3.395 \(\int x^2 (a+b x^3)^3 (c+d x+e x^2+f x^3+g x^4+h x^5) \, dx\)

Optimal. Leaf size=212 \[ \frac{1}{7} a^2 x^7 (a g+3 b d)+\frac{1}{8} a^2 x^8 (a h+3 b e)+\frac{1}{3} a^2 b f x^9+\frac{1}{4} a^3 d x^4+\frac{1}{5} a^3 e x^5+\frac{1}{6} a^3 f x^6+\frac{1}{13} b^2 x^{13} (3 a g+b d)+\frac{1}{14} b^2 x^{14} (3 a h+b e)+\frac{1}{4} a b^2 f x^{12}+\frac{c \left (a+b x^3\right )^4}{12 b}+\frac{3}{10} a b x^{10} (a g+b d)+\frac{3}{11} a b x^{11} (a h+b e)+\frac{1}{15} b^3 f x^{15}+\frac{1}{16} b^3 g x^{16}+\frac{1}{17} b^3 h x^{17} \]

[Out]

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

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Rubi [A]  time = 0.179245, antiderivative size = 212, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {1582, 1850} \[ \frac{1}{7} a^2 x^7 (a g+3 b d)+\frac{1}{8} a^2 x^8 (a h+3 b e)+\frac{1}{3} a^2 b f x^9+\frac{1}{4} a^3 d x^4+\frac{1}{5} a^3 e x^5+\frac{1}{6} a^3 f x^6+\frac{1}{13} b^2 x^{13} (3 a g+b d)+\frac{1}{14} b^2 x^{14} (3 a h+b e)+\frac{1}{4} a b^2 f x^{12}+\frac{c \left (a+b x^3\right )^4}{12 b}+\frac{3}{10} a b x^{10} (a g+b d)+\frac{3}{11} a b x^{11} (a h+b e)+\frac{1}{15} b^3 f x^{15}+\frac{1}{16} b^3 g x^{16}+\frac{1}{17} b^3 h x^{17} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x^3)^3*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5),x]

[Out]

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

Mathematica [A]  time = 0.0480482, size = 223, normalized size = 1.05 \[ \frac{1}{6} a^2 x^6 (a f+3 b c)+\frac{1}{7} a^2 x^7 (a g+3 b d)+\frac{1}{8} a^2 x^8 (a h+3 b e)+\frac{1}{3} a^3 c x^3+\frac{1}{4} a^3 d x^4+\frac{1}{5} a^3 e x^5+\frac{1}{12} b^2 x^{12} (3 a f+b c)+\frac{1}{13} b^2 x^{13} (3 a g+b d)+\frac{1}{14} b^2 x^{14} (3 a h+b e)+\frac{1}{3} a b x^9 (a f+b c)+\frac{3}{10} a b x^{10} (a g+b d)+\frac{3}{11} a b x^{11} (a h+b e)+\frac{1}{15} b^3 f x^{15}+\frac{1}{16} b^3 g x^{16}+\frac{1}{17} b^3 h x^{17} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*x^3)^3*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5),x]

[Out]

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

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Maple [A]  time = 0.001, size = 224, normalized size = 1.1 \begin{align*}{\frac{{b}^{3}h{x}^{17}}{17}}+{\frac{{b}^{3}g{x}^{16}}{16}}+{\frac{{b}^{3}f{x}^{15}}{15}}+{\frac{ \left ( 3\,{b}^{2}ah+{b}^{3}e \right ){x}^{14}}{14}}+{\frac{ \left ( 3\,{b}^{2}ag+{b}^{3}d \right ){x}^{13}}{13}}+{\frac{ \left ( 3\,{b}^{2}af+{b}^{3}c \right ){x}^{12}}{12}}+{\frac{ \left ( 3\,b{a}^{2}h+3\,ae{b}^{2} \right ){x}^{11}}{11}}+{\frac{ \left ( 3\,b{a}^{2}g+3\,a{b}^{2}d \right ){x}^{10}}{10}}+{\frac{ \left ( 3\,b{a}^{2}f+3\,ac{b}^{2} \right ){x}^{9}}{9}}+{\frac{ \left ({a}^{3}h+3\,{a}^{2}be \right ){x}^{8}}{8}}+{\frac{ \left ({a}^{3}g+3\,{a}^{2}bd \right ){x}^{7}}{7}}+{\frac{ \left ({a}^{3}f+3\,b{a}^{2}c \right ){x}^{6}}{6}}+{\frac{{a}^{3}e{x}^{5}}{5}}+{\frac{{a}^{3}d{x}^{4}}{4}}+{\frac{{a}^{3}c{x}^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(b*x^3+a)^3*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c),x)

[Out]

