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

Optimal. Leaf size=130 \[ \frac {1}{3} e^2 x^3 (3 a c f+a d e+b c e)+\frac {1}{9} f^2 x^9 (a d f+b c f+3 b d e)+\frac {1}{7} f x^7 (a f (c f+3 d e)+3 b e (c f+d e))+\frac {1}{5} e x^5 (3 a f (c f+d e)+b e (3 c f+d e))+a c e^3 x+\frac {1}{11} b d f^3 x^{11} \]

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Rubi [A]  time = 0.13, antiderivative size = 130, 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}{3} e^2 x^3 (3 a c f+a d e+b c e)+\frac {1}{9} f^2 x^9 (a d f+b c f+3 b d e)+\frac {1}{7} f x^7 (a f (c f+3 d e)+3 b e (c f+d e))+\frac {1}{5} e x^5 (3 a f (c f+d e)+b e (3 c f+d e))+a c e^3 x+\frac {1}{11} b d f^3 x^{11} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

a*c*e^3*x + (e^2*(b*c*e + a*d*e + 3*a*c*f)*x^3)/3 + (e*(3*a*f*(d*e + c*f) + b*e*(d*e + 3*c*f))*x^5)/5 + (f*(3*
b*e*(d*e + c*f) + a*f*(3*d*e + c*f))*x^7)/7 + (f^2*(3*b*d*e + b*c*f + a*d*f)*x^9)/9 + (b*d*f^3*x^11)/11

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 )^3 \, dx &=\int \left (a c e^3+e^2 (b c e+a d e+3 a c f) x^2+e (3 a f (d e+c f)+b e (d e+3 c f)) x^4+f (3 b e (d e+c f)+a f (3 d e+c f)) x^6+f^2 (3 b d e+b c f+a d f) x^8+b d f^3 x^{10}\right ) \, dx\\ &=a c e^3 x+\frac {1}{3} e^2 (b c e+a d e+3 a c f) x^3+\frac {1}{5} e (3 a f (d e+c f)+b e (d e+3 c f)) x^5+\frac {1}{7} f (3 b e (d e+c f)+a f (3 d e+c f)) x^7+\frac {1}{9} f^2 (3 b d e+b c f+a d f) x^9+\frac {1}{11} b d f^3 x^{11}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 130, normalized size = 1.00 \begin {gather*} \frac {1}{3} e^2 x^3 (3 a c f+a d e+b c e)+\frac {1}{9} f^2 x^9 (a d f+b c f+3 b d e)+\frac {1}{7} f x^7 (a f (c f+3 d e)+3 b e (c f+d e))+\frac {1}{5} e x^5 (3 a f (c f+d e)+b e (3 c f+d e))+a c e^3 x+\frac {1}{11} b d f^3 x^{11} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

a*c*e^3*x + (e^2*(b*c*e + a*d*e + 3*a*c*f)*x^3)/3 + (e*(3*a*f*(d*e + c*f) + b*e*(d*e + 3*c*f))*x^5)/5 + (f*(3*
b*e*(d*e + c*f) + a*f*(3*d*e + c*f))*x^7)/7 + (f^2*(3*b*d*e + b*c*f + a*d*f)*x^9)/9 + (b*d*f^3*x^11)/11

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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 )^3 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.07, size = 165, normalized size = 1.27 \begin {gather*} \frac {1}{11} x^{11} f^{3} d b + \frac {1}{3} x^{9} f^{2} e d b + \frac {1}{9} x^{9} f^{3} c b + \frac {1}{9} x^{9} f^{3} d a + \frac {3}{7} x^{7} f e^{2} d b + \frac {3}{7} x^{7} f^{2} e c b + \frac {3}{7} x^{7} f^{2} e d a + \frac {1}{7} x^{7} f^{3} c a + \frac {1}{5} x^{5} e^{3} d b + \frac {3}{5} x^{5} f e^{2} c b + \frac {3}{5} x^{5} f e^{2} d a + \frac {3}{5} x^{5} f^{2} e c a + \frac {1}{3} x^{3} e^{3} c b + \frac {1}{3} x^{3} e^{3} d a + x^{3} f e^{2} c a + x e^{3} 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)^3,x, algorithm="fricas")

