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

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

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Rubi [A]  time = 0.09, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {1671} \begin {gather*} \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 \end {gather*}

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*}

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Mathematica [A]  time = 0.03, size = 105, normalized size = 1.00 \begin {gather*} \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 \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.86, size = 103, normalized size = 0.98 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.36, size = 106, normalized size = 1.01 \begin {gather*} \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 {gather*}

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

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maple [A]  time = 0.00, size = 90, normalized size = 0.86 \begin {gather*} \frac {c h \,x^{9}}{9}+\frac {c g \,x^{8}}{8}+\frac {\left (b h +c f \right ) x^{7}}{7}+\frac {\left (b g +c e \right ) x^{6}}{6}+\frac {\left (a h +b f +c d \right ) x^{5}}{5}+\frac {a e \,x^{2}}{2}+\frac {\left (a g +b e \right ) x^{4}}{4}+a d x +\frac {\left (a f +b d \right ) x^{3}}{3} \end {gather*}

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 = 1.64, size = 89, normalized size = 0.85 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 0.66, size = 95, normalized size = 0.90 \begin {gather*} \frac {c\,h\,x^9}{9}+\frac {c\,g\,x^8}{8}+\left (\frac {c\,f}{7}+\frac {b\,h}{7}\right )\,x^7+\left (\frac {c\,e}{6}+\frac {b\,g}{6}\right )\,x^6+\left (\frac {c\,d}{5}+\frac {b\,f}{5}+\frac {a\,h}{5}\right )\,x^5+\left (\frac {b\,e}{4}+\frac {a\,g}{4}\right )\,x^4+\left (\frac {b\,d}{3}+\frac {a\,f}{3}\right )\,x^3+\frac {a\,e\,x^2}{2}+a\,d\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.08, size = 102, normalized size = 0.97 \begin {gather*} 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 {gather*}

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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