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

Optimal. Leaf size=106 \[ \frac {1}{9} e^2 x^9 \left (a e^2+6 c d^2\right )+\frac {4}{7} d e x^7 \left (a e^2+c d^2\right )+\frac {1}{5} d^2 x^5 \left (6 a e^2+c d^2\right )+a d^4 x+\frac {4}{3} a d^3 e x^3+\frac {4}{11} c d e^3 x^{11}+\frac {1}{13} c e^4 x^{13} \]

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Rubi [A]  time = 0.08, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {1154} \begin {gather*} \frac {1}{9} e^2 x^9 \left (a e^2+6 c d^2\right )+\frac {4}{7} d e x^7 \left (a e^2+c d^2\right )+\frac {1}{5} d^2 x^5 \left (6 a e^2+c d^2\right )+\frac {4}{3} a d^3 e x^3+a d^4 x+\frac {4}{11} c d e^3 x^{11}+\frac {1}{13} c e^4 x^{13} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x^2)^4*(a + c*x^4),x]

[Out]

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

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Mathematica [A]  time = 0.02, size = 106, normalized size = 1.00 \begin {gather*} \frac {1}{9} e^2 x^9 \left (a e^2+6 c d^2\right )+\frac {4}{7} d e x^7 \left (a e^2+c d^2\right )+\frac {1}{5} d^2 x^5 \left (6 a e^2+c d^2\right )+a d^4 x+\frac {4}{3} a d^3 e x^3+\frac {4}{11} c d e^3 x^{11}+\frac {1}{13} c e^4 x^{13} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

a*d^4*x + (4*a*d^3*e*x^3)/3 + (d^2*(c*d^2 + 6*a*e^2)*x^5)/5 + (4*d*e*(c*d^2 + a*e^2)*x^7)/7 + (e^2*(6*c*d^2 +
a*e^2)*x^9)/9 + (4*c*d*e^3*x^11)/11 + (c*e^4*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 )^4 \left (a+c x^4\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.87, size = 98, normalized size = 0.92 \begin {gather*} \frac {1}{13} x^{13} e^{4} c + \frac {4}{11} x^{11} e^{3} d c + \frac {2}{3} x^{9} e^{2} d^{2} c + \frac {1}{9} x^{9} e^{4} a + \frac {4}{7} x^{7} e d^{3} c + \frac {4}{7} x^{7} e^{3} d a + \frac {1}{5} x^{5} d^{4} c + \frac {6}{5} x^{5} e^{2} d^{2} a + \frac {4}{3} x^{3} e d^{3} a + x d^{4} a \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.15, size = 94, normalized size = 0.89 \begin {gather*} \frac {1}{13} \, c x^{13} e^{4} + \frac {4}{11} \, c d x^{11} e^{3} + \frac {2}{3} \, c d^{2} x^{9} e^{2} + \frac {4}{7} \, c d^{3} x^{7} e + \frac {1}{9} \, a x^{9} e^{4} + \frac {1}{5} \, c d^{4} x^{5} + \frac {4}{7} \, a d x^{7} e^{3} + \frac {6}{5} \, a d^{2} x^{5} e^{2} + \frac {4}{3} \, a d^{3} x^{3} e + a d^{4} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 97, normalized size = 0.92 \begin {gather*} \frac {c \,e^{4} x^{13}}{13}+\frac {4 c d \,e^{3} x^{11}}{11}+\frac {\left (e^{4} a +6 d^{2} e^{2} c \right ) x^{9}}{9}+\frac {4 a \,d^{3} e \,x^{3}}{3}+\frac {\left (4 d \,e^{3} a +4 d^{3} e c \right ) x^{7}}{7}+a \,d^{4} x +\frac {\left (6 d^{2} e^{2} a +d^{4} c \right ) x^{5}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.05, size = 94, normalized size = 0.89 \begin {gather*} \frac {1}{13} \, c e^{4} x^{13} + \frac {4}{11} \, c d e^{3} x^{11} + \frac {1}{9} \, {\left (6 \, c d^{2} e^{2} + a e^{4}\right )} x^{9} + \frac {4}{3} \, a d^{3} e x^{3} + \frac {4}{7} \, {\left (c d^{3} e + a d e^{3}\right )} x^{7} + a d^{4} x + \frac {1}{5} \, {\left (c d^{4} + 6 \, a d^{2} e^{2}\right )} x^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 4.35, size = 95, normalized size = 0.90 \begin {gather*} x^5\,\left (\frac {c\,d^4}{5}+\frac {6\,a\,d^2\,e^2}{5}\right )+x^9\,\left (\frac {2\,c\,d^2\,e^2}{3}+\frac {a\,e^4}{9}\right )+x^7\,\left (\frac {4\,c\,d^3\,e}{7}+\frac {4\,a\,d\,e^3}{7}\right )+\frac {c\,e^4\,x^{13}}{13}+a\,d^4\,x+\frac {4\,a\,d^3\,e\,x^3}{3}+\frac {4\,c\,d\,e^3\,x^{11}}{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^4)*(d + e*x^2)^4,x)

[Out]

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

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sympy [A]  time = 0.09, size = 110, normalized size = 1.04 \begin {gather*} a d^{4} x + \frac {4 a d^{3} e x^{3}}{3} + \frac {4 c d e^{3} x^{11}}{11} + \frac {c e^{4} x^{13}}{13} + x^{9} \left (\frac {a e^{4}}{9} + \frac {2 c d^{2} e^{2}}{3}\right ) + x^{7} \left (\frac {4 a d e^{3}}{7} + \frac {4 c d^{3} e}{7}\right ) + x^{5} \left (\frac {6 a d^{2} e^{2}}{5} + \frac {c d^{4}}{5}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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