Optimal. Leaf size=117 \[ \frac{a (b c-a d)^3}{2 b^5 \left (a+b x^2\right )}+\frac{(b c-4 a d) (b c-a d)^2 \log \left (a+b x^2\right )}{2 b^5}+\frac{3 d x^2 (b c-a d)^2}{2 b^4}+\frac{d^2 x^4 (3 b c-2 a d)}{4 b^3}+\frac{d^3 x^6}{6 b^2} \]
[Out]
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Rubi [A] time = 0.328727, antiderivative size = 117, 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 \[ \frac{a (b c-a d)^3}{2 b^5 \left (a+b x^2\right )}+\frac{(b c-4 a d) (b c-a d)^2 \log \left (a+b x^2\right )}{2 b^5}+\frac{3 d x^2 (b c-a d)^2}{2 b^4}+\frac{d^2 x^4 (3 b c-2 a d)}{4 b^3}+\frac{d^3 x^6}{6 b^2} \]
Antiderivative was successfully verified.
[In] Int[(x^3*(c + d*x^2)^3)/(a + b*x^2)^2,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{a \left (a d - b c\right )^{3}}{2 b^{5} \left (a + b x^{2}\right )} + \frac{d^{3} x^{6}}{6 b^{2}} - \frac{d^{2} \left (2 a d - 3 b c\right ) \int ^{x^{2}} x\, dx}{2 b^{3}} + \frac{3 d x^{2} \left (a d - b c\right )^{2}}{2 b^{4}} - \frac{\left (a d - b c\right )^{2} \left (4 a d - b c\right ) \log{\left (a + b x^{2} \right )}}{2 b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(d*x**2+c)**3/(b*x**2+a)**2,x)
[Out]
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Mathematica [A] time = 0.152384, size = 106, normalized size = 0.91 \[ \frac{3 b^2 d^2 x^4 (3 b c-2 a d)+18 b d x^2 (b c-a d)^2-\frac{6 a (a d-b c)^3}{a+b x^2}+6 (b c-4 a d) (b c-a d)^2 \log \left (a+b x^2\right )+2 b^3 d^3 x^6}{12 b^5} \]
Antiderivative was successfully verified.
[In] Integrate[(x^3*(c + d*x^2)^3)/(a + b*x^2)^2,x]
[Out]
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Maple [B] time = 0.017, size = 229, normalized size = 2. \[{\frac{{d}^{3}{x}^{6}}{6\,{b}^{2}}}-{\frac{{d}^{3}{x}^{4}a}{2\,{b}^{3}}}+{\frac{3\,{d}^{2}{x}^{4}c}{4\,{b}^{2}}}+{\frac{3\,{d}^{3}{x}^{2}{a}^{2}}{2\,{b}^{4}}}-3\,{\frac{{d}^{2}{x}^{2}ac}{{b}^{3}}}+{\frac{3\,d{x}^{2}{c}^{2}}{2\,{b}^{2}}}-2\,{\frac{\ln \left ( b{x}^{2}+a \right ){a}^{3}{d}^{3}}{{b}^{5}}}+{\frac{9\,\ln \left ( b{x}^{2}+a \right ){a}^{2}{d}^{2}c}{2\,{b}^{4}}}-3\,{\frac{\ln \left ( b{x}^{2}+a \right ) ad{c}^{2}}{{b}^{3}}}+{\frac{\ln \left ( b{x}^{2}+a \right ){c}^{3}}{2\,{b}^{2}}}-{\frac{{a}^{4}{d}^{3}}{2\,{b}^{5} \left ( b{x}^{2}+a \right ) }}+{\frac{3\,{a}^{3}{d}^{2}c}{2\,{b}^{4} \left ( b{x}^{2}+a \right ) }}-{\frac{3\,{a}^{2}{c}^{2}d}{2\,{b}^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{a{c}^{3}}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(d*x^2+c)^3/(b*x^2+a)^2,x)
[Out]
