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

Optimal. Leaf size=198 \[ \frac{6}{13} a^2 b^2 d x^{13}+\frac{3}{7} a^2 b^2 e x^{14}+\frac{2}{5} a^2 b^2 f x^{15}+\frac{4}{9} a^3 b d x^9+\frac{2}{5} a^3 b e x^{10}+\frac{4}{11} a^3 b f x^{11}+\frac{1}{5} a^4 d x^5+\frac{1}{6} a^4 e x^6+\frac{1}{7} a^4 f x^7+\frac{4}{17} a b^3 d x^{17}+\frac{2}{9} a b^3 e x^{18}+\frac{4}{19} a b^3 f x^{19}+\frac{c \left (a+b x^4\right )^5}{20 b}+\frac{1}{21} b^4 d x^{21}+\frac{1}{22} b^4 e x^{22}+\frac{1}{23} b^4 f x^{23} \]

[Out]

(a^4*d*x^5)/5 + (a^4*e*x^6)/6 + (a^4*f*x^7)/7 + (4*a^3*b*d*x^9)/9 + (2*a^3*b*e*x^10)/5 + (4*a^3*b*f*x^11)/11 +
(6*a^2*b^2*d*x^13)/13 + (3*a^2*b^2*e*x^14)/7 + (2*a^2*b^2*f*x^15)/5 + (4*a*b^3*d*x^17)/17 + (2*a*b^3*e*x^18)/
9 + (4*a*b^3*f*x^19)/19 + (b^4*d*x^21)/21 + (b^4*e*x^22)/22 + (b^4*f*x^23)/23 + (c*(a + b*x^4)^5)/(20*b)

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Rubi [A]  time = 0.150432, antiderivative size = 198, 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} \[ \frac{6}{13} a^2 b^2 d x^{13}+\frac{3}{7} a^2 b^2 e x^{14}+\frac{2}{5} a^2 b^2 f x^{15}+\frac{4}{9} a^3 b d x^9+\frac{2}{5} a^3 b e x^{10}+\frac{4}{11} a^3 b f x^{11}+\frac{1}{5} a^4 d x^5+\frac{1}{6} a^4 e x^6+\frac{1}{7} a^4 f x^7+\frac{4}{17} a b^3 d x^{17}+\frac{2}{9} a b^3 e x^{18}+\frac{4}{19} a b^3 f x^{19}+\frac{c \left (a+b x^4\right )^5}{20 b}+\frac{1}{21} b^4 d x^{21}+\frac{1}{22} b^4 e x^{22}+\frac{1}{23} b^4 f x^{23} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^4*d*x^5)/5 + (a^4*e*x^6)/6 + (a^4*f*x^7)/7 + (4*a^3*b*d*x^9)/9 + (2*a^3*b*e*x^10)/5 + (4*a^3*b*f*x^11)/11 +
(6*a^2*b^2*d*x^13)/13 + (3*a^2*b^2*e*x^14)/7 + (2*a^2*b^2*f*x^15)/5 + (4*a*b^3*d*x^17)/17 + (2*a*b^3*e*x^18)/
9 + (4*a*b^3*f*x^19)/19 + (b^4*d*x^21)/21 + (b^4*e*x^22)/22 + (b^4*f*x^23)/23 + (c*(a + b*x^4)^5)/(20*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 )^4 \, dx &=\frac{c \left (a+b x^4\right )^5}{20 b}+\int \left (a+b x^4\right )^4 \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 )^5}{20 b}+\int \left (a^4 d x^4+a^4 e x^5+a^4 f x^6+4 a^3 b d x^8+4 a^3 b e x^9+4 a^3 b f x^{10}+6 a^2 b^2 d x^{12}+6 a^2 b^2 e x^{13}+6 a^2 b^2 f x^{14}+4 a b^3 d x^{16}+4 a b^3 e x^{17}+4 a b^3 f x^{18}+b^4 d x^{20}+b^4 e x^{21}+b^4 f x^{22}\right ) \, dx\\ &=\frac{1}{5} a^4 d x^5+\frac{1}{6} a^4 e x^6+\frac{1}{7} a^4 f x^7+\frac{4}{9} a^3 b d x^9+\frac{2}{5} a^3 b e x^{10}+\frac{4}{11} a^3 b f x^{11}+\frac{6}{13} a^2 b^2 d x^{13}+\frac{3}{7} a^2 b^2 e x^{14}+\frac{2}{5} a^2 b^2 f x^{15}+\frac{4}{17} a b^3 d x^{17}+\frac{2}{9} a b^3 e x^{18}+\frac{4}{19} a b^3 f x^{19}+\frac{1}{21} b^4 d x^{21}+\frac{1}{22} b^4 e x^{22}+\frac{1}{23} b^4 f x^{23}+\frac{c \left (a+b x^4\right )^5}{20 b}\\ \end{align*}

