)/14 + (3*a*b^2*f)/14) + x^9*((a^3*g)/9 + (a^2*b*d)/3) + x^15*((
b^3*d)/15 + (a*b^2*g)/5) + x^10*((a^3*h)/10 + (3*a^2*b*e)/10) + x^16*((b^3*e)/16 + (3*a*b^2*h)/16) + (a^3*c*x^
5)/5 + (a^3*d*x^6)/6 + (a^3*e*x^7)/7 + (b^3*f*x^17)/17 + (b^3*g*x^18)/18 + (b^3*h*x^19)/19 + (3*a*b*x^11*(b*c
+ a*f))/11 + (a*b*x^12*(b*d + a*g))/4 + (3*a*b*x^13*(b*e + a*h))/13