3.2.75 \(\int (d+e x^2)^3 (a+b x^2+c x^4)^2 \, dx\)

Optimal. Leaf size=223 \[ \frac {1}{7} x^7 \left (a^2 e^3+6 a b d e^2+6 a c d^2 e+3 b^2 d^2 e+2 b c d^3\right )+a^2 d^3 x+\frac {1}{11} e x^{11} \left (2 c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right )+\frac {1}{5} d x^5 \left (6 a b d e+a \left (3 a e^2+2 c d^2\right )+b^2 d^2\right )+\frac {1}{9} x^9 \left (6 c d e (a e+b d)+b e^2 (2 a e+3 b d)+c^2 d^3\right )+\frac {1}{3} a d^2 x^3 (3 a e+2 b d)+\frac {1}{13} c e^2 x^{13} (2 b e+3 c d)+\frac {1}{15} c^2 e^3 x^{15} \]

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Rubi [A]  time = 0.20, antiderivative size = 223, 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 = {1153} \begin {gather*} \frac {1}{7} x^7 \left (a^2 e^3+6 a b d e^2+6 a c d^2 e+3 b^2 d^2 e+2 b c d^3\right )+a^2 d^3 x+\frac {1}{11} e x^{11} \left (2 c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right )+\frac {1}{5} d x^5 \left (6 a b d e+a \left (3 a e^2+2 c d^2\right )+b^2 d^2\right )+\frac {1}{9} x^9 \left (6 c d e (a e+b d)+b e^2 (2 a e+3 b d)+c^2 d^3\right )+\frac {1}{3} a d^2 x^3 (3 a e+2 b d)+\frac {1}{13} c e^2 x^{13} (2 b e+3 c d)+\frac {1}{15} c^2 e^3 x^{15} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x^2)^3*(a + b*x^2 + c*x^4)^2,x]

[Out]

a^2*d^3*x + (a*d^2*(2*b*d + 3*a*e)*x^3)/3 + (d*(b^2*d^2 + 6*a*b*d*e + a*(2*c*d^2 + 3*a*e^2))*x^5)/5 + ((2*b*c*
d^3 + 3*b^2*d^2*e + 6*a*c*d^2*e + 6*a*b*d*e^2 + a^2*e^3)*x^7)/7 + ((c^2*d^3 + 6*c*d*e*(b*d + a*e) + b*e^2*(3*b
*d + 2*a*e))*x^9)/9 + (e*(3*c^2*d^2 + b^2*e^2 + 2*c*e*(3*b*d + a*e))*x^11)/11 + (c*e^2*(3*c*d + 2*b*e)*x^13)/1
3 + (c^2*e^3*x^15)/15

Rule 1153

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]

Rubi steps

\begin {align*} \int \left (d+e x^2\right )^3 \left (a+b x^2+c x^4\right )^2 \, dx &=\int \left (a^2 d^3+a d^2 (2 b d+3 a e) x^2+d \left (b^2 d^2+6 a b d e+a \left (2 c d^2+3 a e^2\right )\right ) x^4+\left (2 b c d^3+3 b^2 d^2 e+6 a c d^2 e+6 a b d e^2+a^2 e^3\right ) x^6+\left (c^2 d^3+6 c d e (b d+a e)+b e^2 (3 b d+2 a e)\right ) x^8+e \left (3 c^2 d^2+b^2 e^2+2 c e (3 b d+a e)\right ) x^{10}+c e^2 (3 c d+2 b e) x^{12}+c^2 e^3 x^{14}\right ) \, dx\\ &=a^2 d^3 x+\frac {1}{3} a d^2 (2 b d+3 a e) x^3+\frac {1}{5} d \left (b^2 d^2+6 a b d e+a \left (2 c d^2+3 a e^2\right )\right ) x^5+\frac {1}{7} \left (2 b c d^3+3 b^2 d^2 e+6 a c d^2 e+6 a b d e^2+a^2 e^3\right ) x^7+\frac {1}{9} \left (c^2 d^3+6 c d e (b d+a e)+b e^2 (3 b d+2 a e)\right ) x^9+\frac {1}{11} e \left (3 c^2 d^2+b^2 e^2+2 c e (3 b d+a e)\right ) x^{11}+\frac {1}{13} c e^2 (3 c d+2 b e) x^{13}+\frac {1}{15} c^2 e^3 x^{15}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 223, normalized size = 1.00 \begin {gather*} \frac {1}{7} x^7 \left (a^2 e^3+6 a b d e^2+6 a c d^2 e+3 b^2 d^2 e+2 b c d^3\right )+a^2 d^3 x+\frac {1}{11} e x^{11} \left (2 c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right )+\frac {1}{5} d x^5 \left (6 a b d e+a \left (3 a e^2+2 c d^2\right )+b^2 d^2\right )+\frac {1}{9} x^9 \left (6 c d e (a e+b d)+b e^2 (2 a e+3 b d)+c^2 d^3\right )+\frac {1}{3} a d^2 x^3 (3 a e+2 b d)+\frac {1}{13} c e^2 x^{13} (2 b e+3 c d)+\frac {1}{15} c^2 e^3 x^{15} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x^2)^3*(a + b*x^2 + c*x^4)^2,x]

