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

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

[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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Rubi [A]  time = 0.0826949, antiderivative size = 106, normalized size of antiderivative = 1., 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} \[ \frac{1}{9} e^2 x^9 \left (a e^2+6 c d^2\right )+\frac{4}{7} d e x^7 \left (a e^2+c d^2\right )+\frac{1}{5} d^2 x^5 \left (6 a e^2+c d^2\right )+\frac{4}{3} a d^3 e x^3+a d^4 x+\frac{4}{11} c d e^3 x^{11}+\frac{1}{13} c e^4 x^{13} \]

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*}

Mathematica [A]  time = 0.0206913, size = 106, normalized size = 1. \[ \frac{1}{9} e^2 x^9 \left (a e^2+6 c d^2\right )+\frac{4}{7} d e x^7 \left (a e^2+c d^2\right )+\frac{1}{5} d^2 x^5 \left (6 a e^2+c d^2\right )+\frac{4}{3} a d^3 e x^3+a d^4 x+\frac{4}{11} c d e^3 x^{11}+\frac{1}{13} c e^4 x^{13} \]

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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Maple [A]  time = 0.042, size = 97, normalized size = 0.9 \begin{align*}{\frac{c{e}^{4}{x}^{13}}{13}}+{\frac{4\,cd{e}^{3}{x}^{11}}{11}}+{\frac{ \left ({e}^{4}a+6\,{d}^{2}{e}^{2}c \right ){x}^{9}}{9}}+{\frac{ \left ( 4\,ad{e}^{3}+4\,c{d}^{3}e \right ){x}^{7}}{7}}+{\frac{ \left ( 6\,a{d}^{2}{e}^{2}+c{d}^{4} \right ){x}^{5}}{5}}+{\frac{4\,a{d}^{3}e{x}^{3}}{3}}+a{d}^{4}x \end{align*}

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 = 0.995822, size = 127, normalized size = 1.2 \begin{align*} \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{align*}

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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Fricas [A]  time = 1.34702, size = 234, normalized size = 2.21 \begin{align*} \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{align*}

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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Sympy [A]  time = 0.07838, size = 110, normalized size = 1.04 \begin{align*} 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{align*}

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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Giac [A]  time = 1.11698, size = 127, normalized size = 1.2 \begin{align*} \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{align*}

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