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

Optimal. Leaf size=156 \[ \frac{1}{3} a^2 b d x^9+\frac{3}{10} a^2 b e x^{10}+\frac{3}{11} a^2 b f x^{11}+\frac{1}{5} a^3 d x^5+\frac{1}{6} a^3 e x^6+\frac{1}{7} a^3 f x^7+\frac{3}{13} a b^2 d x^{13}+\frac{3}{14} a b^2 e x^{14}+\frac{1}{5} a b^2 f x^{15}+\frac{c \left (a+b x^4\right )^4}{16 b}+\frac{1}{17} b^3 d x^{17}+\frac{1}{18} b^3 e x^{18}+\frac{1}{19} b^3 f x^{19} \]

[Out]

(a^3*d*x^5)/5 + (a^3*e*x^6)/6 + (a^3*f*x^7)/7 + (a^2*b*d*x^9)/3 + (3*a^2*b*e*x^10)/10 + (3*a^2*b*f*x^11)/11 +
(3*a*b^2*d*x^13)/13 + (3*a*b^2*e*x^14)/14 + (a*b^2*f*x^15)/5 + (b^3*d*x^17)/17 + (b^3*e*x^18)/18 + (b^3*f*x^19
)/19 + (c*(a + b*x^4)^4)/(16*b)

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Rubi [A]  time = 0.112745, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {1582, 1850} \[ \frac{1}{3} a^2 b d x^9+\frac{3}{10} a^2 b e x^{10}+\frac{3}{11} a^2 b f x^{11}+\frac{1}{5} a^3 d x^5+\frac{1}{6} a^3 e x^6+\frac{1}{7} a^3 f x^7+\frac{3}{13} a b^2 d x^{13}+\frac{3}{14} a b^2 e x^{14}+\frac{1}{5} a b^2 f x^{15}+\frac{c \left (a+b x^4\right )^4}{16 b}+\frac{1}{17} b^3 d x^{17}+\frac{1}{18} b^3 e x^{18}+\frac{1}{19} b^3 f x^{19} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*d*x^5)/5 + (a^3*e*x^6)/6 + (a^3*f*x^7)/7 + (a^2*b*d*x^9)/3 + (3*a^2*b*e*x^10)/10 + (3*a^2*b*f*x^11)/11 +
(3*a*b^2*d*x^13)/13 + (3*a*b^2*e*x^14)/14 + (a*b^2*f*x^15)/5 + (b^3*d*x^17)/17 + (b^3*e*x^18)/18 + (b^3*f*x^19
)/19 + (c*(a + b*x^4)^4)/(16*b)

Rule 1582

Int[(Px_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(Coeff[Px, x, n - 1]*(a + b*x^n)^(p + 1))/(b*n*(p +
 1)), x] + Int[(Px - Coeff[Px, x, n - 1]*x^(n - 1))*(a + b*x^n)^p, x] /; FreeQ[{a, b}, x] && PolyQ[Px, x] && I
GtQ[p, 1] && IGtQ[n, 1] && NeQ[Coeff[Px, x, n - 1], 0] && NeQ[Px, Coeff[Px, x, n - 1]*x^(n - 1)] &&  !MatchQ[P
x, (Qx_.)*((c_) + (d_.)*x^(m_))^(q_) /; FreeQ[{c, d}, x] && PolyQ[Qx, x] && IGtQ[q, 1] && IGtQ[m, 1] && NeQ[Co
eff[Qx*(a + b*x^n)^p, x, m - 1], 0] && GtQ[m*q, n*p]]

Rule 1850

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, 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 )^3 \, dx &=\frac{c \left (a+b x^4\right )^4}{16 b}+\int \left (a+b x^4\right )^3 \left (-c x^3+x^3 \left (c+d x+e x^2+f x^3\right )\right ) \, dx\\ &=\frac{c \left (a+b x^4\right )^4}{16 b}+\int \left (a^3 d x^4+a^3 e x^5+a^3 f x^6+3 a^2 b d x^8+3 a^2 b e x^9+3 a^2 b f x^{10}+3 a b^2 d x^{12}+3 a b^2 e x^{13}+3 a b^2 f x^{14}+b^3 d x^{16}+b^3 e x^{17}+b^3 f x^{18}\right ) \, dx\\ &=\frac{1}{5} a^3 d x^5+\frac{1}{6} a^3 e x^6+\frac{1}{7} a^3 f x^7+\frac{1}{3} a^2 b d x^9+\frac{3}{10} a^2 b e x^{10}+\frac{3}{11} a^2 b f x^{11}+\frac{3}{13} a b^2 d x^{13}+\frac{3}{14} a b^2 e x^{14}+\frac{1}{5} a b^2 f x^{15}+\frac{1}{17} b^3 d x^{17}+\frac{1}{18} b^3 e x^{18}+\frac{1}{19} b^3 f x^{19}+\frac{c \left (a+b x^4\right )^4}{16 b}\\ \end{align*}

