3.3 \(\int (a+b x^2) (c+d x^2) (e+f x^2)^2 \, dx\)

Optimal. Leaf size=94 \[ \frac{1}{7} f x^7 (a d f+b c f+2 b d e)+\frac{1}{5} x^5 (a f (c f+2 d e)+b e (2 c f+d e))+\frac{1}{3} e x^3 (2 a c f+a d e+b c e)+a c e^2 x+\frac{1}{9} b d f^2 x^9 \]

[Out]

a*c*e^2*x + (e*(b*c*e + a*d*e + 2*a*c*f)*x^3)/3 + ((a*f*(2*d*e + c*f) + b*e*(d*e + 2*c*f))*x^5)/5 + (f*(2*b*d*
e + b*c*f + a*d*f)*x^7)/7 + (b*d*f^2*x^9)/9

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Rubi [A]  time = 0.0826775, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {521} \[ \frac{1}{7} f x^7 (a d f+b c f+2 b d e)+\frac{1}{5} x^5 (a f (c f+2 d e)+b e (2 c f+d e))+\frac{1}{3} e x^3 (2 a c f+a d e+b c e)+a c e^2 x+\frac{1}{9} b d f^2 x^9 \]

Antiderivative was successfully verified.

[In]

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

[Out]

a*c*e^2*x + (e*(b*c*e + a*d*e + 2*a*c*f)*x^3)/3 + ((a*f*(2*d*e + c*f) + b*e*(d*e + 2*c*f))*x^5)/5 + (f*(2*b*d*
e + b*c*f + a*d*f)*x^7)/7 + (b*d*f^2*x^9)/9

Rule 521

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_))^(r_.), x_Symbol] :>
Int[ExpandIntegrand[(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && I
GtQ[p, 0] && IGtQ[q, 0] && IGtQ[r, 0]

Rubi steps

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

Mathematica [A]  time = 0.0358652, size = 96, normalized size = 1.02 \[ \frac{1}{5} x^5 \left (a c f^2+2 a d e f+2 b c e f+b d e^2\right )+\frac{1}{7} f x^7 (a d f+b c f+2 b d e)+\frac{1}{3} e x^3 (2 a c f+a d e+b c e)+a c e^2 x+\frac{1}{9} b d f^2 x^9 \]

Antiderivative was successfully verified.

[In]

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

[Out]

a*c*e^2*x + (e*(b*c*e + a*d*e + 2*a*c*f)*x^3)/3 + ((b*d*e^2 + 2*b*c*e*f + 2*a*d*e*f + a*c*f^2)*x^5)/5 + (f*(2*
b*d*e + b*c*f + a*d*f)*x^7)/7 + (b*d*f^2*x^9)/9

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Maple [A]  time = 0., size = 94, normalized size = 1. \begin{align*}{\frac{bd{f}^{2}{x}^{9}}{9}}+{\frac{ \left ( \left ( ad+bc \right ){f}^{2}+2\,bdef \right ){x}^{7}}{7}}+{\frac{ \left ( ac{f}^{2}+2\, \left ( ad+bc \right ) ef+bd{e}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,acef+ \left ( ad+bc \right ){e}^{2} \right ){x}^{3}}{3}}+ac{e}^{2}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*b*d*f^2*x^9+1/7*((a*d+b*c)*f^2+2*b*d*e*f)*x^7+1/5*(a*c*f^2+2*(a*d+b*c)*e*f+b*d*e^2)*x^5+1/3*(2*a*c*e*f+(a*
d+b*c)*e^2)*x^3+a*c*e^2*x

