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

Optimal. Leaf size=119 \[ \frac{d x^3 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{3 b^3}-\frac{\sqrt{a} (b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{9/2}}+\frac{x (b c-a d)^3}{b^4}+\frac{d^2 x^5 (3 b c-a d)}{5 b^2}+\frac{d^3 x^7}{7 b} \]

[Out]

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

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Rubi [A]  time = 0.191106, antiderivative size = 119, 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 \[ \frac{d x^3 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{3 b^3}-\frac{\sqrt{a} (b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{9/2}}+\frac{x (b c-a d)^3}{b^4}+\frac{d^2 x^5 (3 b c-a d)}{5 b^2}+\frac{d^3 x^7}{7 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\sqrt{a} \left (a d - b c\right )^{3} \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{b^{\frac{9}{2}}} - \left (a d - b c\right )^{3} \int \frac{1}{b^{4}}\, dx + \frac{d^{3} x^{7}}{7 b} - \frac{d^{2} x^{5} \left (a d - 3 b c\right )}{5 b^{2}} + \frac{d x^{3} \left (a^{2} d^{2} - 3 a b c d + 3 b^{2} c^{2}\right )}{3 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(a)*(a*d - b*c)**3*atan(sqrt(b)*x/sqrt(a))/b**(9/2) - (a*d - b*c)**3*Integra
l(b**(-4), x) + d**3*x**7/(7*b) - d**2*x**5*(a*d - 3*b*c)/(5*b**2) + d*x**3*(a**
2*d**2 - 3*a*b*c*d + 3*b**2*c**2)/(3*b**3)

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Mathematica [A]  time = 0.0665046, size = 118, normalized size = 0.99 \[ \frac{d x^3 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{3 b^3}+\frac{\sqrt{a} (a d-b c)^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{9/2}}+\frac{x (b c-a d)^3}{b^4}+\frac{d^2 x^5 (3 b c-a d)}{5 b^2}+\frac{d^3 x^7}{7 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.005, size = 218, normalized size = 1.8 \[{\frac{{d}^{3}{x}^{7}}{7\,b}}-{\frac{{x}^{5}a{d}^{3}}{5\,{b}^{2}}}+{\frac{3\,{x}^{5}c{d}^{2}}{5\,b}}+{\frac{{x}^{3}{a}^{2}{d}^{3}}{3\,{b}^{3}}}-{\frac{{x}^{3}ac{d}^{2}}{{b}^{2}}}+{\frac{{x}^{3}{c}^{2}d}{b}}-{\frac{{a}^{3}{d}^{3}x}{{b}^{4}}}+3\,{\frac{x{a}^{2}c{d}^{2}}{{b}^{3}}}-3\,{\frac{a{c}^{2}dx}{{b}^{2}}}+{\frac{{c}^{3}x}{b}}+{\frac{{a}^{4}{d}^{3}}{{b}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-3\,{\frac{{a}^{3}c{d}^{2}}{{b}^{3}\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }+3\,{\frac{{a}^{2}{c}^{2}d}{{b}^{2}\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }-{\frac{a{c}^{3}}{b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.23907, size = 1, normalized size = 0.01 \[ \left [\frac{30 \, b^{3} d^{3} x^{7} + 42 \,{\left (3 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{5} + 70 \,{\left (3 \, b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{3} - 105 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} + 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) + 210 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x}{210 \, b^{4}}, \frac{15 \, b^{3} d^{3} x^{7} + 21 \,{\left (3 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{5} + 35 \,{\left (3 \, b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{3} - 105 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{x}{\sqrt{\frac{a}{b}}}\right ) + 105 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x}{105 \, b^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/210*(30*b^3*d^3*x^7 + 42*(3*b^3*c*d^2 - a*b^2*d^3)*x^5 + 70*(3*b^3*c^2*d - 3*
a*b^2*c*d^2 + a^2*b*d^3)*x^3 - 105*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^
3*d^3)*sqrt(-a/b)*log((b*x^2 + 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)) + 210*(b^3*c^3
- 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*x)/b^4, 1/105*(15*b^3*d^3*x^7 + 21*(
3*b^3*c*d^2 - a*b^2*d^3)*x^5 + 35*(3*b^3*c^2*d - 3*a*b^2*c*d^2 + a^2*b*d^3)*x^3
- 105*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(a/b)*arctan(x/sqr
t(a/b)) + 105*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*x)/b^4]

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Sympy [A]  time = 2.94718, size = 275, normalized size = 2.31 \[ - \frac{\sqrt{- \frac{a}{b^{9}}} \left (a d - b c\right )^{3} \log{\left (- \frac{b^{4} \sqrt{- \frac{a}{b^{9}}} \left (a d - b c\right )^{3}}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}} + x \right )}}{2} + \frac{\sqrt{- \frac{a}{b^{9}}} \left (a d - b c\right )^{3} \log{\left (\frac{b^{4} \sqrt{- \frac{a}{b^{9}}} \left (a d - b c\right )^{3}}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}} + x \right )}}{2} + \frac{d^{3} x^{7}}{7 b} - \frac{x^{5} \left (a d^{3} - 3 b c d^{2}\right )}{5 b^{2}} + \frac{x^{3} \left (a^{2} d^{3} - 3 a b c d^{2} + 3 b^{2} c^{2} d\right )}{3 b^{3}} - \frac{x \left (a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}\right )}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-a/b**9)*(a*d - b*c)**3*log(-b**4*sqrt(-a/b**9)*(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(-a/b**9)*(a*d - b
*c)**3*log(b**4*sqrt(-a/b**9)*(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**7/(7*b) - x**5*(a*d**3 - 3*b*c*d**2)/(
5*b**2) + x**3*(a**2*d**3 - 3*a*b*c*d**2 + 3*b**2*c**2*d)/(3*b**3) - x*(a**3*d**
3 - 3*a**2*b*c*d**2 + 3*a*b**2*c**2*d - b**3*c**3)/b**4

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GIAC/XCAS [A]  time = 0.229567, size = 248, normalized size = 2.08 \[ -\frac{{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} b^{4}} + \frac{15 \, b^{6} d^{3} x^{7} + 63 \, b^{6} c d^{2} x^{5} - 21 \, a b^{5} d^{3} x^{5} + 105 \, b^{6} c^{2} d x^{3} - 105 \, a b^{5} c d^{2} x^{3} + 35 \, a^{2} b^{4} d^{3} x^{3} + 105 \, b^{6} c^{3} x - 315 \, a b^{5} c^{2} d x + 315 \, a^{2} b^{4} c d^{2} x - 105 \, a^{3} b^{3} d^{3} x}{105 \, b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a*b^3*c^3 - 3*a^2*b^2*c^2*d + 3*a^3*b*c*d^2 - a^4*d^3)*arctan(b*x/sqrt(a*b))/(
sqrt(a*b)*b^4) + 1/105*(15*b^6*d^3*x^7 + 63*b^6*c*d^2*x^5 - 21*a*b^5*d^3*x^5 + 1
05*b^6*c^2*d*x^3 - 105*a*b^5*c*d^2*x^3 + 35*a^2*b^4*d^3*x^3 + 105*b^6*c^3*x - 31
5*a*b^5*c^2*d*x + 315*a^2*b^4*c*d^2*x - 105*a^3*b^3*d^3*x)/b^7