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

Optimal. Leaf size=74 \[ \frac{1}{4} x^4 \left (e (a e+2 b d)+c d^2\right )+\frac{1}{2} d x^2 (2 a e+b d)+a d^2 \log (x)+\frac{1}{6} e x^6 (b e+2 c d)+\frac{1}{8} c e^2 x^8 \]

[Out]

(d*(b*d + 2*a*e)*x^2)/2 + ((c*d^2 + e*(2*b*d + a*e))*x^4)/4 + (e*(2*c*d + b*e)*x^6)/6 + (c*e^2*x^8)/8 + a*d^2*
Log[x]

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Rubi [A]  time = 0.0901146, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {1251, 893} \[ \frac{1}{4} x^4 \left (e (a e+2 b d)+c d^2\right )+\frac{1}{2} d x^2 (2 a e+b d)+a d^2 \log (x)+\frac{1}{6} e x^6 (b e+2 c d)+\frac{1}{8} c e^2 x^8 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d*(b*d + 2*a*e)*x^2)/2 + ((c*d^2 + e*(2*b*d + a*e))*x^4)/4 + (e*(2*c*d + b*e)*x^6)/6 + (c*e^2*x^8)/8 + a*d^2*
Log[x]

Rule 1251

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] &&
IntegerQ[(m - 1)/2]

Rule 893

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rubi steps

\begin{align*} \int \frac{\left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right )}{x} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(d+e x)^2 \left (a+b x+c x^2\right )}{x} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (d (b d+2 a e)+\frac{a d^2}{x}+\left (c d^2+e (2 b d+a e)\right ) x+e (2 c d+b e) x^2+c e^2 x^3\right ) \, dx,x,x^2\right )\\ &=\frac{1}{2} d (b d+2 a e) x^2+\frac{1}{4} \left (c d^2+e (2 b d+a e)\right ) x^4+\frac{1}{6} e (2 c d+b e) x^6+\frac{1}{8} c e^2 x^8+a d^2 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0206325, size = 74, normalized size = 1. \[ \frac{1}{4} x^4 \left (a e^2+2 b d e+c d^2\right )+\frac{1}{2} d x^2 (2 a e+b d)+a d^2 \log (x)+\frac{1}{6} e x^6 (b e+2 c d)+\frac{1}{8} c e^2 x^8 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d*(b*d + 2*a*e)*x^2)/2 + ((c*d^2 + 2*b*d*e + a*e^2)*x^4)/4 + (e*(2*c*d + b*e)*x^6)/6 + (c*e^2*x^8)/8 + a*d^2*
Log[x]

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Maple [A]  time = 0.003, size = 77, normalized size = 1. \begin{align*}{\frac{c{e}^{2}{x}^{8}}{8}}+{\frac{{x}^{6}b{e}^{2}}{6}}+{\frac{{x}^{6}cde}{3}}+{\frac{{x}^{4}a{e}^{2}}{4}}+{\frac{{x}^{4}bde}{2}}+{\frac{{x}^{4}c{d}^{2}}{4}}+{x}^{2}ade+{\frac{{x}^{2}b{d}^{2}}{2}}+a{d}^{2}\ln \left ( x \right ) \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,x)

[Out]

1/8*c*e^2*x^8+1/6*x^6*b*e^2+1/3*x^6*c*d*e+1/4*x^4*a*e^2+1/2*x^4*b*d*e+1/4*x^4*c*d^2+x^2*a*d*e+1/2*x^2*b*d^2+a*
d^2*ln(x)

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Maxima [A]  time = 0.958351, size = 99, normalized size = 1.34 \begin{align*} \frac{1}{8} \, c e^{2} x^{8} + \frac{1}{6} \,{\left (2 \, c d e + b e^{2}\right )} x^{6} + \frac{1}{4} \,{\left (c d^{2} + 2 \, b d e + a e^{2}\right )} x^{4} + \frac{1}{2} \, a d^{2} \log \left (x^{2}\right ) + \frac{1}{2} \,{\left (b d^{2} + 2 \, a d e\right )} x^{2} \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,x, algorithm="maxima")

[Out]

1/8*c*e^2*x^8 + 1/6*(2*c*d*e + b*e^2)*x^6 + 1/4*(c*d^2 + 2*b*d*e + a*e^2)*x^4 + 1/2*a*d^2*log(x^2) + 1/2*(b*d^
2 + 2*a*d*e)*x^2

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Fricas [A]  time = 1.66027, size = 165, normalized size = 2.23 \begin{align*} \frac{1}{8} \, c e^{2} x^{8} + \frac{1}{6} \,{\left (2 \, c d e + b e^{2}\right )} x^{6} + \frac{1}{4} \,{\left (c d^{2} + 2 \, b d e + a e^{2}\right )} x^{4} + a d^{2} \log \left (x\right ) + \frac{1}{2} \,{\left (b d^{2} + 2 \, a d e\right )} x^{2} \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,x, algorithm="fricas")

[Out]

1/8*c*e^2*x^8 + 1/6*(2*c*d*e + b*e^2)*x^6 + 1/4*(c*d^2 + 2*b*d*e + a*e^2)*x^4 + a*d^2*log(x) + 1/2*(b*d^2 + 2*
a*d*e)*x^2

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Sympy [A]  time = 0.316436, size = 73, normalized size = 0.99 \begin{align*} a d^{2} \log{\left (x \right )} + \frac{c e^{2} x^{8}}{8} + x^{6} \left (\frac{b e^{2}}{6} + \frac{c d e}{3}\right ) + x^{4} \left (\frac{a e^{2}}{4} + \frac{b d e}{2} + \frac{c d^{2}}{4}\right ) + x^{2} \left (a d e + \frac{b d^{2}}{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,x)

[Out]

a*d**2*log(x) + c*e**2*x**8/8 + x**6*(b*e**2/6 + c*d*e/3) + x**4*(a*e**2/4 + b*d*e/2 + c*d**2/4) + x**2*(a*d*e
+ b*d**2/2)

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Giac [A]  time = 1.09819, size = 107, normalized size = 1.45 \begin{align*} \frac{1}{8} \, c x^{8} e^{2} + \frac{1}{3} \, c d x^{6} e + \frac{1}{6} \, b x^{6} e^{2} + \frac{1}{4} \, c d^{2} x^{4} + \frac{1}{2} \, b d x^{4} e + \frac{1}{4} \, a x^{4} e^{2} + \frac{1}{2} \, b d^{2} x^{2} + a d x^{2} e + \frac{1}{2} \, a d^{2} \log \left (x^{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,x, algorithm="giac")

[Out]

1/8*c*x^8*e^2 + 1/3*c*d*x^6*e + 1/6*b*x^6*e^2 + 1/4*c*d^2*x^4 + 1/2*b*d*x^4*e + 1/4*a*x^4*e^2 + 1/2*b*d^2*x^2
+ a*d*x^2*e + 1/2*a*d^2*log(x^2)