Mathematica [A]  time = 0.0052431, size = 185, normalized size = 1.19 \[ \frac{3}{8} a^2 b c x^8+\frac{1}{3} a^2 b d x^9+\frac{3}{10} a^2 b e x^{10}+\frac{3}{11} a^2 b f x^{11}+\frac{1}{4} a^3 c x^4+\frac{1}{5} a^3 d x^5+\frac{1}{6} a^3 e x^6+\frac{1}{7} a^3 f x^7+\frac{1}{4} a b^2 c x^{12}+\frac{3}{13} a b^2 d x^{13}+\frac{3}{14} a b^2 e x^{14}+\frac{1}{5} a b^2 f x^{15}+\frac{1}{16} b^3 c x^{16}+\frac{1}{17} b^3 d x^{17}+\frac{1}{18} b^3 e x^{18}+\frac{1}{19} b^3 f x^{19} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*c*x^4)/4 + (a^3*d*x^5)/5 + (a^3*e*x^6)/6 + (a^3*f*x^7)/7 + (3*a^2*b*c*x^8)/8 + (a^2*b*d*x^9)/3 + (3*a^2*b
*e*x^10)/10 + (3*a^2*b*f*x^11)/11 + (a*b^2*c*x^12)/4 + (3*a*b^2*d*x^13)/13 + (3*a*b^2*e*x^14)/14 + (a*b^2*f*x^
15)/5 + (b^3*c*x^16)/16 + (b^3*d*x^17)/17 + (b^3*e*x^18)/18 + (b^3*f*x^19)/19

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Maple [A]  time = 0.001, size = 154, normalized size = 1. \begin{align*}{\frac{{b}^{3}f{x}^{19}}{19}}+{\frac{{b}^{3}e{x}^{18}}{18}}+{\frac{{b}^{3}d{x}^{17}}{17}}+{\frac{{b}^{3}c{x}^{16}}{16}}+{\frac{a{b}^{2}f{x}^{15}}{5}}+{\frac{3\,a{b}^{2}e{x}^{14}}{14}}+{\frac{3\,a{b}^{2}d{x}^{13}}{13}}+{\frac{ac{b}^{2}{x}^{12}}{4}}+{\frac{3\,{a}^{2}bf{x}^{11}}{11}}+{\frac{3\,{a}^{2}be{x}^{10}}{10}}+{\frac{{a}^{2}bd{x}^{9}}{3}}+{\frac{3\,b{a}^{2}c{x}^{8}}{8}}+{\frac{{a}^{3}f{x}^{7}}{7}}+{\frac{{a}^{3}e{x}^{6}}{6}}+{\frac{{a}^{3}d{x}^{5}}{5}}+{\frac{{a}^{3}c{x}^{4}}{4}} \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)^3,x)

[Out]

1/19*b^3*f*x^19+1/18*b^3*e*x^18+1/17*b^3*d*x^17+1/16*b^3*c*x^16+1/5*a*b^2*f*x^15+3/14*a*b^2*e*x^14+3/13*a*b^2*
d*x^13+1/4*a*c*b^2*x^12+3/11*a^2*b*f*x^11+3/10*a^2*b*e*x^10+1/3*a^2*b*d*x^9+3/8*b*a^2*c*x^8+1/7*a^3*f*x^7+1/6*
a^3*e*x^6+1/5*a^3*d*x^5+1/4*a^3*c*x^4

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Maxima [A]  time = 0.916298, size = 207, normalized size = 1.33 \begin{align*} \frac{1}{19} \, b^{3} f x^{19} + \frac{1}{18} \, b^{3} e x^{18} + \frac{1}{17} \, b^{3} d x^{17} + \frac{1}{16} \, b^{3} c x^{16} + \frac{1}{5} \, a b^{2} f x^{15} + \frac{3}{14} \, a b^{2} e x^{14} + \frac{3}{13} \, a b^{2} d x^{13} + \frac{1}{4} \, a b^{2} c x^{12} + \frac{3}{11} \, a^{2} b f x^{11} + \frac{3}{10} \, a^{2} b e x^{10} + \frac{1}{3} \, a^{2} b d x^{9} + \frac{3}{8} \, a^{2} b c x^{8} + \frac{1}{7} \, a^{3} f x^{7} + \frac{1}{6} \, a^{3} e x^{6} + \frac{1}{5} \, a^{3} d x^{5} + \frac{1}{4} \, a^{3} 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)^3,x, algorithm="maxima")

[Out]

1/19*b^3*f*x^19 + 1/18*b^3*e*x^18 + 1/17*b^3*d*x^17 + 1/16*b^3*c*x^16 + 1/5*a*b^2*f*x^15 + 3/14*a*b^2*e*x^14 +
 3/13*a*b^2*d*x^13 + 1/4*a*b^2*c*x^12 + 3/11*a^2*b*f*x^11 + 3/10*a^2*b*e*x^10 + 1/3*a^2*b*d*x^9 + 3/8*a^2*b*c*
x^8 + 1/7*a^3*f*x^7 + 1/6*a^3*e*x^6 + 1/5*a^3*d*x^5 + 1/4*a^3*c*x^4

