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

Optimal. Leaf size=79 \[ \frac{1}{7} e x^7 \left (a e^2+3 c d^2\right )+\frac{1}{5} d x^5 \left (3 a e^2+c d^2\right )+a d^2 e x^3+a d^3 x+\frac{1}{3} c d e^2 x^9+\frac{1}{11} c e^3 x^{11} \]

[Out]

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

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Rubi [A]  time = 0.0561928, antiderivative size = 79, 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} \[ \frac{1}{7} e x^7 \left (a e^2+3 c d^2\right )+\frac{1}{5} d x^5 \left (3 a e^2+c d^2\right )+a d^2 e x^3+a d^3 x+\frac{1}{3} c d e^2 x^9+\frac{1}{11} c e^3 x^{11} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x^2)^3*(a + c*x^4),x]

[Out]

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

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

Mathematica [A]  time = 0.0162709, size = 79, normalized size = 1. \[ \frac{1}{7} e x^7 \left (a e^2+3 c d^2\right )+\frac{1}{5} d x^5 \left (3 a e^2+c d^2\right )+a d^2 e x^3+a d^3 x+\frac{1}{3} c d e^2 x^9+\frac{1}{11} c e^3 x^{11} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.042, size = 72, normalized size = 0.9 \begin{align*}{\frac{c{e}^{3}{x}^{11}}{11}}+{\frac{cd{e}^{2}{x}^{9}}{3}}+{\frac{ \left ( a{e}^{3}+3\,{d}^{2}ec \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,ad{e}^{2}+c{d}^{3} \right ){x}^{5}}{5}}+a{d}^{2}e{x}^{3}+a{d}^{3}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.977206, size = 96, normalized size = 1.22 \begin{align*} \frac{1}{11} \, c e^{3} x^{11} + \frac{1}{3} \, c d e^{2} x^{9} + \frac{1}{7} \,{\left (3 \, c d^{2} e + a e^{3}\right )} x^{7} + a d^{2} e x^{3} + \frac{1}{5} \,{\left (c d^{3} + 3 \, a d e^{2}\right )} x^{5} + a d^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.41715, size = 171, normalized size = 2.16 \begin{align*} \frac{1}{11} x^{11} e^{3} c + \frac{1}{3} x^{9} e^{2} d c + \frac{3}{7} x^{7} e d^{2} c + \frac{1}{7} x^{7} e^{3} a + \frac{1}{5} x^{5} d^{3} c + \frac{3}{5} x^{5} e^{2} d a + x^{3} e d^{2} a + x d^{3} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.073198, size = 78, normalized size = 0.99 \begin{align*} a d^{3} x + a d^{2} e x^{3} + \frac{c d e^{2} x^{9}}{3} + \frac{c e^{3} x^{11}}{11} + x^{7} \left (\frac{a e^{3}}{7} + \frac{3 c d^{2} e}{7}\right ) + x^{5} \left (\frac{3 a d e^{2}}{5} + \frac{c d^{3}}{5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.14096, size = 96, normalized size = 1.22 \begin{align*} \frac{1}{11} \, c x^{11} e^{3} + \frac{1}{3} \, c d x^{9} e^{2} + \frac{3}{7} \, c d^{2} x^{7} e + \frac{1}{5} \, c d^{3} x^{5} + \frac{1}{7} \, a x^{7} e^{3} + \frac{3}{5} \, a d x^{5} e^{2} + a d^{2} x^{3} e + a d^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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