[Out]

a^2*d^3*x + (a*d^2*(2*b*d + 3*a*e)*x^3)/3 + (d*(b^2*d^2 + 6*a*b*d*e + a*(2*c*d^2 + 3*a*e^2))*x^5)/5 + ((2*b*c*
d^3 + 3*b^2*d^2*e + 6*a*c*d^2*e + 6*a*b*d*e^2 + a^2*e^3)*x^7)/7 + ((c^2*d^3 + 6*c*d*e*(b*d + a*e) + b*e^2*(3*b
*d + 2*a*e))*x^9)/9 + (e*(3*c^2*d^2 + b^2*e^2 + 2*c*e*(3*b*d + a*e))*x^11)/11 + (c*e^2*(3*c*d + 2*b*e)*x^13)/1
3 + (c^2*e^3*x^15)/15

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (d+e x^2\right )^3 \left (a+b x^2+c x^4\right )^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(d + e*x^2)^3*(a + b*x^2 + c*x^4)^2,x]

[Out]

IntegrateAlgebraic[(d + e*x^2)^3*(a + b*x^2 + c*x^4)^2, x]

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fricas [A]  time = 0.67, size = 261, normalized size = 1.17 \begin {gather*} \frac {1}{15} x^{15} e^{3} c^{2} + \frac {3}{13} x^{13} e^{2} d c^{2} + \frac {2}{13} x^{13} e^{3} c b + \frac {3}{11} x^{11} e d^{2} c^{2} + \frac {6}{11} x^{11} e^{2} d c b + \frac {1}{11} x^{11} e^{3} b^{2} + \frac {2}{11} x^{11} e^{3} c a + \frac {1}{9} x^{9} d^{3} c^{2} + \frac {2}{3} x^{9} e d^{2} c b + \frac {1}{3} x^{9} e^{2} d b^{2} + \frac {2}{3} x^{9} e^{2} d c a + \frac {2}{9} x^{9} e^{3} b a + \frac {2}{7} x^{7} d^{3} c b + \frac {3}{7} x^{7} e d^{2} b^{2} + \frac {6}{7} x^{7} e d^{2} c a + \frac {6}{7} x^{7} e^{2} d b a + \frac {1}{7} x^{7} e^{3} a^{2} + \frac {1}{5} x^{5} d^{3} b^{2} + \frac {2}{5} x^{5} d^{3} c a + \frac {6}{5} x^{5} e d^{2} b a + \frac {3}{5} x^{5} e^{2} d a^{2} + \frac {2}{3} x^{3} d^{3} b a + x^{3} e d^{2} a^{2} + x d^{3} a^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^3*(c*x^4+b*x^2+a)^2,x, algorithm="fricas")

[Out]

