3.478 \(\int x^3 (c+d x+e x^2+f x^3) (a+b x^4) \, dx\)

Optimal. Leaf size=73 \[ \frac{1}{4} a c x^4+\frac{1}{5} a d x^5+\frac{1}{6} a e x^6+\frac{1}{7} a f x^7+\frac{1}{8} b c x^8+\frac{1}{9} b d x^9+\frac{1}{10} b e x^{10}+\frac{1}{11} b f x^{11} \]

[Out]

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

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Rubi [A]  time = 0.0641713, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038, Rules used = {1820} \[ \frac{1}{4} a c x^4+\frac{1}{5} a d x^5+\frac{1}{6} a e x^6+\frac{1}{7} a f x^7+\frac{1}{8} b c x^8+\frac{1}{9} b d x^9+\frac{1}{10} b e x^{10}+\frac{1}{11} b f x^{11} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0029445, size = 73, normalized size = 1. \[ \frac{1}{4} a c x^4+\frac{1}{5} a d x^5+\frac{1}{6} a e x^6+\frac{1}{7} a f x^7+\frac{1}{8} b c x^8+\frac{1}{9} b d x^9+\frac{1}{10} b e x^{10}+\frac{1}{11} b f x^{11} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0., size = 58, normalized size = 0.8 \begin{align*}{\frac{ac{x}^{4}}{4}}+{\frac{ad{x}^{5}}{5}}+{\frac{ae{x}^{6}}{6}}+{\frac{af{x}^{7}}{7}}+{\frac{bc{x}^{8}}{8}}+{\frac{bd{x}^{9}}{9}}+{\frac{be{x}^{10}}{10}}+{\frac{bf{x}^{11}}{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.918607, size = 77, normalized size = 1.05 \begin{align*} \frac{1}{11} \, b f x^{11} + \frac{1}{10} \, b e x^{10} + \frac{1}{9} \, b d x^{9} + \frac{1}{8} \, b c x^{8} + \frac{1}{7} \, a f x^{7} + \frac{1}{6} \, a e x^{6} + \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*(f*x^3+e*x^2+d*x+c)*(b*x^4+a),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.43254, size = 155, normalized size = 2.12 \begin{align*} \frac{1}{11} x^{11} f b + \frac{1}{10} x^{10} e b + \frac{1}{9} x^{9} d b + \frac{1}{8} x^{8} 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*(f*x^3+e*x^2+d*x+c)*(b*x^4+a),x, algorithm="fricas")

[Out]

1/11*x^11*f*b + 1/10*x^10*e*b + 1/9*x^9*d*b + 1/8*x^8*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 = 0.065535, size = 66, normalized size = 0.9 \begin{align*} \frac{a c x^{4}}{4} + \frac{a d x^{5}}{5} + \frac{a e x^{6}}{6} + \frac{a f x^{7}}{7} + \frac{b c x^{8}}{8} + \frac{b d x^{9}}{9} + \frac{b e x^{10}}{10} + \frac{b f x^{11}}{11} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*c*x**4/4 + a*d*x**5/5 + a*e*x**6/6 + a*f*x**7/7 + b*c*x**8/8 + b*d*x**9/9 + b*e*x**10/10 + b*f*x**11/11

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Giac [A]  time = 1.05782, size = 80, normalized size = 1.1 \begin{align*} \frac{1}{11} \, b f x^{11} + \frac{1}{10} \, b x^{10} e + \frac{1}{9} \, b d x^{9} + \frac{1}{8} \, b c x^{8} + \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*(f*x^3+e*x^2+d*x+c)*(b*x^4+a),x, algorithm="giac")

[Out]

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