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Maxima [A] time = 1.34705, size = 235, normalized size = 2.01 \[ \frac{a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}}{2 \,{\left (b^{6} x^{2} + a b^{5}\right )}} + \frac{2 \, b^{2} d^{3} x^{6} + 3 \,{\left (3 \, b^{2} c d^{2} - 2 \, a b d^{3}\right )} x^{4} + 18 \,{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{2}}{12 \, b^{4}} + \frac{{\left (b^{3} c^{3} - 6 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - 4 \, a^{3} d^{3}\right )} \log \left (b x^{2} + a\right )}{2 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^3*x^3/(b*x^2 + a)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.225204, size = 343, normalized size = 2.93 \[ \frac{2 \, b^{4} d^{3} x^{8} + 6 \, a b^{3} c^{3} - 18 \, a^{2} b^{2} c^{2} d + 18 \, a^{3} b c d^{2} - 6 \, a^{4} d^{3} +{\left (9 \, b^{4} c d^{2} - 4 \, a b^{3} d^{3}\right )} x^{6} + 3 \,{\left (6 \, b^{4} c^{2} d - 9 \, a b^{3} c d^{2} + 4 \, a^{2} b^{2} d^{3}\right )} x^{4} + 18 \,{\left (a b^{3} c^{2} d - 2 \, a^{2} b^{2} c d^{2} + a^{3} b d^{3}\right )} x^{2} + 6 \,{\left (a b^{3} c^{3} - 6 \, a^{2} b^{2} c^{2} d + 9 \, a^{3} b c d^{2} - 4 \, a^{4} d^{3} +{\left (b^{4} c^{3} - 6 \, a b^{3} c^{2} d + 9 \, a^{2} b^{2} c d^{2} - 4 \, a^{3} b d^{3}\right )} x^{2}\right )} \log \left (b x^{2} + a\right )}{12 \,{\left (b^{6} x^{2} + a b^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^3*x^3/(b*x^2 + a)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 6.1864, size = 158, normalized size = 1.35 \[ - \frac{a^{4} d^{3} - 3 a^{3} b c d^{2} + 3 a^{2} b^{2} c^{2} d - a b^{3} c^{3}}{2 a b^{5} + 2 b^{6} x^{2}} + \frac{d^{3} x^{6}}{6 b^{2}} - \frac{x^{4} \left (2 a d^{3} - 3 b c d^{2}\right )}{4 b^{3}} + \frac{x^{2} \left (3 a^{2} d^{3} - 6 a b c d^{2} + 3 b^{2} c^{2} d\right )}{2 b^{4}} - \frac{\left (a d - b c\right )^{2} \left (4 a d - b c\right ) \log{\left (a + b x^{2} \right )}}{2 b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3*(d*x**2+c)**3/(b*x**2+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.247111, size = 336, normalized size = 2.87 \[ \frac{\frac{{\left (2 \, d^{3} + \frac{3 \,{\left (3 \, b^{2} c d^{2} - 4 \, a b d^{3}\right )}}{{\left (b x^{2} + a\right )} b} + \frac{18 \,{\left (b^{4} c^{2} d - 3 \, a b^{3} c d^{2} + 2 \, a^{2} b^{2} d^{3}\right )}}{{\left (b x^{2} + a\right )}^{2} b^{2}}\right )}{\left (b x^{2} + a\right )}^{3}}{b^{4}} - \frac{6 \,{\left (b^{3} c^{3} - 6 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - 4 \, a^{3} d^{3}\right )}{\rm ln}\left (\frac{{\left | b x^{2} + a \right |}}{{\left (b x^{2} + a\right )}^{2}{\left | b \right |}}\right )}{b^{4}} + \frac{6 \,{\left (\frac{a b^{6} c^{3}}{b x^{2} + a} - \frac{3 \, a^{2} b^{5} c^{2} d}{b x^{2} + a} + \frac{3 \, a^{3} b^{4} c d^{2}}{b x^{2} + a} - \frac{a^{4} b^{3} d^{3}}{b x^{2} + a}\right )}}{b^{7}}}{12 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^3*x^3/(b*x^2 + a)^2,x, algorithm="giac")
[Out]