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

Optimal. Leaf size=77 \[ -\frac{(b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} b^{5/2}}+\frac{d^2 x (3 b c-a d)}{b^2}-\frac{c^3}{a x}+\frac{d^3 x^3}{3 b} \]

[Out]

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

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Rubi [A]  time = 0.153554, antiderivative size = 77, 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 \[ -\frac{(b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} b^{5/2}}+\frac{d^2 x (3 b c-a d)}{b^2}-\frac{c^3}{a x}+\frac{d^3 x^3}{3 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0544016, size = 76, normalized size = 0.99 \[ \frac{(a d-b c)^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} b^{5/2}}+\frac{d^2 x (3 b c-a d)}{b^2}-\frac{c^3}{a x}+\frac{d^3 x^3}{3 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 135, normalized size = 1.8 \[{\frac{{d}^{3}{x}^{3}}{3\,b}}-{\frac{{d}^{3}ax}{{b}^{2}}}+3\,{\frac{{d}^{2}xc}{b}}+{\frac{{a}^{2}{d}^{3}}{{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-3\,{\frac{ac{d}^{2}}{b\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }+3\,{\frac{{c}^{2}d}{\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }-{\frac{b{c}^{3}}{a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{{c}^{3}}{ax}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.236956, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) - 2 \,{\left (a b d^{3} x^{4} - 3 \, b^{2} c^{3} + 3 \,{\left (3 \, a b c d^{2} - a^{2} d^{3}\right )} x^{2}\right )} \sqrt{-a b}}{6 \, \sqrt{-a b} a b^{2} x}, -\frac{3 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x \arctan \left (\frac{\sqrt{a b} x}{a}\right ) -{\left (a b d^{3} x^{4} - 3 \, b^{2} c^{3} + 3 \,{\left (3 \, a b c d^{2} - a^{2} d^{3}\right )} x^{2}\right )} \sqrt{a b}}{3 \, \sqrt{a b} a b^{2} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

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Sympy [A]  time = 3.65107, size = 221, normalized size = 2.87 \[ - \frac{\sqrt{- \frac{1}{a^{3} b^{5}}} \left (a d - b c\right )^{3} \log{\left (- \frac{a^{2} b^{2} \sqrt{- \frac{1}{a^{3} b^{5}}} \left (a d - b c\right )^{3}}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}} + x \right )}}{2} + \frac{\sqrt{- \frac{1}{a^{3} b^{5}}} \left (a d - b c\right )^{3} \log{\left (\frac{a^{2} b^{2} \sqrt{- \frac{1}{a^{3} b^{5}}} \left (a d - b c\right )^{3}}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}} + x \right )}}{2} + \frac{d^{3} x^{3}}{3 b} - \frac{x \left (a d^{3} - 3 b c d^{2}\right )}{b^{2}} - \frac{c^{3}}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.234263, size = 140, normalized size = 1.82 \[ -\frac{c^{3}}{a x} - \frac{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} a b^{2}} + \frac{b^{2} d^{3} x^{3} + 9 \, b^{2} c d^{2} x - 3 \, a b d^{3} x}{3 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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