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

Optimal. Leaf size=196 \[ a^2 d x+\frac{1}{2} a^2 e x^2+\frac{1}{7} x^7 \left (2 a c f+b^2 f+2 b c d\right )+\frac{1}{5} x^5 \left (2 a b f+2 a c d+b^2 d\right )+\frac{1}{8} x^8 \left (2 a c g+b^2 g+2 b c e\right )+\frac{1}{6} x^6 \left (2 a b g+2 a c e+b^2 e\right )+\frac{1}{3} a x^3 (a f+2 b d)+\frac{1}{4} a x^4 (a g+2 b e)+\frac{1}{9} c x^9 (2 b f+c d)+\frac{1}{10} c x^{10} (2 b g+c e)+\frac{1}{11} c^2 f x^{11}+\frac{1}{12} c^2 g x^{12} \]

[Out]

a^2*d*x + (a^2*e*x^2)/2 + (a*(2*b*d + a*f)*x^3)/3 + (a*(2*b*e + a*g)*x^4)/4 + ((b^2*d + 2*a*c*d + 2*a*b*f)*x^5
)/5 + ((b^2*e + 2*a*c*e + 2*a*b*g)*x^6)/6 + ((2*b*c*d + b^2*f + 2*a*c*f)*x^7)/7 + ((2*b*c*e + b^2*g + 2*a*c*g)
*x^8)/8 + (c*(c*d + 2*b*f)*x^9)/9 + (c*(c*e + 2*b*g)*x^10)/10 + (c^2*f*x^11)/11 + (c^2*g*x^12)/12

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Rubi [A]  time = 0.167512, antiderivative size = 196, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.033, Rules used = {1671} \[ a^2 d x+\frac{1}{2} a^2 e x^2+\frac{1}{7} x^7 \left (2 a c f+b^2 f+2 b c d\right )+\frac{1}{5} x^5 \left (2 a b f+2 a c d+b^2 d\right )+\frac{1}{8} x^8 \left (2 a c g+b^2 g+2 b c e\right )+\frac{1}{6} x^6 \left (2 a b g+2 a c e+b^2 e\right )+\frac{1}{3} a x^3 (a f+2 b d)+\frac{1}{4} a x^4 (a g+2 b e)+\frac{1}{9} c x^9 (2 b f+c d)+\frac{1}{10} c x^{10} (2 b g+c e)+\frac{1}{11} c^2 f x^{11}+\frac{1}{12} c^2 g x^{12} \]

Antiderivative was successfully verified.

[In]

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

[Out]

a^2*d*x + (a^2*e*x^2)/2 + (a*(2*b*d + a*f)*x^3)/3 + (a*(2*b*e + a*g)*x^4)/4 + ((b^2*d + 2*a*c*d + 2*a*b*f)*x^5
)/5 + ((b^2*e + 2*a*c*e + 2*a*b*g)*x^6)/6 + ((2*b*c*d + b^2*f + 2*a*c*f)*x^7)/7 + ((2*b*c*e + b^2*g + 2*a*c*g)
*x^8)/8 + (c*(c*d + 2*b*f)*x^9)/9 + (c*(c*e + 2*b*g)*x^10)/10 + (c^2*f*x^11)/11 + (c^2*g*x^12)/12

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 (d+e x+f x^2+g x^3\right ) \left (a+b x^2+c x^4\right )^2 \, dx &=\int \left (a^2 d+a^2 e x+a (2 b d+a f) x^2+a (2 b e+a g) x^3+\left (b^2 d+2 a c d+2 a b f\right ) x^4+\left (b^2 e+2 a c e+2 a b g\right ) x^5+\left (2 b c d+b^2 f+2 a c f\right ) x^6+\left (2 b c e+b^2 g+2 a c g\right ) x^7+c (c d+2 b f) x^8+c (c e+2 b g) x^9+c^2 f x^{10}+c^2 g x^{11}\right ) \, dx\\ &=a^2 d x+\frac{1}{2} a^2 e x^2+\frac{1}{3} a (2 b d+a f) x^3+\frac{1}{4} a (2 b e+a g) x^4+\frac{1}{5} \left (b^2 d+2 a c d+2 a b f\right ) x^5+\frac{1}{6} \left (b^2 e+2 a c e+2 a b g\right ) x^6+\frac{1}{7} \left (2 b c d+b^2 f+2 a c f\right ) x^7+\frac{1}{8} \left (2 b c e+b^2 g+2 a c g\right ) x^8+\frac{1}{9} c (c d+2 b f) x^9+\frac{1}{10} c (c e+2 b g) x^{10}+\frac{1}{11} c^2 f x^{11}+\frac{1}{12} c^2 g x^{12}\\ \end{align*}

