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

Optimal. Leaf size=163 \[ \frac{1}{5} a^2 c x^5+\frac{1}{6} a^2 d x^6+\frac{1}{7} a^2 e x^7+\frac{1}{11} b x^{11} (2 a f+b c)+\frac{1}{8} a x^8 (a f+2 b c)+\frac{1}{12} b x^{12} (2 a g+b d)+\frac{1}{9} a x^9 (a g+2 b d)+\frac{1}{13} b x^{13} (2 a h+b e)+\frac{1}{10} a x^{10} (a h+2 b e)+\frac{1}{14} b^2 f x^{14}+\frac{1}{15} b^2 g x^{15}+\frac{1}{16} b^2 h x^{16} \]

[Out]

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

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Rubi [A]  time = 0.210111, antiderivative size = 163, 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} \[ \frac{1}{5} a^2 c x^5+\frac{1}{6} a^2 d x^6+\frac{1}{7} a^2 e x^7+\frac{1}{11} b x^{11} (2 a f+b c)+\frac{1}{8} a x^8 (a f+2 b c)+\frac{1}{12} b x^{12} (2 a g+b d)+\frac{1}{9} a x^9 (a g+2 b d)+\frac{1}{13} b x^{13} (2 a h+b e)+\frac{1}{10} a x^{10} (a h+2 b e)+\frac{1}{14} b^2 f x^{14}+\frac{1}{15} b^2 g x^{15}+\frac{1}{16} b^2 h x^{16} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0395111, size = 163, normalized size = 1. \[ \frac{1}{5} a^2 c x^5+\frac{1}{6} a^2 d x^6+\frac{1}{7} a^2 e x^7+\frac{1}{11} b x^{11} (2 a f+b c)+\frac{1}{8} a x^8 (a f+2 b c)+\frac{1}{12} b x^{12} (2 a g+b d)+\frac{1}{9} a x^9 (a g+2 b d)+\frac{1}{13} b x^{13} (2 a h+b e)+\frac{1}{10} a x^{10} (a h+2 b e)+\frac{1}{14} b^2 f x^{14}+\frac{1}{15} b^2 g x^{15}+\frac{1}{16} b^2 h x^{16} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.001, size = 152, normalized size = 0.9 \begin{align*}{\frac{{b}^{2}h{x}^{16}}{16}}+{\frac{{b}^{2}g{x}^{15}}{15}}+{\frac{{b}^{2}f{x}^{14}}{14}}+{\frac{ \left ( 2\,abh+{b}^{2}e \right ){x}^{13}}{13}}+{\frac{ \left ( 2\,abg+{b}^{2}d \right ){x}^{12}}{12}}+{\frac{ \left ( 2\,abf+{b}^{2}c \right ){x}^{11}}{11}}+{\frac{ \left ({a}^{2}h+2\,aeb \right ){x}^{10}}{10}}+{\frac{ \left ({a}^{2}g+2\,bda \right ){x}^{9}}{9}}+{\frac{ \left ({a}^{2}f+2\,abc \right ){x}^{8}}{8}}+{\frac{{a}^{2}e{x}^{7}}{7}}+{\frac{{a}^{2}d{x}^{6}}{6}}+{\frac{{a}^{2}c{x}^{5}}{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.956301, size = 204, normalized size = 1.25 \begin{align*} \frac{1}{16} \, b^{2} h x^{16} + \frac{1}{15} \, b^{2} g x^{15} + \frac{1}{14} \, b^{2} f x^{14} + \frac{1}{13} \,{\left (b^{2} e + 2 \, a b h\right )} x^{13} + \frac{1}{12} \,{\left (b^{2} d + 2 \, a b g\right )} x^{12} + \frac{1}{11} \,{\left (b^{2} c + 2 \, a b f\right )} x^{11} + \frac{1}{10} \,{\left (2 \, a b e + a^{2} h\right )} x^{10} + \frac{1}{7} \, a^{2} e x^{7} + \frac{1}{9} \,{\left (2 \, a b d + a^{2} g\right )} x^{9} + \frac{1}{6} \, a^{2} d x^{6} + \frac{1}{8} \,{\left (2 \, a b c + a^{2} f\right )} x^{8} + \frac{1}{5} \, a^{2} c x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.11735, size = 414, normalized size = 2.54 \begin{align*} \frac{1}{16} x^{16} h b^{2} + \frac{1}{15} x^{15} g b^{2} + \frac{1}{14} x^{14} f b^{2} + \frac{1}{13} x^{13} e b^{2} + \frac{2}{13} x^{13} h b a + \frac{1}{12} x^{12} d b^{2} + \frac{1}{6} x^{12} g b a + \frac{1}{11} x^{11} c b^{2} + \frac{2}{11} x^{11} f b a + \frac{1}{5} x^{10} e b a + \frac{1}{10} x^{10} h a^{2} + \frac{2}{9} x^{9} d b a + \frac{1}{9} x^{9} g a^{2} + \frac{1}{4} x^{8} c b a + \frac{1}{8} x^{8} f a^{2} + \frac{1}{7} x^{7} e a^{2} + \frac{1}{6} x^{6} d a^{2} + \frac{1}{5} x^{5} c a^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.095411, size = 167, normalized size = 1.02 \begin{align*} \frac{a^{2} c x^{5}}{5} + \frac{a^{2} d x^{6}}{6} + \frac{a^{2} e x^{7}}{7} + \frac{b^{2} f x^{14}}{14} + \frac{b^{2} g x^{15}}{15} + \frac{b^{2} h x^{16}}{16} + x^{13} \left (\frac{2 a b h}{13} + \frac{b^{2} e}{13}\right ) + x^{12} \left (\frac{a b g}{6} + \frac{b^{2} d}{12}\right ) + x^{11} \left (\frac{2 a b f}{11} + \frac{b^{2} c}{11}\right ) + x^{10} \left (\frac{a^{2} h}{10} + \frac{a b e}{5}\right ) + x^{9} \left (\frac{a^{2} g}{9} + \frac{2 a b d}{9}\right ) + x^{8} \left (\frac{a^{2} f}{8} + \frac{a b c}{4}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.07411, size = 216, normalized size = 1.33 \begin{align*} \frac{1}{16} \, b^{2} h x^{16} + \frac{1}{15} \, b^{2} g x^{15} + \frac{1}{14} \, b^{2} f x^{14} + \frac{2}{13} \, a b h x^{13} + \frac{1}{13} \, b^{2} x^{13} e + \frac{1}{12} \, b^{2} d x^{12} + \frac{1}{6} \, a b g x^{12} + \frac{1}{11} \, b^{2} c x^{11} + \frac{2}{11} \, a b f x^{11} + \frac{1}{10} \, a^{2} h x^{10} + \frac{1}{5} \, a b x^{10} e + \frac{2}{9} \, a b d x^{9} + \frac{1}{9} \, a^{2} g x^{9} + \frac{1}{4} \, a b c x^{8} + \frac{1}{8} \, a^{2} f x^{8} + \frac{1}{7} \, a^{2} x^{7} e + \frac{1}{6} \, a^{2} d x^{6} + \frac{1}{5} \, a^{2} c x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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