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

Optimal. Leaf size=193 \[ \frac{2}{3} a^2 b^2 c x^9+\frac{3}{5} a^2 b^2 d x^{10}+\frac{6}{11} a^2 b^2 e x^{11}+\frac{4}{5} a^3 b c x^5+\frac{2}{3} a^3 b d x^6+\frac{4}{7} a^3 b e x^7+a^4 c x+\frac{1}{2} a^4 d x^2+\frac{1}{3} a^4 e x^3+\frac{4}{13} a b^3 c x^{13}+\frac{2}{7} a b^3 d x^{14}+\frac{4}{15} a b^3 e x^{15}+\frac{f \left (a+b x^4\right )^5}{20 b}+\frac{1}{17} b^4 c x^{17}+\frac{1}{18} b^4 d x^{18}+\frac{1}{19} b^4 e x^{19} \]

[Out]

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

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Rubi [A]  time = 0.15563, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {1582, 1657} \[ \frac{2}{3} a^2 b^2 c x^9+\frac{3}{5} a^2 b^2 d x^{10}+\frac{6}{11} a^2 b^2 e x^{11}+\frac{4}{5} a^3 b c x^5+\frac{2}{3} a^3 b d x^6+\frac{4}{7} a^3 b e x^7+a^4 c x+\frac{1}{2} a^4 d x^2+\frac{1}{3} a^4 e x^3+\frac{4}{13} a b^3 c x^{13}+\frac{2}{7} a b^3 d x^{14}+\frac{4}{15} a b^3 e x^{15}+\frac{f \left (a+b x^4\right )^5}{20 b}+\frac{1}{17} b^4 c x^{17}+\frac{1}{18} b^4 d x^{18}+\frac{1}{19} b^4 e x^{19} \]

Antiderivative was successfully verified.

[In]

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

[Out]

a^4*c*x + (a^4*d*x^2)/2 + (a^4*e*x^3)/3 + (4*a^3*b*c*x^5)/5 + (2*a^3*b*d*x^6)/3 + (4*a^3*b*e*x^7)/7 + (2*a^2*b
^2*c*x^9)/3 + (3*a^2*b^2*d*x^10)/5 + (6*a^2*b^2*e*x^11)/11 + (4*a*b^3*c*x^13)/13 + (2*a*b^3*d*x^14)/7 + (4*a*b
^3*e*x^15)/15 + (b^4*c*x^17)/17 + (b^4*d*x^18)/18 + (b^4*e*x^19)/19 + (f*(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 1657

Int[(Pq_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x + c*x^2)^p, x
], x] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

