\(\int (a+b x^3) (c+d x+e x^2+f x^3+g x^4+h x^5) \, dx\) [377]

Optimal result

Integrand size = 33, antiderivative size = 92 \[ \int \left (a+b x^3\right ) \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=a c x+\frac {1}{2} a d x^2+\frac {1}{3} a e x^3+\frac {1}{4} (b c+a f) x^4+\frac {1}{5} (b d+a g) x^5+\frac {1}{6} (b e+a h) x^6+\frac {1}{7} b f x^7+\frac {1}{8} b g x^8+\frac {1}{9} b h x^9 \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {1864} \[ \int \left (a+b x^3\right ) \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {1}{4} x^4 (a f+b c)+\frac {1}{5} x^5 (a g+b d)+\frac {1}{6} x^6 (a h+b e)+a c x+\frac {1}{2} a d x^2+\frac {1}{3} a e x^3+\frac {1}{7} b f x^7+\frac {1}{8} b g x^8+\frac {1}{9} b h x^9 \]

[In]

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

[Out]

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

Rule 1864

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*} \text {integral}& = \int \left (a c+a d x+a e x^2+(b c+a f) x^3+(b d+a g) x^4+(b e+a h) x^5+b f x^6+b g x^7+b h x^8\right ) \, dx \\ & = a c x+\frac {1}{2} a d x^2+\frac {1}{3} a e x^3+\frac {1}{4} (b c+a f) x^4+\frac {1}{5} (b d+a g) x^5+\frac {1}{6} (b e+a h) x^6+\frac {1}{7} b f x^7+\frac {1}{8} b g x^8+\frac {1}{9} b h x^9 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00 \[ \int \left (a+b x^3\right ) \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=a c x+\frac {1}{2} a d x^2+\frac {1}{3} a e x^3+\frac {1}{4} (b c+a f) x^4+\frac {1}{5} (b d+a g) x^5+\frac {1}{6} (b e+a h) x^6+\frac {1}{7} b f x^7+\frac {1}{8} b g x^8+\frac {1}{9} b h x^9 \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.84

