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

Optimal. Leaf size=127 \[ -\frac{\left (-a^2 d^2-6 a b c d+15 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{3/2} d^{7/2}}+\frac{x (b c-a d) (a d+7 b c)}{8 c d^3 \left (c+d x^2\right )}+\frac{x^3 (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2}+\frac{b^2 x}{d^3} \]

[Out]

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

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Rubi [A]  time = 0.32306, antiderivative size = 127, 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 \[ -\frac{\left (-a^2 d^2-6 a b c d+15 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{3/2} d^{7/2}}+\frac{x (b c-a d) (a d+7 b c)}{8 c d^3 \left (c+d x^2\right )}+\frac{x^3 (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2}+\frac{b^2 x}{d^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 70.5266, size = 114, normalized size = 0.9 \[ \frac{b^{2} x}{d^{3}} + \frac{x^{3} \left (a d - b c\right )^{2}}{4 c d^{2} \left (c + d x^{2}\right )^{2}} - \frac{x \left (a d - b c\right ) \left (a d + 7 b c\right )}{8 c d^{3} \left (c + d x^{2}\right )} + \frac{\left (a^{2} d^{2} + 6 a b c d - 15 b^{2} c^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{8 c^{\frac{3}{2}} d^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**2*x/d**3 + x**3*(a*d - b*c)**2/(4*c*d**2*(c + d*x**2)**2) - x*(a*d - b*c)*(a*
d + 7*b*c)/(8*c*d**3*(c + d*x**2)) + (a**2*d**2 + 6*a*b*c*d - 15*b**2*c**2)*atan
(sqrt(d)*x/sqrt(c))/(8*c**(3/2)*d**(7/2))

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Mathematica [A]  time = 0.178515, size = 130, normalized size = 1.02 \[ \frac{x \left (a^2 d^2 \left (d x^2-c\right )-2 a b c d \left (3 c+5 d x^2\right )+b^2 c \left (15 c^2+25 c d x^2+8 d^2 x^4\right )\right )}{8 c d^3 \left (c+d x^2\right )^2}-\frac{\left (-a^2 d^2-6 a b c d+15 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{3/2} d^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(a^2*d^2*(-c + d*x^2) - 2*a*b*c*d*(3*c + 5*d*x^2) + b^2*c*(15*c^2 + 25*c*d*x^
2 + 8*d^2*x^4)))/(8*c*d^3*(c + d*x^2)^2) - ((15*b^2*c^2 - 6*a*b*c*d - a^2*d^2)*A
rcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*c^(3/2)*d^(7/2))

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Maple [A]  time = 0.015, size = 196, normalized size = 1.5 \[{\frac{{b}^{2}x}{{d}^{3}}}+{\frac{{x}^{3}{a}^{2}}{8\, \left ( d{x}^{2}+c \right ) ^{2}c}}-{\frac{5\,ab{x}^{3}}{4\,d \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{9\,{x}^{3}{b}^{2}c}{8\,{d}^{2} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{{a}^{2}x}{8\,d \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{3\,xabc}{4\,{d}^{2} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{7\,{b}^{2}{c}^{2}x}{8\,{d}^{3} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{{a}^{2}}{8\,cd}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{3\,ab}{4\,{d}^{2}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{15\,{b}^{2}c}{8\,{d}^{3}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.240286, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (15 \, b^{2} c^{4} - 6 \, a b c^{3} d - a^{2} c^{2} d^{2} +{\left (15 \, b^{2} c^{2} d^{2} - 6 \, a b c d^{3} - a^{2} d^{4}\right )} x^{4} + 2 \,{\left (15 \, b^{2} c^{3} d - 6 \, a b c^{2} d^{2} - a^{2} c d^{3}\right )} x^{2}\right )} \log \left (\frac{2 \, c d x +{\left (d x^{2} - c\right )} \sqrt{-c d}}{d x^{2} + c}\right ) - 2 \,{\left (8 \, b^{2} c d^{2} x^{5} +{\left (25 \, b^{2} c^{2} d - 10 \, a b c d^{2} + a^{2} d^{3}\right )} x^{3} +{\left (15 \, b^{2} c^{3} - 6 \, a b c^{2} d - a^{2} c d^{2}\right )} x\right )} \sqrt{-c d}}{16 \,{\left (c d^{5} x^{4} + 2 \, c^{2} d^{4} x^{2} + c^{3} d^{3}\right )} \sqrt{-c d}}, -\frac{{\left (15 \, b^{2} c^{4} - 6 \, a b c^{3} d - a^{2} c^{2} d^{2} +{\left (15 \, b^{2} c^{2} d^{2} - 6 \, a b c d^{3} - a^{2} d^{4}\right )} x^{4} + 2 \,{\left (15 \, b^{2} c^{3} d - 6 \, a b c^{2} d^{2} - a^{2} c d^{3}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{c d} x}{c}\right ) -{\left (8 \, b^{2} c d^{2} x^{5} +{\left (25 \, b^{2} c^{2} d - 10 \, a b c d^{2} + a^{2} d^{3}\right )} x^{3} +{\left (15 \, b^{2} c^{3} - 6 \, a b c^{2} d - a^{2} c d^{2}\right )} x\right )} \sqrt{c d}}{8 \,{\left (c d^{5} x^{4} + 2 \, c^{2} d^{4} x^{2} + c^{3} d^{3}\right )} \sqrt{c d}}\right ] \]