Mathematica [A]  time = 0.0586498, size = 196, normalized size = 1. \[ a^2 d x+\frac{1}{2} a^2 e x^2+\frac{1}{7} x^7 \left (2 a c f+b^2 f+2 b c d\right )+\frac{1}{5} x^5 \left (2 a b f+2 a c d+b^2 d\right )+\frac{1}{8} x^8 \left (2 a c g+b^2 g+2 b c e\right )+\frac{1}{6} x^6 \left (2 a b g+2 a c e+b^2 e\right )+\frac{1}{3} a x^3 (a f+2 b d)+\frac{1}{4} a x^4 (a g+2 b e)+\frac{1}{9} c x^9 (2 b f+c d)+\frac{1}{10} c x^{10} (2 b g+c e)+\frac{1}{11} c^2 f x^{11}+\frac{1}{12} c^2 g x^{12} \]

Antiderivative was successfully verified.

[In]

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

[Out]

a^2*d*x + (a^2*e*x^2)/2 + (a*(2*b*d + a*f)*x^3)/3 + (a*(2*b*e + a*g)*x^4)/4 + ((b^2*d + 2*a*c*d + 2*a*b*f)*x^5
)/5 + ((b^2*e + 2*a*c*e + 2*a*b*g)*x^6)/6 + ((2*b*c*d + b^2*f + 2*a*c*f)*x^7)/7 + ((2*b*c*e + b^2*g + 2*a*c*g)
*x^8)/8 + (c*(c*d + 2*b*f)*x^9)/9 + (c*(c*e + 2*b*g)*x^10)/10 + (c^2*f*x^11)/11 + (c^2*g*x^12)/12

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Maple [A]  time = 0., size = 183, normalized size = 0.9 \begin{align*}{\frac{{c}^{2}g{x}^{12}}{12}}+{\frac{{c}^{2}f{x}^{11}}{11}}+{\frac{ \left ( 2\,gbc+e{c}^{2} \right ){x}^{10}}{10}}+{\frac{ \left ( 2\,fbc+{c}^{2}d \right ){x}^{9}}{9}}+{\frac{ \left ( 2\,bce+g \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{8}}{8}}+{\frac{ \left ( 2\,bcd+f \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{7}}{7}}+{\frac{ \left ( e \left ( 2\,ac+{b}^{2} \right ) +2\,abg \right ){x}^{6}}{6}}+{\frac{ \left ( d \left ( 2\,ac+{b}^{2} \right ) +2\,abf \right ){x}^{5}}{5}}+{\frac{ \left ( g{a}^{2}+2\,eab \right ){x}^{4}}{4}}+{\frac{ \left ( f{a}^{2}+2\,dab \right ){x}^{3}}{3}}+{\frac{{a}^{2}e{x}^{2}}{2}}+{a}^{2}dx \end{align*}

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)^2,x)

[Out]

1/12*c^2*g*x^12+1/11*c^2*f*x^11+1/10*(2*b*c*g+c^2*e)*x^10+1/9*(2*b*c*f+c^2*d)*x^9+1/8*(2*b*c*e+g*(2*a*c+b^2))*
x^8+1/7*(2*b*c*d+f*(2*a*c+b^2))*x^7+1/6*(e*(2*a*c+b^2)+2*a*b*g)*x^6+1/5*(d*(2*a*c+b^2)+2*a*b*f)*x^5+1/4*(a^2*g
+2*a*b*e)*x^4+1/3*(a^2*f+2*a*b*d)*x^3+1/2*a^2*e*x^2+a^2*d*x

