3.197 \(\int \frac{\left (a+b x^2\right )^2}{x^4 \left (c+d x^2\right )^3} \, dx\)

Optimal. Leaf size=161 \[ \frac{\left (35 a^2 d^2-30 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{9/2} \sqrt{d}}+\frac{x \left (7 a^2 d^2-6 a b c d+3 b^2 c^2\right )}{12 c^3 \left (c+d x^2\right )^2}-\frac{a^2}{3 c x^3 \left (c+d x^2\right )^2}+\frac{x (3 b c-7 a d)^2}{24 c^4 \left (c+d x^2\right )}-\frac{a (6 b c-7 a d)}{3 c^4 x} \]

[Out]

-(a*(6*b*c - 7*a*d))/(3*c^4*x) - a^2/(3*c*x^3*(c + d*x^2)^2) + ((3*b^2*c^2 - 6*a
*b*c*d + 7*a^2*d^2)*x)/(12*c^3*(c + d*x^2)^2) + ((3*b*c - 7*a*d)^2*x)/(24*c^4*(c
+ d*x^2)) + ((3*b^2*c^2 - 30*a*b*c*d + 35*a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])
/(8*c^(9/2)*Sqrt[d])

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Rubi [A]  time = 0.474135, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{\left (35 a^2 d^2-30 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{9/2} \sqrt{d}}+\frac{x \left (7 a^2 d^2-6 a b c d+3 b^2 c^2\right )}{12 c^3 \left (c+d x^2\right )^2}-\frac{a^2}{3 c x^3 \left (c+d x^2\right )^2}+\frac{x (3 b c-7 a d)^2}{24 c^4 \left (c+d x^2\right )}-\frac{a (6 b c-7 a d)}{3 c^4 x} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^2/(x^4*(c + d*x^2)^3),x]

[Out]

-(a*(6*b*c - 7*a*d))/(3*c^4*x) - a^2/(3*c*x^3*(c + d*x^2)^2) + ((3*b^2*c^2 - 6*a
*b*c*d + 7*a^2*d^2)*x)/(12*c^3*(c + d*x^2)^2) + ((3*b*c - 7*a*d)^2*x)/(24*c^4*(c
+ d*x^2)) + ((3*b^2*c^2 - 30*a*b*c*d + 35*a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])
/(8*c^(9/2)*Sqrt[d])

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Rubi in Sympy [A]  time = 68.002, size = 150, normalized size = 0.93 \[ - \frac{a^{2}}{3 c x^{3} \left (c + d x^{2}\right )^{2}} + \frac{a \left (7 a d - 6 b c\right )}{3 c^{4} x} + \frac{x \left (a d \left (7 a d - 6 b c\right ) + 3 b^{2} c^{2}\right )}{12 c^{3} \left (c + d x^{2}\right )^{2}} + \frac{x \left (7 a d - 3 b c\right )^{2}}{24 c^{4} \left (c + d x^{2}\right )} + \frac{\left (35 a^{2} d^{2} - 30 a b c d + 3 b^{2} c^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{8 c^{\frac{9}{2}} \sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**2/x**4/(d*x**2+c)**3,x)

[Out]

-a**2/(3*c*x**3*(c + d*x**2)**2) + a*(7*a*d - 6*b*c)/(3*c**4*x) + x*(a*d*(7*a*d
- 6*b*c) + 3*b**2*c**2)/(12*c**3*(c + d*x**2)**2) + x*(7*a*d - 3*b*c)**2/(24*c**
4*(c + d*x**2)) + (35*a**2*d**2 - 30*a*b*c*d + 3*b**2*c**2)*atan(sqrt(d)*x/sqrt(
c))/(8*c**(9/2)*sqrt(d))

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Mathematica [A]  time = 0.119381, size = 148, normalized size = 0.92 \[ \frac{\left (35 a^2 d^2-30 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{9/2} \sqrt{d}}+\frac{x \left (11 a^2 d^2-14 a b c d+3 b^2 c^2\right )}{8 c^4 \left (c+d x^2\right )}-\frac{a^2}{3 c^3 x^3}+\frac{a (3 a d-2 b c)}{c^4 x}+\frac{x (b c-a d)^2}{4 c^3 \left (c+d x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^2/(x^4*(c + d*x^2)^3),x]

[Out]

-a^2/(3*c^3*x^3) + (a*(-2*b*c + 3*a*d))/(c^4*x) + ((b*c - a*d)^2*x)/(4*c^3*(c +
d*x^2)^2) + ((3*b^2*c^2 - 14*a*b*c*d + 11*a^2*d^2)*x)/(8*c^4*(c + d*x^2)) + ((3*
b^2*c^2 - 30*a*b*c*d + 35*a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*c^(9/2)*Sqrt[
d])

