\(\int x^3 (a+b x^3) (c+d x+e x^2+f x^3+g x^4+h x^5) \, dx\) [374]

Optimal result

Integrand size = 36, antiderivative size = 97 \[ \int x^3 \left (a+b x^3\right ) \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {1}{4} a c x^4+\frac {1}{5} a d x^5+\frac {1}{6} a e x^6+\frac {1}{7} (b c+a f) x^7+\frac {1}{8} (b d+a g) x^8+\frac {1}{9} (b e+a h) x^9+\frac {1}{10} b f x^{10}+\frac {1}{11} b g x^{11}+\frac {1}{12} b h x^{12} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {1834} \[ \int x^3 \left (a+b x^3\right ) \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {1}{7} x^7 (a f+b c)+\frac {1}{8} x^8 (a g+b d)+\frac {1}{9} x^9 (a h+b e)+\frac {1}{4} a c x^4+\frac {1}{5} a d x^5+\frac {1}{6} a e x^6+\frac {1}{10} b f x^{10}+\frac {1}{11} b g x^{11}+\frac {1}{12} b h x^{12} \]

[In]

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

[Out]

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

Rule 1834

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a +
b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rubi steps \begin{align*} \text {integral}& = \int \left (a c x^3+a d x^4+a e x^5+(b c+a f) x^6+(b d+a g) x^7+(b e+a h) x^8+b f x^9+b g x^{10}+b h x^{11}\right ) \, dx \\ & = \frac {1}{4} a c x^4+\frac {1}{5} a d x^5+\frac {1}{6} a e x^6+\frac {1}{7} (b c+a f) x^7+\frac {1}{8} (b d+a g) x^8+\frac {1}{9} (b e+a h) x^9+\frac {1}{10} b f x^{10}+\frac {1}{11} b g x^{11}+\frac {1}{12} b h x^{12} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00 \[ \int x^3 \left (a+b x^3\right ) \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {1}{4} a c x^4+\frac {1}{5} a d x^5+\frac {1}{6} a e x^6+\frac {1}{7} (b c+a f) x^7+\frac {1}{8} (b d+a g) x^8+\frac {1}{9} (b e+a h) x^9+\frac {1}{10} b f x^{10}+\frac {1}{11} b g x^{11}+\frac {1}{12} b h x^{12} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.16 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.82

