3.3 \(\int \left (d+e x+f x^2+g x^3\right ) \left (a+b x^2+c x^4\right ) \, dx\)

Optimal. Leaf size=88 \[ \frac{1}{3} x^3 (a f+b d)+\frac{1}{4} x^4 (a g+b e)+a d x+\frac{1}{2} a e x^2+\frac{1}{5} x^5 (b f+c d)+\frac{1}{6} x^6 (b g+c e)+\frac{1}{7} c f x^7+\frac{1}{8} c g x^8 \]

[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)*x
^5)/5 + ((c*e + b*g)*x^6)/6 + (c*f*x^7)/7 + (c*g*x^8)/8

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Rubi [A]  time = 0.144957, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036 \[ \frac{1}{3} x^3 (a f+b d)+\frac{1}{4} x^4 (a g+b e)+a d x+\frac{1}{2} a e x^2+\frac{1}{5} x^5 (b f+c d)+\frac{1}{6} x^6 (b g+c e)+\frac{1}{7} c f x^7+\frac{1}{8} c g x^8 \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x + f*x^2 + g*x^3)*(a + b*x^2 + c*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)*x
^5)/5 + ((c*e + b*g)*x^6)/6 + (c*f*x^7)/7 + (c*g*x^8)/8

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ a e \int x\, dx + \frac{c f x^{7}}{7} + \frac{c g x^{8}}{8} + d \int a\, dx + x^{6} \left (\frac{b g}{6} + \frac{c e}{6}\right ) + x^{5} \left (\frac{b f}{5} + \frac{c d}{5}\right ) + x^{4} \left (\frac{a g}{4} + \frac{b e}{4}\right ) + x^{3} \left (\frac{a f}{3} + \frac{b d}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((g*x**3+f*x**2+e*x+d)*(c*x**4+b*x**2+a),x)

[Out]

a*e*Integral(x, x) + c*f*x**7/7 + c*g*x**8/8 + d*Integral(a, x) + x**6*(b*g/6 +
c*e/6) + x**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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Mathematica [A]  time = 0.0365296, size = 88, normalized size = 1. \[ \frac{1}{3} x^3 (a f+b d)+\frac{1}{4} x^4 (a g+b e)+a d x+\frac{1}{2} a e x^2+\frac{1}{5} x^5 (b f+c d)+\frac{1}{6} x^6 (b g+c e)+\frac{1}{7} c f x^7+\frac{1}{8} c g x^8 \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x + f*x^2 + g*x^3)*(a + b*x^2 + c*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)*x
^5)/5 + ((c*e + b*g)*x^6)/6 + (c*f*x^7)/7 + (c*g*x^8)/8

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Maple [A]  time = 0.001, size = 75, normalized size = 0.9 \[ adx+{\frac{ae{x}^{2}}{2}}+{\frac{ \left ( fa+bd \right ){x}^{3}}{3}}+{\frac{ \left ( ag+be \right ){x}^{4}}{4}}+{\frac{ \left ( bf+cd \right ){x}^{5}}{5}}+{\frac{ \left ( bg+ce \right ){x}^{6}}{6}}+{\frac{cf{x}^{7}}{7}}+{\frac{cg{x}^{8}}{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((g*x^3+f*x^2+e*x+d)*(c*x^4+b*x^2+a),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*(b*f+c*d)*x^5+1/6*(b*g
+c*e)*x^6+1/7*c*f*x^7+1/8*c*g*x^8

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Maxima [A]  time = 0.702221, size = 100, normalized size = 1.14 \[ \frac{1}{8} \, c g x^{8} + \frac{1}{7} \, c f x^{7} + \frac{1}{6} \,{\left (c e + b g\right )} x^{6} + \frac{1}{5} \,{\left (c d + b f\right )} x^{5} + \frac{1}{4} \,{\left (b e + a g\right )} x^{4} + \frac{1}{2} \, a e x^{2} + \frac{1}{3} \,{\left (b d + a f\right )} x^{3} + a d x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*c*g*x^8 + 1/7*c*f*x^7 + 1/6*(c*e + b*g)*x^6 + 1/5*(c*d + b*f)*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 = 0.241215, size = 1, normalized size = 0.01 \[ \frac{1}{8} x^{8} g c + \frac{1}{7} x^{7} f c + \frac{1}{6} x^{6} e c + \frac{1}{6} x^{6} g b + \frac{1}{5} x^{5} d c + \frac{1}{5} x^{5} f b + \frac{1}{4} x^{4} e b + \frac{1}{4} x^{4} g a + \frac{1}{3} x^{3} d b + \frac{1}{3} x^{3} f a + \frac{1}{2} x^{2} e a + x d a \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*x^8*g*c + 1/7*x^7*f*c + 1/6*x^6*e*c + 1/6*x^6*g*b + 1/5*x^5*d*c + 1/5*x^5*f*
b + 1/4*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 = 0.120193, size = 83, normalized size = 0.94 \[ a d x + \frac{a e x^{2}}{2} + \frac{c f x^{7}}{7} + \frac{c g x^{8}}{8} + x^{6} \left (\frac{b g}{6} + \frac{c e}{6}\right ) + x^{5} \left (\frac{b f}{5} + \frac{c d}{5}\right ) + x^{4} \left (\frac{a g}{4} + \frac{b e}{4}\right ) + x^{3} \left (\frac{a f}{3} + \frac{b d}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*d*x + a*e*x**2/2 + c*f*x**7/7 + c*g*x**8/8 + x**6*(b*g/6 + c*e/6) + x**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/XCAS [A]  time = 0.287478, size = 115, normalized size = 1.31 \[ \frac{1}{8} \, c g x^{8} + \frac{1}{7} \, c f x^{7} + \frac{1}{6} \, b g x^{6} + \frac{1}{6} \, c x^{6} e + \frac{1}{5} \, c d x^{5} + \frac{1}{5} \, b f x^{5} + \frac{1}{4} \, a g x^{4} + \frac{1}{4} \, b x^{4} e + \frac{1}{3} \, b d x^{3} + \frac{1}{3} \, a f x^{3} + \frac{1}{2} \, a x^{2} e + a d x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*c*g*x^8 + 1/7*c*f*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 + 1/4*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