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

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

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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} c^2 x^3 (a c f+3 a d e+b c e)+\frac {1}{9} d^2 x^9 (a d f+3 b c f+b d e)+\frac {1}{7} d x^7 (a d (3 c f+d e)+3 b c (c f+d e))+\frac {1}{5} c x^5 (3 a d (c f+d e)+b c (c f+3 d e))+a c^3 e x+\frac {1}{11} b d^3 f x^{11} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.06, size = 130, normalized size = 1.00 \begin {gather*} \frac {1}{3} c^2 x^3 (a c f+3 a d e+b c e)+\frac {1}{9} d^2 x^9 (a d f+3 b c f+b d e)+\frac {1}{7} d x^7 (a d (3 c f+d e)+3 b c (c f+d e))+\frac {1}{5} c x^5 (3 a d (c f+d e)+b c (c f+3 d e))+a c^3 e x+\frac {1}{11} b d^3 f x^{11} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

[Out]

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

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fricas [A]  time = 0.85, size = 165, normalized size = 1.27 \begin {gather*} \frac {1}{11} x^{11} f d^{3} b + \frac {1}{9} x^{9} e d^{3} b + \frac {1}{3} x^{9} f d^{2} c b + \frac {1}{9} x^{9} f d^{3} a + \frac {3}{7} x^{7} e d^{2} c b + \frac {3}{7} x^{7} f d c^{2} b + \frac {1}{7} x^{7} e d^{3} a + \frac {3}{7} x^{7} f d^{2} c a + \frac {3}{5} x^{5} e d c^{2} b + \frac {1}{5} x^{5} f c^{3} b + \frac {3}{5} x^{5} e d^{2} c a + \frac {3}{5} x^{5} f d c^{2} a + \frac {1}{3} x^{3} e c^{3} b + x^{3} e d c^{2} a + \frac {1}{3} x^{3} f c^{3} a + x e c^{3} a \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)*(d*x^2+c)^3*(f*x^2+e),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.35, size = 173, normalized size = 1.33 \begin {gather*} \frac {1}{11} \, b d^{3} f x^{11} + \frac {1}{3} \, b c d^{2} f x^{9} + \frac {1}{9} \, a d^{3} f x^{9} + \frac {1}{9} \, b d^{3} x^{9} e + \frac {3}{7} \, b c^{2} d f x^{7} + \frac {3}{7} \, a c d^{2} f x^{7} + \frac {3}{7} \, b c d^{2} x^{7} e + \frac {1}{7} \, a d^{3} x^{7} e + \frac {1}{5} \, b c^{3} f x^{5} + \frac {3}{5} \, a c^{2} d f x^{5} + \frac {3}{5} \, b c^{2} d x^{5} e + \frac {3}{5} \, a c d^{2} x^{5} e + \frac {1}{3} \, a c^{3} f x^{3} + \frac {1}{3} \, b c^{3} x^{3} e + a c^{2} d x^{3} e + a c^{3} x e \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)*(d*x^2+c)^3*(f*x^2+e),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.00, size = 149, normalized size = 1.15 \begin {gather*} \frac {b \,d^{3} f \,x^{11}}{11}+\frac {\left (b \,d^{3} e +\left (a \,d^{3}+3 b c \,d^{2}\right ) f \right ) x^{9}}{9}+\frac {\left (\left (a \,d^{3}+3 b c \,d^{2}\right ) e +\left (3 a c \,d^{2}+3 b \,c^{2} d \right ) f \right ) x^{7}}{7}+a \,c^{3} e x +\frac {\left (\left (3 a c \,d^{2}+3 b \,c^{2} d \right ) e +\left (3 a \,c^{2} d +b \,c^{3}\right ) f \right ) x^{5}}{5}+\frac {\left (a \,c^{3} f +\left (3 a \,c^{2} d +b \,c^{3}\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)^3*(f*x^2+e),x)

[Out]

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

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maxima [A]  time = 0.84, size = 146, normalized size = 1.12 \begin {gather*} \frac {1}{11} \, b d^{3} f x^{11} + \frac {1}{9} \, {\left (b d^{3} e + {\left (3 \, b c d^{2} + a d^{3}\right )} f\right )} x^{9} + \frac {1}{7} \, {\left ({\left (3 \, b c d^{2} + a d^{3}\right )} e + 3 \, {\left (b c^{2} d + a c d^{2}\right )} f\right )} x^{7} + a c^{3} e x + \frac {1}{5} \, {\left (3 \, {\left (b c^{2} d + a c d^{2}\right )} e + {\left (b c^{3} + 3 \, a c^{2} d\right )} f\right )} x^{5} + \frac {1}{3} \, {\left (a c^{3} f + {\left (b c^{3} + 3 \, a c^{2} 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)^3*(f*x^2+e),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.81, size = 143, normalized size = 1.10 \begin {gather*} x^5\,\left (\frac {b\,c^3\,f}{5}+\frac {3\,a\,c\,d^2\,e}{5}+\frac {3\,a\,c^2\,d\,f}{5}+\frac {3\,b\,c^2\,d\,e}{5}\right )+x^7\,\left (\frac {a\,d^3\,e}{7}+\frac {3\,a\,c\,d^2\,f}{7}+\frac {3\,b\,c\,d^2\,e}{7}+\frac {3\,b\,c^2\,d\,f}{7}\right )+x^3\,\left (\frac {a\,c^3\,f}{3}+\frac {b\,c^3\,e}{3}+a\,c^2\,d\,e\right )+x^9\,\left (\frac {a\,d^3\,f}{9}+\frac {b\,d^3\,e}{9}+\frac {b\,c\,d^2\,f}{3}\right )+a\,c^3\,e\,x+\frac {b\,d^3\,f\,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)^3*(e + f*x^2),x)

[Out]

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

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

[Out]

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

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