3.2.10 \(\int (d+e x^2)^2 (a+c x^4)^2 \, dx\)

Optimal. Leaf size=97 \[ a^2 d^2 x+\frac {2}{3} a^2 d e x^3+\frac {1}{9} c x^9 \left (2 a e^2+c d^2\right )+\frac {1}{5} a x^5 \left (a e^2+2 c d^2\right )+\frac {4}{7} a c d e x^7+\frac {2}{11} c^2 d e x^{11}+\frac {1}{13} c^2 e^2 x^{13} \]

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Rubi [A]  time = 0.07, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {1154} \begin {gather*} a^2 d^2 x+\frac {2}{3} a^2 d e x^3+\frac {1}{9} c x^9 \left (2 a e^2+c d^2\right )+\frac {1}{5} a x^5 \left (a e^2+2 c d^2\right )+\frac {4}{7} a c d e x^7+\frac {2}{11} c^2 d e x^{11}+\frac {1}{13} c^2 e^2 x^{13} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x^2)^2*(a + c*x^4)^2,x]

[Out]

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

Rule 1154

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

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 97, normalized size = 1.00 \begin {gather*} a^2 d^2 x+\frac {2}{3} a^2 d e x^3+\frac {1}{9} c x^9 \left (2 a e^2+c d^2\right )+\frac {1}{5} a x^5 \left (a e^2+2 c d^2\right )+\frac {4}{7} a c d e x^7+\frac {2}{11} c^2 d e x^{11}+\frac {1}{13} c^2 e^2 x^{13} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.75, size = 91, normalized size = 0.94 \begin {gather*} \frac {1}{13} x^{13} e^{2} c^{2} + \frac {2}{11} x^{11} e d c^{2} + \frac {1}{9} x^{9} d^{2} c^{2} + \frac {2}{9} x^{9} e^{2} c a + \frac {4}{7} x^{7} e d c a + \frac {2}{5} x^{5} d^{2} c a + \frac {1}{5} x^{5} e^{2} a^{2} + \frac {2}{3} x^{3} e d a^{2} + x d^{2} a^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.15, size = 91, normalized size = 0.94 \begin {gather*} \frac {1}{13} \, c^{2} x^{13} e^{2} + \frac {2}{11} \, c^{2} d x^{11} e + \frac {1}{9} \, c^{2} d^{2} x^{9} + \frac {2}{9} \, a c x^{9} e^{2} + \frac {4}{7} \, a c d x^{7} e + \frac {2}{5} \, a c d^{2} x^{5} + \frac {1}{5} \, a^{2} x^{5} e^{2} + \frac {2}{3} \, a^{2} d x^{3} e + a^{2} d^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 90, normalized size = 0.93 \begin {gather*} \frac {c^{2} e^{2} x^{13}}{13}+\frac {2 c^{2} d e \,x^{11}}{11}+\frac {4 a c d e \,x^{7}}{7}+\frac {\left (2 e^{2} a c +c^{2} d^{2}\right ) x^{9}}{9}+\frac {2 a^{2} d e \,x^{3}}{3}+a^{2} d^{2} x +\frac {\left (e^{2} a^{2}+2 d^{2} a c \right ) x^{5}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.03, size = 89, normalized size = 0.92 \begin {gather*} \frac {1}{13} \, c^{2} e^{2} x^{13} + \frac {2}{11} \, c^{2} d e x^{11} + \frac {4}{7} \, a c d e x^{7} + \frac {1}{9} \, {\left (c^{2} d^{2} + 2 \, a c e^{2}\right )} x^{9} + \frac {2}{3} \, a^{2} d e x^{3} + \frac {1}{5} \, {\left (2 \, a c d^{2} + a^{2} e^{2}\right )} x^{5} + a^{2} d^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.05, size = 89, normalized size = 0.92 \begin {gather*} x^5\,\left (\frac {a^2\,e^2}{5}+\frac {2\,c\,a\,d^2}{5}\right )+x^9\,\left (\frac {c^2\,d^2}{9}+\frac {2\,a\,c\,e^2}{9}\right )+a^2\,d^2\,x+\frac {c^2\,e^2\,x^{13}}{13}+\frac {2\,a^2\,d\,e\,x^3}{3}+\frac {2\,c^2\,d\,e\,x^{11}}{11}+\frac {4\,a\,c\,d\,e\,x^7}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^4)^2*(d + e*x^2)^2,x)

[Out]

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

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sympy [A]  time = 0.09, size = 104, normalized size = 1.07 \begin {gather*} a^{2} d^{2} x + \frac {2 a^{2} d e x^{3}}{3} + \frac {4 a c d e x^{7}}{7} + \frac {2 c^{2} d e x^{11}}{11} + \frac {c^{2} e^{2} x^{13}}{13} + x^{9} \left (\frac {2 a c e^{2}}{9} + \frac {c^{2} d^{2}}{9}\right ) + x^{5} \left (\frac {a^{2} e^{2}}{5} + \frac {2 a c d^{2}}{5}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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