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

Optimal. Leaf size=172 \[ \frac {1}{3} e^3 x^3 (4 a c f+a d e+b c e)+\frac {1}{5} e^2 x^5 (2 a f (3 c f+2 d e)+b e (4 c f+d e))+\frac {1}{11} f^3 x^{11} (a d f+b c f+4 b d e)+\frac {1}{9} f^2 x^9 (a f (c f+4 d e)+2 b e (2 c f+3 d e))+\frac {2}{7} e f x^7 (a f (2 c f+3 d e)+b e (3 c f+2 d e))+a c e^4 x+\frac {1}{13} b d f^4 x^{13} \]

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Rubi [A]  time = 0.19, antiderivative size = 172, 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}{5} e^2 x^5 (2 a f (3 c f+2 d e)+b e (4 c f+d e))+\frac {1}{3} e^3 x^3 (4 a c f+a d e+b c e)+\frac {1}{11} f^3 x^{11} (a d f+b c f+4 b d e)+\frac {1}{9} f^2 x^9 (a f (c f+4 d e)+2 b e (2 c f+3 d e))+\frac {2}{7} e f x^7 (a f (2 c f+3 d e)+b e (3 c f+2 d e))+a c e^4 x+\frac {1}{13} b d f^4 x^{13} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

a*c*e^4*x + (e^3*(b*c*e + a*d*e + 4*a*c*f)*x^3)/3 + (e^2*(2*a*f*(2*d*e + 3*c*f) + b*e*(d*e + 4*c*f))*x^5)/5 +
(2*e*f*(a*f*(3*d*e + 2*c*f) + b*e*(2*d*e + 3*c*f))*x^7)/7 + (f^2*(a*f*(4*d*e + c*f) + 2*b*e*(3*d*e + 2*c*f))*x
^9)/9 + (f^3*(4*b*d*e + b*c*f + a*d*f)*x^11)/11 + (b*d*f^4*x^13)/13

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 )^4 \, dx &=\int \left (a c e^4+e^3 (b c e+a d e+4 a c f) x^2+e^2 (2 a f (2 d e+3 c f)+b e (d e+4 c f)) x^4+2 e f (a f (3 d e+2 c f)+b e (2 d e+3 c f)) x^6+f^2 (a f (4 d e+c f)+2 b e (3 d e+2 c f)) x^8+f^3 (4 b d e+b c f+a d f) x^{10}+b d f^4 x^{12}\right ) \, dx\\ &=a c e^4 x+\frac {1}{3} e^3 (b c e+a d e+4 a c f) x^3+\frac {1}{5} e^2 (2 a f (2 d e+3 c f)+b e (d e+4 c f)) x^5+\frac {2}{7} e f (a f (3 d e+2 c f)+b e (2 d e+3 c f)) x^7+\frac {1}{9} f^2 (a f (4 d e+c f)+2 b e (3 d e+2 c f)) x^9+\frac {1}{11} f^3 (4 b d e+b c f+a d f) x^{11}+\frac {1}{13} b d f^4 x^{13}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 172, normalized size = 1.00 \begin {gather*} \frac {1}{3} e^3 x^3 (4 a c f+a d e+b c e)+\frac {1}{5} e^2 x^5 (2 a f (3 c f+2 d e)+b e (4 c f+d e))+\frac {1}{11} f^3 x^{11} (a d f+b c f+4 b d e)+\frac {1}{9} f^2 x^9 (a f (c f+4 d e)+2 b e (2 c f+3 d e))+\frac {2}{7} e f x^7 (a f (2 c f+3 d e)+b e (3 c f+2 d e))+a c e^4 x+\frac {1}{13} b d f^4 x^{13} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

a*c*e^4*x + (e^3*(b*c*e + a*d*e + 4*a*c*f)*x^3)/3 + (e^2*(2*a*f*(2*d*e + 3*c*f) + b*e*(d*e + 4*c*f))*x^5)/5 +
(2*e*f*(a*f*(3*d*e + 2*c*f) + b*e*(2*d*e + 3*c*f))*x^7)/7 + (f^2*(a*f*(4*d*e + c*f) + 2*b*e*(3*d*e + 2*c*f))*x
^9)/9 + (f^3*(4*b*d*e + b*c*f + a*d*f)*x^11)/11 + (b*d*f^4*x^13)/13