method	result	size
default	\(a c x +\frac {a d \,x^{2}}{2}+\frac {a e \,x^{3}}{3}+\frac {\left (a f +b c \right ) x^{4}}{4}+\frac {\left (a g +b d \right ) x^{5}}{5}+\frac {\left (a h +b e \right ) x^{6}}{6}+\frac {b f \,x^{7}}{7}+\frac {b g \,x^{8}}{8}+\frac {b h \,x^{9}}{9}\)	\(77\)
norman	\(\frac {b h \,x^{9}}{9}+\frac {b g \,x^{8}}{8}+\frac {b f \,x^{7}}{7}+\left (\frac {a h}{6}+\frac {b e}{6}\right ) x^{6}+\left (\frac {a g}{5}+\frac {b d}{5}\right ) x^{5}+\left (\frac {a f}{4}+\frac {b c}{4}\right ) x^{4}+\frac {a e \,x^{3}}{3}+\frac {a d \,x^{2}}{2}+a c x\)	\(80\)
gosper	\(\frac {1}{9} b h \,x^{9}+\frac {1}{8} b g \,x^{8}+\frac {1}{7} b f \,x^{7}+\frac {1}{6} x^{6} a h +\frac {1}{6} b e \,x^{6}+\frac {1}{5} x^{5} a g +\frac {1}{5} b d \,x^{5}+\frac {1}{4} a f \,x^{4}+\frac {1}{4} b c \,x^{4}+\frac {1}{3} a e \,x^{3}+\frac {1}{2} a d \,x^{2}+a c x\)	\(83\)
risch	\(\frac {1}{9} b h \,x^{9}+\frac {1}{8} b g \,x^{8}+\frac {1}{7} b f \,x^{7}+\frac {1}{6} x^{6} a h +\frac {1}{6} b e \,x^{6}+\frac {1}{5} x^{5} a g +\frac {1}{5} b d \,x^{5}+\frac {1}{4} a f \,x^{4}+\frac {1}{4} b c \,x^{4}+\frac {1}{3} a e \,x^{3}+\frac {1}{2} a d \,x^{2}+a c x\)	\(83\)
parallelrisch	\(\frac {1}{9} b h \,x^{9}+\frac {1}{8} b g \,x^{8}+\frac {1}{7} b f \,x^{7}+\frac {1}{6} x^{6} a h +\frac {1}{6} b e \,x^{6}+\frac {1}{5} x^{5} a g +\frac {1}{5} b d \,x^{5}+\frac {1}{4} a f \,x^{4}+\frac {1}{4} b c \,x^{4}+\frac {1}{3} a e \,x^{3}+\frac {1}{2} a d \,x^{2}+a c x\)	\(83\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.83 \[ \int \left (a+b x^3\right ) \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {1}{9} \, b h x^{9} + \frac {1}{8} \, b g x^{8} + \frac {1}{7} \, b f x^{7} + \frac {1}{6} \, {\left (b e + a h\right )} x^{6} + \frac {1}{5} \, {\left (b d + a g\right )} x^{5} + \frac {1}{3} \, a e x^{3} + \frac {1}{4} \, {\left (b c + a f\right )} x^{4} + \frac {1}{2} \, a d x^{2} + a c x \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.95 \[ \int \left (a+b x^3\right ) \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=a c x + \frac {a d x^{2}}{2} + \frac {a e x^{3}}{3} + \frac {b f x^{7}}{7} + \frac {b g x^{8}}{8} + \frac {b h x^{9}}{9} + x^{6} \left (\frac {a h}{6} + \frac {b e}{6}\right ) + x^{5} \left (\frac {a g}{5} + \frac {b d}{5}\right ) + x^{4} \left (\frac {a f}{4} + \frac {b c}{4}\right ) \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.83 \[ \int \left (a+b x^3\right ) \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {1}{9} \, b h x^{9} + \frac {1}{8} \, b g x^{8} + \frac {1}{7} \, b f x^{7} + \frac {1}{6} \, {\left (b e + a h\right )} x^{6} + \frac {1}{5} \, {\left (b d + a g\right )} x^{5} + \frac {1}{3} \, a e x^{3} + \frac {1}{4} \, {\left (b c + a f\right )} x^{4} + \frac {1}{2} \, a d x^{2} + a c x \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.89 \[ \int \left (a+b x^3\right ) \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {1}{9} \, b h x^{9} + \frac {1}{8} \, b g x^{8} + \frac {1}{7} \, b f x^{7} + \frac {1}{6} \, b e x^{6} + \frac {1}{6} \, a h x^{6} + \frac {1}{5} \, b d x^{5} + \frac {1}{5} \, a g x^{5} + \frac {1}{4} \, b c x^{4} + \frac {1}{4} \, a f x^{4} + \frac {1}{3} \, a e x^{3} + \frac {1}{2} \, a d x^{2} + a c x \]

[In]

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

[Out]

1/9*b*h*x^9 + 1/8*b*g*x^8 + 1/7*b*f*x^7 + 1/6*b*e*x^6 + 1/6*a*h*x^6 + 1/5*b*d*x^5 + 1/5*a*g*x^5 + 1/4*b*c*x^4
+ 1/4*a*f*x^4 + 1/3*a*e*x^3 + 1/2*a*d*x^2 + a*c*x

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.86 \[ \int \left (a+b x^3\right ) \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {b\,h\,x^9}{9}+\frac {b\,g\,x^8}{8}+\frac {b\,f\,x^7}{7}+\left (\frac {b\,e}{6}+\frac {a\,h}{6}\right )\,x^6+\left (\frac {b\,d}{5}+\frac {a\,g}{5}\right )\,x^5+\left (\frac {b\,c}{4}+\frac {a\,f}{4}\right )\,x^4+\frac {a\,e\,x^3}{3}+\frac {a\,d\,x^2}{2}+a\,c\,x \]

[In]

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

[Out]

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