1/15*x^15*e^3*c^2 + 3/13*x^13*e^2*d*c^2 + 2/13*x^13*e^3*c*b + 3/11*x^11*e*d^2*c^2 + 6/11*x^11*e^2*d*c*b + 1/11
*x^11*e^3*b^2 + 2/11*x^11*e^3*c*a + 1/9*x^9*d^3*c^2 + 2/3*x^9*e*d^2*c*b + 1/3*x^9*e^2*d*b^2 + 2/3*x^9*e^2*d*c*
a + 2/9*x^9*e^3*b*a + 2/7*x^7*d^3*c*b + 3/7*x^7*e*d^2*b^2 + 6/7*x^7*e*d^2*c*a + 6/7*x^7*e^2*d*b*a + 1/7*x^7*e^
3*a^2 + 1/5*x^5*d^3*b^2 + 2/5*x^5*d^3*c*a + 6/5*x^5*e*d^2*b*a + 3/5*x^5*e^2*d*a^2 + 2/3*x^3*d^3*b*a + x^3*e*d^
2*a^2 + x*d^3*a^2

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giac [A]  time = 0.16, size = 255, normalized size = 1.14 \begin {gather*} \frac {1}{15} \, c^{2} x^{15} e^{3} + \frac {3}{13} \, c^{2} d x^{13} e^{2} + \frac {2}{13} \, b c x^{13} e^{3} + \frac {3}{11} \, c^{2} d^{2} x^{11} e + \frac {6}{11} \, b c d x^{11} e^{2} + \frac {1}{9} \, c^{2} d^{3} x^{9} + \frac {1}{11} \, b^{2} x^{11} e^{3} + \frac {2}{11} \, a c x^{11} e^{3} + \frac {2}{3} \, b c d^{2} x^{9} e + \frac {1}{3} \, b^{2} d x^{9} e^{2} + \frac {2}{3} \, a c d x^{9} e^{2} + \frac {2}{7} \, b c d^{3} x^{7} + \frac {2}{9} \, a b x^{9} e^{3} + \frac {3}{7} \, b^{2} d^{2} x^{7} e + \frac {6}{7} \, a c d^{2} x^{7} e + \frac {6}{7} \, a b d x^{7} e^{2} + \frac {1}{5} \, b^{2} d^{3} x^{5} + \frac {2}{5} \, a c d^{3} x^{5} + \frac {1}{7} \, a^{2} x^{7} e^{3} + \frac {6}{5} \, a b d^{2} x^{5} e + \frac {3}{5} \, a^{2} d x^{5} e^{2} + \frac {2}{3} \, a b d^{3} x^{3} + a^{2} d^{2} x^{3} e + a^{2} d^{3} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^3*(c*x^4+b*x^2+a)^2,x, algorithm="giac")

[Out]

1/15*c^2*x^15*e^3 + 3/13*c^2*d*x^13*e^2 + 2/13*b*c*x^13*e^3 + 3/11*c^2*d^2*x^11*e + 6/11*b*c*d*x^11*e^2 + 1/9*
c^2*d^3*x^9 + 1/11*b^2*x^11*e^3 + 2/11*a*c*x^11*e^3 + 2/3*b*c*d^2*x^9*e + 1/3*b^2*d*x^9*e^2 + 2/3*a*c*d*x^9*e^
2 + 2/7*b*c*d^3*x^7 + 2/9*a*b*x^9*e^3 + 3/7*b^2*d^2*x^7*e + 6/7*a*c*d^2*x^7*e + 6/7*a*b*d*x^7*e^2 + 1/5*b^2*d^
3*x^5 + 2/5*a*c*d^3*x^5 + 1/7*a^2*x^7*e^3 + 6/5*a*b*d^2*x^5*e + 3/5*a^2*d*x^5*e^2 + 2/3*a*b*d^3*x^3 + a^2*d^2*
x^3*e + a^2*d^3*x

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maple [A]  time = 0.00, size = 219, normalized size = 0.98 \begin {gather*} \frac {c^{2} e^{3} x^{15}}{15}+\frac {\left (2 e^{3} b c +3 d \,e^{2} c^{2}\right ) x^{13}}{13}+\frac {\left (6 b c d \,e^{2}+3 c^{2} d^{2} e +\left (2 a c +b^{2}\right ) e^{3}\right ) x^{11}}{11}+\frac {\left (2 a b \,e^{3}+6 b c \,d^{2} e +c^{2} d^{3}+3 \left (2 a c +b^{2}\right ) d \,e^{2}\right ) x^{9}}{9}+\frac {\left (a^{2} e^{3}+6 a b d \,e^{2}+2 b c \,d^{3}+3 \left (2 a c +b^{2}\right ) d^{2} e \right ) x^{7}}{7}+a^{2} d^{3} x +\frac {\left (3 a^{2} d \,e^{2}+6 a b \,d^{2} e +\left (2 a c +b^{2}\right ) d^{3}\right ) x^{5}}{5}+\frac {\left (3 d^{2} e \,a^{2}+2 d^{3} a b \right ) x^{3}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)^3*(c*x^4+b*x^2+a)^2,x)

