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

Optimal. Leaf size=97 \[ \frac{1}{7} x^7 (a f+b c)+\frac{1}{8} x^8 (a g+b d)+\frac{1}{9} x^9 (a h+b e)+\frac{1}{4} a c x^4+\frac{1}{5} a d x^5+\frac{1}{6} a e x^6+\frac{1}{10} b f x^{10}+\frac{1}{11} b g x^{11}+\frac{1}{12} b h x^{12} \]

[Out]

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

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Rubi [A]  time = 0.100829, 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} \[ \frac{1}{7} x^7 (a f+b c)+\frac{1}{8} x^8 (a g+b d)+\frac{1}{9} x^9 (a h+b e)+\frac{1}{4} a c x^4+\frac{1}{5} a d x^5+\frac{1}{6} a e x^6+\frac{1}{10} b f x^{10}+\frac{1}{11} b g x^{11}+\frac{1}{12} b h x^{12} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.027894, size = 97, normalized size = 1. \[ \frac{1}{7} x^7 (a f+b c)+\frac{1}{8} x^8 (a g+b d)+\frac{1}{9} x^9 (a h+b e)+\frac{1}{4} a c x^4+\frac{1}{5} a d x^5+\frac{1}{6} a e x^6+\frac{1}{10} b f x^{10}+\frac{1}{11} b g x^{11}+\frac{1}{12} b h x^{12} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.001, size = 80, normalized size = 0.8 \begin{align*}{\frac{ac{x}^{4}}{4}}+{\frac{ad{x}^{5}}{5}}+{\frac{ae{x}^{6}}{6}}+{\frac{ \left ( af+bc \right ){x}^{7}}{7}}+{\frac{ \left ( ag+bd \right ){x}^{8}}{8}}+{\frac{ \left ( ah+be \right ){x}^{9}}{9}}+{\frac{bf{x}^{10}}{10}}+{\frac{bg{x}^{11}}{11}}+{\frac{bh{x}^{12}}{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.936407, size = 107, normalized size = 1.1 \begin{align*} \frac{1}{12} \, b h x^{12} + \frac{1}{11} \, b g x^{11} + \frac{1}{10} \, b f x^{10} + \frac{1}{9} \,{\left (b e + a h\right )} x^{9} + \frac{1}{8} \,{\left (b d + a g\right )} x^{8} + \frac{1}{6} \, a e x^{6} + \frac{1}{7} \,{\left (b c + a f\right )} x^{7} + \frac{1}{5} \, a d x^{5} + \frac{1}{4} \, a c x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.875431, size = 234, normalized size = 2.41 \begin{align*} \frac{1}{12} x^{12} h b + \frac{1}{11} x^{11} g b + \frac{1}{10} x^{10} f b + \frac{1}{9} x^{9} e b + \frac{1}{9} x^{9} h a + \frac{1}{8} x^{8} d b + \frac{1}{8} x^{8} g a + \frac{1}{7} x^{7} c b + \frac{1}{7} x^{7} f a + \frac{1}{6} x^{6} e a + \frac{1}{5} x^{5} d a + \frac{1}{4} x^{4} c a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.072214, size = 90, normalized size = 0.93 \begin{align*} \frac{a c x^{4}}{4} + \frac{a d x^{5}}{5} + \frac{a e x^{6}}{6} + \frac{b f x^{10}}{10} + \frac{b g x^{11}}{11} + \frac{b h x^{12}}{12} + x^{9} \left (\frac{a h}{9} + \frac{b e}{9}\right ) + x^{8} \left (\frac{a g}{8} + \frac{b d}{8}\right ) + x^{7} \left (\frac{a f}{7} + \frac{b c}{7}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.06642, size = 117, normalized size = 1.21 \begin{align*} \frac{1}{12} \, b h x^{12} + \frac{1}{11} \, b g x^{11} + \frac{1}{10} \, b f x^{10} + \frac{1}{9} \, a h x^{9} + \frac{1}{9} \, b x^{9} e + \frac{1}{8} \, b d x^{8} + \frac{1}{8} \, a g x^{8} + \frac{1}{7} \, b c x^{7} + \frac{1}{7} \, a f x^{7} + \frac{1}{6} \, a x^{6} e + \frac{1}{5} \, a d x^{5} + \frac{1}{4} \, a c x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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