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

Optimal. Leaf size=163 \[ \frac{1}{4} a^2 c x^4+\frac{1}{5} a^2 d x^5+\frac{1}{6} a^2 e x^6+\frac{1}{10} b x^{10} (2 a f+b c)+\frac{1}{7} a x^7 (a f+2 b c)+\frac{1}{11} b x^{11} (2 a g+b d)+\frac{1}{8} a x^8 (a g+2 b d)+\frac{1}{12} b x^{12} (2 a h+b e)+\frac{1}{9} a x^9 (a h+2 b e)+\frac{1}{13} b^2 f x^{13}+\frac{1}{14} b^2 g x^{14}+\frac{1}{15} b^2 h x^{15} \]

[Out]

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

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Rubi [A]  time = 0.15901, 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} \[ \frac{1}{4} a^2 c x^4+\frac{1}{5} a^2 d x^5+\frac{1}{6} a^2 e x^6+\frac{1}{10} b x^{10} (2 a f+b c)+\frac{1}{7} a x^7 (a f+2 b c)+\frac{1}{11} b x^{11} (2 a g+b d)+\frac{1}{8} a x^8 (a g+2 b d)+\frac{1}{12} b x^{12} (2 a h+b e)+\frac{1}{9} a x^9 (a h+2 b e)+\frac{1}{13} b^2 f x^{13}+\frac{1}{14} b^2 g x^{14}+\frac{1}{15} b^2 h x^{15} \]

Antiderivative was successfully verified.

[In]

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

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

Mathematica [A]  time = 0.027851, size = 163, normalized size = 1. \[ \frac{1}{4} a^2 c x^4+\frac{1}{5} a^2 d x^5+\frac{1}{6} a^2 e x^6+\frac{1}{10} b x^{10} (2 a f+b c)+\frac{1}{7} a x^7 (a f+2 b c)+\frac{1}{11} b x^{11} (2 a g+b d)+\frac{1}{8} a x^8 (a g+2 b d)+\frac{1}{12} b x^{12} (2 a h+b e)+\frac{1}{9} a x^9 (a h+2 b e)+\frac{1}{13} b^2 f x^{13}+\frac{1}{14} b^2 g x^{14}+\frac{1}{15} b^2 h x^{15} \]

Antiderivative was successfully verified.

[In]

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

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Maple [A]  time = 0.001, size = 152, normalized size = 0.9 \begin{align*}{\frac{{b}^{2}h{x}^{15}}{15}}+{\frac{{b}^{2}g{x}^{14}}{14}}+{\frac{{b}^{2}f{x}^{13}}{13}}+{\frac{ \left ( 2\,abh+{b}^{2}e \right ){x}^{12}}{12}}+{\frac{ \left ( 2\,abg+{b}^{2}d \right ){x}^{11}}{11}}+{\frac{ \left ( 2\,abf+{b}^{2}c \right ){x}^{10}}{10}}+{\frac{ \left ({a}^{2}h+2\,aeb \right ){x}^{9}}{9}}+{\frac{ \left ({a}^{2}g+2\,bda \right ){x}^{8}}{8}}+{\frac{ \left ({a}^{2}f+2\,abc \right ){x}^{7}}{7}}+{\frac{{a}^{2}e{x}^{6}}{6}}+{\frac{{a}^{2}d{x}^{5}}{5}}+{\frac{{a}^{2}c{x}^{4}}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.935679, size = 204, normalized size = 1.25 \begin{align*} \frac{1}{15} \, b^{2} h x^{15} + \frac{1}{14} \, b^{2} g x^{14} + \frac{1}{13} \, b^{2} f x^{13} + \frac{1}{12} \,{\left (b^{2} e + 2 \, a b h\right )} x^{12} + \frac{1}{11} \,{\left (b^{2} d + 2 \, a b g\right )} x^{11} + \frac{1}{10} \,{\left (b^{2} c + 2 \, a b f\right )} x^{10} + \frac{1}{9} \,{\left (2 \, a b e + a^{2} h\right )} x^{9} + \frac{1}{6} \, a^{2} e x^{6} + \frac{1}{8} \,{\left (2 \, a b d + a^{2} g\right )} x^{8} + \frac{1}{5} \, a^{2} d x^{5} + \frac{1}{7} \,{\left (2 \, a b c + a^{2} f\right )} x^{7} + \frac{1}{4} \, a^{2} c x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.0839, size = 409, normalized size = 2.51 \begin{align*} \frac{1}{15} x^{15} h b^{2} + \frac{1}{14} x^{14} g b^{2} + \frac{1}{13} x^{13} f b^{2} + \frac{1}{12} x^{12} e b^{2} + \frac{1}{6} x^{12} h b a + \frac{1}{11} x^{11} d b^{2} + \frac{2}{11} x^{11} g b a + \frac{1}{10} x^{10} c b^{2} + \frac{1}{5} x^{10} f b a + \frac{2}{9} x^{9} e b a + \frac{1}{9} x^{9} h a^{2} + \frac{1}{4} x^{8} d b a + \frac{1}{8} x^{8} g a^{2} + \frac{2}{7} x^{7} c b a + \frac{1}{7} x^{7} f a^{2} + \frac{1}{6} x^{6} e a^{2} + \frac{1}{5} x^{5} d a^{2} + \frac{1}{4} x^{4} c a^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.13967, size = 167, normalized size = 1.02 \begin{align*} \frac{a^{2} c x^{4}}{4} + \frac{a^{2} d x^{5}}{5} + \frac{a^{2} e x^{6}}{6} + \frac{b^{2} f x^{13}}{13} + \frac{b^{2} g x^{14}}{14} + \frac{b^{2} h x^{15}}{15} + x^{12} \left (\frac{a b h}{6} + \frac{b^{2} e}{12}\right ) + x^{11} \left (\frac{2 a b g}{11} + \frac{b^{2} d}{11}\right ) + x^{10} \left (\frac{a b f}{5} + \frac{b^{2} c}{10}\right ) + x^{9} \left (\frac{a^{2} h}{9} + \frac{2 a b e}{9}\right ) + x^{8} \left (\frac{a^{2} g}{8} + \frac{a b d}{4}\right ) + x^{7} \left (\frac{a^{2} f}{7} + \frac{2 a b c}{7}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Giac [A]  time = 1.06638, size = 216, normalized size = 1.33 \begin{align*} \frac{1}{15} \, b^{2} h x^{15} + \frac{1}{14} \, b^{2} g x^{14} + \frac{1}{13} \, b^{2} f x^{13} + \frac{1}{6} \, a b h x^{12} + \frac{1}{12} \, b^{2} x^{12} e + \frac{1}{11} \, b^{2} d x^{11} + \frac{2}{11} \, a b g x^{11} + \frac{1}{10} \, b^{2} c x^{10} + \frac{1}{5} \, a b f x^{10} + \frac{1}{9} \, a^{2} h x^{9} + \frac{2}{9} \, a b x^{9} e + \frac{1}{4} \, a b d x^{8} + \frac{1}{8} \, a^{2} g x^{8} + \frac{2}{7} \, a b c x^{7} + \frac{1}{7} \, a^{2} f x^{7} + \frac{1}{6} \, a^{2} x^{6} e + \frac{1}{5} \, a^{2} d x^{5} + \frac{1}{4} \, a^{2} c x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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