[Out]

1/15*c^2*e^3*x^15+1/13*(2*b*c*e^3+3*c^2*d*e^2)*x^13+1/11*(3*d^2*e*c^2+6*d*e^2*b*c+e^3*(2*a*c+b^2))*x^11+1/9*(c
^2*d^3+6*d^2*e*b*c+3*d*e^2*(2*a*c+b^2)+2*e^3*a*b)*x^9+1/7*(2*b*c*d^3+3*d^2*e*(2*a*c+b^2)+6*a*b*d*e^2+a^2*e^3)*
x^7+1/5*(d^3*(2*a*c+b^2)+6*d^2*e*a*b+3*d*e^2*a^2)*x^5+1/3*(3*a^2*d^2*e+2*a*b*d^3)*x^3+a^2*d^3*x

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maxima [A]  time = 1.04, size = 218, normalized size = 0.98 \begin {gather*} \frac {1}{15} \, c^{2} e^{3} x^{15} + \frac {1}{13} \, {\left (3 \, c^{2} d e^{2} + 2 \, b c e^{3}\right )} x^{13} + \frac {1}{11} \, {\left (3 \, c^{2} d^{2} e + 6 \, b c d e^{2} + {\left (b^{2} + 2 \, a c\right )} e^{3}\right )} x^{11} + \frac {1}{9} \, {\left (c^{2} d^{3} + 6 \, b c d^{2} e + 2 \, a b e^{3} + 3 \, {\left (b^{2} + 2 \, a c\right )} d e^{2}\right )} x^{9} + \frac {1}{7} \, {\left (2 \, b c d^{3} + 6 \, a b d e^{2} + a^{2} e^{3} + 3 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e\right )} x^{7} + a^{2} d^{3} x + \frac {1}{5} \, {\left (6 \, a b d^{2} e + 3 \, a^{2} d e^{2} + {\left (b^{2} + 2 \, a c\right )} d^{3}\right )} x^{5} + \frac {1}{3} \, {\left (2 \, a b d^{3} + 3 \, a^{2} d^{2} e\right )} x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^3*(c*x^4+b*x^2+a)^2,x, algorithm="maxima")

[Out]

1/15*c^2*e^3*x^15 + 1/13*(3*c^2*d*e^2 + 2*b*c*e^3)*x^13 + 1/11*(3*c^2*d^2*e + 6*b*c*d*e^2 + (b^2 + 2*a*c)*e^3)
*x^11 + 1/9*(c^2*d^3 + 6*b*c*d^2*e + 2*a*b*e^3 + 3*(b^2 + 2*a*c)*d*e^2)*x^9 + 1/7*(2*b*c*d^3 + 6*a*b*d*e^2 + a
^2*e^3 + 3*(b^2 + 2*a*c)*d^2*e)*x^7 + a^2*d^3*x + 1/5*(6*a*b*d^2*e + 3*a^2*d*e^2 + (b^2 + 2*a*c)*d^3)*x^5 + 1/
3*(2*a*b*d^3 + 3*a^2*d^2*e)*x^3

