3.6 \(\int (d+e x) (a+b x^2+c x^4)^2 \, dx\)

Optimal. Leaf size=112 \[ a^2 d x+\frac{1}{2} a^2 e x^2+\frac{1}{5} d x^5 \left (2 a c+b^2\right )+\frac{1}{6} e x^6 \left (2 a c+b^2\right )+\frac{2}{3} a b d x^3+\frac{1}{2} a b e x^4+\frac{2}{7} b c d x^7+\frac{1}{4} b c e x^8+\frac{1}{9} c^2 d x^9+\frac{1}{10} c^2 e x^{10} \]

[Out]

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

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Rubi [A]  time = 0.125778, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {1671} \[ a^2 d x+\frac{1}{2} a^2 e x^2+\frac{1}{5} d x^5 \left (2 a c+b^2\right )+\frac{1}{6} e x^6 \left (2 a c+b^2\right )+\frac{2}{3} a b d x^3+\frac{1}{2} a b e x^4+\frac{2}{7} b c d x^7+\frac{1}{4} b c e x^8+\frac{1}{9} c^2 d x^9+\frac{1}{10} c^2 e x^{10} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0506573, size = 97, normalized size = 0.87 \[ \frac{630 a^2 x (2 d+e x)+42 a \left (5 b x^3 (4 d+3 e x)+2 c x^5 (6 d+5 e x)\right )+42 b^2 x^5 (6 d+5 e x)+45 b c x^7 (8 d+7 e x)+14 c^2 x^9 (10 d+9 e x)}{1260} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(630*a^2*x*(2*d + e*x) + 42*b^2*x^5*(6*d + 5*e*x) + 45*b*c*x^7*(8*d + 7*e*x) + 14*c^2*x^9*(10*d + 9*e*x) + 42*
a*(5*b*x^3*(4*d + 3*e*x) + 2*c*x^5*(6*d + 5*e*x)))/1260

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Maple [A]  time = 0.001, size = 95, normalized size = 0.9 \begin{align*}{a}^{2}dx+{\frac{{a}^{2}e{x}^{2}}{2}}+{\frac{2\,abd{x}^{3}}{3}}+{\frac{abe{x}^{4}}{2}}+{\frac{ \left ( 2\,ac+{b}^{2} \right ) d{x}^{5}}{5}}+{\frac{ \left ( 2\,ac+{b}^{2} \right ) e{x}^{6}}{6}}+{\frac{2\,bcd{x}^{7}}{7}}+{\frac{bce{x}^{8}}{4}}+{\frac{{c}^{2}d{x}^{9}}{9}}+{\frac{{c}^{2}e{x}^{10}}{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.951583, size = 127, normalized size = 1.13 \begin{align*} \frac{1}{10} \, c^{2} e x^{10} + \frac{1}{9} \, c^{2} d x^{9} + \frac{1}{4} \, b c e x^{8} + \frac{2}{7} \, b c d x^{7} + \frac{1}{6} \,{\left (b^{2} + 2 \, a c\right )} e x^{6} + \frac{1}{2} \, a b e x^{4} + \frac{1}{5} \,{\left (b^{2} + 2 \, a c\right )} d x^{5} + \frac{2}{3} \, a b d x^{3} + \frac{1}{2} \, a^{2} e x^{2} + a^{2} d x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.47995, size = 252, normalized size = 2.25 \begin{align*} \frac{1}{10} x^{10} e c^{2} + \frac{1}{9} x^{9} d c^{2} + \frac{1}{4} x^{8} e c b + \frac{2}{7} x^{7} d c b + \frac{1}{6} x^{6} e b^{2} + \frac{1}{3} x^{6} e c a + \frac{1}{5} x^{5} d b^{2} + \frac{2}{5} x^{5} d c a + \frac{1}{2} x^{4} e b a + \frac{2}{3} x^{3} d b a + \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((e*x+d)*(c*x^4+b*x^2+a)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.08321, size = 116, normalized size = 1.04 \begin{align*} a^{2} d x + \frac{a^{2} e x^{2}}{2} + \frac{2 a b d x^{3}}{3} + \frac{a b e x^{4}}{2} + \frac{2 b c d x^{7}}{7} + \frac{b c e x^{8}}{4} + \frac{c^{2} d x^{9}}{9} + \frac{c^{2} e x^{10}}{10} + x^{6} \left (\frac{a c e}{3} + \frac{b^{2} e}{6}\right ) + x^{5} \left (\frac{2 a c d}{5} + \frac{b^{2} d}{5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(c*x**4+b*x**2+a)**2,x)

[Out]

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

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Giac [A]  time = 1.0943, size = 143, normalized size = 1.28 \begin{align*} \frac{1}{10} \, c^{2} x^{10} e + \frac{1}{9} \, c^{2} d x^{9} + \frac{1}{4} \, b c x^{8} e + \frac{2}{7} \, b c d x^{7} + \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{1}{2} \, a b x^{4} e + \frac{2}{3} \, a b d 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((e*x+d)*(c*x^4+b*x^2+a)^2,x, algorithm="giac")

[Out]

1/10*c^2*x^10*e + 1/9*c^2*d*x^9 + 1/4*b*c*x^8*e + 2/7*b*c*d*x^7 + 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 + 1/2*a*b*x^4*e + 2/3*a*b*d*x^3 + 1/2*a^2*x^2*e + a^2*d*x