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

Optimal. Leaf size=56 \[ \frac {1}{5} x^5 (a d f+b c f+b d e)+\frac {1}{3} x^3 (a c f+a d e+b c e)+a c e x+\frac {1}{7} b d f x^7 \]

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Rubi [A]  time = 0.04, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {521} \begin {gather*} \frac {1}{5} x^5 (a d f+b c f+b d e)+\frac {1}{3} x^3 (a c f+a d e+b c e)+a c e x+\frac {1}{7} b d f x^7 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.01, size = 56, normalized size = 1.00 \begin {gather*} \frac {1}{5} x^5 (a d f+b c f+b d e)+\frac {1}{3} x^3 (a c f+a d e+b c e)+a c e x+\frac {1}{7} b d f x^7 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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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 ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.63, size = 62, normalized size = 1.11 \begin {gather*} \frac {1}{7} x^{7} f d b + \frac {1}{5} x^{5} e d b + \frac {1}{5} x^{5} f c b + \frac {1}{5} x^{5} f d a + \frac {1}{3} x^{3} e c b + \frac {1}{3} x^{3} e d a + \frac {1}{3} x^{3} f c a + x e 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),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.34, size = 66, normalized size = 1.18 \begin {gather*} \frac {1}{7} \, b d f x^{7} + \frac {1}{5} \, b c f x^{5} + \frac {1}{5} \, a d f x^{5} + \frac {1}{5} \, b d x^{5} e + \frac {1}{3} \, a c f x^{3} + \frac {1}{3} \, b c x^{3} e + \frac {1}{3} \, a d x^{3} e + a c x e \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),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.00, size = 53, normalized size = 0.95 \begin {gather*} \frac {b d f \,x^{7}}{7}+\frac {\left (b d e +\left (a d +b c \right ) f \right ) x^{5}}{5}+a c e x +\frac {\left (a c f +\left (a d +b c \right ) e \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),x)

[Out]

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

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maxima [A]  time = 0.55, size = 52, normalized size = 0.93 \begin {gather*} \frac {1}{7} \, b d f x^{7} + \frac {1}{5} \, {\left (b d e + {\left (b c + a d\right )} f\right )} x^{5} + a c e x + \frac {1}{3} \, {\left (a c f + {\left (b c + a d\right )} e\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),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.05, size = 54, normalized size = 0.96 \begin {gather*} \frac {b\,d\,f\,x^7}{7}+\left (\frac {a\,d\,f}{5}+\frac {b\,c\,f}{5}+\frac {b\,d\,e}{5}\right )\,x^5+\left (\frac {a\,c\,f}{3}+\frac {a\,d\,e}{3}+\frac {b\,c\,e}{3}\right )\,x^3+a\,c\,e\,x \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),x)

[Out]

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

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sympy [A]  time = 0.07, size = 63, normalized size = 1.12 \begin {gather*} a c e x + \frac {b d f x^{7}}{7} + x^{5} \left (\frac {a d f}{5} + \frac {b c f}{5} + \frac {b d e}{5}\right ) + x^{3} \left (\frac {a c f}{3} + \frac {a d e}{3} + \frac {b c e}{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),x)

[Out]

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

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