3.195 \(\int \frac{(a+b x^2)^2}{x^2 (c+d x^2)^3} \, dx\)

Optimal. Leaf size=152 \[ -\frac{x \left (5 a^2 d^2-2 a b c d+b^2 c^2\right )}{4 c^2 d \left (c+d x^2\right )^2}-\frac{a^2}{c x \left (c+d x^2\right )^2}+\frac{\left (3 a d (2 b c-5 a d)+b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{7/2} d^{3/2}}+\frac{x \left (3 a d (2 b c-5 a d)+b^2 c^2\right )}{8 c^3 d \left (c+d x^2\right )} \]

[Out]

-(a^2/(c*x*(c + d*x^2)^2)) - ((b^2*c^2 - 2*a*b*c*d + 5*a^2*d^2)*x)/(4*c^2*d*(c + d*x^2)^2) + ((b^2*c^2 + 3*a*d
*(2*b*c - 5*a*d))*x)/(8*c^3*d*(c + d*x^2)) + ((b^2*c^2 + 3*a*d*(2*b*c - 5*a*d))*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(
8*c^(7/2)*d^(3/2))

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Rubi [A]  time = 0.108841, antiderivative size = 151, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {462, 385, 199, 205} \[ -\frac{x \left (5 a^2 d^2-2 a b c d+b^2 c^2\right )}{4 c^2 d \left (c+d x^2\right )^2}-\frac{a^2}{c x \left (c+d x^2\right )^2}+\frac{\left (3 a d (2 b c-5 a d)+b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{7/2} d^{3/2}}+\frac{x \left (\frac{3 a (2 b c-5 a d)}{c^2}+\frac{b^2}{d}\right )}{8 c \left (c+d x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^2/(c*x*(c + d*x^2)^2)) - ((b^2*c^2 - 2*a*b*c*d + 5*a^2*d^2)*x)/(4*c^2*d*(c + d*x^2)^2) + ((b^2/d + (3*a*(2
*b*c - 5*a*d))/c^2)*x)/(8*c*(c + d*x^2)) + ((b^2*c^2 + 3*a*d*(2*b*c - 5*a*d))*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*
c^(7/2)*d^(3/2))

Rule 462

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^2, x_Symbol] :> Simp[(c^2*(e*x)^(
m + 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] - Dist[1/(a*e^n*(m + 1)), Int[(e*x)^(m + n)*(a + b*x^n)^p*Simp[b
*c^2*n*(p + 1) + c*(b*c - 2*a*d)*(m + 1) - a*(m + 1)*d^2*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && Ne
Q[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[m, -1] && GtQ[n, 0]

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> -Simp[((b*c - a*d)*x*(a + b*x^n)^(p +
1))/(a*b*n*(p + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /
; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

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^2 \left (c+d x^2\right )^3} \, dx &=-\frac{a^2}{c x \left (c+d x^2\right )^2}+\frac{\int \frac{a (2 b c-5 a d)+b^2 c x^2}{\left (c+d x^2\right )^3} \, dx}{c}\\ &=-\frac{a^2}{c x \left (c+d x^2\right )^2}-\frac{\left (b^2 c^2-2 a b c d+5 a^2 d^2\right ) x}{4 c^2 d \left (c+d x^2\right )^2}+\frac{1}{4} \left (\frac{b^2}{d}+\frac{3 a (2 b c-5 a d)}{c^2}\right ) \int \frac{1}{\left (c+d x^2\right )^2} \, dx\\ &=-\frac{a^2}{c x \left (c+d x^2\right )^2}-\frac{\left (b^2 c^2-2 a b c d+5 a^2 d^2\right ) x}{4 c^2 d \left (c+d x^2\right )^2}+\frac{\left (\frac{b^2}{d}+\frac{3 a (2 b c-5 a d)}{c^2}\right ) x}{8 c \left (c+d x^2\right )}+\frac{\left (\frac{b^2}{d}+\frac{3 a (2 b c-5 a d)}{c^2}\right ) \int \frac{1}{c+d x^2} \, dx}{8 c}\\ &=-\frac{a^2}{c x \left (c+d x^2\right )^2}-\frac{\left (b^2 c^2-2 a b c d+5 a^2 d^2\right ) x}{4 c^2 d \left (c+d x^2\right )^2}+\frac{\left (\frac{b^2}{d}+\frac{3 a (2 b c-5 a d)}{c^2}\right ) x}{8 c \left (c+d x^2\right )}+\frac{\left (b^2 c^2+6 a b c d-15 a^2 d^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{7/2} d^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0880691, size = 133, normalized size = 0.88 \[ \frac{x \left (-7 a^2 d^2+6 a b c d+b^2 c^2\right )}{8 c^3 d \left (c+d x^2\right )}+\frac{\left (-15 a^2 d^2+6 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{7/2} d^{3/2}}-\frac{a^2}{c^3 x}-\frac{x (b c-a d)^2}{4 c^2 d \left (c+d x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^2/(c^3*x)) - ((b*c - a*d)^2*x)/(4*c^2*d*(c + d*x^2)^2) + ((b^2*c^2 + 6*a*b*c*d - 7*a^2*d^2)*x)/(8*c^3*d*(c
+ d*x^2)) + ((b^2*c^2 + 6*a*b*c*d - 15*a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*c^(7/2)*d^(3/2))

