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

Optimal. Leaf size=97 \[ \frac{1}{6} x^6 (a f+b c)+\frac{1}{7} x^7 (a g+b d)+\frac{1}{8} x^8 (a h+b e)+\frac{1}{3} a c x^3+\frac{1}{4} a d x^4+\frac{1}{5} a e x^5+\frac{1}{9} b f x^9+\frac{1}{10} b g x^{10}+\frac{1}{11} b h x^{11} \]

[Out]

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

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Rubi [A]  time = 0.0914639, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.028, Rules used = {1820} \[ \frac{1}{6} x^6 (a f+b c)+\frac{1}{7} x^7 (a g+b d)+\frac{1}{8} x^8 (a h+b e)+\frac{1}{3} a c x^3+\frac{1}{4} a d x^4+\frac{1}{5} a e x^5+\frac{1}{9} b f x^9+\frac{1}{10} b g x^{10}+\frac{1}{11} b h x^{11} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0242234, size = 97, normalized size = 1. \[ \frac{1}{6} x^6 (a f+b c)+\frac{1}{7} x^7 (a g+b d)+\frac{1}{8} x^8 (a h+b e)+\frac{1}{3} a c x^3+\frac{1}{4} a d x^4+\frac{1}{5} a e x^5+\frac{1}{9} b f x^9+\frac{1}{10} b g x^{10}+\frac{1}{11} b h x^{11} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.001, size = 80, normalized size = 0.8 \begin{align*}{\frac{ac{x}^{3}}{3}}+{\frac{ad{x}^{4}}{4}}+{\frac{ae{x}^{5}}{5}}+{\frac{ \left ( af+bc \right ){x}^{6}}{6}}+{\frac{ \left ( ag+bd \right ){x}^{7}}{7}}+{\frac{ \left ( ah+be \right ){x}^{8}}{8}}+{\frac{bf{x}^{9}}{9}}+{\frac{bg{x}^{10}}{10}}+{\frac{bh{x}^{11}}{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.941309, size = 107, normalized size = 1.1 \begin{align*} \frac{1}{11} \, b h x^{11} + \frac{1}{10} \, b g x^{10} + \frac{1}{9} \, b f x^{9} + \frac{1}{8} \,{\left (b e + a h\right )} x^{8} + \frac{1}{7} \,{\left (b d + a g\right )} x^{7} + \frac{1}{5} \, a e x^{5} + \frac{1}{6} \,{\left (b c + a f\right )} x^{6} + \frac{1}{4} \, a d x^{4} + \frac{1}{3} \, a c x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.838869, size = 231, normalized size = 2.38 \begin{align*} \frac{1}{11} x^{11} h b + \frac{1}{10} x^{10} g b + \frac{1}{9} x^{9} f b + \frac{1}{8} x^{8} e b + \frac{1}{8} x^{8} h a + \frac{1}{7} x^{7} d b + \frac{1}{7} x^{7} g a + \frac{1}{6} x^{6} c b + \frac{1}{6} x^{6} f a + \frac{1}{5} x^{5} e a + \frac{1}{4} x^{4} d a + \frac{1}{3} x^{3} c a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*x^11*h*b + 1/10*x^10*g*b + 1/9*x^9*f*b + 1/8*x^8*e*b + 1/8*x^8*h*a + 1/7*x^7*d*b + 1/7*x^7*g*a + 1/6*x^6*
c*b + 1/6*x^6*f*a + 1/5*x^5*e*a + 1/4*x^4*d*a + 1/3*x^3*c*a

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Sympy [A]  time = 0.071216, size = 90, normalized size = 0.93 \begin{align*} \frac{a c x^{3}}{3} + \frac{a d x^{4}}{4} + \frac{a e x^{5}}{5} + \frac{b f x^{9}}{9} + \frac{b g x^{10}}{10} + \frac{b h x^{11}}{11} + x^{8} \left (\frac{a h}{8} + \frac{b e}{8}\right ) + x^{7} \left (\frac{a g}{7} + \frac{b d}{7}\right ) + x^{6} \left (\frac{a f}{6} + \frac{b c}{6}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*c*x**3/3 + a*d*x**4/4 + a*e*x**5/5 + b*f*x**9/9 + b*g*x**10/10 + b*h*x**11/11 + x**8*(a*h/8 + b*e/8) + x**7*
(a*g/7 + b*d/7) + x**6*(a*f/6 + b*c/6)

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Giac [A]  time = 1.32095, size = 117, normalized size = 1.21 \begin{align*} \frac{1}{11} \, b h x^{11} + \frac{1}{10} \, b g x^{10} + \frac{1}{9} \, b f x^{9} + \frac{1}{8} \, a h x^{8} + \frac{1}{8} \, b x^{8} e + \frac{1}{7} \, b d x^{7} + \frac{1}{7} \, a g x^{7} + \frac{1}{6} \, b c x^{6} + \frac{1}{6} \, a f x^{6} + \frac{1}{5} \, a x^{5} e + \frac{1}{4} \, a d x^{4} + \frac{1}{3} \, a c x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*b*h*x^11 + 1/10*b*g*x^10 + 1/9*b*f*x^9 + 1/8*a*h*x^8 + 1/8*b*x^8*e + 1/7*b*d*x^7 + 1/7*a*g*x^7 + 1/6*b*c*
x^6 + 1/6*a*f*x^6 + 1/5*a*x^5*e + 1/4*a*d*x^4 + 1/3*a*c*x^3