Mathematica [A]  time = 0.0062771, size = 241, normalized size = 1.22 \[ \frac{1}{2} a^2 b^2 c x^{12}+\frac{6}{13} a^2 b^2 d x^{13}+\frac{3}{7} a^2 b^2 e x^{14}+\frac{2}{5} a^2 b^2 f x^{15}+\frac{1}{2} a^3 b c x^8+\frac{4}{9} a^3 b d x^9+\frac{2}{5} a^3 b e x^{10}+\frac{4}{11} a^3 b f x^{11}+\frac{1}{4} a^4 c x^4+\frac{1}{5} a^4 d x^5+\frac{1}{6} a^4 e x^6+\frac{1}{7} a^4 f x^7+\frac{1}{4} a b^3 c x^{16}+\frac{4}{17} a b^3 d x^{17}+\frac{2}{9} a b^3 e x^{18}+\frac{4}{19} a b^3 f x^{19}+\frac{1}{20} b^4 c x^{20}+\frac{1}{21} b^4 d x^{21}+\frac{1}{22} b^4 e x^{22}+\frac{1}{23} b^4 f x^{23} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^4*c*x^4)/4 + (a^4*d*x^5)/5 + (a^4*e*x^6)/6 + (a^4*f*x^7)/7 + (a^3*b*c*x^8)/2 + (4*a^3*b*d*x^9)/9 + (2*a^3*b
*e*x^10)/5 + (4*a^3*b*f*x^11)/11 + (a^2*b^2*c*x^12)/2 + (6*a^2*b^2*d*x^13)/13 + (3*a^2*b^2*e*x^14)/7 + (2*a^2*
b^2*f*x^15)/5 + (a*b^3*c*x^16)/4 + (4*a*b^3*d*x^17)/17 + (2*a*b^3*e*x^18)/9 + (4*a*b^3*f*x^19)/19 + (b^4*c*x^2
0)/20 + (b^4*d*x^21)/21 + (b^4*e*x^22)/22 + (b^4*f*x^23)/23

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Maple [A]  time = 0.001, size = 202, normalized size = 1. \begin{align*}{\frac{{b}^{4}f{x}^{23}}{23}}+{\frac{{b}^{4}e{x}^{22}}{22}}+{\frac{{b}^{4}d{x}^{21}}{21}}+{\frac{c{b}^{4}{x}^{20}}{20}}+{\frac{4\,a{b}^{3}f{x}^{19}}{19}}+{\frac{2\,a{b}^{3}e{x}^{18}}{9}}+{\frac{4\,a{b}^{3}d{x}^{17}}{17}}+{\frac{ac{b}^{3}{x}^{16}}{4}}+{\frac{2\,{a}^{2}{b}^{2}f{x}^{15}}{5}}+{\frac{3\,{a}^{2}{b}^{2}e{x}^{14}}{7}}+{\frac{6\,{a}^{2}{b}^{2}d{x}^{13}}{13}}+{\frac{{a}^{2}{b}^{2}c{x}^{12}}{2}}+{\frac{4\,{a}^{3}bf{x}^{11}}{11}}+{\frac{2\,{a}^{3}be{x}^{10}}{5}}+{\frac{4\,{a}^{3}bd{x}^{9}}{9}}+{\frac{c{a}^{3}b{x}^{8}}{2}}+{\frac{{a}^{4}f{x}^{7}}{7}}+{\frac{{a}^{4}e{x}^{6}}{6}}+{\frac{{a}^{4}d{x}^{5}}{5}}+{\frac{{a}^{4}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)^4,x)

[Out]

1/23*b^4*f*x^23+1/22*b^4*e*x^22+1/21*b^4*d*x^21+1/20*c*b^4*x^20+4/19*a*b^3*f*x^19+2/9*a*b^3*e*x^18+4/17*a*b^3*
d*x^17+1/4*a*c*b^3*x^16+2/5*a^2*b^2*f*x^15+3/7*a^2*b^2*e*x^14+6/13*a^2*b^2*d*x^13+1/2*a^2*b^2*c*x^12+4/11*a^3*
b*f*x^11+2/5*a^3*b*e*x^10+4/9*a^3*b*d*x^9+1/2*c*a^3*b*x^8+1/7*a^4*f*x^7+1/6*a^4*e*x^6+1/5*a^4*d*x^5+1/4*a^4*c*
x^4

