3.178 \(\int \frac{(a+b x^2)^2}{x^6 (c+d x^2)} \, dx\)

Optimal. Leaf size=87 \[ -\frac{a^2}{5 c x^5}-\frac{a (2 b c-a d)}{3 c^2 x^3}-\frac{(b c-a d)^2}{c^3 x}-\frac{\sqrt{d} (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{7/2}} \]

[Out]

-a^2/(5*c*x^5) - (a*(2*b*c - a*d))/(3*c^2*x^3) - (b*c - a*d)^2/(c^3*x) - (Sqrt[d]*(b*c - a*d)^2*ArcTan[(Sqrt[d
]*x)/Sqrt[c]])/c^(7/2)

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Rubi [A]  time = 0.0649296, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {461, 205} \[ -\frac{a^2}{5 c x^5}-\frac{a (2 b c-a d)}{3 c^2 x^3}-\frac{(b c-a d)^2}{c^3 x}-\frac{\sqrt{d} (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^2/(x^6*(c + d*x^2)),x]

[Out]

-a^2/(5*c*x^5) - (a*(2*b*c - a*d))/(3*c^2*x^3) - (b*c - a*d)^2/(c^3*x) - (Sqrt[d]*(b*c - a*d)^2*ArcTan[(Sqrt[d
]*x)/Sqrt[c]])/c^(7/2)

Rule 461

Int[(((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegr
and[((e*x)^m*(a + b*x^n)^p)/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n
, 0] && IGtQ[p, 0] && (IntegerQ[m] || IGtQ[2*(m + 1), 0] ||  !RationalQ[m])

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

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

Mathematica [A]  time = 0.0695702, size = 86, normalized size = 0.99 \[ -\frac{a^2}{5 c x^5}+\frac{a (a d-2 b c)}{3 c^2 x^3}-\frac{(b c-a d)^2}{c^3 x}-\frac{\sqrt{d} (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^2/(x^6*(c + d*x^2)),x]

[Out]

-a^2/(5*c*x^5) + (a*(-2*b*c + a*d))/(3*c^2*x^3) - (b*c - a*d)^2/(c^3*x) - (Sqrt[d]*(b*c - a*d)^2*ArcTan[(Sqrt[
d]*x)/Sqrt[c]])/c^(7/2)

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Maple [A]  time = 0.006, size = 143, normalized size = 1.6 \begin{align*} -{\frac{{a}^{2}{d}^{3}}{{c}^{3}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+2\,{\frac{ab{d}^{2}}{{c}^{2}\sqrt{cd}}\arctan \left ({\frac{dx}{\sqrt{cd}}} \right ) }-{\frac{{b}^{2}d}{c}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{{a}^{2}}{5\,c{x}^{5}}}-{\frac{{a}^{2}{d}^{2}}{{c}^{3}x}}+2\,{\frac{abd}{{c}^{2}x}}-{\frac{{b}^{2}}{cx}}+{\frac{{a}^{2}d}{3\,{c}^{2}{x}^{3}}}-{\frac{2\,ab}{3\,c{x}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^2/x^6/(d*x^2+c),x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/x^6/(d*x^2+c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.2616, size = 506, normalized size = 5.82 \begin{align*} \left [\frac{15 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{5} \sqrt{-\frac{d}{c}} \log \left (\frac{d x^{2} - 2 \, c x \sqrt{-\frac{d}{c}} - c}{d x^{2} + c}\right ) - 30 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{4} - 6 \, a^{2} c^{2} - 10 \,{\left (2 \, a b c^{2} - a^{2} c d\right )} x^{2}}{30 \, c^{3} x^{5}}, -\frac{15 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{5} \sqrt{\frac{d}{c}} \arctan \left (x \sqrt{\frac{d}{c}}\right ) + 15 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{4} + 3 \, a^{2} c^{2} + 5 \,{\left (2 \, a b c^{2} - a^{2} c d\right )} x^{2}}{15 \, c^{3} x^{5}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/x^6/(d*x^2+c),x, algorithm="fricas")

[Out]

[1/30*(15*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x^5*sqrt(-d/c)*log((d*x^2 - 2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)) - 30*
(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x^4 - 6*a^2*c^2 - 10*(2*a*b*c^2 - a^2*c*d)*x^2)/(c^3*x^5), -1/15*(15*(b^2*c^2
- 2*a*b*c*d + a^2*d^2)*x^5*sqrt(d/c)*arctan(x*sqrt(d/c)) + 15*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x^4 + 3*a^2*c^2
+ 5*(2*a*b*c^2 - a^2*c*d)*x^2)/(c^3*x^5)]

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Sympy [B]  time = 0.950197, size = 207, normalized size = 2.38 \begin{align*} \frac{\sqrt{- \frac{d}{c^{7}}} \left (a d - b c\right )^{2} \log{\left (- \frac{c^{4} \sqrt{- \frac{d}{c^{7}}} \left (a d - b c\right )^{2}}{a^{2} d^{3} - 2 a b c d^{2} + b^{2} c^{2} d} + x \right )}}{2} - \frac{\sqrt{- \frac{d}{c^{7}}} \left (a d - b c\right )^{2} \log{\left (\frac{c^{4} \sqrt{- \frac{d}{c^{7}}} \left (a d - b c\right )^{2}}{a^{2} d^{3} - 2 a b c d^{2} + b^{2} c^{2} d} + x \right )}}{2} - \frac{3 a^{2} c^{2} + x^{4} \left (15 a^{2} d^{2} - 30 a b c d + 15 b^{2} c^{2}\right ) + x^{2} \left (- 5 a^{2} c d + 10 a b c^{2}\right )}{15 c^{3} x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**2/x**6/(d*x**2+c),x)

[Out]

sqrt(-d/c**7)*(a*d - b*c)**2*log(-c**4*sqrt(-d/c**7)*(a*d - b*c)**2/(a**2*d**3 - 2*a*b*c*d**2 + b**2*c**2*d) +
x)/2 - sqrt(-d/c**7)*(a*d - b*c)**2*log(c**4*sqrt(-d/c**7)*(a*d - b*c)**2/(a**2*d**3 - 2*a*b*c*d**2 + b**2*c*
*2*d) + x)/2 - (3*a**2*c**2 + x**4*(15*a**2*d**2 - 30*a*b*c*d + 15*b**2*c**2) + x**2*(-5*a**2*c*d + 10*a*b*c**
2))/(15*c**3*x**5)

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Giac [A]  time = 1.18369, size = 151, normalized size = 1.74 \begin{align*} -\frac{{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{\sqrt{c d} c^{3}} - \frac{15 \, b^{2} c^{2} x^{4} - 30 \, a b c d x^{4} + 15 \, a^{2} d^{2} x^{4} + 10 \, a b c^{2} x^{2} - 5 \, a^{2} c d x^{2} + 3 \, a^{2} c^{2}}{15 \, c^{3} x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/x^6/(d*x^2+c),x, algorithm="giac")

[Out]

-(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*arctan(d*x/sqrt(c*d))/(sqrt(c*d)*c^3) - 1/15*(15*b^2*c^2*x^4 - 30*a*b*c*d
*x^4 + 15*a^2*d^2*x^4 + 10*a*b*c^2*x^2 - 5*a^2*c*d*x^2 + 3*a^2*c^2)/(c^3*x^5)