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

Optimal. Leaf size=207 \[ \frac{1}{4} a^2 x^4 (a f+3 b c)+\frac{1}{5} a^2 x^5 (a g+3 b d)+\frac{1}{3} a^2 b h x^9+a^3 c x+\frac{1}{2} a^3 d x^2+\frac{1}{6} a^3 h x^6+\frac{1}{10} b^2 x^{10} (3 a f+b c)+\frac{1}{11} b^2 x^{11} (3 a g+b d)+\frac{1}{4} a b^2 h x^{12}+\frac{3}{7} a b x^7 (a f+b c)+\frac{3}{8} a b x^8 (a g+b d)+\frac{e \left (a+b x^3\right )^4}{12 b}+\frac{1}{13} b^3 f x^{13}+\frac{1}{14} b^3 g x^{14}+\frac{1}{15} b^3 h x^{15} \]

[Out]

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

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Rubi [A]  time = 0.177319, antiderivative size = 207, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057, Rules used = {1582, 1850} \[ \frac{1}{4} a^2 x^4 (a f+3 b c)+\frac{1}{5} a^2 x^5 (a g+3 b d)+\frac{1}{3} a^2 b h x^9+a^3 c x+\frac{1}{2} a^3 d x^2+\frac{1}{6} a^3 h x^6+\frac{1}{10} b^2 x^{10} (3 a f+b c)+\frac{1}{11} b^2 x^{11} (3 a g+b d)+\frac{1}{4} a b^2 h x^{12}+\frac{3}{7} a b x^7 (a f+b c)+\frac{3}{8} a b x^8 (a g+b d)+\frac{e \left (a+b x^3\right )^4}{12 b}+\frac{1}{13} b^3 f x^{13}+\frac{1}{14} b^3 g x^{14}+\frac{1}{15} b^3 h x^{15} \]

Antiderivative was successfully verified.

[In]

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

Mathematica [A]  time = 0.101428, size = 170, normalized size = 0.82 \[ \frac{x \left (143 a^2 b x^3 (630 c+x (504 d+5 x (84 e+x (72 f+7 x (9 g+8 h x)))))+2002 a^3 \left (60 c+x \left (30 d+x \left (20 e+15 f x+12 g x^2+10 h x^3\right )\right )\right )+13 a b^2 x^6 \left (3960 c+7 x \left (495 d+440 e x+6 x^2 \left (66 f+60 g x+55 h x^2\right )\right )\right )+2 b^3 x^9 \left (6006 c+x \left (5460 d+11 x \left (455 e+420 f x+390 g x^2+364 h x^3\right )\right )\right )\right )}{120120} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(13*a*b^2*x^6*(3960*c + 7*x*(495*d + 440*e*x + 6*x^2*(66*f + 60*g*x + 55*h*x^2))) + 2002*a^3*(60*c + x*(30*
d + x*(20*e + 15*f*x + 12*g*x^2 + 10*h*x^3))) + 2*b^3*x^9*(6006*c + x*(5460*d + 11*x*(455*e + 420*f*x + 390*g*
x^2 + 364*h*x^3))) + 143*a^2*b*x^3*(630*c + x*(504*d + 5*x*(84*e + x*(72*f + 7*x*(9*g + 8*h*x)))))))/120120