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Maxima [A]  time = 0.920182, size = 271, normalized size = 1.37 \begin{align*} \frac{1}{23} \, b^{4} f x^{23} + \frac{1}{22} \, b^{4} e x^{22} + \frac{1}{21} \, b^{4} d x^{21} + \frac{1}{20} \, b^{4} c x^{20} + \frac{4}{19} \, a b^{3} f x^{19} + \frac{2}{9} \, a b^{3} e x^{18} + \frac{4}{17} \, a b^{3} d x^{17} + \frac{1}{4} \, a b^{3} c x^{16} + \frac{2}{5} \, a^{2} b^{2} f x^{15} + \frac{3}{7} \, a^{2} b^{2} e x^{14} + \frac{6}{13} \, a^{2} b^{2} d x^{13} + \frac{1}{2} \, a^{2} b^{2} c x^{12} + \frac{4}{11} \, a^{3} b f x^{11} + \frac{2}{5} \, a^{3} b e x^{10} + \frac{4}{9} \, a^{3} b d x^{9} + \frac{1}{2} \, a^{3} b c x^{8} + \frac{1}{7} \, a^{4} f x^{7} + \frac{1}{6} \, a^{4} e x^{6} + \frac{1}{5} \, a^{4} d x^{5} + \frac{1}{4} \, a^{4} 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)^4,x, algorithm="maxima")

[Out]

1/23*b^4*f*x^23 + 1/22*b^4*e*x^22 + 1/21*b^4*d*x^21 + 1/20*b^4*c*x^20 + 4/19*a*b^3*f*x^19 + 2/9*a*b^3*e*x^18 +
4/17*a*b^3*d*x^17 + 1/4*a*b^3*c*x^16 + 2/5*a^2*b^2*f*x^15 + 3/7*a^2*b^2*e*x^14 + 6/13*a^2*b^2*d*x^13 + 1/2*a^
2*b^2*c*x^12 + 4/11*a^3*b*f*x^11 + 2/5*a^3*b*e*x^10 + 4/9*a^3*b*d*x^9 + 1/2*a^3*b*c*x^8 + 1/7*a^4*f*x^7 + 1/6*
a^4*e*x^6 + 1/5*a^4*d*x^5 + 1/4*a^4*c*x^4

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Fricas [A]  time = 1.49758, size = 504, normalized size = 2.55 \begin{align*} \frac{1}{23} x^{23} f b^{4} + \frac{1}{22} x^{22} e b^{4} + \frac{1}{21} x^{21} d b^{4} + \frac{1}{20} x^{20} c b^{4} + \frac{4}{19} x^{19} f b^{3} a + \frac{2}{9} x^{18} e b^{3} a + \frac{4}{17} x^{17} d b^{3} a + \frac{1}{4} x^{16} c b^{3} a + \frac{2}{5} x^{15} f b^{2} a^{2} + \frac{3}{7} x^{14} e b^{2} a^{2} + \frac{6}{13} x^{13} d b^{2} a^{2} + \frac{1}{2} x^{12} c b^{2} a^{2} + \frac{4}{11} x^{11} f b a^{3} + \frac{2}{5} x^{10} e b a^{3} + \frac{4}{9} x^{9} d b a^{3} + \frac{1}{2} x^{8} c b a^{3} + \frac{1}{7} x^{7} f a^{4} + \frac{1}{6} x^{6} e a^{4} + \frac{1}{5} x^{5} d a^{4} + \frac{1}{4} x^{4} c a^{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)^4,x, algorithm="fricas")

[Out]

1/23*x^23*f*b^4 + 1/22*x^22*e*b^4 + 1/21*x^21*d*b^4 + 1/20*x^20*c*b^4 + 4/19*x^19*f*b^3*a + 2/9*x^18*e*b^3*a +
4/17*x^17*d*b^3*a + 1/4*x^16*c*b^3*a + 2/5*x^15*f*b^2*a^2 + 3/7*x^14*e*b^2*a^2 + 6/13*x^13*d*b^2*a^2 + 1/2*x^
12*c*b^2*a^2 + 4/11*x^11*f*b*a^3 + 2/5*x^10*e*b*a^3 + 4/9*x^9*d*b*a^3 + 1/2*x^8*c*b*a^3 + 1/7*x^7*f*a^4 + 1/6*
x^6*e*a^4 + 1/5*x^5*d*a^4 + 1/4*x^4*c*a^4