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Maple [A]  time = 0.012, size = 199, normalized size = 1.3 \begin{align*} -{\frac{7\,{x}^{3}{a}^{2}{d}^{2}}{8\,{c}^{3} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{3\,{x}^{3}abd}{4\,{c}^{2} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{{b}^{2}{x}^{3}}{8\,c \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{9\,{a}^{2}dx}{8\,{c}^{2} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{5\,abx}{4\,c \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{x{b}^{2}}{8\, \left ( d{x}^{2}+c \right ) ^{2}d}}-{\frac{15\,{a}^{2}d}{8\,{c}^{3}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{3\,ab}{4\,{c}^{2}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{{b}^{2}}{8\,cd}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{{a}^{2}}{{c}^{3}x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^2/x^2/(d*x^2+c)^3,x)

[Out]

-7/8/c^3/(d*x^2+c)^2*x^3*a^2*d^2+3/4/c^2/(d*x^2+c)^2*x^3*a*b*d+1/8/c/(d*x^2+c)^2*x^3*b^2-9/8/c^2/(d*x^2+c)^2*a
^2*d*x+5/4/c/(d*x^2+c)^2*a*b*x-1/8/(d*x^2+c)^2/d*x*b^2-15/8/c^3*d/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))*a^2+3/4/
c^2/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))*a*b+1/8/c/d/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))*b^2-a^2/c^3/x

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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^2/(d*x^2+c)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.53838, size = 976, normalized size = 6.42 \begin{align*} \left [-\frac{16 \, a^{2} c^{3} d^{2} - 2 \,{\left (b^{2} c^{3} d^{2} + 6 \, a b c^{2} d^{3} - 15 \, a^{2} c d^{4}\right )} x^{4} + 2 \,{\left (b^{2} c^{4} d - 10 \, a b c^{3} d^{2} + 25 \, a^{2} c^{2} d^{3}\right )} x^{2} -{\left ({\left (b^{2} c^{2} d^{2} + 6 \, a b c d^{3} - 15 \, a^{2} d^{4}\right )} x^{5} + 2 \,{\left (b^{2} c^{3} d + 6 \, a b c^{2} d^{2} - 15 \, a^{2} c d^{3}\right )} x^{3} +{\left (b^{2} c^{4} + 6 \, a b c^{3} d - 15 \, a^{2} c^{2} d^{2}\right )} x\right )} \sqrt{-c d} \log \left (\frac{d x^{2} + 2 \, \sqrt{-c d} x - c}{d x^{2} + c}\right )}{16 \,{\left (c^{4} d^{4} x^{5} + 2 \, c^{5} d^{3} x^{3} + c^{6} d^{2} x\right )}}, -\frac{8 \, a^{2} c^{3} d^{2} -{\left (b^{2} c^{3} d^{2} + 6 \, a b c^{2} d^{3} - 15 \, a^{2} c d^{4}\right )} x^{4} +{\left (b^{2} c^{4} d - 10 \, a b c^{3} d^{2} + 25 \, a^{2} c^{2} d^{3}\right )} x^{2} -{\left ({\left (b^{2} c^{2} d^{2} + 6 \, a b c d^{3} - 15 \, a^{2} d^{4}\right )} x^{5} + 2 \,{\left (b^{2} c^{3} d + 6 \, a b c^{2} d^{2} - 15 \, a^{2} c d^{3}\right )} x^{3} +{\left (b^{2} c^{4} + 6 \, a b c^{3} d - 15 \, a^{2} c^{2} d^{2}\right )} x\right )} \sqrt{c d} \arctan \left (\frac{\sqrt{c d} x}{c}\right )}{8 \,{\left (c^{4} d^{4} x^{5} + 2 \, c^{5} d^{3} x^{3} + c^{6} d^{2} x\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/16*(16*a^2*c^3*d^2 - 2*(b^2*c^3*d^2 + 6*a*b*c^2*d^3 - 15*a^2*c*d^4)*x^4 + 2*(b^2*c^4*d - 10*a*b*c^3*d^2 +
25*a^2*c^2*d^3)*x^2 - ((b^2*c^2*d^2 + 6*a*b*c*d^3 - 15*a^2*d^4)*x^5 + 2*(b^2*c^3*d + 6*a*b*c^2*d^2 - 15*a^2*c*
d^3)*x^3 + (b^2*c^4 + 6*a*b*c^3*d - 15*a^2*c^2*d^2)*x)*sqrt(-c*d)*log((d*x^2 + 2*sqrt(-c*d)*x - c)/(d*x^2 + c)
))/(c^4*d^4*x^5 + 2*c^5*d^3*x^3 + c^6*d^2*x), -1/8*(8*a^2*c^3*d^2 - (b^2*c^3*d^2 + 6*a*b*c^2*d^3 - 15*a^2*c*d^
4)*x^4 + (b^2*c^4*d - 10*a*b*c^3*d^2 + 25*a^2*c^2*d^3)*x^2 - ((b^2*c^2*d^2 + 6*a*b*c*d^3 - 15*a^2*d^4)*x^5 + 2
*(b^2*c^3*d + 6*a*b*c^2*d^2 - 15*a^2*c*d^3)*x^3 + (b^2*c^4 + 6*a*b*c^3*d - 15*a^2*c^2*d^2)*x)*sqrt(c*d)*arctan
(sqrt(c*d)*x/c))/(c^4*d^4*x^5 + 2*c^5*d^3*x^3 + c^6*d^2*x)]

