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

Optimal. Leaf size=163 \[ \frac{\left (3 a^2 d^2-30 a b c d+35 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 \sqrt{c} d^{9/2}}-\frac{x \left (a^2 d^2-10 a b c d+13 b^2 c^2\right )}{4 c d^4}-\frac{x (b c-a d) (9 b c-a d)}{8 d^4 \left (c+d x^2\right )}+\frac{x^5 (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2}+\frac{b^2 x^3}{3 d^3} \]

[Out]

-((13*b^2*c^2 - 10*a*b*c*d + a^2*d^2)*x)/(4*c*d^4) + (b^2*x^3)/(3*d^3) + ((b*c -
a*d)^2*x^5)/(4*c*d^2*(c + d*x^2)^2) - ((b*c - a*d)*(9*b*c - a*d)*x)/(8*d^4*(c +
d*x^2)) + ((35*b^2*c^2 - 30*a*b*c*d + 3*a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(
8*Sqrt[c]*d^(9/2))

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Rubi [A]  time = 0.392188, antiderivative size = 163, 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 \[ \frac{\left (3 a^2 d^2-30 a b c d+35 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 \sqrt{c} d^{9/2}}-\frac{x \left (a^2 d^2-10 a b c d+13 b^2 c^2\right )}{4 c d^4}-\frac{x (b c-a d) (9 b c-a d)}{8 d^4 \left (c+d x^2\right )}+\frac{x^5 (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2}+\frac{b^2 x^3}{3 d^3} \]

Antiderivative was successfully verified.

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

[Out]

-((13*b^2*c^2 - 10*a*b*c*d + a^2*d^2)*x)/(4*c*d^4) + (b^2*x^3)/(3*d^3) + ((b*c -
a*d)^2*x^5)/(4*c*d^2*(c + d*x^2)^2) - ((b*c - a*d)*(9*b*c - a*d)*x)/(8*d^4*(c +
d*x^2)) + ((35*b^2*c^2 - 30*a*b*c*d + 3*a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(
8*Sqrt[c]*d^(9/2))

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Rubi in Sympy [A]  time = 106.856, size = 150, normalized size = 0.92 \[ \frac{b^{2} x^{3}}{3 d^{3}} - \frac{x \left (a d - 9 b c\right ) \left (a d - b c\right )}{8 d^{4} \left (c + d x^{2}\right )} + \frac{x^{5} \left (a d - b c\right )^{2}}{4 c d^{2} \left (c + d x^{2}\right )^{2}} - \frac{x \left (a^{2} d^{2} - 10 a b c d + 13 b^{2} c^{2}\right )}{4 c d^{4}} + \frac{\left (3 a^{2} d^{2} - 30 a b c d + 35 b^{2} c^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{8 \sqrt{c} d^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**2*x**3/(3*d**3) - x*(a*d - 9*b*c)*(a*d - b*c)/(8*d**4*(c + d*x**2)) + x**5*(a
*d - b*c)**2/(4*c*d**2*(c + d*x**2)**2) - x*(a**2*d**2 - 10*a*b*c*d + 13*b**2*c*
*2)/(4*c*d**4) + (3*a**2*d**2 - 30*a*b*c*d + 35*b**2*c**2)*atan(sqrt(d)*x/sqrt(c
))/(8*sqrt(c)*d**(9/2))

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Mathematica [A]  time = 0.153554, size = 148, normalized size = 0.91 \[ \frac{\left (3 a^2 d^2-30 a b c d+35 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 \sqrt{c} d^{9/2}}-\frac{x \left (5 a^2 d^2-18 a b c d+13 b^2 c^2\right )}{8 d^4 \left (c+d x^2\right )}+\frac{c x (b c-a d)^2}{4 d^4 \left (c+d x^2\right )^2}-\frac{b x (3 b c-2 a d)}{d^4}+\frac{b^2 x^3}{3 d^3} \]

Antiderivative was successfully verified.

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

[Out]

-((b*(3*b*c - 2*a*d)*x)/d^4) + (b^2*x^3)/(3*d^3) + (c*(b*c - a*d)^2*x)/(4*d^4*(c
+ d*x^2)^2) - ((13*b^2*c^2 - 18*a*b*c*d + 5*a^2*d^2)*x)/(8*d^4*(c + d*x^2)) + (
(35*b^2*c^2 - 30*a*b*c*d + 3*a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*Sqrt[c]*d^
(9/2))

