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

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

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Rubi [A]  time = 0.08, antiderivative size = 94, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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 \end {gather*}

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*}

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Mathematica [A]  time = 0.04, size = 96, normalized size = 1.02 \begin {gather*} \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 \end {gather*}

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.17, size = 114, normalized size = 1.21 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.28, size = 114, normalized size = 1.21 \begin {gather*} \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 {gather*}

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

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maple [A]  time = 0.00, size = 94, normalized size = 1.00 \begin {gather*} \frac {b d \,f^{2} x^{9}}{9}+\frac {\left (2 b d e f +\left (a d +b c \right ) f^{2}\right ) x^{7}}{7}+a c \,e^{2} x +\frac {\left (a c \,f^{2}+b d \,e^{2}+2 \left (a d +b c \right ) e f \right ) x^{5}}{5}+\frac {\left (2 a c e f +\left (a d +b c \right ) e^{2}\right ) x^{3}}{3} \end {gather*}

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 = 0.57, size = 93, normalized size = 0.99 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 0.05, size = 99, normalized size = 1.05 \begin {gather*} x^5\,\left (\frac {a\,c\,f^2}{5}+\frac {b\,d\,e^2}{5}+\frac {2\,a\,d\,e\,f}{5}+\frac {2\,b\,c\,e\,f}{5}\right )+x^3\,\left (\frac {a\,d\,e^2}{3}+\frac {b\,c\,e^2}{3}+\frac {2\,a\,c\,e\,f}{3}\right )+x^7\,\left (\frac {a\,d\,f^2}{7}+\frac {b\,c\,f^2}{7}+\frac {2\,b\,d\,e\,f}{7}\right )+a\,c\,e^2\,x+\frac {b\,d\,f^2\,x^9}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.08, size = 121, normalized size = 1.29 \begin {gather*} 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 {gather*}

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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