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

Optimal. Leaf size=130 \[ \frac{6}{7} a^2 b^2 c x^7+\frac{3}{4} a^2 b^2 d x^8+a^3 b c x^4+\frac{4}{5} a^3 b d x^5+a^4 c x+\frac{1}{2} a^4 d x^2+\frac{2}{5} a b^3 c x^{10}+\frac{4}{11} a b^3 d x^{11}+\frac{e \left (a+b x^3\right )^5}{15 b}+\frac{1}{13} b^4 c x^{13}+\frac{1}{14} b^4 d x^{14} \]

[Out]

a^4*c*x + (a^4*d*x^2)/2 + a^3*b*c*x^4 + (4*a^3*b*d*x^5)/5 + (6*a^2*b^2*c*x^7)/7 + (3*a^2*b^2*d*x^8)/4 + (2*a*b
^3*c*x^10)/5 + (4*a*b^3*d*x^11)/11 + (b^4*c*x^13)/13 + (b^4*d*x^14)/14 + (e*(a + b*x^3)^5)/(15*b)

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Rubi [A]  time = 0.144909, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {1582, 1850} \[ \frac{6}{7} a^2 b^2 c x^7+\frac{3}{4} a^2 b^2 d x^8+a^3 b c x^4+\frac{4}{5} a^3 b d x^5+a^4 c x+\frac{1}{2} a^4 d x^2+\frac{2}{5} a b^3 c x^{10}+\frac{4}{11} a b^3 d x^{11}+\frac{e \left (a+b x^3\right )^5}{15 b}+\frac{1}{13} b^4 c x^{13}+\frac{1}{14} b^4 d x^{14} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0038972, size = 173, normalized size = 1.33 \[ \frac{6}{7} a^2 b^2 c x^7+\frac{3}{4} a^2 b^2 d x^8+\frac{2}{3} a^2 b^2 e x^9+a^3 b c x^4+\frac{4}{5} a^3 b d x^5+\frac{2}{3} a^3 b e x^6+a^4 c x+\frac{1}{2} a^4 d x^2+\frac{1}{3} a^4 e x^3+\frac{2}{5} a b^3 c x^{10}+\frac{4}{11} a b^3 d x^{11}+\frac{1}{3} a b^3 e x^{12}+\frac{1}{13} b^4 c x^{13}+\frac{1}{14} b^4 d x^{14}+\frac{1}{15} b^4 e x^{15} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.001, size = 148, normalized size = 1.1 \begin{align*}{\frac{{b}^{4}e{x}^{15}}{15}}+{\frac{{b}^{4}d{x}^{14}}{14}}+{\frac{{b}^{4}c{x}^{13}}{13}}+{\frac{a{b}^{3}e{x}^{12}}{3}}+{\frac{4\,a{b}^{3}d{x}^{11}}{11}}+{\frac{2\,a{b}^{3}c{x}^{10}}{5}}+{\frac{2\,{a}^{2}{b}^{2}e{x}^{9}}{3}}+{\frac{3\,{a}^{2}{b}^{2}d{x}^{8}}{4}}+{\frac{6\,{a}^{2}{b}^{2}c{x}^{7}}{7}}+{\frac{2\,{a}^{3}be{x}^{6}}{3}}+{\frac{4\,{a}^{3}bd{x}^{5}}{5}}+{a}^{3}bc{x}^{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((e*x^2+d*x+c)*(b*x^3+a)^4,x)

[Out]

1/15*b^4*e*x^15+1/14*b^4*d*x^14+1/13*b^4*c*x^13+1/3*a*b^3*e*x^12+4/11*a*b^3*d*x^11+2/5*a*b^3*c*x^10+2/3*a^2*b^
2*e*x^9+3/4*a^2*b^2*d*x^8+6/7*a^2*b^2*c*x^7+2/3*a^3*b*e*x^6+4/5*a^3*b*d*x^5+a^3*b*c*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 = 0.935401, size = 198, normalized size = 1.52 \begin{align*} \frac{1}{15} \, b^{4} e x^{15} + \frac{1}{14} \, b^{4} d x^{14} + \frac{1}{13} \, b^{4} c x^{13} + \frac{1}{3} \, a b^{3} e x^{12} + \frac{4}{11} \, a b^{3} d x^{11} + \frac{2}{5} \, a b^{3} c x^{10} + \frac{2}{3} \, a^{2} b^{2} e x^{9} + \frac{3}{4} \, a^{2} b^{2} d x^{8} + \frac{6}{7} \, a^{2} b^{2} c x^{7} + \frac{2}{3} \, a^{3} b e x^{6} + \frac{4}{5} \, a^{3} b d x^{5} + a^{3} b c 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((e*x^2+d*x+c)*(b*x^3+a)^4,x, algorithm="maxima")