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Maxima [A]  time = 0.949298, size = 246, normalized size = 1.26 \begin{align*} \frac{1}{12} \, c^{2} g x^{12} + \frac{1}{11} \, c^{2} f x^{11} + \frac{1}{10} \,{\left (c^{2} e + 2 \, b c g\right )} x^{10} + \frac{1}{9} \,{\left (c^{2} d + 2 \, b c f\right )} x^{9} + \frac{1}{8} \,{\left (2 \, b c e +{\left (b^{2} + 2 \, a c\right )} g\right )} x^{8} + \frac{1}{7} \,{\left (2 \, b c d +{\left (b^{2} + 2 \, a c\right )} f\right )} x^{7} + \frac{1}{6} \,{\left (2 \, a b g +{\left (b^{2} + 2 \, a c\right )} e\right )} x^{6} + \frac{1}{5} \,{\left (2 \, a b f +{\left (b^{2} + 2 \, a c\right )} d\right )} x^{5} + \frac{1}{2} \, a^{2} e x^{2} + \frac{1}{4} \,{\left (2 \, a b e + a^{2} g\right )} x^{4} + a^{2} d x + \frac{1}{3} \,{\left (2 \, a b d + a^{2} f\right )} x^{3} \end{align*}

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)^2,x, algorithm="maxima")

[Out]

1/12*c^2*g*x^12 + 1/11*c^2*f*x^11 + 1/10*(c^2*e + 2*b*c*g)*x^10 + 1/9*(c^2*d + 2*b*c*f)*x^9 + 1/8*(2*b*c*e + (
b^2 + 2*a*c)*g)*x^8 + 1/7*(2*b*c*d + (b^2 + 2*a*c)*f)*x^7 + 1/6*(2*a*b*g + (b^2 + 2*a*c)*e)*x^6 + 1/5*(2*a*b*f
+ (b^2 + 2*a*c)*d)*x^5 + 1/2*a^2*e*x^2 + 1/4*(2*a*b*e + a^2*g)*x^4 + a^2*d*x + 1/3*(2*a*b*d + a^2*f)*x^3

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Fricas [A]  time = 1.73068, size = 518, normalized size = 2.64 \begin{align*} \frac{1}{12} x^{12} g c^{2} + \frac{1}{11} x^{11} f c^{2} + \frac{1}{10} x^{10} e c^{2} + \frac{1}{5} x^{10} g c b + \frac{1}{9} x^{9} d c^{2} + \frac{2}{9} x^{9} f c b + \frac{1}{4} x^{8} e c b + \frac{1}{8} x^{8} g b^{2} + \frac{1}{4} x^{8} g c a + \frac{2}{7} x^{7} d c b + \frac{1}{7} x^{7} f b^{2} + \frac{2}{7} x^{7} f c a + \frac{1}{6} x^{6} e b^{2} + \frac{1}{3} x^{6} e c a + \frac{1}{3} x^{6} g b a + \frac{1}{5} x^{5} d b^{2} + \frac{2}{5} x^{5} d c a + \frac{2}{5} x^{5} f b a + \frac{1}{2} x^{4} e b a + \frac{1}{4} x^{4} g a^{2} + \frac{2}{3} x^{3} d b a + \frac{1}{3} x^{3} f a^{2} + \frac{1}{2} x^{2} e a^{2} + x d a^{2} \end{align*}

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)^2,x, algorithm="fricas")

[Out]

1/12*x^12*g*c^2 + 1/11*x^11*f*c^2 + 1/10*x^10*e*c^2 + 1/5*x^10*g*c*b + 1/9*x^9*d*c^2 + 2/9*x^9*f*c*b + 1/4*x^8
*e*c*b + 1/8*x^8*g*b^2 + 1/4*x^8*g*c*a + 2/7*x^7*d*c*b + 1/7*x^7*f*b^2 + 2/7*x^7*f*c*a + 1/6*x^6*e*b^2 + 1/3*x
^6*e*c*a + 1/3*x^6*g*b*a + 1/5*x^5*d*b^2 + 2/5*x^5*d*c*a + 2/5*x^5*f*b*a + 1/2*x^4*e*b*a + 1/4*x^4*g*a^2 + 2/3
*x^3*d*b*a + 1/3*x^3*f*a^2 + 1/2*x^2*e*a^2 + x*d*a^2

