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

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

[Out]

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

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Rubi [A]  time = 0.227574, 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} \[ \frac{1}{7} a^2 x^7 (a f+3 b c)+\frac{1}{8} a^2 x^8 (a g+3 b d)+\frac{1}{9} a^2 x^9 (a h+3 b e)+\frac{1}{4} a^3 c x^4+\frac{1}{5} a^3 d x^5+\frac{1}{6} a^3 e x^6+\frac{1}{13} b^2 x^{13} (3 a f+b c)+\frac{1}{14} b^2 x^{14} (3 a g+b d)+\frac{1}{15} b^2 x^{15} (3 a h+b e)+\frac{3}{10} a b x^{10} (a f+b c)+\frac{3}{11} a b x^{11} (a g+b d)+\frac{1}{4} a b x^{12} (a h+b e)+\frac{1}{16} b^3 f x^{16}+\frac{1}{17} b^3 g x^{17}+\frac{1}{18} b^3 h x^{18} \]

Antiderivative was successfully verified.

[In]

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

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

Mathematica [A]  time = 0.0468082, size = 223, normalized size = 1. \[ \frac{1}{7} a^2 x^7 (a f+3 b c)+\frac{1}{8} a^2 x^8 (a g+3 b d)+\frac{1}{9} a^2 x^9 (a h+3 b e)+\frac{1}{4} a^3 c x^4+\frac{1}{5} a^3 d x^5+\frac{1}{6} a^3 e x^6+\frac{1}{13} b^2 x^{13} (3 a f+b c)+\frac{1}{14} b^2 x^{14} (3 a g+b d)+\frac{1}{15} b^2 x^{15} (3 a h+b e)+\frac{3}{10} a b x^{10} (a f+b c)+\frac{3}{11} a b x^{11} (a g+b d)+\frac{1}{4} a b x^{12} (a h+b e)+\frac{1}{16} b^3 f x^{16}+\frac{1}{17} b^3 g x^{17}+\frac{1}{18} b^3 h x^{18} \]

Antiderivative was successfully verified.

[In]

