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

Optimal. Leaf size=126 \[ \frac{x \left (5 a^2 d^2-6 a b c d+3 b^2 c^2\right )}{6 c^3 \left (c+d x^2\right )}-\frac{a^2}{3 c x^3 \left (c+d x^2\right )}-\frac{a (6 b c-5 a d)}{3 c^3 x}+\frac{(b c-5 a d) (b c-a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{7/2} \sqrt{d}} \]

[Out]

-(a*(6*b*c - 5*a*d))/(3*c^3*x) - a^2/(3*c*x^3*(c + d*x^2)) + ((3*b^2*c^2 - 6*a*b*c*d + 5*a^2*d^2)*x)/(6*c^3*(c
+ d*x^2)) + ((b*c - 5*a*d)*(b*c - a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(2*c^(7/2)*Sqrt[d])

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Rubi [A]  time = 0.134679, antiderivative size = 126, normalized size of antiderivative = 1., 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, 456, 453, 205} \[ \frac{x \left (5 a^2 d^2-6 a b c d+3 b^2 c^2\right )}{6 c^3 \left (c+d x^2\right )}-\frac{a^2}{3 c x^3 \left (c+d x^2\right )}-\frac{a (6 b c-5 a d)}{3 c^3 x}+\frac{(b c-5 a d) (b c-a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{7/2} \sqrt{d}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*(6*b*c - 5*a*d))/(3*c^3*x) - a^2/(3*c*x^3*(c + d*x^2)) + ((3*b^2*c^2 - 6*a*b*c*d + 5*a^2*d^2)*x)/(6*c^3*(c
+ d*x^2)) + ((b*c - 5*a*d)*(b*c - a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(2*c^(7/2)*Sqrt[d])

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 456

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[((-a)^(m/2 - 1)*(b*c - a*d)*
x*(a + b*x^2)^(p + 1))/(2*b^(m/2 + 1)*(p + 1)), x] + Dist[1/(2*b^(m/2 + 1)*(p + 1)), Int[x^m*(a + b*x^2)^(p +
1)*ExpandToSum[2*b*(p + 1)*Together[(b^(m/2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d)*x^(-m + 2))/(a + b*x^2)]
- ((-a)^(m/2 - 1)*(b*c - a*d))/x^m, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &
& ILtQ[m/2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])

Rule 453

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

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

Mathematica [A]  time = 0.0638684, size = 107, normalized size = 0.85 \[ \frac{\left (5 a^2 d^2-6 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{7/2} \sqrt{d}}-\frac{a^2}{3 c^2 x^3}+\frac{x (b c-a d)^2}{2 c^3 \left (c+d x^2\right )}+\frac{2 a (a d-b c)}{c^3 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-a^2/(3*c^2*x^3) + (2*a*(-(b*c) + a*d))/(c^3*x) + ((b*c - a*d)^2*x)/(2*c^3*(c + d*x^2)) + ((b^2*c^2 - 6*a*b*c*
d + 5*a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(2*c^(7/2)*Sqrt[d])

