Optimal. Leaf size=115 \[ \frac{d x^4 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{4 b^3}-\frac{a (b c-a d)^3 \log \left (a+b x^2\right )}{2 b^5}+\frac{x^2 (b c-a d)^3}{2 b^4}+\frac{d^2 x^6 (3 b c-a d)}{6 b^2}+\frac{d^3 x^8}{8 b} \]
[Out]
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Rubi [A] time = 0.276325, antiderivative size = 115, 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 \[ \frac{d x^4 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{4 b^3}-\frac{a (b c-a d)^3 \log \left (a+b x^2\right )}{2 b^5}+\frac{x^2 (b c-a d)^3}{2 b^4}+\frac{d^2 x^6 (3 b c-a d)}{6 b^2}+\frac{d^3 x^8}{8 b} \]
Antiderivative was successfully verified.
[In] Int[(x^3*(c + d*x^2)^3)/(a + b*x^2),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{a \left (a d - b c\right )^{3} \log{\left (a + b x^{2} \right )}}{2 b^{5}} - \frac{\left (a d - b c\right )^{3} \int ^{x^{2}} \frac{1}{b^{4}}\, dx}{2} + \frac{d^{3} x^{8}}{8 b} - \frac{d^{2} x^{6} \left (a d - 3 b c\right )}{6 b^{2}} + \frac{d \left (a^{2} d^{2} - 3 a b c d + 3 b^{2} c^{2}\right ) \int ^{x^{2}} x\, dx}{2 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(d*x**2+c)**3/(b*x**2+a),x)
[Out]
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Mathematica [A] time = 0.093547, size = 125, normalized size = 1.09 \[ \frac{b x^2 \left (-12 a^3 d^3+6 a^2 b d^2 \left (6 c+d x^2\right )-2 a b^2 d \left (18 c^2+9 c d x^2+2 d^2 x^4\right )+3 b^3 \left (4 c^3+6 c^2 d x^2+4 c d^2 x^4+d^3 x^6\right )\right )+12 a (a d-b c)^3 \log \left (a+b x^2\right )}{24 b^5} \]
Antiderivative was successfully verified.
[In] Integrate[(x^3*(c + d*x^2)^3)/(a + b*x^2),x]
[Out]
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Maple [A] time = 0.005, size = 205, normalized size = 1.8 \[{\frac{{d}^{3}{x}^{8}}{8\,b}}-{\frac{{x}^{6}a{d}^{3}}{6\,{b}^{2}}}+{\frac{{x}^{6}c{d}^{2}}{2\,b}}+{\frac{{x}^{4}{a}^{2}{d}^{3}}{4\,{b}^{3}}}-{\frac{3\,{x}^{4}ac{d}^{2}}{4\,{b}^{2}}}+{\frac{3\,{x}^{4}{c}^{2}d}{4\,b}}-{\frac{{a}^{3}{d}^{3}{x}^{2}}{2\,{b}^{4}}}+{\frac{3\,{x}^{2}{a}^{2}c{d}^{2}}{2\,{b}^{3}}}-{\frac{3\,a{c}^{2}d{x}^{2}}{2\,{b}^{2}}}+{\frac{{c}^{3}{x}^{2}}{2\,b}}+{\frac{{a}^{4}\ln \left ( b{x}^{2}+a \right ){d}^{3}}{2\,{b}^{5}}}-{\frac{3\,{a}^{3}\ln \left ( b{x}^{2}+a \right ) c{d}^{2}}{2\,{b}^{4}}}+{\frac{3\,{a}^{2}\ln \left ( b{x}^{2}+a \right ){c}^{2}d}{2\,{b}^{3}}}-{\frac{a\ln \left ( b{x}^{2}+a \right ){c}^{3}}{2\,{b}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(d*x^2+c)^3/(b*x^2+a),x)
[Out]
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Maxima [A] time = 1.34921, size = 227, normalized size = 1.97 \[ \frac{3 \, b^{3} d^{3} x^{8} + 4 \,{\left (3 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{6} + 6 \,{\left (3 \, b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{4} + 12 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{2}}{24 \, b^{4}} - \frac{{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} \log \left (b x^{2} + a\right )}{2 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^3*x^3/(b*x^2 + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.221415, size = 228, normalized size = 1.98 \[ \frac{3 \, b^{4} d^{3} x^{8} + 4 \,{\left (3 \, b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{6} + 6 \,{\left (3 \, b^{4} c^{2} d - 3 \, a b^{3} c d^{2} + a^{2} b^{2} d^{3}\right )} x^{4} + 12 \,{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{2} - 12 \,{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} \log \left (b x^{2} + a\right )}{24 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^3*x^3/(b*x^2 + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.65603, size = 136, normalized size = 1.18 \[ \frac{a \left (a d - b c\right )^{3} \log{\left (a + b x^{2} \right )}}{2 b^{5}} + \frac{d^{3} x^{8}}{8 b} - \frac{x^{6} \left (a d^{3} - 3 b c d^{2}\right )}{6 b^{2}} + \frac{x^{4} \left (a^{2} d^{3} - 3 a b c d^{2} + 3 b^{2} c^{2} d\right )}{4 b^{3}} - \frac{x^{2} \left (a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}\right )}{2 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3*(d*x**2+c)**3/(b*x**2+a),x)
[Out]
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GIAC/XCAS [A] time = 0.225778, size = 243, normalized size = 2.11 \[ \frac{3 \, b^{3} d^{3} x^{8} + 12 \, b^{3} c d^{2} x^{6} - 4 \, a b^{2} d^{3} x^{6} + 18 \, b^{3} c^{2} d x^{4} - 18 \, a b^{2} c d^{2} x^{4} + 6 \, a^{2} b d^{3} x^{4} + 12 \, b^{3} c^{3} x^{2} - 36 \, a b^{2} c^{2} d x^{2} + 36 \, a^{2} b c d^{2} x^{2} - 12 \, a^{3} d^{3} x^{2}}{24 \, b^{4}} - \frac{{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^3*x^3/(b*x^2 + a),x, algorithm="giac")
[Out]