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Maple [A]  time = 0.02, size = 227, normalized size = 1.4 \[ -{\frac{{a}^{2}}{3\,{c}^{3}{x}^{3}}}+3\,{\frac{{a}^{2}d}{{c}^{4}x}}-2\,{\frac{ab}{{c}^{3}x}}+{\frac{11\,{x}^{3}{a}^{2}{d}^{3}}{8\,{c}^{4} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{7\,{x}^{3}ab{d}^{2}}{4\,{c}^{3} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{3\,{b}^{2}d{x}^{3}}{8\,{c}^{2} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{13\,{a}^{2}{d}^{2}x}{8\,{c}^{3} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{9\,xabd}{4\,{c}^{2} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{5\,x{b}^{2}}{8\,c \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{35\,{a}^{2}{d}^{2}}{8\,{c}^{4}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{15\,abd}{4\,{c}^{3}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{3\,{b}^{2}}{8\,{c}^{2}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2/x^4/(d*x^2+c)^3,x)

[Out]

-1/3*a^2/c^3/x^3+3*a^2/c^4/x*d-2*a/c^3/x*b+11/8/c^4/(d*x^2+c)^2*x^3*a^2*d^3-7/4/
c^3/(d*x^2+c)^2*x^3*a*b*d^2+3/8/c^2/(d*x^2+c)^2*x^3*b^2*d+13/8/c^3/(d*x^2+c)^2*x
*a^2*d^2-9/4/c^2/(d*x^2+c)^2*x*a*b*d+5/8/c/(d*x^2+c)^2*x*b^2+35/8/c^4/(c*d)^(1/2
)*arctan(x*d/(c*d)^(1/2))*a^2*d^2-15/4/c^3/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))*a
*b*d+3/8/c^2/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))*b^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2/((d*x^2 + c)^3*x^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.248185, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left ({\left (3 \, b^{2} c^{2} d^{2} - 30 \, a b c d^{3} + 35 \, a^{2} d^{4}\right )} x^{7} + 2 \,{\left (3 \, b^{2} c^{3} d - 30 \, a b c^{2} d^{2} + 35 \, a^{2} c d^{3}\right )} x^{5} +{\left (3 \, b^{2} c^{4} - 30 \, a b c^{3} d + 35 \, a^{2} c^{2} d^{2}\right )} x^{3}\right )} \log \left (\frac{2 \, c d x +{\left (d x^{2} - c\right )} \sqrt{-c d}}{d x^{2} + c}\right ) + 2 \,{\left (3 \,{\left (3 \, b^{2} c^{2} d - 30 \, a b c d^{2} + 35 \, a^{2} d^{3}\right )} x^{6} - 8 \, a^{2} c^{3} + 5 \,{\left (3 \, b^{2} c^{3} - 30 \, a b c^{2} d + 35 \, a^{2} c d^{2}\right )} x^{4} - 8 \,{\left (6 \, a b c^{3} - 7 \, a^{2} c^{2} d\right )} x^{2}\right )} \sqrt{-c d}}{48 \,{\left (c^{4} d^{2} x^{7} + 2 \, c^{5} d x^{5} + c^{6} x^{3}\right )} \sqrt{-c d}}, \frac{3 \,{\left ({\left (3 \, b^{2} c^{2} d^{2} - 30 \, a b c d^{3} + 35 \, a^{2} d^{4}\right )} x^{7} + 2 \,{\left (3 \, b^{2} c^{3} d - 30 \, a b c^{2} d^{2} + 35 \, a^{2} c d^{3}\right )} x^{5} +{\left (3 \, b^{2} c^{4} - 30 \, a b c^{3} d + 35 \, a^{2} c^{2} d^{2}\right )} x^{3}\right )} \arctan \left (\frac{\sqrt{c d} x}{c}\right ) +{\left (3 \,{\left (3 \, b^{2} c^{2} d - 30 \, a b c d^{2} + 35 \, a^{2} d^{3}\right )} x^{6} - 8 \, a^{2} c^{3} + 5 \,{\left (3 \, b^{2} c^{3} - 30 \, a b c^{2} d + 35 \, a^{2} c d^{2}\right )} x^{4} - 8 \,{\left (6 \, a b c^{3} - 7 \, a^{2} c^{2} d\right )} x^{2}\right )} \sqrt{c d}}{24 \,{\left (c^{4} d^{2} x^{7} + 2 \, c^{5} d x^{5} + c^{6} x^{3}\right )} \sqrt{c d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2/((d*x^2 + c)^3*x^4),x, algorithm="fricas")

[Out]