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Sympy [A]  time = 0.094753, size = 245, normalized size = 1.24 \begin{align*} \frac{a^{4} c x^{4}}{4} + \frac{a^{4} d x^{5}}{5} + \frac{a^{4} e x^{6}}{6} + \frac{a^{4} f x^{7}}{7} + \frac{a^{3} b c x^{8}}{2} + \frac{4 a^{3} b d x^{9}}{9} + \frac{2 a^{3} b e x^{10}}{5} + \frac{4 a^{3} b f x^{11}}{11} + \frac{a^{2} b^{2} c x^{12}}{2} + \frac{6 a^{2} b^{2} d x^{13}}{13} + \frac{3 a^{2} b^{2} e x^{14}}{7} + \frac{2 a^{2} b^{2} f x^{15}}{5} + \frac{a b^{3} c x^{16}}{4} + \frac{4 a b^{3} d x^{17}}{17} + \frac{2 a b^{3} e x^{18}}{9} + \frac{4 a b^{3} f x^{19}}{19} + \frac{b^{4} c x^{20}}{20} + \frac{b^{4} d x^{21}}{21} + \frac{b^{4} e x^{22}}{22} + \frac{b^{4} f x^{23}}{23} \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)**4,x)

[Out]

a**4*c*x**4/4 + a**4*d*x**5/5 + a**4*e*x**6/6 + a**4*f*x**7/7 + a**3*b*c*x**8/2 + 4*a**3*b*d*x**9/9 + 2*a**3*b
*e*x**10/5 + 4*a**3*b*f*x**11/11 + a**2*b**2*c*x**12/2 + 6*a**2*b**2*d*x**13/13 + 3*a**2*b**2*e*x**14/7 + 2*a*
*2*b**2*f*x**15/5 + a*b**3*c*x**16/4 + 4*a*b**3*d*x**17/17 + 2*a*b**3*e*x**18/9 + 4*a*b**3*f*x**19/19 + b**4*c
*x**20/20 + b**4*d*x**21/21 + b**4*e*x**22/22 + b**4*f*x**23/23

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Giac [A]  time = 1.08794, size = 278, normalized size = 1.4 \begin{align*} \frac{1}{23} \, b^{4} f x^{23} + \frac{1}{22} \, b^{4} x^{22} e + \frac{1}{21} \, b^{4} d x^{21} + \frac{1}{20} \, b^{4} c x^{20} + \frac{4}{19} \, a b^{3} f x^{19} + \frac{2}{9} \, a b^{3} x^{18} e + \frac{4}{17} \, a b^{3} d x^{17} + \frac{1}{4} \, a b^{3} c x^{16} + \frac{2}{5} \, a^{2} b^{2} f x^{15} + \frac{3}{7} \, a^{2} b^{2} x^{14} e + \frac{6}{13} \, a^{2} b^{2} d x^{13} + \frac{1}{2} \, a^{2} b^{2} c x^{12} + \frac{4}{11} \, a^{3} b f x^{11} + \frac{2}{5} \, a^{3} b x^{10} e + \frac{4}{9} \, a^{3} b d x^{9} + \frac{1}{2} \, a^{3} b c x^{8} + \frac{1}{7} \, a^{4} f x^{7} + \frac{1}{6} \, a^{4} x^{6} e + \frac{1}{5} \, a^{4} d x^{5} + \frac{1}{4} \, a^{4} 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)^4,x, algorithm="giac")

[Out]

1/23*b^4*f*x^23 + 1/22*b^4*x^22*e + 1/21*b^4*d*x^21 + 1/20*b^4*c*x^20 + 4/19*a*b^3*f*x^19 + 2/9*a*b^3*x^18*e +
4/17*a*b^3*d*x^17 + 1/4*a*b^3*c*x^16 + 2/5*a^2*b^2*f*x^15 + 3/7*a^2*b^2*x^14*e + 6/13*a^2*b^2*d*x^13 + 1/2*a^
2*b^2*c*x^12 + 4/11*a^3*b*f*x^11 + 2/5*a^3*b*x^10*e + 4/9*a^3*b*d*x^9 + 1/2*a^3*b*c*x^8 + 1/7*a^4*f*x^7 + 1/6*
a^4*x^6*e + 1/5*a^4*d*x^5 + 1/4*a^4*c*x^4