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mupad [B]  time = 4.48, size = 220, normalized size = 0.99 \begin {gather*} x^7\,\left (\frac {a^2\,e^3}{7}+\frac {6\,a\,b\,d\,e^2}{7}+\frac {6\,c\,a\,d^2\,e}{7}+\frac {3\,b^2\,d^2\,e}{7}+\frac {2\,c\,b\,d^3}{7}\right )+x^9\,\left (\frac {b^2\,d\,e^2}{3}+\frac {2\,b\,c\,d^2\,e}{3}+\frac {2\,a\,b\,e^3}{9}+\frac {c^2\,d^3}{9}+\frac {2\,a\,c\,d\,e^2}{3}\right )+x^5\,\left (\frac {3\,a^2\,d\,e^2}{5}+\frac {6\,a\,b\,d^2\,e}{5}+\frac {2\,c\,a\,d^3}{5}+\frac {b^2\,d^3}{5}\right )+x^{11}\,\left (\frac {b^2\,e^3}{11}+\frac {6\,b\,c\,d\,e^2}{11}+\frac {3\,c^2\,d^2\,e}{11}+\frac {2\,a\,c\,e^3}{11}\right )+a^2\,d^3\,x+\frac {c^2\,e^3\,x^{15}}{15}+\frac {a\,d^2\,x^3\,\left (3\,a\,e+2\,b\,d\right )}{3}+\frac {c\,e^2\,x^{13}\,\left (2\,b\,e+3\,c\,d\right )}{13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x^2)^3*(a + b*x^2 + c*x^4)^2,x)

[Out]

x^7*((a^2*e^3)/7 + (3*b^2*d^2*e)/7 + (2*b*c*d^3)/7 + (6*a*b*d*e^2)/7 + (6*a*c*d^2*e)/7) + x^9*((c^2*d^3)/9 + (
b^2*d*e^2)/3 + (2*a*b*e^3)/9 + (2*a*c*d*e^2)/3 + (2*b*c*d^2*e)/3) + x^5*((b^2*d^3)/5 + (3*a^2*d*e^2)/5 + (2*a*
c*d^3)/5 + (6*a*b*d^2*e)/5) + x^11*((b^2*e^3)/11 + (3*c^2*d^2*e)/11 + (2*a*c*e^3)/11 + (6*b*c*d*e^2)/11) + a^2
*d^3*x + (c^2*e^3*x^15)/15 + (a*d^2*x^3*(3*a*e + 2*b*d))/3 + (c*e^2*x^13*(2*b*e + 3*c*d))/13

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sympy [A]  time = 0.22, size = 272, normalized size = 1.22 \begin {gather*} a^{2} d^{3} x + \frac {c^{2} e^{3} x^{15}}{15} + x^{13} \left (\frac {2 b c e^{3}}{13} + \frac {3 c^{2} d e^{2}}{13}\right ) + x^{11} \left (\frac {2 a c e^{3}}{11} + \frac {b^{2} e^{3}}{11} + \frac {6 b c d e^{2}}{11} + \frac {3 c^{2} d^{2} e}{11}\right ) + x^{9} \left (\frac {2 a b e^{3}}{9} + \frac {2 a c d e^{2}}{3} + \frac {b^{2} d e^{2}}{3} + \frac {2 b c d^{2} e}{3} + \frac {c^{2} d^{3}}{9}\right ) + x^{7} \left (\frac {a^{2} e^{3}}{7} + \frac {6 a b d e^{2}}{7} + \frac {6 a c d^{2} e}{7} + \frac {3 b^{2} d^{2} e}{7} + \frac {2 b c d^{3}}{7}\right ) + x^{5} \left (\frac {3 a^{2} d e^{2}}{5} + \frac {6 a b d^{2} e}{5} + \frac {2 a c d^{3}}{5} + \frac {b^{2} d^{3}}{5}\right ) + x^{3} \left (a^{2} d^{2} e + \frac {2 a b d^{3}}{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**2+d)**3*(c*x**4+b*x**2+a)**2,x)

[Out]

a**2*d**3*x + c**2*e**3*x**15/15 + x**13*(2*b*c*e**3/13 + 3*c**2*d*e**2/13) + x**11*(2*a*c*e**3/11 + b**2*e**3
/11 + 6*b*c*d*e**2/11 + 3*c**2*d**2*e/11) + x**9*(2*a*b*e**3/9 + 2*a*c*d*e**2/3 + b**2*d*e**2/3 + 2*b*c*d**2*e
/3 + c**2*d**3/9) + x**7*(a**2*e**3/7 + 6*a*b*d*e**2/7 + 6*a*c*d**2*e/7 + 3*b**2*d**2*e/7 + 2*b*c*d**3/7) + x*
*5*(3*a**2*d*e**2/5 + 6*a*b*d**2*e/5 + 2*a*c*d**3/5 + b**2*d**3/5) + x**3*(a**2*d**2*e + 2*a*b*d**3/3)

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