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

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

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Rubi [A]  time = 0.17, antiderivative size = 196, normalized size of antiderivative = 1.00, 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} \begin {gather*} 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} \end {gather*}

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*}

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Mathematica [A]  time = 0.06, size = 196, normalized size = 1.00 \begin {gather*} 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} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (d+e x+f x^2+g x^3\right ) \left (a+b x^2+c x^4\right )^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.63, size = 202, normalized size = 1.03 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.30, size = 208, normalized size = 1.06 \begin {gather*} \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 {gather*}

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

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maple [A]  time = 0.00, size = 183, normalized size = 0.93 \begin {gather*} \frac {c^{2} g \,x^{12}}{12}+\frac {c^{2} f \,x^{11}}{11}+\frac {\left (2 g b c +e \,c^{2}\right ) x^{10}}{10}+\frac {\left (2 f b c +c^{2} d \right ) x^{9}}{9}+\frac {\left (2 b c e +\left (2 a c +b^{2}\right ) g \right ) x^{8}}{8}+\frac {\left (2 b c d +\left (2 a c +b^{2}\right ) f \right ) x^{7}}{7}+\frac {\left (2 a b g +\left (2 a c +b^{2}\right ) e \right ) x^{6}}{6}+\frac {a^{2} e \,x^{2}}{2}+\frac {\left (2 a b f +\left (2 a c +b^{2}\right ) d \right ) x^{5}}{5}+a^{2} d x +\frac {\left (g \,a^{2}+2 a b e \right ) x^{4}}{4}+\frac {\left (f \,a^{2}+2 a b d \right ) x^{3}}{3} \end {gather*}

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+(2*a*c+b^2)*f)*x^7+1/6*(e*(2*a*c+b^2)+2*a*b*g)*x^6+1/5*(2*a*b*f+(2*a*c+b^2)*d)*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 = 1.40, size = 182, normalized size = 0.93 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 0.72, size = 182, normalized size = 0.93 \begin {gather*} x^5\,\left (\frac {d\,b^2}{5}+\frac {2\,a\,f\,b}{5}+\frac {2\,a\,c\,d}{5}\right )+x^6\,\left (\frac {e\,b^2}{6}+\frac {a\,g\,b}{3}+\frac {a\,c\,e}{3}\right )+x^7\,\left (\frac {f\,b^2}{7}+\frac {2\,c\,d\,b}{7}+\frac {2\,a\,c\,f}{7}\right )+x^8\,\left (\frac {g\,b^2}{8}+\frac {c\,e\,b}{4}+\frac {a\,c\,g}{4}\right )+x^3\,\left (\frac {f\,a^2}{3}+\frac {2\,b\,d\,a}{3}\right )+x^4\,\left (\frac {g\,a^2}{4}+\frac {b\,e\,a}{2}\right )+x^9\,\left (\frac {d\,c^2}{9}+\frac {2\,b\,f\,c}{9}\right )+x^{10}\,\left (\frac {e\,c^2}{10}+\frac {b\,g\,c}{5}\right )+\frac {a^2\,e\,x^2}{2}+\frac {c^2\,f\,x^{11}}{11}+\frac {c^2\,g\,x^{12}}{12}+a^2\,d\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.10, size = 209, normalized size = 1.07 \begin {gather*} 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 {gather*}

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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