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

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Maxima [A]  time = 0.942734, size = 293, normalized size = 1.38 \begin{align*} \frac{1}{17} \, b^{3} h x^{17} + \frac{1}{16} \, b^{3} g x^{16} + \frac{1}{15} \, b^{3} f x^{15} + \frac{1}{14} \,{\left (b^{3} e + 3 \, a b^{2} h\right )} x^{14} + \frac{1}{13} \,{\left (b^{3} d + 3 \, a b^{2} g\right )} x^{13} + \frac{1}{12} \,{\left (b^{3} c + 3 \, a b^{2} f\right )} x^{12} + \frac{3}{11} \,{\left (a b^{2} e + a^{2} b h\right )} x^{11} + \frac{3}{10} \,{\left (a b^{2} d + a^{2} b g\right )} x^{10} + \frac{1}{3} \,{\left (a b^{2} c + a^{2} b f\right )} x^{9} + \frac{1}{5} \, a^{3} e x^{5} + \frac{1}{8} \,{\left (3 \, a^{2} b e + a^{3} h\right )} x^{8} + \frac{1}{4} \, a^{3} d x^{4} + \frac{1}{7} \,{\left (3 \, a^{2} b d + a^{3} g\right )} x^{7} + \frac{1}{3} \, a^{3} c x^{3} + \frac{1}{6} \,{\left (3 \, a^{2} b c + a^{3} f\right )} x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x^3+a)^3*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.828252, size = 583, normalized size = 2.75 \begin{align*} \frac{1}{17} x^{17} h b^{3} + \frac{1}{16} x^{16} g b^{3} + \frac{1}{15} x^{15} f b^{3} + \frac{1}{14} x^{14} e b^{3} + \frac{3}{14} x^{14} h b^{2} a + \frac{1}{13} x^{13} d b^{3} + \frac{3}{13} x^{13} g b^{2} a + \frac{1}{12} x^{12} c b^{3} + \frac{1}{4} x^{12} f b^{2} a + \frac{3}{11} x^{11} e b^{2} a + \frac{3}{11} x^{11} h b a^{2} + \frac{3}{10} x^{10} d b^{2} a + \frac{3}{10} x^{10} g b a^{2} + \frac{1}{3} x^{9} c b^{2} a + \frac{1}{3} x^{9} f b a^{2} + \frac{3}{8} x^{8} e b a^{2} + \frac{1}{8} x^{8} h a^{3} + \frac{3}{7} x^{7} d b a^{2} + \frac{1}{7} x^{7} g a^{3} + \frac{1}{2} x^{6} c b a^{2} + \frac{1}{6} x^{6} f a^{3} + \frac{1}{5} x^{5} e a^{3} + \frac{1}{4} x^{4} d a^{3} + \frac{1}{3} x^{3} c a^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x^3+a)^3*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.098678, size = 246, normalized size = 1.16 \begin{align*} \frac{a^{3} c x^{3}}{3} + \frac{a^{3} d x^{4}}{4} + \frac{a^{3} e x^{5}}{5} + \frac{b^{3} f x^{15}}{15} + \frac{b^{3} g x^{16}}{16} + \frac{b^{3} h x^{17}}{17} + x^{14} \left (\frac{3 a b^{2} h}{14} + \frac{b^{3} e}{14}\right ) + x^{13} \left (\frac{3 a b^{2} g}{13} + \frac{b^{3} d}{13}\right ) + x^{12} \left (\frac{a b^{2} f}{4} + \frac{b^{3} c}{12}\right ) + x^{11} \left (\frac{3 a^{2} b h}{11} + \frac{3 a b^{2} e}{11}\right ) + x^{10} \left (\frac{3 a^{2} b g}{10} + \frac{3 a b^{2} d}{10}\right ) + x^{9} \left (\frac{a^{2} b f}{3} + \frac{a b^{2} c}{3}\right ) + x^{8} \left (\frac{a^{3} h}{8} + \frac{3 a^{2} b e}{8}\right ) + x^{7} \left (\frac{a^{3} g}{7} + \frac{3 a^{2} b d}{7}\right ) + x^{6} \left (\frac{a^{3} f}{6} + \frac{a^{2} b c}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(b*x**3+a)**3*(h*x**5+g*x**4+f*x**3+e*x**2+d*x+c),x)

[Out]

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

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Giac [A]  time = 1.07915, size = 315, normalized size = 1.49 \begin{align*} \frac{1}{17} \, b^{3} h x^{17} + \frac{1}{16} \, b^{3} g x^{16} + \frac{1}{15} \, b^{3} f x^{15} + \frac{3}{14} \, a b^{2} h x^{14} + \frac{1}{14} \, b^{3} x^{14} e + \frac{1}{13} \, b^{3} d x^{13} + \frac{3}{13} \, a b^{2} g x^{13} + \frac{1}{12} \, b^{3} c x^{12} + \frac{1}{4} \, a b^{2} f x^{12} + \frac{3}{11} \, a^{2} b h x^{11} + \frac{3}{11} \, a b^{2} x^{11} e + \frac{3}{10} \, a b^{2} d x^{10} + \frac{3}{10} \, a^{2} b g x^{10} + \frac{1}{3} \, a b^{2} c x^{9} + \frac{1}{3} \, a^{2} b f x^{9} + \frac{1}{8} \, a^{3} h x^{8} + \frac{3}{8} \, a^{2} b x^{8} e + \frac{3}{7} \, a^{2} b d x^{7} + \frac{1}{7} \, a^{3} g x^{7} + \frac{1}{2} \, a^{2} b c x^{6} + \frac{1}{6} \, a^{3} f x^{6} + \frac{1}{5} \, a^{3} x^{5} e + \frac{1}{4} \, a^{3} d x^{4} + \frac{1}{3} \, a^{3} c x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x^3+a)^3*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c),x, algorithm="giac")

[Out]

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