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

Optimal. Leaf size=97 \[ \frac{1}{5} x^5 (a f+b c)+\frac{1}{6} x^6 (a g+b d)+\frac{1}{7} x^7 (a h+b e)+\frac{1}{2} a c x^2+\frac{1}{3} a d x^3+\frac{1}{4} a e x^4+\frac{1}{8} b f x^8+\frac{1}{9} b g x^9+\frac{1}{10} b h x^{10} \]

[Out]

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

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Rubi [A]  time = 0.0802172, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.029, Rules used = {1820} \[ \frac{1}{5} x^5 (a f+b c)+\frac{1}{6} x^6 (a g+b d)+\frac{1}{7} x^7 (a h+b e)+\frac{1}{2} a c x^2+\frac{1}{3} a d x^3+\frac{1}{4} a e x^4+\frac{1}{8} b f x^8+\frac{1}{9} b g x^9+\frac{1}{10} b h x^{10} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0173656, size = 97, normalized size = 1. \[ \frac{1}{5} x^5 (a f+b c)+\frac{1}{6} x^6 (a g+b d)+\frac{1}{7} x^7 (a h+b e)+\frac{1}{2} a c x^2+\frac{1}{3} a d x^3+\frac{1}{4} a e x^4+\frac{1}{8} b f x^8+\frac{1}{9} b g x^9+\frac{1}{10} b h x^{10} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.002, size = 80, normalized size = 0.8 \begin{align*}{\frac{ac{x}^{2}}{2}}+{\frac{ad{x}^{3}}{3}}+{\frac{ae{x}^{4}}{4}}+{\frac{ \left ( af+bc \right ){x}^{5}}{5}}+{\frac{ \left ( ag+bd \right ){x}^{6}}{6}}+{\frac{ \left ( ah+be \right ){x}^{7}}{7}}+{\frac{bf{x}^{8}}{8}}+{\frac{bg{x}^{9}}{9}}+{\frac{bh{x}^{10}}{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.951365, size = 107, normalized size = 1.1 \begin{align*} \frac{1}{10} \, b h x^{10} + \frac{1}{9} \, b g x^{9} + \frac{1}{8} \, b f x^{8} + \frac{1}{7} \,{\left (b e + a h\right )} x^{7} + \frac{1}{6} \,{\left (b d + a g\right )} x^{6} + \frac{1}{4} \, a e x^{4} + \frac{1}{5} \,{\left (b c + a f\right )} x^{5} + \frac{1}{3} \, a d x^{3} + \frac{1}{2} \, a c x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.893111, size = 228, normalized size = 2.35 \begin{align*} \frac{1}{10} x^{10} h b + \frac{1}{9} x^{9} g b + \frac{1}{8} x^{8} f b + \frac{1}{7} x^{7} e b + \frac{1}{7} x^{7} h a + \frac{1}{6} x^{6} d b + \frac{1}{6} x^{6} g a + \frac{1}{5} x^{5} c b + \frac{1}{5} x^{5} f a + \frac{1}{4} x^{4} e a + \frac{1}{3} x^{3} d a + \frac{1}{2} x^{2} c a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*x^10*h*b + 1/9*x^9*g*b + 1/8*x^8*f*b + 1/7*x^7*e*b + 1/7*x^7*h*a + 1/6*x^6*d*b + 1/6*x^6*g*a + 1/5*x^5*c*
b + 1/5*x^5*f*a + 1/4*x^4*e*a + 1/3*x^3*d*a + 1/2*x^2*c*a

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Sympy [A]  time = 0.070509, size = 90, normalized size = 0.93 \begin{align*} \frac{a c x^{2}}{2} + \frac{a d x^{3}}{3} + \frac{a e x^{4}}{4} + \frac{b f x^{8}}{8} + \frac{b g x^{9}}{9} + \frac{b h x^{10}}{10} + x^{7} \left (\frac{a h}{7} + \frac{b e}{7}\right ) + x^{6} \left (\frac{a g}{6} + \frac{b d}{6}\right ) + x^{5} \left (\frac{a f}{5} + \frac{b c}{5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*c*x**2/2 + a*d*x**3/3 + a*e*x**4/4 + b*f*x**8/8 + b*g*x**9/9 + b*h*x**10/10 + x**7*(a*h/7 + b*e/7) + x**6*(a
*g/6 + b*d/6) + x**5*(a*f/5 + b*c/5)

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Giac [A]  time = 1.05363, size = 117, normalized size = 1.21 \begin{align*} \frac{1}{10} \, b h x^{10} + \frac{1}{9} \, b g x^{9} + \frac{1}{8} \, b f x^{8} + \frac{1}{7} \, a h x^{7} + \frac{1}{7} \, b x^{7} e + \frac{1}{6} \, b d x^{6} + \frac{1}{6} \, a g x^{6} + \frac{1}{5} \, b c x^{5} + \frac{1}{5} \, a f x^{5} + \frac{1}{4} \, a x^{4} e + \frac{1}{3} \, a d x^{3} + \frac{1}{2} \, a c x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*b*h*x^10 + 1/9*b*g*x^9 + 1/8*b*f*x^8 + 1/7*a*h*x^7 + 1/7*b*x^7*e + 1/6*b*d*x^6 + 1/6*a*g*x^6 + 1/5*b*c*x^
5 + 1/5*a*f*x^5 + 1/4*a*x^4*e + 1/3*a*d*x^3 + 1/2*a*c*x^2