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

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Sympy [A]  time = 6.68556, size = 223, normalized size = 1.76 \[ \frac{b^{2} x}{d^{3}} - \frac{\sqrt{- \frac{1}{c^{3} d^{7}}} \left (a^{2} d^{2} + 6 a b c d - 15 b^{2} c^{2}\right ) \log{\left (- c^{2} d^{3} \sqrt{- \frac{1}{c^{3} d^{7}}} + x \right )}}{16} + \frac{\sqrt{- \frac{1}{c^{3} d^{7}}} \left (a^{2} d^{2} + 6 a b c d - 15 b^{2} c^{2}\right ) \log{\left (c^{2} d^{3} \sqrt{- \frac{1}{c^{3} d^{7}}} + x \right )}}{16} + \frac{x^{3} \left (a^{2} d^{3} - 10 a b c d^{2} + 9 b^{2} c^{2} d\right ) + x \left (- a^{2} c d^{2} - 6 a b c^{2} d + 7 b^{2} c^{3}\right )}{8 c^{3} d^{3} + 16 c^{2} d^{4} x^{2} + 8 c d^{5} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**2*x/d**3 - sqrt(-1/(c**3*d**7))*(a**2*d**2 + 6*a*b*c*d - 15*b**2*c**2)*log(-c
**2*d**3*sqrt(-1/(c**3*d**7)) + x)/16 + sqrt(-1/(c**3*d**7))*(a**2*d**2 + 6*a*b*
c*d - 15*b**2*c**2)*log(c**2*d**3*sqrt(-1/(c**3*d**7)) + x)/16 + (x**3*(a**2*d**
3 - 10*a*b*c*d**2 + 9*b**2*c**2*d) + x*(-a**2*c*d**2 - 6*a*b*c**2*d + 7*b**2*c**
3))/(8*c**3*d**3 + 16*c**2*d**4*x**2 + 8*c*d**5*x**4)

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GIAC/XCAS [A]  time = 0.229788, size = 180, normalized size = 1.42 \[ \frac{b^{2} x}{d^{3}} - \frac{{\left (15 \, b^{2} c^{2} - 6 \, a b c d - a^{2} d^{2}\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{8 \, \sqrt{c d} c d^{3}} + \frac{9 \, b^{2} c^{2} d x^{3} - 10 \, a b c d^{2} x^{3} + a^{2} d^{3} x^{3} + 7 \, b^{2} c^{3} x - 6 \, a b c^{2} d x - a^{2} c d^{2} x}{8 \,{\left (d x^{2} + c\right )}^{2} c d^{3}} \]

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]

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