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

Optimal. Leaf size=114 \[ \frac{1}{5} a^2 d x^5+\frac{1}{6} a^2 e x^6+\frac{1}{7} a^2 f x^7+\frac{c \left (a+b x^4\right )^3}{12 b}+\frac{2}{9} a b d x^9+\frac{1}{5} a b e x^{10}+\frac{2}{11} a b f x^{11}+\frac{1}{13} b^2 d x^{13}+\frac{1}{14} b^2 e x^{14}+\frac{1}{15} b^2 f x^{15} \]

[Out]

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

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Rubi [A]  time = 0.0837339, antiderivative size = 114, 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} \[ \frac{1}{5} a^2 d x^5+\frac{1}{6} a^2 e x^6+\frac{1}{7} a^2 f x^7+\frac{c \left (a+b x^4\right )^3}{12 b}+\frac{2}{9} a b d x^9+\frac{1}{5} a b e x^{10}+\frac{2}{11} a b f x^{11}+\frac{1}{13} b^2 d x^{13}+\frac{1}{14} b^2 e x^{14}+\frac{1}{15} b^2 f x^{15} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0041538, size = 129, normalized size = 1.13 \[ \frac{1}{4} a^2 c x^4+\frac{1}{5} a^2 d x^5+\frac{1}{6} a^2 e x^6+\frac{1}{7} a^2 f x^7+\frac{1}{4} a b c x^8+\frac{2}{9} a b d x^9+\frac{1}{5} a b e x^{10}+\frac{2}{11} a b f x^{11}+\frac{1}{12} b^2 c x^{12}+\frac{1}{13} b^2 d x^{13}+\frac{1}{14} b^2 e x^{14}+\frac{1}{15} b^2 f x^{15} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.001, size = 106, normalized size = 0.9 \begin{align*}{\frac{{b}^{2}f{x}^{15}}{15}}+{\frac{{b}^{2}e{x}^{14}}{14}}+{\frac{{b}^{2}d{x}^{13}}{13}}+{\frac{{b}^{2}c{x}^{12}}{12}}+{\frac{2\,abf{x}^{11}}{11}}+{\frac{abe{x}^{10}}{5}}+{\frac{2\,abd{x}^{9}}{9}}+{\frac{abc{x}^{8}}{4}}+{\frac{{a}^{2}f{x}^{7}}{7}}+{\frac{{a}^{2}e{x}^{6}}{6}}+{\frac{{a}^{2}d{x}^{5}}{5}}+{\frac{{a}^{2}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)^2,x)

[Out]

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

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Maxima [A]  time = 0.91656, size = 142, normalized size = 1.25 \begin{align*} \frac{1}{15} \, b^{2} f x^{15} + \frac{1}{14} \, b^{2} e x^{14} + \frac{1}{13} \, b^{2} d x^{13} + \frac{1}{12} \, b^{2} c x^{12} + \frac{2}{11} \, a b f x^{11} + \frac{1}{5} \, a b e x^{10} + \frac{2}{9} \, a b d x^{9} + \frac{1}{4} \, a b c x^{8} + \frac{1}{7} \, a^{2} f x^{7} + \frac{1}{6} \, a^{2} e x^{6} + \frac{1}{5} \, a^{2} d x^{5} + \frac{1}{4} \, a^{2} 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)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.43756, size = 273, normalized size = 2.39 \begin{align*} \frac{1}{15} x^{15} f b^{2} + \frac{1}{14} x^{14} e b^{2} + \frac{1}{13} x^{13} d b^{2} + \frac{1}{12} x^{12} c b^{2} + \frac{2}{11} x^{11} f b a + \frac{1}{5} x^{10} e b a + \frac{2}{9} x^{9} d b a + \frac{1}{4} x^{8} c b a + \frac{1}{7} x^{7} f a^{2} + \frac{1}{6} x^{6} e a^{2} + \frac{1}{5} x^{5} d a^{2} + \frac{1}{4} x^{4} c a^{2} \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)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.079195, size = 124, normalized size = 1.09 \begin{align*} \frac{a^{2} c x^{4}}{4} + \frac{a^{2} d x^{5}}{5} + \frac{a^{2} e x^{6}}{6} + \frac{a^{2} f x^{7}}{7} + \frac{a b c x^{8}}{4} + \frac{2 a b d x^{9}}{9} + \frac{a b e x^{10}}{5} + \frac{2 a b f x^{11}}{11} + \frac{b^{2} c x^{12}}{12} + \frac{b^{2} d x^{13}}{13} + \frac{b^{2} e x^{14}}{14} + \frac{b^{2} f x^{15}}{15} \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)**2,x)

[Out]

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

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Giac [A]  time = 1.08274, size = 146, normalized size = 1.28 \begin{align*} \frac{1}{15} \, b^{2} f x^{15} + \frac{1}{14} \, b^{2} x^{14} e + \frac{1}{13} \, b^{2} d x^{13} + \frac{1}{12} \, b^{2} c x^{12} + \frac{2}{11} \, a b f x^{11} + \frac{1}{5} \, a b x^{10} e + \frac{2}{9} \, a b d x^{9} + \frac{1}{4} \, a b c x^{8} + \frac{1}{7} \, a^{2} f x^{7} + \frac{1}{6} \, a^{2} x^{6} e + \frac{1}{5} \, a^{2} d x^{5} + \frac{1}{4} \, a^{2} 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)^2,x, algorithm="giac")

[Out]

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