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

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

[Out]

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

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Rubi [A]  time = 0.18, antiderivative size = 212, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {1582, 1850} \[ \frac {1}{5} a^2 x^5 (a f+3 b c)+\frac {1}{7} a^2 x^7 (a h+3 b e)+\frac {1}{3} a^2 b g x^9+\frac {1}{2} a^3 c x^2+\frac {1}{4} a^3 e x^4+\frac {1}{6} a^3 g x^6+\frac {1}{11} b^2 x^{11} (3 a f+b c)+\frac {1}{13} b^2 x^{13} (3 a h+b e)+\frac {1}{4} a b^2 g x^{12}+\frac {3}{8} a b x^8 (a f+b c)+\frac {d \left (a+b x^3\right )^4}{12 b}+\frac {3}{10} a b x^{10} (a h+b e)+\frac {1}{14} b^3 f x^{14}+\frac {1}{15} b^3 g x^{15}+\frac {1}{16} b^3 h x^{16} \]

Antiderivative was successfully verified.

[In]

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

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Mathematica [A]  time = 0.04, size = 223, normalized size = 1.05 \[ \frac {1}{2} a^3 c x^2+\frac {1}{3} a^3 d x^3+\frac {1}{4} a^3 e x^4+\frac {1}{5} a^2 x^5 (a f+3 b c)+\frac {1}{6} a^2 x^6 (a g+3 b d)+\frac {1}{7} a^2 x^7 (a h+3 b e)+\frac {1}{11} b^2 x^{11} (3 a f+b c)+\frac {1}{12} b^2 x^{12} (3 a g+b d)+\frac {1}{13} b^2 x^{13} (3 a h+b e)+\frac {3}{8} a b x^8 (a f+b c)+\frac {1}{3} a b x^9 (a g+b d)+\frac {3}{10} a b x^{10} (a h+b e)+\frac {1}{14} b^3 f x^{14}+\frac {1}{15} b^3 g x^{15}+\frac {1}{16} b^3 h x^{16} \]

Antiderivative was successfully verified.

[In]

