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

Optimal. Leaf size=74 \[ \frac{(b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2} b^{3/2}}+\frac{c^2 (b c-3 a d)}{a^2 x}-\frac{c^3}{3 a x^3}+\frac{d^3 x}{b} \]

[Out]

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

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Rubi [A]  time = 0.157758, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{(b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2} b^{3/2}}+\frac{c^2 (b c-3 a d)}{a^2 x}-\frac{c^3}{3 a x^3}+\frac{d^3 x}{b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ d^{3} \int \frac{1}{b}\, dx - \frac{c^{3}}{3 a x^{3}} - \frac{c^{2} \left (3 a d - b c\right )}{a^{2} x} - \frac{\left (a d - b c\right )^{3} \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{a^{\frac{5}{2}} b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**3*Integral(1/b, x) - c**3/(3*a*x**3) - c**2*(3*a*d - b*c)/(a**2*x) - (a*d - b
*c)**3*atan(sqrt(b)*x/sqrt(a))/(a**(5/2)*b**(3/2))

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Mathematica [A]  time = 0.0633176, size = 74, normalized size = 1. \[ \frac{(b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2} b^{3/2}}+\frac{c^2 (b c-3 a d)}{a^2 x}-\frac{c^3}{3 a x^3}+\frac{d^3 x}{b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.01, size = 135, normalized size = 1.8 \[{\frac{{d}^{3}x}{b}}-{\frac{{c}^{3}}{3\,a{x}^{3}}}-3\,{\frac{{c}^{2}d}{ax}}+{\frac{b{c}^{3}}{{a}^{2}x}}-{\frac{a{d}^{3}}{b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+3\,{\frac{c{d}^{2}}{\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }-3\,{\frac{b{c}^{2}d}{a\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }+{\frac{{b}^{2}{c}^{3}}{{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.242014, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{3} \log \left (-\frac{2 \, a b x -{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) - 2 \,{\left (3 \, a^{2} d^{3} x^{4} - a b c^{3} + 3 \,{\left (b^{2} c^{3} - 3 \, a b c^{2} d\right )} x^{2}\right )} \sqrt{-a b}}{6 \, \sqrt{-a b} a^{2} b x^{3}}, \frac{3 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{3} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left (3 \, a^{2} d^{3} x^{4} - a b c^{3} + 3 \,{\left (b^{2} c^{3} - 3 \, a b c^{2} d\right )} x^{2}\right )} \sqrt{a b}}{3 \, \sqrt{a b} a^{2} b x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.92188, size = 221, normalized size = 2.99 \[ \frac{\sqrt{- \frac{1}{a^{5} b^{3}}} \left (a d - b c\right )^{3} \log{\left (- \frac{a^{3} b \sqrt{- \frac{1}{a^{5} b^{3}}} \left (a d - b c\right )^{3}}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}} + x \right )}}{2} - \frac{\sqrt{- \frac{1}{a^{5} b^{3}}} \left (a d - b c\right )^{3} \log{\left (\frac{a^{3} b \sqrt{- \frac{1}{a^{5} b^{3}}} \left (a d - b c\right )^{3}}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}} + x \right )}}{2} + \frac{d^{3} x}{b} - \frac{a c^{3} + x^{2} \left (9 a c^{2} d - 3 b c^{3}\right )}{3 a^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(-1/(a**5*b**3))*(a*d - b*c)**3*log(-a**3*b*sqrt(-1/(a**5*b**3))*(a*d - b*c)
**3/(a**3*d**3 - 3*a**2*b*c*d**2 + 3*a*b**2*c**2*d - b**3*c**3) + x)/2 - sqrt(-1
/(a**5*b**3))*(a*d - b*c)**3*log(a**3*b*sqrt(-1/(a**5*b**3))*(a*d - b*c)**3/(a**
3*d**3 - 3*a**2*b*c*d**2 + 3*a*b**2*c**2*d - b**3*c**3) + x)/2 + d**3*x/b - (a*c
**3 + x**2*(9*a*c**2*d - 3*b*c**3))/(3*a**2*x**3)

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GIAC/XCAS [A]  time = 0.230552, size = 135, normalized size = 1.82 \[ \frac{d^{3} x}{b} + \frac{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} a^{2} b} + \frac{3 \, b c^{3} x^{2} - 9 \, a c^{2} d x^{2} - a c^{3}}{3 \, a^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d^3*x/b + (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*arctan(b*x/sqrt(a*
b))/(sqrt(a*b)*a^2*b) + 1/3*(3*b*c^3*x^2 - 9*a*c^2*d*x^2 - a*c^3)/(a^2*x^3)