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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 )^4 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.10, size = 218, normalized size = 1.27 \begin {gather*} \frac {1}{13} x^{13} f^{4} d b + \frac {4}{11} x^{11} f^{3} e d b + \frac {1}{11} x^{11} f^{4} c b + \frac {1}{11} x^{11} f^{4} d a + \frac {2}{3} x^{9} f^{2} e^{2} d b + \frac {4}{9} x^{9} f^{3} e c b + \frac {4}{9} x^{9} f^{3} e d a + \frac {1}{9} x^{9} f^{4} c a + \frac {4}{7} x^{7} f e^{3} d b + \frac {6}{7} x^{7} f^{2} e^{2} c b + \frac {6}{7} x^{7} f^{2} e^{2} d a + \frac {4}{7} x^{7} f^{3} e c a + \frac {1}{5} x^{5} e^{4} d b + \frac {4}{5} x^{5} f e^{3} c b + \frac {4}{5} x^{5} f e^{3} d a + \frac {6}{5} x^{5} f^{2} e^{2} c a + \frac {1}{3} x^{3} e^{4} c b + \frac {1}{3} x^{3} e^{4} d a + \frac {4}{3} x^{3} f e^{3} c a + x e^{4} 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)^4,x, algorithm="fricas")

[Out]

1/13*x^13*f^4*d*b + 4/11*x^11*f^3*e*d*b + 1/11*x^11*f^4*c*b + 1/11*x^11*f^4*d*a + 2/3*x^9*f^2*e^2*d*b + 4/9*x^
9*f^3*e*c*b + 4/9*x^9*f^3*e*d*a + 1/9*x^9*f^4*c*a + 4/7*x^7*f*e^3*d*b + 6/7*x^7*f^2*e^2*c*b + 6/7*x^7*f^2*e^2*
d*a + 4/7*x^7*f^3*e*c*a + 1/5*x^5*e^4*d*b + 4/5*x^5*f*e^3*c*b + 4/5*x^5*f*e^3*d*a + 6/5*x^5*f^2*e^2*c*a + 1/3*
x^3*e^4*c*b + 1/3*x^3*e^4*d*a + 4/3*x^3*f*e^3*c*a + x*e^4*c*a

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giac [A]  time = 0.31, size = 210, normalized size = 1.22 \begin {gather*} \frac {1}{13} \, b d f^{4} x^{13} + \frac {1}{11} \, b c f^{4} x^{11} + \frac {1}{11} \, a d f^{4} x^{11} + \frac {4}{11} \, b d f^{3} x^{11} e + \frac {1}{9} \, a c f^{4} x^{9} + \frac {4}{9} \, b c f^{3} x^{9} e + \frac {4}{9} \, a d f^{3} x^{9} e + \frac {2}{3} \, b d f^{2} x^{9} e^{2} + \frac {4}{7} \, a c f^{3} x^{7} e + \frac {6}{7} \, b c f^{2} x^{7} e^{2} + \frac {6}{7} \, a d f^{2} x^{7} e^{2} + \frac {4}{7} \, b d f x^{7} e^{3} + \frac {6}{5} \, a c f^{2} x^{5} e^{2} + \frac {4}{5} \, b c f x^{5} e^{3} + \frac {4}{5} \, a d f x^{5} e^{3} + \frac {1}{5} \, b d x^{5} e^{4} + \frac {4}{3} \, a c f x^{3} e^{3} + \frac {1}{3} \, b c x^{3} e^{4} + \frac {1}{3} \, a d x^{3} e^{4} + a c x e^{4} \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)^4,x, algorithm="giac")

[Out]