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

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fricas [A]  time = 0.37, size = 229, normalized size = 1.08 \[ \frac {1}{16} x^{16} h b^{3} + \frac {1}{15} x^{15} g b^{3} + \frac {1}{14} x^{14} f b^{3} + \frac {1}{13} x^{13} e b^{3} + \frac {3}{13} x^{13} h b^{2} a + \frac {1}{12} x^{12} d b^{3} + \frac {1}{4} x^{12} g b^{2} a + \frac {1}{11} x^{11} c b^{3} + \frac {3}{11} x^{11} f b^{2} a + \frac {3}{10} x^{10} e b^{2} a + \frac {3}{10} x^{10} h b a^{2} + \frac {1}{3} x^{9} d b^{2} a + \frac {1}{3} x^{9} g b a^{2} + \frac {3}{8} x^{8} c b^{2} a + \frac {3}{8} x^{8} f b a^{2} + \frac {3}{7} x^{7} e b a^{2} + \frac {1}{7} x^{7} h a^{3} + \frac {1}{2} x^{6} d b a^{2} + \frac {1}{6} x^{6} g a^{3} + \frac {3}{5} x^{5} c b a^{2} + \frac {1}{5} x^{5} f a^{3} + \frac {1}{4} x^{4} e a^{3} + \frac {1}{3} x^{3} d a^{3} + \frac {1}{2} x^{2} c a^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.17, size = 233, normalized size = 1.10 \[ \frac {1}{16} \, b^{3} h x^{16} + \frac {1}{15} \, b^{3} g x^{15} + \frac {1}{14} \, b^{3} f x^{14} + \frac {3}{13} \, a b^{2} h x^{13} + \frac {1}{13} \, b^{3} x^{13} e + \frac {1}{12} \, b^{3} d x^{12} + \frac {1}{4} \, a b^{2} g x^{12} + \frac {1}{11} \, b^{3} c x^{11} + \frac {3}{11} \, a b^{2} f x^{11} + \frac {3}{10} \, a^{2} b h x^{10} + \frac {3}{10} \, a b^{2} x^{10} e + \frac {1}{3} \, a b^{2} d x^{9} + \frac {1}{3} \, a^{2} b g x^{9} + \frac {3}{8} \, a b^{2} c x^{8} + \frac {3}{8} \, a^{2} b f x^{8} + \frac {1}{7} \, a^{3} h x^{7} + \frac {3}{7} \, a^{2} b x^{7} e + \frac {1}{2} \, a^{2} b d x^{6} + \frac {1}{6} \, a^{3} g x^{6} + \frac {3}{5} \, a^{2} b c x^{5} + \frac {1}{5} \, a^{3} f x^{5} + \frac {1}{4} \, a^{3} x^{4} e + \frac {1}{3} \, a^{3} d x^{3} + \frac {1}{2} \, a^{3} c x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.04, size = 224, normalized size = 1.06 \[ \frac {b^{3} h \,x^{16}}{16}+\frac {b^{3} g \,x^{15}}{15}+\frac {b^{3} f \,x^{14}}{14}+\frac {\left (3 a \,b^{2} h +b^{3} e \right ) x^{13}}{13}+\frac {\left (3 a \,b^{2} g +b^{3} d \right ) x^{12}}{12}+\frac {\left (3 a \,b^{2} f +b^{3} c \right ) x^{11}}{11}+\frac {\left (3 a^{2} b h +3 a e \,b^{2}\right ) x^{10}}{10}+\frac {\left (3 a^{2} b g +3 a \,b^{2} d \right ) x^{9}}{9}+\frac {a^{3} e \,x^{4}}{4}+\frac {\left (3 a^{2} b f +3 a \,b^{2} c \right ) x^{8}}{8}+\frac {a^{3} d \,x^{3}}{3}+\frac {\left (a^{3} h +3 a^{2} b e \right ) x^{7}}{7}+\frac {a^{3} c \,x^{2}}{2}+\frac {\left (a^{3} g +3 a^{2} d b \right ) x^{6}}{6}+\frac {\left (a^{3} f +3 a^{2} c b \right ) x^{5}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.35, size = 217, normalized size = 1.02 \[ \frac {1}{16} \, b^{3} h x^{16} + \frac {1}{15} \, b^{3} g x^{15} + \frac {1}{14} \, b^{3} f x^{14} + \frac {1}{13} \, {\left (b^{3} e + 3 \, a b^{2} h\right )} x^{13} + \frac {1}{12} \, {\left (b^{3} d + 3 \, a b^{2} g\right )} x^{12} + \frac {1}{11} \, {\left (b^{3} c + 3 \, a b^{2} f\right )} x^{11} + \frac {3}{10} \, {\left (a b^{2} e + a^{2} b h\right )} x^{10} + \frac {1}{3} \, {\left (a b^{2} d + a^{2} b g\right )} x^{9} + \frac {3}{8} \, {\left (a b^{2} c + a^{2} b f\right )} x^{8} + \frac {1}{4} \, a^{3} e x^{4} + \frac {1}{7} \, {\left (3 \, a^{2} b e + a^{3} h\right )} x^{7} + \frac {1}{3} \, a^{3} d x^{3} + \frac {1}{6} \, {\left (3 \, a^{2} b d + a^{3} g\right )} x^{6} + \frac {1}{2} \, a^{3} c x^{2} + \frac {1}{5} \, {\left (3 \, a^{2} b c + a^{3} f\right )} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.16, size = 205, normalized size = 0.97 \[ x^5\,\left (\frac {f\,a^3}{5}+\frac {3\,b\,c\,a^2}{5}\right )+x^{11}\,\left (\frac {c\,b^3}{11}+\frac {3\,a\,f\,b^2}{11}\right )+x^6\,\left (\frac {g\,a^3}{6}+\frac {b\,d\,a^2}{2}\right )+x^{12}\,\left (\frac {d\,b^3}{12}+\frac {a\,g\,b^2}{4}\right )+x^7\,\left (\frac {h\,a^3}{7}+\frac {3\,b\,e\,a^2}{7}\right )+x^{13}\,\left (\frac {e\,b^3}{13}+\frac {3\,a\,h\,b^2}{13}\right )+\frac {a^3\,c\,x^2}{2}+\frac {a^3\,d\,x^3}{3}+\frac {a^3\,e\,x^4}{4}+\frac {b^3\,f\,x^{14}}{14}+\frac {b^3\,g\,x^{15}}{15}+\frac {b^3\,h\,x^{16}}{16}+\frac {3\,a\,b\,x^8\,\left (b\,c+a\,f\right )}{8}+\frac {a\,b\,x^9\,\left (b\,d+a\,g\right )}{3}+\frac {3\,a\,b\,x^{10}\,\left (b\,e+a\,h\right )}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.11, size = 246, normalized size = 1.16 \[ \frac {a^{3} c x^{2}}{2} + \frac {a^{3} d x^{3}}{3} + \frac {a^{3} e x^{4}}{4} + \frac {b^{3} f x^{14}}{14} + \frac {b^{3} g x^{15}}{15} + \frac {b^{3} h x^{16}}{16} + x^{13} \left (\frac {3 a b^{2} h}{13} + \frac {b^{3} e}{13}\right ) + x^{12} \left (\frac {a b^{2} g}{4} + \frac {b^{3} d}{12}\right ) + x^{11} \left (\frac {3 a b^{2} f}{11} + \frac {b^{3} c}{11}\right ) + x^{10} \left (\frac {3 a^{2} b h}{10} + \frac {3 a b^{2} e}{10}\right ) + x^{9} \left (\frac {a^{2} b g}{3} + \frac {a b^{2} d}{3}\right ) + x^{8} \left (\frac {3 a^{2} b f}{8} + \frac {3 a b^{2} c}{8}\right ) + x^{7} \left (\frac {a^{3} h}{7} + \frac {3 a^{2} b e}{7}\right ) + x^{6} \left (\frac {a^{3} g}{6} + \frac {a^{2} b d}{2}\right ) + x^{5} \left (\frac {a^{3} f}{5} + \frac {3 a^{2} b c}{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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