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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.18, antiderivative size = 207, normalized size of antiderivative = 1.00, 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 = 0.11, size = 170, normalized size = 0.82 \[ \frac {x \left (2002 a^3 \left (60 c+x \left (30 d+x \left (20 e+15 f x+12 g x^2+10 h x^3\right )\right )\right )+143 a^2 b x^3 (630 c+x (504 d+5 x (84 e+x (72 f+7 x (9 g+8 h x)))))+13 a b^2 x^6 \left (3960 c+7 x \left (495 d+440 e x+6 x^2 \left (66 f+60 g x+55 h x^2\right )\right )\right )+2 b^3 x^9 \left (6006 c+x \left (5460 d+11 x \left (455 e+420 f x+390 g x^2+364 h x^3\right )\right )\right )\right )}{120120} \]

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

________________________________________________________________________________________

fricas [A]  time = 0.37, size = 226, normalized size = 1.09 \[ \frac {1}{15} x^{15} h b^{3} + \frac {1}{14} x^{14} g b^{3} + \frac {1}{13} x^{13} f b^{3} + \frac {1}{12} x^{12} e b^{3} + \frac {1}{4} x^{12} h b^{2} a + \frac {1}{11} x^{11} d b^{3} + \frac {3}{11} x^{11} g b^{2} a + \frac {1}{10} x^{10} c b^{3} + \frac {3}{10} x^{10} f b^{2} a + \frac {1}{3} x^{9} e b^{2} a + \frac {1}{3} x^{9} h b a^{2} + \frac {3}{8} x^{8} d b^{2} a + \frac {3}{8} x^{8} g b a^{2} + \frac {3}{7} x^{7} c b^{2} a + \frac {3}{7} x^{7} f b a^{2} + \frac {1}{2} x^{6} e b a^{2} + \frac {1}{6} x^{6} h a^{3} + \frac {3}{5} x^{5} d b a^{2} + \frac {1}{5} x^{5} g a^{3} + \frac {3}{4} x^{4} c b a^{2} + \frac {1}{4} x^{4} f a^{3} + \frac {1}{3} x^{3} e a^{3} + \frac {1}{2} x^{2} d a^{3} + x c a^{3} \]

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

________________________________________________________________________________________

giac [A]  time = 0.15, size = 230, normalized size = 1.11 \[ \frac {1}{15} \, b^{3} h x^{15} + \frac {1}{14} \, b^{3} g x^{14} + \frac {1}{13} \, b^{3} f x^{13} + \frac {1}{4} \, a b^{2} h x^{12} + \frac {1}{12} \, b^{3} x^{12} e + \frac {1}{11} \, b^{3} d x^{11} + \frac {3}{11} \, a b^{2} g x^{11} + \frac {1}{10} \, b^{3} c x^{10} + \frac {3}{10} \, a b^{2} f x^{10} + \frac {1}{3} \, a^{2} b h x^{9} + \frac {1}{3} \, a b^{2} x^{9} e + \frac {3}{8} \, a b^{2} d x^{8} + \frac {3}{8} \, a^{2} b g x^{8} + \frac {3}{7} \, a b^{2} c x^{7} + \frac {3}{7} \, a^{2} b f x^{7} + \frac {1}{6} \, a^{3} h x^{6} + \frac {1}{2} \, a^{2} b x^{6} e + \frac {3}{5} \, a^{2} b d x^{5} + \frac {1}{5} \, a^{3} g x^{5} + \frac {3}{4} \, a^{2} b c x^{4} + \frac {1}{4} \, a^{3} f x^{4} + \frac {1}{3} \, a^{3} x^{3} e + \frac {1}{2} \, a^{3} d x^{2} + a^{3} c x \]

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

________________________________________________________________________________________

maple [A]  time = 0.04, size = 221, normalized size = 1.07 \[ \frac {b^{3} h \,x^{15}}{15}+\frac {b^{3} g \,x^{14}}{14}+\frac {b^{3} f \,x^{13}}{13}+\frac {\left (3 a \,b^{2} h +b^{3} e \right ) x^{12}}{12}+\frac {\left (3 a \,b^{2} g +b^{3} d \right ) x^{11}}{11}+\frac {\left (3 a \,b^{2} f +b^{3} c \right ) x^{10}}{10}+\frac {\left (3 a^{2} b h +3 a e \,b^{2}\right ) x^{9}}{9}+\frac {\left (3 a^{2} b g +3 a \,b^{2} d \right ) x^{8}}{8}+\frac {a^{3} e \,x^{3}}{3}+\frac {\left (3 a^{2} b f +3 a \,b^{2} c \right ) x^{7}}{7}+\frac {a^{3} d \,x^{2}}{2}+\frac {\left (a^{3} h +3 a^{2} b e \right ) x^{6}}{6}+a^{3} c x +\frac {\left (a^{3} g +3 a^{2} d b \right ) x^{5}}{5}+\frac {\left (a^{3} f +3 a^{2} c b \right ) x^{4}}{4} \]