[Out]

1/11*x^11*f^3*d*b + 1/3*x^9*f^2*e*d*b + 1/9*x^9*f^3*c*b + 1/9*x^9*f^3*d*a + 3/7*x^7*f*e^2*d*b + 3/7*x^7*f^2*e*
c*b + 3/7*x^7*f^2*e*d*a + 1/7*x^7*f^3*c*a + 1/5*x^5*e^3*d*b + 3/5*x^5*f*e^2*c*b + 3/5*x^5*f*e^2*d*a + 3/5*x^5*
f^2*e*c*a + 1/3*x^3*e^3*c*b + 1/3*x^3*e^3*d*a + x^3*f*e^2*c*a + x*e^3*c*a

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giac [A]  time = 0.29, size = 161, normalized size = 1.24 \begin {gather*} \frac {1}{11} \, b d f^{3} x^{11} + \frac {1}{9} \, b c f^{3} x^{9} + \frac {1}{9} \, a d f^{3} x^{9} + \frac {1}{3} \, b d f^{2} x^{9} e + \frac {1}{7} \, a c f^{3} x^{7} + \frac {3}{7} \, b c f^{2} x^{7} e + \frac {3}{7} \, a d f^{2} x^{7} e + \frac {3}{7} \, b d f x^{7} e^{2} + \frac {3}{5} \, a c f^{2} x^{5} e + \frac {3}{5} \, b c f x^{5} e^{2} + \frac {3}{5} \, a d f x^{5} e^{2} + \frac {1}{5} \, b d x^{5} e^{3} + a c f x^{3} e^{2} + \frac {1}{3} \, b c x^{3} e^{3} + \frac {1}{3} \, a d x^{3} e^{3} + a c x e^{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)^3,x, algorithm="giac")

[Out]

1/11*b*d*f^3*x^11 + 1/9*b*c*f^3*x^9 + 1/9*a*d*f^3*x^9 + 1/3*b*d*f^2*x^9*e + 1/7*a*c*f^3*x^7 + 3/7*b*c*f^2*x^7*
e + 3/7*a*d*f^2*x^7*e + 3/7*b*d*f*x^7*e^2 + 3/5*a*c*f^2*x^5*e + 3/5*b*c*f*x^5*e^2 + 3/5*a*d*f*x^5*e^2 + 1/5*b*
d*x^5*e^3 + a*c*f*x^3*e^2 + 1/3*b*c*x^3*e^3 + 1/3*a*d*x^3*e^3 + a*c*x*e^3

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maple [A]  time = 0.00, size = 135, normalized size = 1.04 \begin {gather*} \frac {b d \,f^{3} x^{11}}{11}+\frac {\left (3 b d e \,f^{2}+\left (a d +b c \right ) f^{3}\right ) x^{9}}{9}+\frac {\left (a c \,f^{3}+3 b d \,e^{2} f +3 \left (a d +b c \right ) e \,f^{2}\right ) x^{7}}{7}+a c \,e^{3} x +\frac {\left (3 a c e \,f^{2}+b d \,e^{3}+3 \left (a d +b c \right ) e^{2} f \right ) x^{5}}{5}+\frac {\left (3 a c \,e^{2} f +\left (a d +b c \right ) e^{3}\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)^3,x)

[Out]

1/11*b*d*f^3*x^11+1/9*((a*d+b*c)*f^3+3*b*d*e*f^2)*x^9+1/7*(a*c*f^3+3*(a*d+b*c)*e*f^2+3*b*d*e^2*f)*x^7+1/5*(3*a
*c*e*f^2+3*(a*d+b*c)*e^2*f+b*d*e^3)*x^5+1/3*(3*a*c*e^2*f+(a*d+b*c)*e^3)*x^3+a*c*e^3*x