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Sympy [A]  time = 1.49247, size = 224, normalized size = 1.47 \begin{align*} \frac{\sqrt{- \frac{1}{c^{7} d^{3}}} \left (15 a^{2} d^{2} - 6 a b c d - b^{2} c^{2}\right ) \log{\left (- c^{4} d \sqrt{- \frac{1}{c^{7} d^{3}}} + x \right )}}{16} - \frac{\sqrt{- \frac{1}{c^{7} d^{3}}} \left (15 a^{2} d^{2} - 6 a b c d - b^{2} c^{2}\right ) \log{\left (c^{4} d \sqrt{- \frac{1}{c^{7} d^{3}}} + x \right )}}{16} - \frac{8 a^{2} c^{2} d + x^{4} \left (15 a^{2} d^{3} - 6 a b c d^{2} - b^{2} c^{2} d\right ) + x^{2} \left (25 a^{2} c d^{2} - 10 a b c^{2} d + b^{2} c^{3}\right )}{8 c^{5} d x + 16 c^{4} d^{2} x^{3} + 8 c^{3} d^{3} x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(-1/(c**7*d**3))*(15*a**2*d**2 - 6*a*b*c*d - b**2*c**2)*log(-c**4*d*sqrt(-1/(c**7*d**3)) + x)/16 - sqrt(-1
/(c**7*d**3))*(15*a**2*d**2 - 6*a*b*c*d - b**2*c**2)*log(c**4*d*sqrt(-1/(c**7*d**3)) + x)/16 - (8*a**2*c**2*d
+ x**4*(15*a**2*d**3 - 6*a*b*c*d**2 - b**2*c**2*d) + x**2*(25*a**2*c*d**2 - 10*a*b*c**2*d + b**2*c**3))/(8*c**
5*d*x + 16*c**4*d**2*x**3 + 8*c**3*d**3*x**5)

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Giac [A]  time = 1.16719, size = 182, normalized size = 1.2 \begin{align*} -\frac{a^{2}}{c^{3} x} + \frac{{\left (b^{2} c^{2} + 6 \, a b c d - 15 \, a^{2} d^{2}\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{8 \, \sqrt{c d} c^{3} d} + \frac{b^{2} c^{2} d x^{3} + 6 \, a b c d^{2} x^{3} - 7 \, a^{2} d^{3} x^{3} - b^{2} c^{3} x + 10 \, a b c^{2} d x - 9 \, a^{2} c d^{2} x}{8 \,{\left (d x^{2} + c\right )}^{2} c^{3} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a^2/(c^3*x) + 1/8*(b^2*c^2 + 6*a*b*c*d - 15*a^2*d^2)*arctan(d*x/sqrt(c*d))/(sqrt(c*d)*c^3*d) + 1/8*(b^2*c^2*d
*x^3 + 6*a*b*c*d^2*x^3 - 7*a^2*d^3*x^3 - b^2*c^3*x + 10*a*b*c^2*d*x - 9*a^2*c*d^2*x)/((d*x^2 + c)^2*c^3*d)