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

Optimal. Leaf size=223 \[ \frac{1}{8} a^2 x^8 (a f+3 b c)+\frac{1}{9} a^2 x^9 (a g+3 b d)+\frac{1}{10} a^2 x^{10} (a h+3 b e)+\frac{1}{5} a^3 c x^5+\frac{1}{6} a^3 d x^6+\frac{1}{7} a^3 e x^7+\frac{1}{14} b^2 x^{14} (3 a f+b c)+\frac{1}{15} b^2 x^{15} (3 a g+b d)+\frac{1}{16} b^2 x^{16} (3 a h+b e)+\frac{3}{11} a b x^{11} (a f+b c)+\frac{1}{4} a b x^{12} (a g+b d)+\frac{3}{13} a b x^{13} (a h+b e)+\frac{1}{17} b^3 f x^{17}+\frac{1}{18} b^3 g x^{18}+\frac{1}{19} b^3 h x^{19} \]

[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

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Rubi [A]  time = 0.292612, antiderivative size = 223, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.026, Rules used = {1820} \[ \frac{1}{8} a^2 x^8 (a f+3 b c)+\frac{1}{9} a^2 x^9 (a g+3 b d)+\frac{1}{10} a^2 x^{10} (a h+3 b e)+\frac{1}{5} a^3 c x^5+\frac{1}{6} a^3 d x^6+\frac{1}{7} a^3 e x^7+\frac{1}{14} b^2 x^{14} (3 a f+b c)+\frac{1}{15} b^2 x^{15} (3 a g+b d)+\frac{1}{16} b^2 x^{16} (3 a h+b e)+\frac{3}{11} a b x^{11} (a f+b c)+\frac{1}{4} a b x^{12} (a g+b d)+\frac{3}{13} a b x^{13} (a h+b e)+\frac{1}{17} b^3 f x^{17}+\frac{1}{18} b^3 g x^{18}+\frac{1}{19} b^3 h x^{19} \]

Antiderivative was successfully verified.

[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 1820

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*} \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 &=\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]  time = 0.0509596, size = 223, normalized size = 1. \[ \frac{1}{8} a^2 x^8 (a f+3 b c)+\frac{1}{9} a^2 x^9 (a g+3 b d)+\frac{1}{10} a^2 x^{10} (a h+3 b e)+\frac{1}{5} a^3 c x^5+\frac{1}{6} a^3 d x^6+\frac{1}{7} a^3 e x^7+\frac{1}{14} b^2 x^{14} (3 a f+b c)+\frac{1}{15} b^2 x^{15} (3 a g+b d)+\frac{1}{16} b^2 x^{16} (3 a h+b e)+\frac{3}{11} a b x^{11} (a f+b c)+\frac{1}{4} a b x^{12} (a g+b d)+\frac{3}{13} a b x^{13} (a h+b e)+\frac{1}{17} b^3 f x^{17}+\frac{1}{18} b^3 g x^{18}+\frac{1}{19} b^3 h x^{19} \]

Antiderivative was successfully verified.

[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

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Maple [A]  time = 0.001, size = 224, normalized size = 1. \begin{align*}{\frac{{b}^{3}h{x}^{19}}{19}}+{\frac{{b}^{3}g{x}^{18}}{18}}+{\frac{{b}^{3}f{x}^{17}}{17}}+{\frac{ \left ( 3\,{b}^{2}ah+{b}^{3}e \right ){x}^{16}}{16}}+{\frac{ \left ( 3\,{b}^{2}ag+{b}^{3}d \right ){x}^{15}}{15}}+{\frac{ \left ( 3\,{b}^{2}af+{b}^{3}c \right ){x}^{14}}{14}}+{\frac{ \left ( 3\,b{a}^{2}h+3\,ae{b}^{2} \right ){x}^{13}}{13}}+{\frac{ \left ( 3\,b{a}^{2}g+3\,a{b}^{2}d \right ){x}^{12}}{12}}+{\frac{ \left ( 3\,b{a}^{2}f+3\,ac{b}^{2} \right ){x}^{11}}{11}}+{\frac{ \left ({a}^{3}h+3\,{a}^{2}be \right ){x}^{10}}{10}}+{\frac{ \left ({a}^{3}g+3\,{a}^{2}bd \right ){x}^{9}}{9}}+{\frac{ \left ({a}^{3}f+3\,b{a}^{2}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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

[Out]

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

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Maxima [A]  time = 0.957263, size = 293, normalized size = 1.31 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 0.873177, size = 591, normalized size = 2.65 \begin{align*} \frac{1}{19} x^{19} h b^{3} + \frac{1}{18} x^{18} g b^{3} + \frac{1}{17} x^{17} f b^{3} + \frac{1}{16} x^{16} e b^{3} + \frac{3}{16} x^{16} h b^{2} a + \frac{1}{15} x^{15} d b^{3} + \frac{1}{5} x^{15} g b^{2} a + \frac{1}{14} x^{14} c b^{3} + \frac{3}{14} x^{14} f b^{2} a + \frac{3}{13} x^{13} e b^{2} a + \frac{3}{13} x^{13} h b a^{2} + \frac{1}{4} x^{12} d b^{2} a + \frac{1}{4} x^{12} g b a^{2} + \frac{3}{11} x^{11} c b^{2} a + \frac{3}{11} x^{11} f b a^{2} + \frac{3}{10} x^{10} e b a^{2} + \frac{1}{10} x^{10} h a^{3} + \frac{1}{3} x^{9} d b a^{2} + \frac{1}{9} x^{9} g a^{3} + \frac{3}{8} x^{8} c b a^{2} + \frac{1}{8} x^{8} f a^{3} + \frac{1}{7} x^{7} e a^{3} + \frac{1}{6} x^{6} d a^{3} + \frac{1}{5} x^{5} c a^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

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

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Sympy [A]  time = 0.099741, size = 246, normalized size = 1.1 \begin{align*} \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} \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} \left (\frac{3 a b^{2} f}{14} + \frac{b^{3} c}{14}\right ) + x^{13} \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} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [A]  time = 1.06827, size = 315, normalized size = 1.41 \begin{align*} \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{3}{16} \, a b^{2} h x^{16} + \frac{1}{16} \, b^{3} x^{16} e + \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^{2} b h x^{13} + \frac{3}{13} \, a b^{2} x^{13} e + \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{1}{10} \, a^{3} h x^{10} + \frac{3}{10} \, a^{2} b x^{10} e + \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} x^{7} e + \frac{1}{6} \, a^{3} d x^{6} + \frac{1}{5} \, a^{3} c x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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 + 3/16*a*b^2*h*x^16 + 1/16*b^3*x^16*e + 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^2*b*h*x^13 + 3/13*a*b^2*x^13*e + 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 + 1/10*a^3*h*x^10 + 3/10*a^2*b*x^10*e + 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*x^7*e + 1/6*a^3*d*x^6 + 1/5*a^3*c*x^5