[Out]

1/15*b^4*e*x^15 + 1/14*b^4*d*x^14 + 1/13*b^4*c*x^13 + 1/3*a*b^3*e*x^12 + 4/11*a*b^3*d*x^11 + 2/5*a*b^3*c*x^10
+ 2/3*a^2*b^2*e*x^9 + 3/4*a^2*b^2*d*x^8 + 6/7*a^2*b^2*c*x^7 + 2/3*a^3*b*e*x^6 + 4/5*a^3*b*d*x^5 + a^3*b*c*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 = 1.28289, size = 355, normalized size = 2.73 \begin{align*} \frac{1}{15} x^{15} e b^{4} + \frac{1}{14} x^{14} d b^{4} + \frac{1}{13} x^{13} c b^{4} + \frac{1}{3} x^{12} e b^{3} a + \frac{4}{11} x^{11} d b^{3} a + \frac{2}{5} x^{10} c b^{3} a + \frac{2}{3} x^{9} e b^{2} a^{2} + \frac{3}{4} x^{8} d b^{2} a^{2} + \frac{6}{7} x^{7} c b^{2} a^{2} + \frac{2}{3} x^{6} e b a^{3} + \frac{4}{5} x^{5} d b a^{3} + x^{4} c b a^{3} + \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((e*x^2+d*x+c)*(b*x^3+a)^4,x, algorithm="fricas")

[Out]

1/15*x^15*e*b^4 + 1/14*x^14*d*b^4 + 1/13*x^13*c*b^4 + 1/3*x^12*e*b^3*a + 4/11*x^11*d*b^3*a + 2/5*x^10*c*b^3*a
+ 2/3*x^9*e*b^2*a^2 + 3/4*x^8*d*b^2*a^2 + 6/7*x^7*c*b^2*a^2 + 2/3*x^6*e*b*a^3 + 4/5*x^5*d*b*a^3 + x^4*c*b*a^3
+ 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 = 0.083989, size = 178, normalized size = 1.37 \begin{align*} a^{4} c x + \frac{a^{4} d x^{2}}{2} + \frac{a^{4} e x^{3}}{3} + a^{3} b c x^{4} + \frac{4 a^{3} b d x^{5}}{5} + \frac{2 a^{3} b e x^{6}}{3} + \frac{6 a^{2} b^{2} c x^{7}}{7} + \frac{3 a^{2} b^{2} d x^{8}}{4} + \frac{2 a^{2} b^{2} e x^{9}}{3} + \frac{2 a b^{3} c x^{10}}{5} + \frac{4 a b^{3} d x^{11}}{11} + \frac{a b^{3} e x^{12}}{3} + \frac{b^{4} c x^{13}}{13} + \frac{b^{4} d x^{14}}{14} + \frac{b^{4} e x^{15}}{15} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.06435, size = 205, normalized size = 1.58 \begin{align*} \frac{1}{15} \, b^{4} x^{15} e + \frac{1}{14} \, b^{4} d x^{14} + \frac{1}{13} \, b^{4} c x^{13} + \frac{1}{3} \, a b^{3} x^{12} e + \frac{4}{11} \, a b^{3} d x^{11} + \frac{2}{5} \, a b^{3} c x^{10} + \frac{2}{3} \, a^{2} b^{2} x^{9} e + \frac{3}{4} \, a^{2} b^{2} d x^{8} + \frac{6}{7} \, a^{2} b^{2} c x^{7} + \frac{2}{3} \, a^{3} b x^{6} e + \frac{4}{5} \, a^{3} b d x^{5} + a^{3} b c 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((e*x^2+d*x+c)*(b*x^3+a)^4,x, algorithm="giac")

[Out]

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