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

Optimal. Leaf size=103 \[ \frac{1}{7} e x^7 \left (e (a e+3 b d)+3 c d^2\right )+\frac{1}{5} d x^5 \left (3 e (a e+b d)+c d^2\right )+\frac{1}{3} d^2 x^3 (3 a e+b d)+a d^3 x+\frac{1}{9} e^2 x^9 (b e+3 c d)+\frac{1}{11} c e^3 x^{11} \]

[Out]

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

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Rubi [A]  time = 0.0954183, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {1153} \[ \frac{1}{7} e x^7 \left (e (a e+3 b d)+3 c d^2\right )+\frac{1}{5} d x^5 \left (3 e (a e+b d)+c d^2\right )+\frac{1}{3} d^2 x^3 (3 a e+b d)+a d^3 x+\frac{1}{9} e^2 x^9 (b e+3 c d)+\frac{1}{11} c e^3 x^{11} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0286084, size = 104, normalized size = 1.01 \[ \frac{1}{7} e x^7 \left (a e^2+3 b d e+3 c d^2\right )+\frac{1}{5} d x^5 \left (3 a e^2+3 b d e+c d^2\right )+\frac{1}{3} d^2 x^3 (3 a e+b d)+a d^3 x+\frac{1}{9} e^2 x^9 (b e+3 c d)+\frac{1}{11} c e^3 x^{11} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0., size = 103, normalized size = 1. \begin{align*}{\frac{c{e}^{3}{x}^{11}}{11}}+{\frac{ \left ({e}^{3}b+3\,d{e}^{2}c \right ){x}^{9}}{9}}+{\frac{ \left ( a{e}^{3}+3\,d{e}^{2}b+3\,c{d}^{2}e \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,d{e}^{2}a+3\,{d}^{2}eb+{d}^{3}c \right ){x}^{5}}{5}}+{\frac{ \left ( 3\,{d}^{2}ea+{d}^{3}b \right ){x}^{3}}{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+b*x^2+a),x)

[Out]

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

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Maxima [A]  time = 0.982227, size = 138, normalized size = 1.34 \begin{align*} \frac{1}{11} \, c e^{3} x^{11} + \frac{1}{9} \,{\left (3 \, c d e^{2} + b e^{3}\right )} x^{9} + \frac{1}{7} \,{\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} x^{7} + \frac{1}{5} \,{\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} x^{5} + a d^{3} x + \frac{1}{3} \,{\left (b d^{3} + 3 \, a d^{2} e\right )} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.38829, size = 263, normalized size = 2.55 \begin{align*} \frac{1}{11} x^{11} e^{3} c + \frac{1}{3} x^{9} e^{2} d c + \frac{1}{9} x^{9} e^{3} b + \frac{3}{7} x^{7} e d^{2} c + \frac{3}{7} x^{7} e^{2} d b + \frac{1}{7} x^{7} e^{3} a + \frac{1}{5} x^{5} d^{3} c + \frac{3}{5} x^{5} e d^{2} b + \frac{3}{5} x^{5} e^{2} d a + \frac{1}{3} x^{3} d^{3} b + 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+b*x^2+a),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.083789, size = 112, normalized size = 1.09 \begin{align*} a d^{3} x + \frac{c e^{3} x^{11}}{11} + x^{9} \left (\frac{b e^{3}}{9} + \frac{c d e^{2}}{3}\right ) + x^{7} \left (\frac{a e^{3}}{7} + \frac{3 b d e^{2}}{7} + \frac{3 c d^{2} e}{7}\right ) + x^{5} \left (\frac{3 a d e^{2}}{5} + \frac{3 b d^{2} e}{5} + \frac{c d^{3}}{5}\right ) + x^{3} \left (a d^{2} e + \frac{b d^{3}}{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.13618, size = 146, normalized size = 1.42 \begin{align*} \frac{1}{11} \, c x^{11} e^{3} + \frac{1}{3} \, c d x^{9} e^{2} + \frac{1}{9} \, b x^{9} e^{3} + \frac{3}{7} \, c d^{2} x^{7} e + \frac{3}{7} \, b d x^{7} e^{2} + \frac{1}{5} \, c d^{3} x^{5} + \frac{1}{7} \, a x^{7} e^{3} + \frac{3}{5} \, b d^{2} x^{5} e + \frac{3}{5} \, a d x^{5} e^{2} + \frac{1}{3} \, b d^{3} x^{3} + 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+b*x^2+a),x, algorithm="giac")

[Out]

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