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

________________________________________________________________________________________

maxima [A]  time = 1.34, size = 214, normalized size = 1.03 \[ \frac {1}{15} \, b^{3} h x^{15} + \frac {1}{14} \, b^{3} g x^{14} + \frac {1}{13} \, b^{3} f x^{13} + \frac {1}{12} \, {\left (b^{3} e + 3 \, a b^{2} h\right )} x^{12} + \frac {1}{11} \, {\left (b^{3} d + 3 \, a b^{2} g\right )} x^{11} + \frac {1}{10} \, {\left (b^{3} c + 3 \, a b^{2} f\right )} x^{10} + \frac {1}{3} \, {\left (a b^{2} e + a^{2} b h\right )} x^{9} + \frac {3}{8} \, {\left (a b^{2} d + a^{2} b g\right )} x^{8} + \frac {3}{7} \, {\left (a b^{2} c + a^{2} b f\right )} x^{7} + \frac {1}{3} \, a^{3} e x^{3} + \frac {1}{6} \, {\left (3 \, a^{2} b e + a^{3} h\right )} x^{6} + \frac {1}{2} \, a^{3} d x^{2} + \frac {1}{5} \, {\left (3 \, a^{2} b d + a^{3} g\right )} x^{5} + a^{3} c x + \frac {1}{4} \, {\left (3 \, a^{2} b c + a^{3} f\right )} x^{4} \]

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

________________________________________________________________________________________

mupad [B]  time = 0.16, size = 202, normalized size = 0.98 \[ x^4\,\left (\frac {f\,a^3}{4}+\frac {3\,b\,c\,a^2}{4}\right )+x^{10}\,\left (\frac {c\,b^3}{10}+\frac {3\,a\,f\,b^2}{10}\right )+x^5\,\left (\frac {g\,a^3}{5}+\frac {3\,b\,d\,a^2}{5}\right )+x^{11}\,\left (\frac {d\,b^3}{11}+\frac {3\,a\,g\,b^2}{11}\right )+x^6\,\left (\frac {h\,a^3}{6}+\frac {b\,e\,a^2}{2}\right )+x^{12}\,\left (\frac {e\,b^3}{12}+\frac {a\,h\,b^2}{4}\right )+\frac {a^3\,d\,x^2}{2}+\frac {a^3\,e\,x^3}{3}+\frac {b^3\,f\,x^{13}}{13}+\frac {b^3\,g\,x^{14}}{14}+\frac {b^3\,h\,x^{15}}{15}+a^3\,c\,x+\frac {3\,a\,b\,x^7\,\left (b\,c+a\,f\right )}{7}+\frac {3\,a\,b\,x^8\,\left (b\,d+a\,g\right )}{8}+\frac {a\,b\,x^9\,\left (b\,e+a\,h\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.12, size = 243, normalized size = 1.17 \[ a^{3} c x + \frac {a^{3} d x^{2}}{2} + \frac {a^{3} e x^{3}}{3} + \frac {b^{3} f x^{13}}{13} + \frac {b^{3} g x^{14}}{14} + \frac {b^{3} h x^{15}}{15} + x^{12} \left (\frac {a b^{2} h}{4} + \frac {b^{3} e}{12}\right ) + x^{11} \left (\frac {3 a b^{2} g}{11} + \frac {b^{3} d}{11}\right ) + x^{10} \left (\frac {3 a b^{2} f}{10} + \frac {b^{3} c}{10}\right ) + x^{9} \left (\frac {a^{2} b h}{3} + \frac {a b^{2} e}{3}\right ) + x^{8} \left (\frac {3 a^{2} b g}{8} + \frac {3 a b^{2} d}{8}\right ) + x^{7} \left (\frac {3 a^{2} b f}{7} + \frac {3 a b^{2} c}{7}\right ) + x^{6} \left (\frac {a^{3} h}{6} + \frac {a^{2} b e}{2}\right ) + x^{5} \left (\frac {a^{3} g}{5} + \frac {3 a^{2} b d}{5}\right ) + x^{4} \left (\frac {a^{3} f}{4} + \frac {3 a^{2} b c}{4}\right ) \]

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)

________________________________________________________________________________________