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

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

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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} d x^7 (a d f+2 b c f+b d e)+\frac {1}{5} x^5 (a d (2 c f+d e)+b c (c f+2 d e))+\frac {1}{3} c x^3 (a c f+2 a d e+b c e)+a c^2 e x+\frac {1}{9} b d^2 f x^9 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.03, size = 96, normalized size = 1.02 \begin {gather*} \frac {1}{5} x^5 \left (2 a c d f+a d^2 e+b c^2 f+2 b c d e\right )+\frac {1}{7} d x^7 (a d f+2 b c f+b d e)+\frac {1}{3} c x^3 (a c f+2 a d e+b c e)+a c^2 e x+\frac {1}{9} b d^2 f x^9 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

[Out]

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

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fricas [A]  time = 1.05, size = 114, normalized size = 1.21 \begin {gather*} \frac {1}{9} x^{9} f d^{2} b + \frac {1}{7} x^{7} e d^{2} b + \frac {2}{7} x^{7} f d c b + \frac {1}{7} x^{7} f d^{2} a + \frac {2}{5} x^{5} e d c b + \frac {1}{5} x^{5} f c^{2} b + \frac {1}{5} x^{5} e d^{2} a + \frac {2}{5} x^{5} f d c a + \frac {1}{3} x^{3} e c^{2} b + \frac {2}{3} x^{3} e d c a + \frac {1}{3} x^{3} f c^{2} a + x e c^{2} a \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.29, size = 120, normalized size = 1.28 \begin {gather*} \frac {1}{9} \, b d^{2} f x^{9} + \frac {2}{7} \, b c d f x^{7} + \frac {1}{7} \, a d^{2} f x^{7} + \frac {1}{7} \, b d^{2} x^{7} e + \frac {1}{5} \, b c^{2} f x^{5} + \frac {2}{5} \, a c d f x^{5} + \frac {2}{5} \, b c d x^{5} e + \frac {1}{5} \, a d^{2} x^{5} e + \frac {1}{3} \, a c^{2} f x^{3} + \frac {1}{3} \, b c^{2} x^{3} e + \frac {2}{3} \, a c d x^{3} e + a c^{2} x e \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 101, normalized size = 1.07 \begin {gather*} \frac {b \,d^{2} f \,x^{9}}{9}+\frac {\left (b \,d^{2} e +\left (a \,d^{2}+2 b c d \right ) f \right ) x^{7}}{7}+a \,c^{2} e x +\frac {\left (\left (a \,d^{2}+2 b c d \right ) e +\left (2 a c d +b \,c^{2}\right ) f \right ) x^{5}}{5}+\frac {\left (a \,c^{2} f +\left (2 a c d +b \,c^{2}\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)^2*(f*x^2+e),x)

[Out]

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

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maxima [A]  time = 0.76, size = 100, normalized size = 1.06 \begin {gather*} \frac {1}{9} \, b d^{2} f x^{9} + \frac {1}{7} \, {\left (b d^{2} e + {\left (2 \, b c d + a d^{2}\right )} f\right )} x^{7} + \frac {1}{5} \, {\left ({\left (2 \, b c d + a d^{2}\right )} e + {\left (b c^{2} + 2 \, a c d\right )} f\right )} x^{5} + a c^{2} e x + \frac {1}{3} \, {\left (a c^{2} f + {\left (b c^{2} + 2 \, a c 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)^2*(f*x^2+e),x, algorithm="maxima")

[Out]

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

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

[Out]

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

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sympy [A]  time = 0.08, size = 121, normalized size = 1.29 \begin {gather*} a c^{2} e x + \frac {b d^{2} f x^{9}}{9} + x^{7} \left (\frac {a d^{2} f}{7} + \frac {2 b c d f}{7} + \frac {b d^{2} e}{7}\right ) + x^{5} \left (\frac {2 a c d f}{5} + \frac {a d^{2} e}{5} + \frac {b c^{2} f}{5} + \frac {2 b c d e}{5}\right ) + x^{3} \left (\frac {a c^{2} f}{3} + \frac {2 a c d e}{3} + \frac {b c^{2} 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)**2*(f*x**2+e),x)

[Out]

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

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