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Maple [A]  time = 0.001, size = 221, normalized size = 1.1 \begin{align*}{\frac{{b}^{3}h{x}^{15}}{15}}+{\frac{{b}^{3}g{x}^{14}}{14}}+{\frac{{b}^{3}f{x}^{13}}{13}}+{\frac{ \left ( 3\,{b}^{2}ah+{b}^{3}e \right ){x}^{12}}{12}}+{\frac{ \left ( 3\,{b}^{2}ag+{b}^{3}d \right ){x}^{11}}{11}}+{\frac{ \left ( 3\,{b}^{2}af+{b}^{3}c \right ){x}^{10}}{10}}+{\frac{ \left ( 3\,b{a}^{2}h+3\,ae{b}^{2} \right ){x}^{9}}{9}}+{\frac{ \left ( 3\,b{a}^{2}g+3\,a{b}^{2}d \right ){x}^{8}}{8}}+{\frac{ \left ( 3\,b{a}^{2}f+3\,ac{b}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ({a}^{3}h+3\,{a}^{2}be \right ){x}^{6}}{6}}+{\frac{ \left ({a}^{3}g+3\,{a}^{2}bd \right ){x}^{5}}{5}}+{\frac{ \left ({a}^{3}f+3\,b{a}^{2}c \right ){x}^{4}}{4}}+{\frac{{a}^{3}e{x}^{3}}{3}}+{\frac{{a}^{3}d{x}^{2}}{2}}+{a}^{3}cx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.948258, size = 289, normalized size = 1.4 \begin{align*} \frac{1}{15} \, b^{3} h x^{15} + \frac{1}{14} \, b^{3} g x^{14} + \frac{1}{13} \, b^{3} f x^{13} + \frac{1}{12} \,{\left (b^{3} e + 3 \, a b^{2} h\right )} x^{12} + \frac{1}{11} \,{\left (b^{3} d + 3 \, a b^{2} g\right )} x^{11} + \frac{1}{10} \,{\left (b^{3} c + 3 \, a b^{2} f\right )} x^{10} + \frac{1}{3} \,{\left (a b^{2} e + a^{2} b h\right )} x^{9} + \frac{3}{8} \,{\left (a b^{2} d + a^{2} b g\right )} x^{8} + \frac{3}{7} \,{\left (a b^{2} c + a^{2} b f\right )} x^{7} + \frac{1}{3} \, a^{3} e x^{3} + \frac{1}{6} \,{\left (3 \, a^{2} b e + a^{3} h\right )} x^{6} + \frac{1}{2} \, a^{3} d x^{2} + \frac{1}{5} \,{\left (3 \, a^{2} b d + a^{3} g\right )} x^{5} + a^{3} c x + \frac{1}{4} \,{\left (3 \, a^{2} b c + a^{3} f\right )} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.832962, size = 564, normalized size = 2.72 \begin{align*} \frac{1}{15} x^{15} h b^{3} + \frac{1}{14} x^{14} g b^{3} + \frac{1}{13} x^{13} f b^{3} + \frac{1}{12} x^{12} e b^{3} + \frac{1}{4} x^{12} h b^{2} a + \frac{1}{11} x^{11} d b^{3} + \frac{3}{11} x^{11} g b^{2} a + \frac{1}{10} x^{10} c b^{3} + \frac{3}{10} x^{10} f b^{2} a + \frac{1}{3} x^{9} e b^{2} a + \frac{1}{3} x^{9} h b a^{2} + \frac{3}{8} x^{8} d b^{2} a + \frac{3}{8} x^{8} g b a^{2} + \frac{3}{7} x^{7} c b^{2} a + \frac{3}{7} x^{7} f b a^{2} + \frac{1}{2} x^{6} e b a^{2} + \frac{1}{6} x^{6} h a^{3} + \frac{3}{5} x^{5} d b a^{2} + \frac{1}{5} x^{5} g a^{3} + \frac{3}{4} x^{4} c b a^{2} + \frac{1}{4} x^{4} f a^{3} + \frac{1}{3} x^{3} e a^{3} + \frac{1}{2} x^{2} d a^{3} + x c a^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.095226, size = 243, normalized size = 1.17 \begin{align*} a^{3} c x + \frac{a^{3} d x^{2}}{2} + \frac{a^{3} e x^{3}}{3} + \frac{b^{3} f x^{13}}{13} + \frac{b^{3} g x^{14}}{14} + \frac{b^{3} h x^{15}}{15} + x^{12} \left (\frac{a b^{2} h}{4} + \frac{b^{3} e}{12}\right ) + x^{11} \left (\frac{3 a b^{2} g}{11} + \frac{b^{3} d}{11}\right ) + x^{10} \left (\frac{3 a b^{2} f}{10} + \frac{b^{3} c}{10}\right ) + x^{9} \left (\frac{a^{2} b h}{3} + \frac{a b^{2} e}{3}\right ) + x^{8} \left (\frac{3 a^{2} b g}{8} + \frac{3 a b^{2} d}{8}\right ) + x^{7} \left (\frac{3 a^{2} b f}{7} + \frac{3 a b^{2} c}{7}\right ) + x^{6} \left (\frac{a^{3} h}{6} + \frac{a^{2} b e}{2}\right ) + x^{5} \left (\frac{a^{3} g}{5} + \frac{3 a^{2} b d}{5}\right ) + x^{4} \left (\frac{a^{3} f}{4} + \frac{3 a^{2} b c}{4}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Giac [A]  time = 1.06531, size = 311, normalized size = 1.5 \begin{align*} \frac{1}{15} \, b^{3} h x^{15} + \frac{1}{14} \, b^{3} g x^{14} + \frac{1}{13} \, b^{3} f x^{13} + \frac{1}{4} \, a b^{2} h x^{12} + \frac{1}{12} \, b^{3} x^{12} e + \frac{1}{11} \, b^{3} d x^{11} + \frac{3}{11} \, a b^{2} g x^{11} + \frac{1}{10} \, b^{3} c x^{10} + \frac{3}{10} \, a b^{2} f x^{10} + \frac{1}{3} \, a^{2} b h x^{9} + \frac{1}{3} \, a b^{2} x^{9} e + \frac{3}{8} \, a b^{2} d x^{8} + \frac{3}{8} \, a^{2} b g x^{8} + \frac{3}{7} \, a b^{2} c x^{7} + \frac{3}{7} \, a^{2} b f x^{7} + \frac{1}{6} \, a^{3} h x^{6} + \frac{1}{2} \, a^{2} b x^{6} e + \frac{3}{5} \, a^{2} b d x^{5} + \frac{1}{5} \, a^{3} g x^{5} + \frac{3}{4} \, a^{2} b c x^{4} + \frac{1}{4} \, a^{3} f x^{4} + \frac{1}{3} \, a^{3} x^{3} e + \frac{1}{2} \, a^{3} d x^{2} + a^{3} c x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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