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

Optimal. Leaf size=98 \[ \frac{d x \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{b^3}+\frac{(b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{7/2}}+\frac{d^2 x^3 (3 b c-a d)}{3 b^2}+\frac{d^3 x^5}{5 b} \]

[Out]

(d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*x)/b^3 + (d^2*(3*b*c - a*d)*x^3)/(3*b^2) +
(d^3*x^5)/(5*b) + ((b*c - a*d)^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*b^(7/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.130333, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{d x \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{b^3}+\frac{(b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{7/2}}+\frac{d^2 x^3 (3 b c-a d)}{3 b^2}+\frac{d^3 x^5}{5 b} \]

Antiderivative was successfully verified.

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

[Out]

(d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*x)/b^3 + (d^2*(3*b*c - a*d)*x^3)/(3*b^2) +
(d^3*x^5)/(5*b) + ((b*c - a*d)^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*b^(7/2))

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{d^{3} x^{5}}{5 b} - \frac{d^{2} x^{3} \left (a d - 3 b c\right )}{3 b^{2}} + \frac{\left (a^{2} d^{2} - 3 a b c d + 3 b^{2} c^{2}\right ) \int d\, dx}{b^{3}} - \frac{\left (a d - b c\right )^{3} \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{\sqrt{a} b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**3*x**5/(5*b) - d**2*x**3*(a*d - 3*b*c)/(3*b**2) + (a**2*d**2 - 3*a*b*c*d + 3*
b**2*c**2)*Integral(d, x)/b**3 - (a*d - b*c)**3*atan(sqrt(b)*x/sqrt(a))/(sqrt(a)
*b**(7/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.114231, size = 92, normalized size = 0.94 \[ \frac{d x \left (15 a^2 d^2-5 a b d \left (9 c+d x^2\right )+3 b^2 \left (15 c^2+5 c d x^2+d^2 x^4\right )\right )}{15 b^3}+\frac{(b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(d*x*(15*a^2*d^2 - 5*a*b*d*(9*c + d*x^2) + 3*b^2*(15*c^2 + 5*c*d*x^2 + d^2*x^4))
)/(15*b^3) + ((b*c - a*d)^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*b^(7/2))

_______________________________________________________________________________________

Maple [A]  time = 0., size = 161, normalized size = 1.6 \[{\frac{{d}^{3}{x}^{5}}{5\,b}}-{\frac{{d}^{3}{x}^{3}a}{3\,{b}^{2}}}+{\frac{{d}^{2}{x}^{3}c}{b}}+{\frac{{a}^{2}{d}^{3}x}{{b}^{3}}}-3\,{\frac{a{d}^{2}cx}{{b}^{2}}}+3\,{\frac{d{c}^{2}x}{b}}-{\frac{{a}^{3}{d}^{3}}{{b}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+3\,{\frac{{a}^{2}c{d}^{2}}{{b}^{2}\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }-3\,{\frac{a{c}^{2}d}{b\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }+{{c}^{3}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^2+c)^3/(b*x^2+a),x)

[Out]

1/5*d^3*x^5/b-1/3*d^3/b^2*x^3*a+d^2/b*x^3*c+d^3/b^3*a^2*x-3*d^2/b^2*a*c*x+3*d/b*
c^2*x-1/b^3/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*a^3*d^3+3/b^2/(a*b)^(1/2)*arctan
(x*b/(a*b)^(1/2))*a^2*c*d^2-3/b/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*a*c^2*d+1/(a
*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*c^3

_______________________________________________________________________________________

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, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.233497, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (-\frac{2 \, a b x -{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) - 2 \,{\left (3 \, b^{2} d^{3} x^{5} + 5 \,{\left (3 \, b^{2} c d^{2} - a b d^{3}\right )} x^{3} + 15 \,{\left (3 \, b^{2} c^{2} d - 3 \, a b c d^{2} + a^{2} d^{3}\right )} x\right )} \sqrt{-a b}}{30 \, \sqrt{-a b} b^{3}}, \frac{15 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left (3 \, b^{2} d^{3} x^{5} + 5 \,{\left (3 \, b^{2} c d^{2} - a b d^{3}\right )} x^{3} + 15 \,{\left (3 \, b^{2} c^{2} d - 3 \, a b c d^{2} + a^{2} d^{3}\right )} x\right )} \sqrt{a b}}{15 \, \sqrt{a b} b^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/30*(15*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*log(-(2*a*b*x - (
b*x^2 - a)*sqrt(-a*b))/(b*x^2 + a)) - 2*(3*b^2*d^3*x^5 + 5*(3*b^2*c*d^2 - a*b*d^
3)*x^3 + 15*(3*b^2*c^2*d - 3*a*b*c*d^2 + a^2*d^3)*x)*sqrt(-a*b))/(sqrt(-a*b)*b^3
), 1/15*(15*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*arctan(sqrt(a*b)
*x/a) + (3*b^2*d^3*x^5 + 5*(3*b^2*c*d^2 - a*b*d^3)*x^3 + 15*(3*b^2*c^2*d - 3*a*b
*c*d^2 + a^2*d^3)*x)*sqrt(a*b))/(sqrt(a*b)*b^3)]

_______________________________________________________________________________________

Sympy [A]  time = 2.78379, size = 240, normalized size = 2.45 \[ \frac{\sqrt{- \frac{1}{a b^{7}}} \left (a d - b c\right )^{3} \log{\left (- \frac{a b^{3} \sqrt{- \frac{1}{a b^{7}}} \left (a d - b c\right )^{3}}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}} + x \right )}}{2} - \frac{\sqrt{- \frac{1}{a b^{7}}} \left (a d - b c\right )^{3} \log{\left (\frac{a b^{3} \sqrt{- \frac{1}{a b^{7}}} \left (a d - b c\right )^{3}}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}} + x \right )}}{2} + \frac{d^{3} x^{5}}{5 b} - \frac{x^{3} \left (a d^{3} - 3 b c d^{2}\right )}{3 b^{2}} + \frac{x \left (a^{2} d^{3} - 3 a b c d^{2} + 3 b^{2} c^{2} d\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(-1/(a*b**7))*(a*d - b*c)**3*log(-a*b**3*sqrt(-1/(a*b**7))*(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*b*
*7))*(a*d - b*c)**3*log(a*b**3*sqrt(-1/(a*b**7))*(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**5/(5*b) - x**3*(a*d
**3 - 3*b*c*d**2)/(3*b**2) + x*(a**2*d**3 - 3*a*b*c*d**2 + 3*b**2*c**2*d)/b**3

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.226119, size = 174, normalized size = 1.78 \[ \frac{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} b^{3}} + \frac{3 \, b^{4} d^{3} x^{5} + 15 \, b^{4} c d^{2} x^{3} - 5 \, a b^{3} d^{3} x^{3} + 45 \, b^{4} c^{2} d x - 45 \, a b^{3} c d^{2} x + 15 \, a^{2} b^{2} d^{3} x}{15 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(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)*b^3) + 1/15*(3*b^4*d^3*x^5 + 15*b^4*c*d^2*x^3 - 5*a*b^3*d^3*x^3 + 45*b^4*c^
2*d*x - 45*a*b^3*c*d^2*x + 15*a^2*b^2*d^3*x)/b^5