method	result	size
default	\(\frac {a \,x^{4} c}{4}+\frac {a d \,x^{5}}{5}+\frac {a e \,x^{6}}{6}+\frac {\left (a f +b c \right ) x^{7}}{7}+\frac {\left (a g +b d \right ) x^{8}}{8}+\frac {\left (a h +b e \right ) x^{9}}{9}+\frac {b f \,x^{10}}{10}+\frac {b g \,x^{11}}{11}+\frac {b h \,x^{12}}{12}\)	\(80\)
norman	\(\frac {b h \,x^{12}}{12}+\frac {b g \,x^{11}}{11}+\frac {b f \,x^{10}}{10}+\left (\frac {a h}{9}+\frac {b e}{9}\right ) x^{9}+\left (\frac {a g}{8}+\frac {b d}{8}\right ) x^{8}+\left (\frac {a f}{7}+\frac {b c}{7}\right ) x^{7}+\frac {a e \,x^{6}}{6}+\frac {a d \,x^{5}}{5}+\frac {a \,x^{4} c}{4}\)	\(83\)
gosper	\(\frac {1}{12} b h \,x^{12}+\frac {1}{11} b g \,x^{11}+\frac {1}{10} b f \,x^{10}+\frac {1}{9} x^{9} a h +\frac {1}{9} x^{9} b e +\frac {1}{8} x^{8} a g +\frac {1}{8} b d \,x^{8}+\frac {1}{7} a f \,x^{7}+\frac {1}{7} b \,x^{7} c +\frac {1}{6} a e \,x^{6}+\frac {1}{5} a d \,x^{5}+\frac {1}{4} a \,x^{4} c\)	\(86\)
risch	\(\frac {1}{12} b h \,x^{12}+\frac {1}{11} b g \,x^{11}+\frac {1}{10} b f \,x^{10}+\frac {1}{9} x^{9} a h +\frac {1}{9} x^{9} b e +\frac {1}{8} x^{8} a g +\frac {1}{8} b d \,x^{8}+\frac {1}{7} a f \,x^{7}+\frac {1}{7} b \,x^{7} c +\frac {1}{6} a e \,x^{6}+\frac {1}{5} a d \,x^{5}+\frac {1}{4} a \,x^{4} c\)	\(86\)
parallelrisch	\(\frac {1}{12} b h \,x^{12}+\frac {1}{11} b g \,x^{11}+\frac {1}{10} b f \,x^{10}+\frac {1}{9} x^{9} a h +\frac {1}{9} x^{9} b e +\frac {1}{8} x^{8} a g +\frac {1}{8} b d \,x^{8}+\frac {1}{7} a f \,x^{7}+\frac {1}{7} b \,x^{7} c +\frac {1}{6} a e \,x^{6}+\frac {1}{5} a d \,x^{5}+\frac {1}{4} a \,x^{4} c\)	\(86\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.81 \[ \int x^3 \left (a+b x^3\right ) \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {1}{12} \, b h x^{12} + \frac {1}{11} \, b g x^{11} + \frac {1}{10} \, b f x^{10} + \frac {1}{9} \, {\left (b e + a h\right )} x^{9} + \frac {1}{8} \, {\left (b d + a g\right )} x^{8} + \frac {1}{6} \, a e x^{6} + \frac {1}{7} \, {\left (b c + a f\right )} x^{7} + \frac {1}{5} \, a d x^{5} + \frac {1}{4} \, a c x^{4} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.93 \[ \int x^3 \left (a+b x^3\right ) \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {a c x^{4}}{4} + \frac {a d x^{5}}{5} + \frac {a e x^{6}}{6} + \frac {b f x^{10}}{10} + \frac {b g x^{11}}{11} + \frac {b h x^{12}}{12} + x^{9} \left (\frac {a h}{9} + \frac {b e}{9}\right ) + x^{8} \left (\frac {a g}{8} + \frac {b d}{8}\right ) + x^{7} \left (\frac {a f}{7} + \frac {b c}{7}\right ) \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.81 \[ \int x^3 \left (a+b x^3\right ) \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {1}{12} \, b h x^{12} + \frac {1}{11} \, b g x^{11} + \frac {1}{10} \, b f x^{10} + \frac {1}{9} \, {\left (b e + a h\right )} x^{9} + \frac {1}{8} \, {\left (b d + a g\right )} x^{8} + \frac {1}{6} \, a e x^{6} + \frac {1}{7} \, {\left (b c + a f\right )} x^{7} + \frac {1}{5} \, a d x^{5} + \frac {1}{4} \, a c x^{4} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.88 \[ \int x^3 \left (a+b x^3\right ) \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {1}{12} \, b h x^{12} + \frac {1}{11} \, b g x^{11} + \frac {1}{10} \, b f x^{10} + \frac {1}{9} \, b e x^{9} + \frac {1}{9} \, a h x^{9} + \frac {1}{8} \, b d x^{8} + \frac {1}{8} \, a g x^{8} + \frac {1}{7} \, b c x^{7} + \frac {1}{7} \, a f x^{7} + \frac {1}{6} \, a e x^{6} + \frac {1}{5} \, a d x^{5} + \frac {1}{4} \, a c x^{4} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.85 \[ \int x^3 \left (a+b x^3\right ) \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {b\,h\,x^{12}}{12}+\frac {b\,g\,x^{11}}{11}+\frac {b\,f\,x^{10}}{10}+\left (\frac {b\,e}{9}+\frac {a\,h}{9}\right )\,x^9+\left (\frac {b\,d}{8}+\frac {a\,g}{8}\right )\,x^8+\left (\frac {b\,c}{7}+\frac {a\,f}{7}\right )\,x^7+\frac {a\,e\,x^6}{6}+\frac {a\,d\,x^5}{5}+\frac {a\,c\,x^4}{4} \]

[In]

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

[Out]

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