[1/48*(3*((3*b^2*c^2*d^2 - 30*a*b*c*d^3 + 35*a^2*d^4)*x^7 + 2*(3*b^2*c^3*d - 30*
a*b*c^2*d^2 + 35*a^2*c*d^3)*x^5 + (3*b^2*c^4 - 30*a*b*c^3*d + 35*a^2*c^2*d^2)*x^
3)*log((2*c*d*x + (d*x^2 - c)*sqrt(-c*d))/(d*x^2 + c)) + 2*(3*(3*b^2*c^2*d - 30*
a*b*c*d^2 + 35*a^2*d^3)*x^6 - 8*a^2*c^3 + 5*(3*b^2*c^3 - 30*a*b*c^2*d + 35*a^2*c
*d^2)*x^4 - 8*(6*a*b*c^3 - 7*a^2*c^2*d)*x^2)*sqrt(-c*d))/((c^4*d^2*x^7 + 2*c^5*d
*x^5 + c^6*x^3)*sqrt(-c*d)), 1/24*(3*((3*b^2*c^2*d^2 - 30*a*b*c*d^3 + 35*a^2*d^4
)*x^7 + 2*(3*b^2*c^3*d - 30*a*b*c^2*d^2 + 35*a^2*c*d^3)*x^5 + (3*b^2*c^4 - 30*a*
b*c^3*d + 35*a^2*c^2*d^2)*x^3)*arctan(sqrt(c*d)*x/c) + (3*(3*b^2*c^2*d - 30*a*b*
c*d^2 + 35*a^2*d^3)*x^6 - 8*a^2*c^3 + 5*(3*b^2*c^3 - 30*a*b*c^2*d + 35*a^2*c*d^2
)*x^4 - 8*(6*a*b*c^3 - 7*a^2*c^2*d)*x^2)*sqrt(c*d))/((c^4*d^2*x^7 + 2*c^5*d*x^5
+ c^6*x^3)*sqrt(c*d))]

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Sympy [A]  time = 6.92507, size = 240, normalized size = 1.49 \[ - \frac{\sqrt{- \frac{1}{c^{9} d}} \left (35 a^{2} d^{2} - 30 a b c d + 3 b^{2} c^{2}\right ) \log{\left (- c^{5} \sqrt{- \frac{1}{c^{9} d}} + x \right )}}{16} + \frac{\sqrt{- \frac{1}{c^{9} d}} \left (35 a^{2} d^{2} - 30 a b c d + 3 b^{2} c^{2}\right ) \log{\left (c^{5} \sqrt{- \frac{1}{c^{9} d}} + x \right )}}{16} + \frac{- 8 a^{2} c^{3} + x^{6} \left (105 a^{2} d^{3} - 90 a b c d^{2} + 9 b^{2} c^{2} d\right ) + x^{4} \left (175 a^{2} c d^{2} - 150 a b c^{2} d + 15 b^{2} c^{3}\right ) + x^{2} \left (56 a^{2} c^{2} d - 48 a b c^{3}\right )}{24 c^{6} x^{3} + 48 c^{5} d x^{5} + 24 c^{4} d^{2} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**2/x**4/(d*x**2+c)**3,x)

[Out]

-sqrt(-1/(c**9*d))*(35*a**2*d**2 - 30*a*b*c*d + 3*b**2*c**2)*log(-c**5*sqrt(-1/(
c**9*d)) + x)/16 + sqrt(-1/(c**9*d))*(35*a**2*d**2 - 30*a*b*c*d + 3*b**2*c**2)*l
og(c**5*sqrt(-1/(c**9*d)) + x)/16 + (-8*a**2*c**3 + x**6*(105*a**2*d**3 - 90*a*b
*c*d**2 + 9*b**2*c**2*d) + x**4*(175*a**2*c*d**2 - 150*a*b*c**2*d + 15*b**2*c**3
) + x**2*(56*a**2*c**2*d - 48*a*b*c**3))/(24*c**6*x**3 + 48*c**5*d*x**5 + 24*c**
4*d**2*x**7)

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GIAC/XCAS [A]  time = 0.226923, size = 204, normalized size = 1.27 \[ \frac{{\left (3 \, b^{2} c^{2} - 30 \, a b c d + 35 \, a^{2} d^{2}\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{8 \, \sqrt{c d} c^{4}} + \frac{3 \, b^{2} c^{2} d x^{3} - 14 \, a b c d^{2} x^{3} + 11 \, a^{2} d^{3} x^{3} + 5 \, b^{2} c^{3} x - 18 \, a b c^{2} d x + 13 \, a^{2} c d^{2} x}{8 \,{\left (d x^{2} + c\right )}^{2} c^{4}} - \frac{6 \, a b c x^{2} - 9 \, a^{2} d x^{2} + a^{2} c}{3 \, c^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2/((d*x^2 + c)^3*x^4),x, algorithm="giac")

[Out]

1/8*(3*b^2*c^2 - 30*a*b*c*d + 35*a^2*d^2)*arctan(d*x/sqrt(c*d))/(sqrt(c*d)*c^4)
+ 1/8*(3*b^2*c^2*d*x^3 - 14*a*b*c*d^2*x^3 + 11*a^2*d^3*x^3 + 5*b^2*c^3*x - 18*a*
b*c^2*d*x + 13*a^2*c*d^2*x)/((d*x^2 + c)^2*c^4) - 1/3*(6*a*b*c*x^2 - 9*a^2*d*x^2
+ a^2*c)/(c^4*x^3)