3.278 \(\int \frac{(d+e x^2)^2 (a+b x^2+c x^4)}{x^2} \, dx\)

Optimal. Leaf size=71 \[ \frac{1}{3} x^3 \left (e (a e+2 b d)+c d^2\right )+d x (2 a e+b d)-\frac{a d^2}{x}+\frac{1}{5} e x^5 (b e+2 c d)+\frac{1}{7} c e^2 x^7 \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0483196, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {1261} \[ \frac{1}{3} x^3 \left (e (a e+2 b d)+c d^2\right )+d x (2 a e+b d)-\frac{a d^2}{x}+\frac{1}{5} e x^5 (b e+2 c d)+\frac{1}{7} c e^2 x^7 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1261

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

Rubi steps

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

Mathematica [A]  time = 0.0332107, size = 71, normalized size = 1. \[ \frac{1}{3} x^3 \left (a e^2+2 b d e+c d^2\right )+d x (2 a e+b d)-\frac{a d^2}{x}+\frac{1}{5} e x^5 (b e+2 c d)+\frac{1}{7} c e^2 x^7 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 75, normalized size = 1.1 \begin{align*}{\frac{c{e}^{2}{x}^{7}}{7}}+{\frac{{x}^{5}b{e}^{2}}{5}}+{\frac{2\,{x}^{5}cde}{5}}+{\frac{{x}^{3}a{e}^{2}}{3}}+{\frac{2\,{x}^{3}bde}{3}}+{\frac{{x}^{3}c{d}^{2}}{3}}+2\,deax+{d}^{2}bx-{\frac{a{d}^{2}}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.947018, size = 93, normalized size = 1.31 \begin{align*} \frac{1}{7} \, c e^{2} x^{7} + \frac{1}{5} \,{\left (2 \, c d e + b e^{2}\right )} x^{5} + \frac{1}{3} \,{\left (c d^{2} + 2 \, b d e + a e^{2}\right )} x^{3} - \frac{a d^{2}}{x} +{\left (b d^{2} + 2 \, a d e\right )} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.6627, size = 170, normalized size = 2.39 \begin{align*} \frac{15 \, c e^{2} x^{8} + 21 \,{\left (2 \, c d e + b e^{2}\right )} x^{6} + 35 \,{\left (c d^{2} + 2 \, b d e + a e^{2}\right )} x^{4} - 105 \, a d^{2} + 105 \,{\left (b d^{2} + 2 \, a d e\right )} x^{2}}{105 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*(15*c*e^2*x^8 + 21*(2*c*d*e + b*e^2)*x^6 + 35*(c*d^2 + 2*b*d*e + a*e^2)*x^4 - 105*a*d^2 + 105*(b*d^2 + 2
*a*d*e)*x^2)/x

________________________________________________________________________________________

Sympy [A]  time = 0.315117, size = 73, normalized size = 1.03 \begin{align*} - \frac{a d^{2}}{x} + \frac{c e^{2} x^{7}}{7} + x^{5} \left (\frac{b e^{2}}{5} + \frac{2 c d e}{5}\right ) + x^{3} \left (\frac{a e^{2}}{3} + \frac{2 b d e}{3} + \frac{c d^{2}}{3}\right ) + x \left (2 a d e + b d^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.08033, size = 100, normalized size = 1.41 \begin{align*} \frac{1}{7} \, c x^{7} e^{2} + \frac{2}{5} \, c d x^{5} e + \frac{1}{5} \, b x^{5} e^{2} + \frac{1}{3} \, c d^{2} x^{3} + \frac{2}{3} \, b d x^{3} e + \frac{1}{3} \, a x^{3} e^{2} + b d^{2} x + 2 \, a d x e - \frac{a d^{2}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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