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Fricas [A]  time = 1.44069, size = 390, normalized size = 2.5 \begin{align*} \frac{1}{19} x^{19} f b^{3} + \frac{1}{18} x^{18} e b^{3} + \frac{1}{17} x^{17} d b^{3} + \frac{1}{16} x^{16} c b^{3} + \frac{1}{5} x^{15} f b^{2} a + \frac{3}{14} x^{14} e b^{2} a + \frac{3}{13} x^{13} d b^{2} a + \frac{1}{4} x^{12} c b^{2} a + \frac{3}{11} x^{11} f b a^{2} + \frac{3}{10} x^{10} e b a^{2} + \frac{1}{3} x^{9} d b a^{2} + \frac{3}{8} x^{8} c b a^{2} + \frac{1}{7} x^{7} f a^{3} + \frac{1}{6} x^{6} e a^{3} + \frac{1}{5} x^{5} d a^{3} + \frac{1}{4} x^{4} c a^{3} \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)^3,x, algorithm="fricas")

[Out]

1/19*x^19*f*b^3 + 1/18*x^18*e*b^3 + 1/17*x^17*d*b^3 + 1/16*x^16*c*b^3 + 1/5*x^15*f*b^2*a + 3/14*x^14*e*b^2*a +
 3/13*x^13*d*b^2*a + 1/4*x^12*c*b^2*a + 3/11*x^11*f*b*a^2 + 3/10*x^10*e*b*a^2 + 1/3*x^9*d*b*a^2 + 3/8*x^8*c*b*
a^2 + 1/7*x^7*f*a^3 + 1/6*x^6*e*a^3 + 1/5*x^5*d*a^3 + 1/4*x^4*c*a^3

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Sympy [A]  time = 0.084306, size = 184, normalized size = 1.18 \begin{align*} \frac{a^{3} c x^{4}}{4} + \frac{a^{3} d x^{5}}{5} + \frac{a^{3} e x^{6}}{6} + \frac{a^{3} f x^{7}}{7} + \frac{3 a^{2} b c x^{8}}{8} + \frac{a^{2} b d x^{9}}{3} + \frac{3 a^{2} b e x^{10}}{10} + \frac{3 a^{2} b f x^{11}}{11} + \frac{a b^{2} c x^{12}}{4} + \frac{3 a b^{2} d x^{13}}{13} + \frac{3 a b^{2} e x^{14}}{14} + \frac{a b^{2} f x^{15}}{5} + \frac{b^{3} c x^{16}}{16} + \frac{b^{3} d x^{17}}{17} + \frac{b^{3} e x^{18}}{18} + \frac{b^{3} f x^{19}}{19} \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)**3,x)

[Out]

a**3*c*x**4/4 + a**3*d*x**5/5 + a**3*e*x**6/6 + a**3*f*x**7/7 + 3*a**2*b*c*x**8/8 + a**2*b*d*x**9/3 + 3*a**2*b
*e*x**10/10 + 3*a**2*b*f*x**11/11 + a*b**2*c*x**12/4 + 3*a*b**2*d*x**13/13 + 3*a*b**2*e*x**14/14 + a*b**2*f*x*
*15/5 + b**3*c*x**16/16 + b**3*d*x**17/17 + b**3*e*x**18/18 + b**3*f*x**19/19

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Giac [A]  time = 1.07847, size = 212, normalized size = 1.36 \begin{align*} \frac{1}{19} \, b^{3} f x^{19} + \frac{1}{18} \, b^{3} x^{18} e + \frac{1}{17} \, b^{3} d x^{17} + \frac{1}{16} \, b^{3} c x^{16} + \frac{1}{5} \, a b^{2} f x^{15} + \frac{3}{14} \, a b^{2} x^{14} e + \frac{3}{13} \, a b^{2} d x^{13} + \frac{1}{4} \, a b^{2} c x^{12} + \frac{3}{11} \, a^{2} b f x^{11} + \frac{3}{10} \, a^{2} b x^{10} e + \frac{1}{3} \, a^{2} b d x^{9} + \frac{3}{8} \, a^{2} b c x^{8} + \frac{1}{7} \, a^{3} f x^{7} + \frac{1}{6} \, a^{3} x^{6} e + \frac{1}{5} \, a^{3} d x^{5} + \frac{1}{4} \, a^{3} 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)^3,x, algorithm="giac")

[Out]

1/19*b^3*f*x^19 + 1/18*b^3*x^18*e + 1/17*b^3*d*x^17 + 1/16*b^3*c*x^16 + 1/5*a*b^2*f*x^15 + 3/14*a*b^2*x^14*e +
 3/13*a*b^2*d*x^13 + 1/4*a*b^2*c*x^12 + 3/11*a^2*b*f*x^11 + 3/10*a^2*b*x^10*e + 1/3*a^2*b*d*x^9 + 3/8*a^2*b*c*
x^8 + 1/7*a^3*f*x^7 + 1/6*a^3*x^6*e + 1/5*a^3*d*x^5 + 1/4*a^3*c*x^4