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Maple [A]  time = 0.011, size = 161, normalized size = 1.3 \begin{align*}{\frac{{a}^{2}{d}^{2}x}{2\,{c}^{3} \left ( d{x}^{2}+c \right ) }}-{\frac{abdx}{{c}^{2} \left ( d{x}^{2}+c \right ) }}+{\frac{x{b}^{2}}{2\,c \left ( d{x}^{2}+c \right ) }}+{\frac{5\,{a}^{2}{d}^{2}}{2\,{c}^{3}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-3\,{\frac{abd}{{c}^{2}\sqrt{cd}}\arctan \left ({\frac{dx}{\sqrt{cd}}} \right ) }+{\frac{{b}^{2}}{2\,c}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{{a}^{2}}{3\,{c}^{2}{x}^{3}}}+2\,{\frac{{a}^{2}d}{{c}^{3}x}}-2\,{\frac{ab}{{c}^{2}x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/c^3*x/(d*x^2+c)*a^2*d^2-1/c^2*x/(d*x^2+c)*a*b*d+1/2/c*x/(d*x^2+c)*b^2+5/2/c^3/(c*d)^(1/2)*arctan(x*d/(c*d)
^(1/2))*a^2*d^2-3/c^2/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))*a*b*d+1/2/c/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))*b^2-
1/3*a^2/c^2/x^3+2*a^2/c^3/x*d-2*a/c^2/x*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^4/(d*x^2+c)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.48407, size = 737, normalized size = 5.85 \begin{align*} \left [-\frac{4 \, a^{2} c^{3} d - 6 \,{\left (b^{2} c^{3} d - 6 \, a b c^{2} d^{2} + 5 \, a^{2} c d^{3}\right )} x^{4} + 4 \,{\left (6 \, a b c^{3} d - 5 \, a^{2} c^{2} d^{2}\right )} x^{2} + 3 \,{\left ({\left (b^{2} c^{2} d - 6 \, a b c d^{2} + 5 \, a^{2} d^{3}\right )} x^{5} +{\left (b^{2} c^{3} - 6 \, a b c^{2} d + 5 \, a^{2} c d^{2}\right )} x^{3}\right )} \sqrt{-c d} \log \left (\frac{d x^{2} - 2 \, \sqrt{-c d} x - c}{d x^{2} + c}\right )}{12 \,{\left (c^{4} d^{2} x^{5} + c^{5} d x^{3}\right )}}, -\frac{2 \, a^{2} c^{3} d - 3 \,{\left (b^{2} c^{3} d - 6 \, a b c^{2} d^{2} + 5 \, a^{2} c d^{3}\right )} x^{4} + 2 \,{\left (6 \, a b c^{3} d - 5 \, a^{2} c^{2} d^{2}\right )} x^{2} - 3 \,{\left ({\left (b^{2} c^{2} d - 6 \, a b c d^{2} + 5 \, a^{2} d^{3}\right )} x^{5} +{\left (b^{2} c^{3} - 6 \, a b c^{2} d + 5 \, a^{2} c d^{2}\right )} x^{3}\right )} \sqrt{c d} \arctan \left (\frac{\sqrt{c d} x}{c}\right )}{6 \,{\left (c^{4} d^{2} x^{5} + c^{5} d x^{3}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/12*(4*a^2*c^3*d - 6*(b^2*c^3*d - 6*a*b*c^2*d^2 + 5*a^2*c*d^3)*x^4 + 4*(6*a*b*c^3*d - 5*a^2*c^2*d^2)*x^2 +
3*((b^2*c^2*d - 6*a*b*c*d^2 + 5*a^2*d^3)*x^5 + (b^2*c^3 - 6*a*b*c^2*d + 5*a^2*c*d^2)*x^3)*sqrt(-c*d)*log((d*x^
2 - 2*sqrt(-c*d)*x - c)/(d*x^2 + c)))/(c^4*d^2*x^5 + c^5*d*x^3), -1/6*(2*a^2*c^3*d - 3*(b^2*c^3*d - 6*a*b*c^2*
d^2 + 5*a^2*c*d^3)*x^4 + 2*(6*a*b*c^3*d - 5*a^2*c^2*d^2)*x^2 - 3*((b^2*c^2*d - 6*a*b*c*d^2 + 5*a^2*d^3)*x^5 +
(b^2*c^3 - 6*a*b*c^2*d + 5*a^2*c*d^2)*x^3)*sqrt(c*d)*arctan(sqrt(c*d)*x/c))/(c^4*d^2*x^5 + c^5*d*x^3)]

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Sympy [B]  time = 1.17905, size = 248, normalized size = 1.97 \begin{align*} - \frac{\sqrt{- \frac{1}{c^{7} d}} \left (a d - b c\right ) \left (5 a d - b c\right ) \log{\left (- \frac{c^{4} \sqrt{- \frac{1}{c^{7} d}} \left (a d - b c\right ) \left (5 a d - b c\right )}{5 a^{2} d^{2} - 6 a b c d + b^{2} c^{2}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{c^{7} d}} \left (a d - b c\right ) \left (5 a d - b c\right ) \log{\left (\frac{c^{4} \sqrt{- \frac{1}{c^{7} d}} \left (a d - b c\right ) \left (5 a d - b c\right )}{5 a^{2} d^{2} - 6 a b c d + b^{2} c^{2}} + x \right )}}{4} + \frac{- 2 a^{2} c^{2} + x^{4} \left (15 a^{2} d^{2} - 18 a b c d + 3 b^{2} c^{2}\right ) + x^{2} \left (10 a^{2} c d - 12 a b c^{2}\right )}{6 c^{4} x^{3} + 6 c^{3} d x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-1/(c**7*d))*(a*d - b*c)*(5*a*d - b*c)*log(-c**4*sqrt(-1/(c**7*d))*(a*d - b*c)*(5*a*d - b*c)/(5*a**2*d**
2 - 6*a*b*c*d + b**2*c**2) + x)/4 + sqrt(-1/(c**7*d))*(a*d - b*c)*(5*a*d - b*c)*log(c**4*sqrt(-1/(c**7*d))*(a*
d - b*c)*(5*a*d - b*c)/(5*a**2*d**2 - 6*a*b*c*d + b**2*c**2) + x)/4 + (-2*a**2*c**2 + x**4*(15*a**2*d**2 - 18*
a*b*c*d + 3*b**2*c**2) + x**2*(10*a**2*c*d - 12*a*b*c**2))/(6*c**4*x**3 + 6*c**3*d*x**5)

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Giac [A]  time = 1.12711, size = 150, normalized size = 1.19 \begin{align*} \frac{{\left (b^{2} c^{2} - 6 \, a b c d + 5 \, a^{2} d^{2}\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{2 \, \sqrt{c d} c^{3}} + \frac{b^{2} c^{2} x - 2 \, a b c d x + a^{2} d^{2} x}{2 \,{\left (d x^{2} + c\right )} c^{3}} - \frac{6 \, a b c x^{2} - 6 \, a^{2} d x^{2} + a^{2} c}{3 \, c^{3} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b^2*c^2 - 6*a*b*c*d + 5*a^2*d^2)*arctan(d*x/sqrt(c*d))/(sqrt(c*d)*c^3) + 1/2*(b^2*c^2*x - 2*a*b*c*d*x + a
^2*d^2*x)/((d*x^2 + c)*c^3) - 1/3*(6*a*b*c*x^2 - 6*a^2*d*x^2 + a^2*c)/(c^3*x^3)