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

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Maple [A]  time = 0.002, size = 224, normalized size = 1. \begin{align*}{\frac{{b}^{3}h{x}^{18}}{18}}+{\frac{{b}^{3}g{x}^{17}}{17}}+{\frac{{b}^{3}f{x}^{16}}{16}}+{\frac{ \left ( 3\,{b}^{2}ah+{b}^{3}e \right ){x}^{15}}{15}}+{\frac{ \left ( 3\,{b}^{2}ag+{b}^{3}d \right ){x}^{14}}{14}}+{\frac{ \left ( 3\,{b}^{2}af+{b}^{3}c \right ){x}^{13}}{13}}+{\frac{ \left ( 3\,b{a}^{2}h+3\,ae{b}^{2} \right ){x}^{12}}{12}}+{\frac{ \left ( 3\,b{a}^{2}g+3\,a{b}^{2}d \right ){x}^{11}}{11}}+{\frac{ \left ( 3\,b{a}^{2}f+3\,ac{b}^{2} \right ){x}^{10}}{10}}+{\frac{ \left ({a}^{3}h+3\,{a}^{2}be \right ){x}^{9}}{9}}+{\frac{ \left ({a}^{3}g+3\,{a}^{2}bd \right ){x}^{8}}{8}}+{\frac{ \left ({a}^{3}f+3\,b{a}^{2}c \right ){x}^{7}}{7}}+{\frac{{a}^{3}e{x}^{6}}{6}}+{\frac{{a}^{3}d{x}^{5}}{5}}+{\frac{{a}^{3}c{x}^{4}}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.931833, size = 293, normalized size = 1.31 \begin{align*} \frac{1}{18} \, b^{3} h x^{18} + \frac{1}{17} \, b^{3} g x^{17} + \frac{1}{16} \, b^{3} f x^{16} + \frac{1}{15} \,{\left (b^{3} e + 3 \, a b^{2} h\right )} x^{15} + \frac{1}{14} \,{\left (b^{3} d + 3 \, a b^{2} g\right )} x^{14} + \frac{1}{13} \,{\left (b^{3} c + 3 \, a b^{2} f\right )} x^{13} + \frac{1}{4} \,{\left (a b^{2} e + a^{2} b h\right )} x^{12} + \frac{3}{11} \,{\left (a b^{2} d + a^{2} b g\right )} x^{11} + \frac{3}{10} \,{\left (a b^{2} c + a^{2} b f\right )} x^{10} + \frac{1}{6} \, a^{3} e x^{6} + \frac{1}{9} \,{\left (3 \, a^{2} b e + a^{3} h\right )} x^{9} + \frac{1}{5} \, a^{3} d x^{5} + \frac{1}{8} \,{\left (3 \, a^{2} b d + a^{3} g\right )} x^{8} + \frac{1}{4} \, a^{3} c x^{4} + \frac{1}{7} \,{\left (3 \, a^{2} b c + a^{3} f\right )} x^{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.869605, size = 586, normalized size = 2.63 \begin{align*} \frac{1}{18} x^{18} h b^{3} + \frac{1}{17} x^{17} g b^{3} + \frac{1}{16} x^{16} f b^{3} + \frac{1}{15} x^{15} e b^{3} + \frac{1}{5} x^{15} h b^{2} a + \frac{1}{14} x^{14} d b^{3} + \frac{3}{14} x^{14} g b^{2} a + \frac{1}{13} x^{13} c b^{3} + \frac{3}{13} x^{13} f b^{2} a + \frac{1}{4} x^{12} e b^{2} a + \frac{1}{4} x^{12} h b a^{2} + \frac{3}{11} x^{11} d b^{2} a + \frac{3}{11} x^{11} g b a^{2} + \frac{3}{10} x^{10} c b^{2} a + \frac{3}{10} x^{10} f b a^{2} + \frac{1}{3} x^{9} e b a^{2} + \frac{1}{9} x^{9} h a^{3} + \frac{3}{8} x^{8} d b a^{2} + \frac{1}{8} x^{8} g a^{3} + \frac{3}{7} x^{7} c b a^{2} + \frac{1}{7} x^{7} f a^{3} + \frac{1}{6} x^{6} e a^{3} + \frac{1}{5} x^{5} d a^{3} + \frac{1}{4} x^{4} c a^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.100897, size = 246, normalized size = 1.1 \begin{align*} \frac{a^{3} c x^{4}}{4} + \frac{a^{3} d x^{5}}{5} + \frac{a^{3} e x^{6}}{6} + \frac{b^{3} f x^{16}}{16} + \frac{b^{3} g x^{17}}{17} + \frac{b^{3} h x^{18}}{18} + x^{15} \left (\frac{a b^{2} h}{5} + \frac{b^{3} e}{15}\right ) + x^{14} \left (\frac{3 a b^{2} g}{14} + \frac{b^{3} d}{14}\right ) + x^{13} \left (\frac{3 a b^{2} f}{13} + \frac{b^{3} c}{13}\right ) + x^{12} \left (\frac{a^{2} b h}{4} + \frac{a b^{2} e}{4}\right ) + x^{11} \left (\frac{3 a^{2} b g}{11} + \frac{3 a b^{2} d}{11}\right ) + x^{10} \left (\frac{3 a^{2} b f}{10} + \frac{3 a b^{2} c}{10}\right ) + x^{9} \left (\frac{a^{3} h}{9} + \frac{a^{2} b e}{3}\right ) + x^{8} \left (\frac{a^{3} g}{8} + \frac{3 a^{2} b d}{8}\right ) + x^{7} \left (\frac{a^{3} f}{7} + \frac{3 a^{2} b c}{7}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Giac [A]  time = 1.07661, size = 315, normalized size = 1.41 \begin{align*} \frac{1}{18} \, b^{3} h x^{18} + \frac{1}{17} \, b^{3} g x^{17} + \frac{1}{16} \, b^{3} f x^{16} + \frac{1}{5} \, a b^{2} h x^{15} + \frac{1}{15} \, b^{3} x^{15} e + \frac{1}{14} \, b^{3} d x^{14} + \frac{3}{14} \, a b^{2} g x^{14} + \frac{1}{13} \, b^{3} c x^{13} + \frac{3}{13} \, a b^{2} f x^{13} + \frac{1}{4} \, a^{2} b h x^{12} + \frac{1}{4} \, a b^{2} x^{12} e + \frac{3}{11} \, a b^{2} d x^{11} + \frac{3}{11} \, a^{2} b g x^{11} + \frac{3}{10} \, a b^{2} c x^{10} + \frac{3}{10} \, a^{2} b f x^{10} + \frac{1}{9} \, a^{3} h x^{9} + \frac{1}{3} \, a^{2} b x^{9} e + \frac{3}{8} \, a^{2} b d x^{8} + \frac{1}{8} \, a^{3} g x^{8} + \frac{3}{7} \, a^{2} b c x^{7} + \frac{1}{7} \, a^{3} f x^{7} + \frac{1}{6} \, a^{3} x^{6} e + \frac{1}{5} \, a^{3} d x^{5} + \frac{1}{4} \, a^{3} c x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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