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

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

[Out]

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

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Rubi [A]  time = 0.146696, 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 \[ \frac{b x \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )}{d^3}-\frac{b^2 x^3 (b c-3 a d)}{3 d^2}-\frac{(b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{c} d^{7/2}}+\frac{b^3 x^5}{5 d} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0987528, size = 93, normalized size = 0.95 \[ \frac{b x \left (45 a^2 d^2+15 a b d \left (d x^2-3 c\right )+b^2 \left (15 c^2-5 c d x^2+3 d^2 x^4\right )\right )}{15 d^3}-\frac{(b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{c} d^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(b*x*(45*a^2*d^2 + 15*a*b*d*(-3*c + d*x^2) + b^2*(15*c^2 - 5*c*d*x^2 + 3*d^2*x^4
)))/(15*d^3) - ((b*c - a*d)^3*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[c]*d^(7/2))

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Maple [A]  time = 0.004, size = 161, normalized size = 1.6 \[{\frac{{b}^{3}{x}^{5}}{5\,d}}+{\frac{a{b}^{2}{x}^{3}}{d}}-{\frac{{b}^{3}{x}^{3}c}{3\,{d}^{2}}}+3\,{\frac{{a}^{2}bx}{d}}-3\,{\frac{a{b}^{2}cx}{{d}^{2}}}+{\frac{{b}^{3}{c}^{2}x}{{d}^{3}}}+{{a}^{3}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-3\,{\frac{{a}^{2}bc}{d\sqrt{cd}}\arctan \left ({\frac{dx}{\sqrt{cd}}} \right ) }+3\,{\frac{a{b}^{2}{c}^{2}}{{d}^{2}\sqrt{cd}}\arctan \left ({\frac{dx}{\sqrt{cd}}} \right ) }-{\frac{{b}^{3}{c}^{3}}{{d}^{3}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^3/(d*x^2+c),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.209798, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (\frac{2 \, c d x +{\left (d x^{2} - c\right )} \sqrt{-c d}}{d x^{2} + c}\right ) - 2 \,{\left (3 \, b^{3} d^{2} x^{5} - 5 \,{\left (b^{3} c d - 3 \, a b^{2} d^{2}\right )} x^{3} + 15 \,{\left (b^{3} c^{2} - 3 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} x\right )} \sqrt{-c d}}{30 \, \sqrt{-c d} d^{3}}, -\frac{15 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac{\sqrt{c d} x}{c}\right ) -{\left (3 \, b^{3} d^{2} x^{5} - 5 \,{\left (b^{3} c d - 3 \, a b^{2} d^{2}\right )} x^{3} + 15 \,{\left (b^{3} c^{2} - 3 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} x\right )} \sqrt{c d}}{15 \, \sqrt{c d} d^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^3/(d*x^2 + c),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*c*d*x + (d
*x^2 - c)*sqrt(-c*d))/(d*x^2 + c)) - 2*(3*b^3*d^2*x^5 - 5*(b^3*c*d - 3*a*b^2*d^2
)*x^3 + 15*(b^3*c^2 - 3*a*b^2*c*d + 3*a^2*b*d^2)*x)*sqrt(-c*d))/(sqrt(-c*d)*d^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(c*d)
*x/c) - (3*b^3*d^2*x^5 - 5*(b^3*c*d - 3*a*b^2*d^2)*x^3 + 15*(b^3*c^2 - 3*a*b^2*c
*d + 3*a^2*b*d^2)*x)*sqrt(c*d))/(sqrt(c*d)*d^3)]

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Sympy [A]  time = 2.65285, size = 240, normalized size = 2.45 \[ \frac{b^{3} x^{5}}{5 d} - \frac{\sqrt{- \frac{1}{c d^{7}}} \left (a d - b c\right )^{3} \log{\left (- \frac{c d^{3} \sqrt{- \frac{1}{c d^{7}}} \left (a d - b c\right )^{3}}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}} + x \right )}}{2} + \frac{\sqrt{- \frac{1}{c d^{7}}} \left (a d - b c\right )^{3} \log{\left (\frac{c d^{3} \sqrt{- \frac{1}{c d^{7}}} \left (a d - b c\right )^{3}}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}} + x \right )}}{2} + \frac{x^{3} \left (3 a b^{2} d - b^{3} c\right )}{3 d^{2}} + \frac{x \left (3 a^{2} b d^{2} - 3 a b^{2} c d + b^{3} c^{2}\right )}{d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.235751, size = 176, normalized size = 1.8 \[ -\frac{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{\sqrt{c d} d^{3}} + \frac{3 \, b^{3} d^{4} x^{5} - 5 \, b^{3} c d^{3} x^{3} + 15 \, a b^{2} d^{4} x^{3} + 15 \, b^{3} c^{2} d^{2} x - 45 \, a b^{2} c d^{3} x + 45 \, a^{2} b d^{4} x}{15 \, d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^3/(d*x^2 + c),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(d*x/sqrt(c*d))/(sqrt
(c*d)*d^3) + 1/15*(3*b^3*d^4*x^5 - 5*b^3*c*d^3*x^3 + 15*a*b^2*d^4*x^3 + 15*b^3*c
^2*d^2*x - 45*a*b^2*c*d^3*x + 45*a^2*b*d^4*x)/d^5