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maxima [A]  time = 0.75, size = 134, normalized size = 1.03 \begin {gather*} \frac {1}{11} \, b d f^{3} x^{11} + \frac {1}{9} \, {\left (3 \, b d e f^{2} + {\left (b c + a d\right )} f^{3}\right )} x^{9} + \frac {1}{7} \, {\left (3 \, b d e^{2} f + a c f^{3} + 3 \, {\left (b c + a d\right )} e f^{2}\right )} x^{7} + a c e^{3} x + \frac {1}{5} \, {\left (b d e^{3} + 3 \, a c e f^{2} + 3 \, {\left (b c + a d\right )} e^{2} f\right )} x^{5} + \frac {1}{3} \, {\left (3 \, a c e^{2} f + {\left (b c + a d\right )} e^{3}\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)^3,x, algorithm="maxima")

[Out]

1/11*b*d*f^3*x^11 + 1/9*(3*b*d*e*f^2 + (b*c + a*d)*f^3)*x^9 + 1/7*(3*b*d*e^2*f + a*c*f^3 + 3*(b*c + a*d)*e*f^2
)*x^7 + a*c*e^3*x + 1/5*(b*d*e^3 + 3*a*c*e*f^2 + 3*(b*c + a*d)*e^2*f)*x^5 + 1/3*(3*a*c*e^2*f + (b*c + a*d)*e^3
)*x^3

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mupad [B]  time = 0.84, size = 143, normalized size = 1.10 \begin {gather*} x^5\,\left (\frac {b\,d\,e^3}{5}+\frac {3\,a\,c\,e\,f^2}{5}+\frac {3\,a\,d\,e^2\,f}{5}+\frac {3\,b\,c\,e^2\,f}{5}\right )+x^7\,\left (\frac {a\,c\,f^3}{7}+\frac {3\,a\,d\,e\,f^2}{7}+\frac {3\,b\,c\,e\,f^2}{7}+\frac {3\,b\,d\,e^2\,f}{7}\right )+x^3\,\left (\frac {a\,d\,e^3}{3}+\frac {b\,c\,e^3}{3}+a\,c\,e^2\,f\right )+x^9\,\left (\frac {a\,d\,f^3}{9}+\frac {b\,c\,f^3}{9}+\frac {b\,d\,e\,f^2}{3}\right )+a\,c\,e^3\,x+\frac {b\,d\,f^3\,x^{11}}{11} \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)^3,x)

[Out]

x^5*((b*d*e^3)/5 + (3*a*c*e*f^2)/5 + (3*a*d*e^2*f)/5 + (3*b*c*e^2*f)/5) + x^7*((a*c*f^3)/7 + (3*a*d*e*f^2)/7 +
(3*b*c*e*f^2)/7 + (3*b*d*e^2*f)/7) + x^3*((a*d*e^3)/3 + (b*c*e^3)/3 + a*c*e^2*f) + x^9*((a*d*f^3)/9 + (b*c*f^
3)/9 + (b*d*e*f^2)/3) + a*c*e^3*x + (b*d*f^3*x^11)/11

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sympy [A]  time = 0.09, size = 173, normalized size = 1.33 \begin {gather*} a c e^{3} x + \frac {b d f^{3} x^{11}}{11} + x^{9} \left (\frac {a d f^{3}}{9} + \frac {b c f^{3}}{9} + \frac {b d e f^{2}}{3}\right ) + x^{7} \left (\frac {a c f^{3}}{7} + \frac {3 a d e f^{2}}{7} + \frac {3 b c e f^{2}}{7} + \frac {3 b d e^{2} f}{7}\right ) + x^{5} \left (\frac {3 a c e f^{2}}{5} + \frac {3 a d e^{2} f}{5} + \frac {3 b c e^{2} f}{5} + \frac {b d e^{3}}{5}\right ) + x^{3} \left (a c e^{2} f + \frac {a d e^{3}}{3} + \frac {b c e^{3}}{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)**3,x)

[Out]

a*c*e**3*x + b*d*f**3*x**11/11 + x**9*(a*d*f**3/9 + b*c*f**3/9 + b*d*e*f**2/3) + x**7*(a*c*f**3/7 + 3*a*d*e*f*
*2/7 + 3*b*c*e*f**2/7 + 3*b*d*e**2*f/7) + x**5*(3*a*c*e*f**2/5 + 3*a*d*e**2*f/5 + 3*b*c*e**2*f/5 + b*d*e**3/5)
+ x**3*(a*c*e**2*f + a*d*e**3/3 + b*c*e**3/3)

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