1/13*b*d*f^4*x^13 + 1/11*b*c*f^4*x^11 + 1/11*a*d*f^4*x^11 + 4/11*b*d*f^3*x^11*e + 1/9*a*c*f^4*x^9 + 4/9*b*c*f^
3*x^9*e + 4/9*a*d*f^3*x^9*e + 2/3*b*d*f^2*x^9*e^2 + 4/7*a*c*f^3*x^7*e + 6/7*b*c*f^2*x^7*e^2 + 6/7*a*d*f^2*x^7*
e^2 + 4/7*b*d*f*x^7*e^3 + 6/5*a*c*f^2*x^5*e^2 + 4/5*b*c*f*x^5*e^3 + 4/5*a*d*f*x^5*e^3 + 1/5*b*d*x^5*e^4 + 4/3*
a*c*f*x^3*e^3 + 1/3*b*c*x^3*e^4 + 1/3*a*d*x^3*e^4 + a*c*x*e^4

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maple [A]  time = 0.00, size = 176, normalized size = 1.02 \begin {gather*} \frac {b d \,f^{4} x^{13}}{13}+\frac {\left (4 b d e \,f^{3}+\left (a d +b c \right ) f^{4}\right ) x^{11}}{11}+\frac {\left (a c \,f^{4}+6 b d \,e^{2} f^{2}+4 \left (a d +b c \right ) e \,f^{3}\right ) x^{9}}{9}+a c \,e^{4} x +\frac {\left (4 a c e \,f^{3}+4 b d \,e^{3} f +6 \left (a d +b c \right ) e^{2} f^{2}\right ) x^{7}}{7}+\frac {\left (6 a c \,e^{2} f^{2}+b d \,e^{4}+4 \left (a d +b c \right ) e^{3} f \right ) x^{5}}{5}+\frac {\left (4 a c \,e^{3} f +\left (a d +b c \right ) e^{4}\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)^4,x)

[Out]

1/13*b*d*f^4*x^13+1/11*((a*d+b*c)*f^4+4*b*d*e*f^3)*x^11+1/9*(a*c*f^4+4*(a*d+b*c)*e*f^3+6*b*d*e^2*f^2)*x^9+1/7*
(4*a*c*e*f^3+6*(a*d+b*c)*e^2*f^2+4*b*d*e^3*f)*x^7+1/5*(6*a*c*e^2*f^2+4*(a*d+b*c)*e^3*f+b*d*e^4)*x^5+1/3*(4*a*c
*e^3*f+(a*d+b*c)*e^4)*x^3+a*c*e^4*x

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maxima [A]  time = 0.67, size = 175, normalized size = 1.02 \begin {gather*} \frac {1}{13} \, b d f^{4} x^{13} + \frac {1}{11} \, {\left (4 \, b d e f^{3} + {\left (b c + a d\right )} f^{4}\right )} x^{11} + \frac {1}{9} \, {\left (6 \, b d e^{2} f^{2} + a c f^{4} + 4 \, {\left (b c + a d\right )} e f^{3}\right )} x^{9} + \frac {2}{7} \, {\left (2 \, b d e^{3} f + 2 \, a c e f^{3} + 3 \, {\left (b c + a d\right )} e^{2} f^{2}\right )} x^{7} + a c e^{4} x + \frac {1}{5} \, {\left (b d e^{4} + 6 \, a c e^{2} f^{2} + 4 \, {\left (b c + a d\right )} e^{3} f\right )} x^{5} + \frac {1}{3} \, {\left (4 \, a c e^{3} f + {\left (b c + a d\right )} e^{4}\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)^4,x, algorithm="maxima")

[Out]

1/13*b*d*f^4*x^13 + 1/11*(4*b*d*e*f^3 + (b*c + a*d)*f^4)*x^11 + 1/9*(6*b*d*e^2*f^2 + a*c*f^4 + 4*(b*c + a*d)*e
*f^3)*x^9 + 2/7*(2*b*d*e^3*f + 2*a*c*e*f^3 + 3*(b*c + a*d)*e^2*f^2)*x^7 + a*c*e^4*x + 1/5*(b*d*e^4 + 6*a*c*e^2
*f^2 + 4*(b*c + a*d)*e^3*f)*x^5 + 1/3*(4*a*c*e^3*f + (b*c + a*d)*e^4)*x^3