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Maxima [A]  time = 0.990341, size = 126, normalized size = 1.34 \begin{align*} \frac{1}{9} \, b d f^{2} x^{9} + \frac{1}{7} \,{\left (2 \, b d e f +{\left (b c + a d\right )} f^{2}\right )} x^{7} + \frac{1}{5} \,{\left (b d e^{2} + a c f^{2} + 2 \,{\left (b c + a d\right )} e f\right )} x^{5} + a c e^{2} x + \frac{1}{3} \,{\left (2 \, a c e f +{\left (b c + a d\right )} e^{2}\right )} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*b*d*f^2*x^9 + 1/7*(2*b*d*e*f + (b*c + a*d)*f^2)*x^7 + 1/5*(b*d*e^2 + a*c*f^2 + 2*(b*c + a*d)*e*f)*x^5 + a*
c*e^2*x + 1/3*(2*a*c*e*f + (b*c + a*d)*e^2)*x^3

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Fricas [A]  time = 1.24458, size = 282, normalized size = 3. \begin{align*} \frac{1}{9} x^{9} f^{2} d b + \frac{2}{7} x^{7} f e d b + \frac{1}{7} x^{7} f^{2} c b + \frac{1}{7} x^{7} f^{2} d a + \frac{1}{5} x^{5} e^{2} d b + \frac{2}{5} x^{5} f e c b + \frac{2}{5} x^{5} f e d a + \frac{1}{5} x^{5} f^{2} c a + \frac{1}{3} x^{3} e^{2} c b + \frac{1}{3} x^{3} e^{2} d a + \frac{2}{3} x^{3} f e c a + x e^{2} c a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*x^9*f^2*d*b + 2/7*x^7*f*e*d*b + 1/7*x^7*f^2*c*b + 1/7*x^7*f^2*d*a + 1/5*x^5*e^2*d*b + 2/5*x^5*f*e*c*b + 2/
5*x^5*f*e*d*a + 1/5*x^5*f^2*c*a + 1/3*x^3*e^2*c*b + 1/3*x^3*e^2*d*a + 2/3*x^3*f*e*c*a + x*e^2*c*a

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Sympy [A]  time = 0.072183, size = 121, normalized size = 1.29 \begin{align*} a c e^{2} x + \frac{b d f^{2} x^{9}}{9} + x^{7} \left (\frac{a d f^{2}}{7} + \frac{b c f^{2}}{7} + \frac{2 b d e f}{7}\right ) + x^{5} \left (\frac{a c f^{2}}{5} + \frac{2 a d e f}{5} + \frac{2 b c e f}{5} + \frac{b d e^{2}}{5}\right ) + x^{3} \left (\frac{2 a c e f}{3} + \frac{a d e^{2}}{3} + \frac{b c e^{2}}{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*c*e**2*x + b*d*f**2*x**9/9 + x**7*(a*d*f**2/7 + b*c*f**2/7 + 2*b*d*e*f/7) + x**5*(a*c*f**2/5 + 2*a*d*e*f/5 +
2*b*c*e*f/5 + b*d*e**2/5) + x**3*(2*a*c*e*f/3 + a*d*e**2/3 + b*c*e**2/3)

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Giac [A]  time = 1.21591, size = 154, normalized size = 1.64 \begin{align*} \frac{1}{9} \, b d f^{2} x^{9} + \frac{1}{7} \, b c f^{2} x^{7} + \frac{1}{7} \, a d f^{2} x^{7} + \frac{2}{7} \, b d f x^{7} e + \frac{1}{5} \, a c f^{2} x^{5} + \frac{2}{5} \, b c f x^{5} e + \frac{2}{5} \, a d f x^{5} e + \frac{1}{5} \, b d x^{5} e^{2} + \frac{2}{3} \, a c f x^{3} e + \frac{1}{3} \, b c x^{3} e^{2} + \frac{1}{3} \, a d x^{3} e^{2} + a c x e^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*b*d*f^2*x^9 + 1/7*b*c*f^2*x^7 + 1/7*a*d*f^2*x^7 + 2/7*b*d*f*x^7*e + 1/5*a*c*f^2*x^5 + 2/5*b*c*f*x^5*e + 2/
5*a*d*f*x^5*e + 1/5*b*d*x^5*e^2 + 2/3*a*c*f*x^3*e + 1/3*b*c*x^3*e^2 + 1/3*a*d*x^3*e^2 + a*c*x*e^2