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Maple [A]  time = 0.016, size = 223, normalized size = 1.4 \[{\frac{{b}^{2}{x}^{3}}{3\,{d}^{3}}}+2\,{\frac{abx}{{d}^{3}}}-3\,{\frac{x{b}^{2}c}{{d}^{4}}}-{\frac{5\,{x}^{3}{a}^{2}}{8\,d \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{9\,ab{x}^{3}c}{4\,{d}^{2} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{13\,{x}^{3}{b}^{2}{c}^{2}}{8\,{d}^{3} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{3\,{a}^{2}cx}{8\,{d}^{2} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{7\,xab{c}^{2}}{4\,{d}^{3} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{11\,x{b}^{2}{c}^{3}}{8\,{d}^{4} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{3\,{a}^{2}}{8\,{d}^{2}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{15\,abc}{4\,{d}^{3}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{35\,{b}^{2}{c}^{2}}{8\,{d}^{4}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.241915, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (35 \, b^{2} c^{4} - 30 \, a b c^{3} d + 3 \, a^{2} c^{2} d^{2} +{\left (35 \, b^{2} c^{2} d^{2} - 30 \, a b c d^{3} + 3 \, a^{2} d^{4}\right )} x^{4} + 2 \,{\left (35 \, b^{2} c^{3} d - 30 \, a b c^{2} d^{2} + 3 \, a^{2} c d^{3}\right )} x^{2}\right )} \log \left (\frac{2 \, c d x +{\left (d x^{2} - c\right )} \sqrt{-c d}}{d x^{2} + c}\right ) + 2 \,{\left (8 \, b^{2} d^{3} x^{7} - 8 \,{\left (7 \, b^{2} c d^{2} - 6 \, a b d^{3}\right )} x^{5} - 5 \,{\left (35 \, b^{2} c^{2} d - 30 \, a b c d^{2} + 3 \, a^{2} d^{3}\right )} x^{3} - 3 \,{\left (35 \, b^{2} c^{3} - 30 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x\right )} \sqrt{-c d}}{48 \,{\left (d^{6} x^{4} + 2 \, c d^{5} x^{2} + c^{2} d^{4}\right )} \sqrt{-c d}}, \frac{3 \,{\left (35 \, b^{2} c^{4} - 30 \, a b c^{3} d + 3 \, a^{2} c^{2} d^{2} +{\left (35 \, b^{2} c^{2} d^{2} - 30 \, a b c d^{3} + 3 \, a^{2} d^{4}\right )} x^{4} + 2 \,{\left (35 \, b^{2} c^{3} d - 30 \, a b c^{2} d^{2} + 3 \, a^{2} c d^{3}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{c d} x}{c}\right ) +{\left (8 \, b^{2} d^{3} x^{7} - 8 \,{\left (7 \, b^{2} c d^{2} - 6 \, a b d^{3}\right )} x^{5} - 5 \,{\left (35 \, b^{2} c^{2} d - 30 \, a b c d^{2} + 3 \, a^{2} d^{3}\right )} x^{3} - 3 \,{\left (35 \, b^{2} c^{3} - 30 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x\right )} \sqrt{c d}}{24 \,{\left (d^{6} x^{4} + 2 \, c d^{5} x^{2} + c^{2} d^{4}\right )} \sqrt{c d}}\right ] \]

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, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 8.47404, size = 238, normalized size = 1.46 \[ \frac{b^{2} x^{3}}{3 d^{3}} - \frac{\sqrt{- \frac{1}{c d^{9}}} \left (3 a^{2} d^{2} - 30 a b c d + 35 b^{2} c^{2}\right ) \log{\left (- c d^{4} \sqrt{- \frac{1}{c d^{9}}} + x \right )}}{16} + \frac{\sqrt{- \frac{1}{c d^{9}}} \left (3 a^{2} d^{2} - 30 a b c d + 35 b^{2} c^{2}\right ) \log{\left (c d^{4} \sqrt{- \frac{1}{c d^{9}}} + x \right )}}{16} - \frac{x^{3} \left (5 a^{2} d^{3} - 18 a b c d^{2} + 13 b^{2} c^{2} d\right ) + x \left (3 a^{2} c d^{2} - 14 a b c^{2} d + 11 b^{2} c^{3}\right )}{8 c^{2} d^{4} + 16 c d^{5} x^{2} + 8 d^{6} x^{4}} + \frac{x \left (2 a b d - 3 b^{2} c\right )}{d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**2*x**3/(3*d**3) - sqrt(-1/(c*d**9))*(3*a**2*d**2 - 30*a*b*c*d + 35*b**2*c**2)
*log(-c*d**4*sqrt(-1/(c*d**9)) + x)/16 + sqrt(-1/(c*d**9))*(3*a**2*d**2 - 30*a*b
*c*d + 35*b**2*c**2)*log(c*d**4*sqrt(-1/(c*d**9)) + x)/16 - (x**3*(5*a**2*d**3 -
18*a*b*c*d**2 + 13*b**2*c**2*d) + x*(3*a**2*c*d**2 - 14*a*b*c**2*d + 11*b**2*c*
*3))/(8*c**2*d**4 + 16*c*d**5*x**2 + 8*d**6*x**4) + x*(2*a*b*d - 3*b**2*c)/d**4

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GIAC/XCAS [A]  time = 0.227829, size = 208, normalized size = 1.28 \[ \frac{{\left (35 \, b^{2} c^{2} - 30 \, a b c d + 3 \, a^{2} d^{2}\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{8 \, \sqrt{c d} d^{4}} - \frac{13 \, b^{2} c^{2} d x^{3} - 18 \, a b c d^{2} x^{3} + 5 \, a^{2} d^{3} x^{3} + 11 \, b^{2} c^{3} x - 14 \, a b c^{2} d x + 3 \, a^{2} c d^{2} x}{8 \,{\left (d x^{2} + c\right )}^{2} d^{4}} + \frac{b^{2} d^{6} x^{3} - 9 \, b^{2} c d^{5} x + 6 \, a b d^{6} x}{3 \, d^{9}} \]

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, algorithm="giac")

[Out]

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