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mupad [B]  time = 0.10, size = 182, normalized size = 1.06 \begin {gather*} x^3\,\left (\frac {a\,d\,e^4}{3}+\frac {b\,c\,e^4}{3}+\frac {4\,a\,c\,e^3\,f}{3}\right )+x^{11}\,\left (\frac {a\,d\,f^4}{11}+\frac {b\,c\,f^4}{11}+\frac {4\,b\,d\,e\,f^3}{11}\right )+x^5\,\left (\frac {b\,d\,e^4}{5}+\frac {4\,a\,d\,e^3\,f}{5}+\frac {4\,b\,c\,e^3\,f}{5}+\frac {6\,a\,c\,e^2\,f^2}{5}\right )+x^9\,\left (\frac {a\,c\,f^4}{9}+\frac {4\,a\,d\,e\,f^3}{9}+\frac {4\,b\,c\,e\,f^3}{9}+\frac {2\,b\,d\,e^2\,f^2}{3}\right )+\frac {2\,e\,f\,x^7\,\left (2\,a\,c\,f^2+2\,b\,d\,e^2+3\,a\,d\,e\,f+3\,b\,c\,e\,f\right )}{7}+a\,c\,e^4\,x+\frac {b\,d\,f^4\,x^{13}}{13} \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)^4,x)

[Out]

x^3*((a*d*e^4)/3 + (b*c*e^4)/3 + (4*a*c*e^3*f)/3) + x^11*((a*d*f^4)/11 + (b*c*f^4)/11 + (4*b*d*e*f^3)/11) + x^
5*((b*d*e^4)/5 + (4*a*d*e^3*f)/5 + (4*b*c*e^3*f)/5 + (6*a*c*e^2*f^2)/5) + x^9*((a*c*f^4)/9 + (4*a*d*e*f^3)/9 +
(4*b*c*e*f^3)/9 + (2*b*d*e^2*f^2)/3) + (2*e*f*x^7*(2*a*c*f^2 + 2*b*d*e^2 + 3*a*d*e*f + 3*b*c*e*f))/7 + a*c*e^
4*x + (b*d*f^4*x^13)/13

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sympy [A]  time = 0.10, size = 236, normalized size = 1.37 \begin {gather*} a c e^{4} x + \frac {b d f^{4} x^{13}}{13} + x^{11} \left (\frac {a d f^{4}}{11} + \frac {b c f^{4}}{11} + \frac {4 b d e f^{3}}{11}\right ) + x^{9} \left (\frac {a c f^{4}}{9} + \frac {4 a d e f^{3}}{9} + \frac {4 b c e f^{3}}{9} + \frac {2 b d e^{2} f^{2}}{3}\right ) + x^{7} \left (\frac {4 a c e f^{3}}{7} + \frac {6 a d e^{2} f^{2}}{7} + \frac {6 b c e^{2} f^{2}}{7} + \frac {4 b d e^{3} f}{7}\right ) + x^{5} \left (\frac {6 a c e^{2} f^{2}}{5} + \frac {4 a d e^{3} f}{5} + \frac {4 b c e^{3} f}{5} + \frac {b d e^{4}}{5}\right ) + x^{3} \left (\frac {4 a c e^{3} f}{3} + \frac {a d e^{4}}{3} + \frac {b c e^{4}}{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)**4,x)

[Out]

a*c*e**4*x + b*d*f**4*x**13/13 + x**11*(a*d*f**4/11 + b*c*f**4/11 + 4*b*d*e*f**3/11) + x**9*(a*c*f**4/9 + 4*a*
d*e*f**3/9 + 4*b*c*e*f**3/9 + 2*b*d*e**2*f**2/3) + x**7*(4*a*c*e*f**3/7 + 6*a*d*e**2*f**2/7 + 6*b*c*e**2*f**2/
7 + 4*b*d*e**3*f/7) + x**5*(6*a*c*e**2*f**2/5 + 4*a*d*e**3*f/5 + 4*b*c*e**3*f/5 + b*d*e**4/5) + x**3*(4*a*c*e*
*3*f/3 + a*d*e**4/3 + b*c*e**4/3)

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