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Sympy [A]  time = 0.099348, size = 209, normalized size = 1.07 \begin{align*} a^{2} d x + \frac{a^{2} e x^{2}}{2} + \frac{c^{2} f x^{11}}{11} + \frac{c^{2} g x^{12}}{12} + x^{10} \left (\frac{b c g}{5} + \frac{c^{2} e}{10}\right ) + x^{9} \left (\frac{2 b c f}{9} + \frac{c^{2} d}{9}\right ) + x^{8} \left (\frac{a c g}{4} + \frac{b^{2} g}{8} + \frac{b c e}{4}\right ) + x^{7} \left (\frac{2 a c f}{7} + \frac{b^{2} f}{7} + \frac{2 b c d}{7}\right ) + x^{6} \left (\frac{a b g}{3} + \frac{a c e}{3} + \frac{b^{2} e}{6}\right ) + x^{5} \left (\frac{2 a b f}{5} + \frac{2 a c d}{5} + \frac{b^{2} d}{5}\right ) + x^{4} \left (\frac{a^{2} g}{4} + \frac{a b e}{2}\right ) + x^{3} \left (\frac{a^{2} f}{3} + \frac{2 a b d}{3}\right ) \end{align*}

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)**2,x)

[Out]

a**2*d*x + a**2*e*x**2/2 + c**2*f*x**11/11 + c**2*g*x**12/12 + x**10*(b*c*g/5 + c**2*e/10) + x**9*(2*b*c*f/9 +
c**2*d/9) + x**8*(a*c*g/4 + b**2*g/8 + b*c*e/4) + x**7*(2*a*c*f/7 + b**2*f/7 + 2*b*c*d/7) + x**6*(a*b*g/3 + a
*c*e/3 + b**2*e/6) + x**5*(2*a*b*f/5 + 2*a*c*d/5 + b**2*d/5) + x**4*(a**2*g/4 + a*b*e/2) + x**3*(a**2*f/3 + 2*
a*b*d/3)

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Giac [A]  time = 1.09694, size = 281, normalized size = 1.43 \begin{align*} \frac{1}{12} \, c^{2} g x^{12} + \frac{1}{11} \, c^{2} f x^{11} + \frac{1}{5} \, b c g x^{10} + \frac{1}{10} \, c^{2} x^{10} e + \frac{1}{9} \, c^{2} d x^{9} + \frac{2}{9} \, b c f x^{9} + \frac{1}{8} \, b^{2} g x^{8} + \frac{1}{4} \, a c g x^{8} + \frac{1}{4} \, b c x^{8} e + \frac{2}{7} \, b c d x^{7} + \frac{1}{7} \, b^{2} f x^{7} + \frac{2}{7} \, a c f x^{7} + \frac{1}{3} \, a b g x^{6} + \frac{1}{6} \, b^{2} x^{6} e + \frac{1}{3} \, a c x^{6} e + \frac{1}{5} \, b^{2} d x^{5} + \frac{2}{5} \, a c d x^{5} + \frac{2}{5} \, a b f x^{5} + \frac{1}{4} \, a^{2} g x^{4} + \frac{1}{2} \, a b x^{4} e + \frac{2}{3} \, a b d x^{3} + \frac{1}{3} \, a^{2} f x^{3} + \frac{1}{2} \, a^{2} x^{2} e + a^{2} d x \end{align*}

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)^2,x, algorithm="giac")

[Out]

1/12*c^2*g*x^12 + 1/11*c^2*f*x^11 + 1/5*b*c*g*x^10 + 1/10*c^2*x^10*e + 1/9*c^2*d*x^9 + 2/9*b*c*f*x^9 + 1/8*b^2
*g*x^8 + 1/4*a*c*g*x^8 + 1/4*b*c*x^8*e + 2/7*b*c*d*x^7 + 1/7*b^2*f*x^7 + 2/7*a*c*f*x^7 + 1/3*a*b*g*x^6 + 1/6*b
^2*x^6*e + 1/3*a*c*x^6*e + 1/5*b^2*d*x^5 + 2/5*a*c*d*x^5 + 2/5*a*b*f*x^5 + 1/4*a^2*g*x^4 + 1/2*a*b*x^4*e + 2/3
*a*b*d*x^3 + 1/3*a^2*f*x^3 + 1/2*a^2*x^2*e + a^2*d*x