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

Optimal. Leaf size=105 \[ \frac{1}{5} x^5 (a h+b f+c d)+\frac{1}{3} x^3 (a f+b d)+\frac{1}{4} x^4 (a g+b e)+a d x+\frac{1}{2} a e x^2+\frac{1}{6} x^6 (b g+c e)+\frac{1}{7} x^7 (b h+c f)+\frac{1}{8} c g x^8+\frac{1}{9} c h x^9 \]

[Out]

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

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Rubi [A]  time = 0.0948779, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.03, Rules used = {1671} \[ \frac{1}{5} x^5 (a h+b f+c d)+\frac{1}{3} x^3 (a f+b d)+\frac{1}{4} x^4 (a g+b e)+a d x+\frac{1}{2} a e x^2+\frac{1}{6} x^6 (b g+c e)+\frac{1}{7} x^7 (b h+c f)+\frac{1}{8} c g x^8+\frac{1}{9} c h x^9 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1671

Int[(Pq_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^2 + c*x^4)^
p, x], x] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && IGtQ[p, 0]

Rubi steps

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

Mathematica [A]  time = 0.035032, size = 105, normalized size = 1. \[ \frac{1}{5} x^5 (a h+b f+c d)+\frac{1}{3} x^3 (a f+b d)+\frac{1}{4} x^4 (a g+b e)+a d x+\frac{1}{2} a e x^2+\frac{1}{6} x^6 (b g+c e)+\frac{1}{7} x^7 (b h+c f)+\frac{1}{8} c g x^8+\frac{1}{9} c h x^9 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0., size = 90, normalized size = 0.9 \begin{align*} adx+{\frac{ae{x}^{2}}{2}}+{\frac{ \left ( af+bd \right ){x}^{3}}{3}}+{\frac{ \left ( ag+be \right ){x}^{4}}{4}}+{\frac{ \left ( ah+bf+cd \right ){x}^{5}}{5}}+{\frac{ \left ( bg+ce \right ){x}^{6}}{6}}+{\frac{ \left ( bh+cf \right ){x}^{7}}{7}}+{\frac{cg{x}^{8}}{8}}+{\frac{ch{x}^{9}}{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.957798, size = 120, normalized size = 1.14 \begin{align*} \frac{1}{9} \, c h x^{9} + \frac{1}{8} \, c g x^{8} + \frac{1}{7} \,{\left (c f + b h\right )} x^{7} + \frac{1}{6} \,{\left (c e + b g\right )} x^{6} + \frac{1}{5} \,{\left (c d + b f + a h\right )} x^{5} + \frac{1}{4} \,{\left (b e + a g\right )} x^{4} + \frac{1}{2} \, a e x^{2} + \frac{1}{3} \,{\left (b d + a f\right )} x^{3} + a d x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.43696, size = 274, normalized size = 2.61 \begin{align*} \frac{1}{9} x^{9} h c + \frac{1}{8} x^{8} g c + \frac{1}{7} x^{7} f c + \frac{1}{7} x^{7} h b + \frac{1}{6} x^{6} e c + \frac{1}{6} x^{6} g b + \frac{1}{5} x^{5} d c + \frac{1}{5} x^{5} f b + \frac{1}{5} x^{5} h a + \frac{1}{4} x^{4} e b + \frac{1}{4} x^{4} g a + \frac{1}{3} x^{3} d b + \frac{1}{3} x^{3} f a + \frac{1}{2} x^{2} e a + x d a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.077167, size = 102, normalized size = 0.97 \begin{align*} a d x + \frac{a e x^{2}}{2} + \frac{c g x^{8}}{8} + \frac{c h x^{9}}{9} + x^{7} \left (\frac{b h}{7} + \frac{c f}{7}\right ) + x^{6} \left (\frac{b g}{6} + \frac{c e}{6}\right ) + x^{5} \left (\frac{a h}{5} + \frac{b f}{5} + \frac{c d}{5}\right ) + x^{4} \left (\frac{a g}{4} + \frac{b e}{4}\right ) + x^{3} \left (\frac{a f}{3} + \frac{b d}{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.09345, size = 143, normalized size = 1.36 \begin{align*} \frac{1}{9} \, c h x^{9} + \frac{1}{8} \, c g x^{8} + \frac{1}{7} \, c f x^{7} + \frac{1}{7} \, b h x^{7} + \frac{1}{6} \, b g x^{6} + \frac{1}{6} \, c x^{6} e + \frac{1}{5} \, c d x^{5} + \frac{1}{5} \, b f x^{5} + \frac{1}{5} \, a h x^{5} + \frac{1}{4} \, a g x^{4} + \frac{1}{4} \, b x^{4} e + \frac{1}{3} \, b d x^{3} + \frac{1}{3} \, a f x^{3} + \frac{1}{2} \, a x^{2} e + a d x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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