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

Mathematica [A]  time = 0.0055653, size = 236, normalized size = 1.22 \[ \frac{2}{3} a^2 b^2 c x^9+\frac{3}{5} a^2 b^2 d x^{10}+\frac{6}{11} a^2 b^2 e x^{11}+\frac{1}{2} a^2 b^2 f x^{12}+\frac{4}{5} a^3 b c x^5+\frac{2}{3} a^3 b d x^6+\frac{4}{7} a^3 b e x^7+\frac{1}{2} a^3 b f x^8+a^4 c x+\frac{1}{2} a^4 d x^2+\frac{1}{3} a^4 e x^3+\frac{1}{4} a^4 f x^4+\frac{4}{13} a b^3 c x^{13}+\frac{2}{7} a b^3 d x^{14}+\frac{4}{15} a b^3 e x^{15}+\frac{1}{4} a b^3 f x^{16}+\frac{1}{17} b^4 c x^{17}+\frac{1}{18} b^4 d x^{18}+\frac{1}{19} b^4 e x^{19}+\frac{1}{20} b^4 f x^{20} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.001, size = 199, normalized size = 1. \begin{align*}{\frac{f{b}^{4}{x}^{20}}{20}}+{\frac{{b}^{4}e{x}^{19}}{19}}+{\frac{{b}^{4}d{x}^{18}}{18}}+{\frac{{b}^{4}c{x}^{17}}{17}}+{\frac{fa{b}^{3}{x}^{16}}{4}}+{\frac{4\,a{b}^{3}e{x}^{15}}{15}}+{\frac{2\,a{b}^{3}d{x}^{14}}{7}}+{\frac{4\,a{b}^{3}c{x}^{13}}{13}}+{\frac{f{b}^{2}{a}^{2}{x}^{12}}{2}}+{\frac{6\,{a}^{2}{b}^{2}e{x}^{11}}{11}}+{\frac{3\,{a}^{2}{b}^{2}d{x}^{10}}{5}}+{\frac{2\,{a}^{2}{b}^{2}c{x}^{9}}{3}}+{\frac{bf{a}^{3}{x}^{8}}{2}}+{\frac{4\,{a}^{3}be{x}^{7}}{7}}+{\frac{2\,{a}^{3}bd{x}^{6}}{3}}+{\frac{4\,{a}^{3}bc{x}^{5}}{5}}+{\frac{{a}^{4}f{x}^{4}}{4}}+{\frac{{a}^{4}e{x}^{3}}{3}}+{\frac{{a}^{4}d{x}^{2}}{2}}+{a}^{4}cx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.931103, size = 267, normalized size = 1.38 \begin{align*} \frac{1}{20} \, b^{4} f x^{20} + \frac{1}{19} \, b^{4} e x^{19} + \frac{1}{18} \, b^{4} d x^{18} + \frac{1}{17} \, b^{4} c x^{17} + \frac{1}{4} \, a b^{3} f x^{16} + \frac{4}{15} \, a b^{3} e x^{15} + \frac{2}{7} \, a b^{3} d x^{14} + \frac{4}{13} \, a b^{3} c x^{13} + \frac{1}{2} \, a^{2} b^{2} f x^{12} + \frac{6}{11} \, a^{2} b^{2} e x^{11} + \frac{3}{5} \, a^{2} b^{2} d x^{10} + \frac{2}{3} \, a^{2} b^{2} c x^{9} + \frac{1}{2} \, a^{3} b f x^{8} + \frac{4}{7} \, a^{3} b e x^{7} + \frac{2}{3} \, a^{3} b d x^{6} + \frac{4}{5} \, a^{3} b c x^{5} + \frac{1}{4} \, a^{4} f x^{4} + \frac{1}{3} \, a^{4} e x^{3} + \frac{1}{2} \, a^{4} d x^{2} + a^{4} c x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.48056, size = 490, normalized size = 2.54 \begin{align*} \frac{1}{20} x^{20} f b^{4} + \frac{1}{19} x^{19} e b^{4} + \frac{1}{18} x^{18} d b^{4} + \frac{1}{17} x^{17} c b^{4} + \frac{1}{4} x^{16} f b^{3} a + \frac{4}{15} x^{15} e b^{3} a + \frac{2}{7} x^{14} d b^{3} a + \frac{4}{13} x^{13} c b^{3} a + \frac{1}{2} x^{12} f b^{2} a^{2} + \frac{6}{11} x^{11} e b^{2} a^{2} + \frac{3}{5} x^{10} d b^{2} a^{2} + \frac{2}{3} x^{9} c b^{2} a^{2} + \frac{1}{2} x^{8} f b a^{3} + \frac{4}{7} x^{7} e b a^{3} + \frac{2}{3} x^{6} d b a^{3} + \frac{4}{5} x^{5} c b a^{3} + \frac{1}{4} x^{4} f a^{4} + \frac{1}{3} x^{3} e a^{4} + \frac{1}{2} x^{2} d a^{4} + x c a^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.089925, size = 241, normalized size = 1.25 \begin{align*} a^{4} c x + \frac{a^{4} d x^{2}}{2} + \frac{a^{4} e x^{3}}{3} + \frac{a^{4} f x^{4}}{4} + \frac{4 a^{3} b c x^{5}}{5} + \frac{2 a^{3} b d x^{6}}{3} + \frac{4 a^{3} b e x^{7}}{7} + \frac{a^{3} b f x^{8}}{2} + \frac{2 a^{2} b^{2} c x^{9}}{3} + \frac{3 a^{2} b^{2} d x^{10}}{5} + \frac{6 a^{2} b^{2} e x^{11}}{11} + \frac{a^{2} b^{2} f x^{12}}{2} + \frac{4 a b^{3} c x^{13}}{13} + \frac{2 a b^{3} d x^{14}}{7} + \frac{4 a b^{3} e x^{15}}{15} + \frac{a b^{3} f x^{16}}{4} + \frac{b^{4} c x^{17}}{17} + \frac{b^{4} d x^{18}}{18} + \frac{b^{4} e x^{19}}{19} + \frac{b^{4} f x^{20}}{20} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.10567, size = 274, normalized size = 1.42 \begin{align*} \frac{1}{20} \, b^{4} f x^{20} + \frac{1}{19} \, b^{4} x^{19} e + \frac{1}{18} \, b^{4} d x^{18} + \frac{1}{17} \, b^{4} c x^{17} + \frac{1}{4} \, a b^{3} f x^{16} + \frac{4}{15} \, a b^{3} x^{15} e + \frac{2}{7} \, a b^{3} d x^{14} + \frac{4}{13} \, a b^{3} c x^{13} + \frac{1}{2} \, a^{2} b^{2} f x^{12} + \frac{6}{11} \, a^{2} b^{2} x^{11} e + \frac{3}{5} \, a^{2} b^{2} d x^{10} + \frac{2}{3} \, a^{2} b^{2} c x^{9} + \frac{1}{2} \, a^{3} b f x^{8} + \frac{4}{7} \, a^{3} b x^{7} e + \frac{2}{3} \, a^{3} b d x^{6} + \frac{4}{5} \, a^{3} b c x^{5} + \frac{1}{4} \, a^{4} f x^{4} + \frac{1}{3} \, a^{4} x^{3} e + \frac{1}{2} \, a^{4} d x^{2} + a^{4} c x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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