Optimal. Leaf size=87 \[ \frac{(b c-a d)^3 \log \left (a+b x^2\right )}{2 b^4}+\frac{d x^2 (b c-a d)^2}{2 b^3}+\frac{\left (c+d x^2\right )^2 (b c-a d)}{4 b^2}+\frac{\left (c+d x^2\right )^3}{6 b} \]
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Rubi [A] time = 0.168154, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{(b c-a d)^3 \log \left (a+b x^2\right )}{2 b^4}+\frac{d x^2 (b c-a d)^2}{2 b^3}+\frac{\left (c+d x^2\right )^2 (b c-a d)}{4 b^2}+\frac{\left (c+d x^2\right )^3}{6 b} \]
Antiderivative was successfully verified.
[In] Int[(x*(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. \[ \frac{\left (c + d x^{2}\right )^{3}}{6 b} - \frac{\left (c + d x^{2}\right )^{2} \left (a d - b c\right )}{4 b^{2}} + \frac{\left (a d - b c\right )^{2} \int ^{x^{2}} d\, dx}{2 b^{3}} - \frac{\left (a d - b c\right )^{3} \log{\left (a + b x^{2} \right )}}{2 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(d*x**2+c)**3/(b*x**2+a),x)
[Out]
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Mathematica [A] time = 0.0537773, size = 82, normalized size = 0.94 \[ \frac{b d x^2 \left (6 a^2 d^2-3 a b d \left (6 c+d x^2\right )+b^2 \left (18 c^2+9 c d x^2+2 d^2 x^4\right )\right )+6 (b c-a d)^3 \log \left (a+b x^2\right )}{12 b^4} \]
Antiderivative was successfully verified.
[In] Integrate[(x*(c + d*x^2)^3)/(a + b*x^2),x]
[Out]
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Maple [A] time = 0.005, size = 149, normalized size = 1.7 \[{\frac{{d}^{3}{x}^{6}}{6\,b}}-{\frac{{d}^{3}{x}^{4}a}{4\,{b}^{2}}}+{\frac{3\,{d}^{2}{x}^{4}c}{4\,b}}+{\frac{{d}^{3}{x}^{2}{a}^{2}}{2\,{b}^{3}}}-{\frac{3\,{d}^{2}{x}^{2}ac}{2\,{b}^{2}}}+{\frac{3\,d{x}^{2}{c}^{2}}{2\,b}}-{\frac{\ln \left ( b{x}^{2}+a \right ){a}^{3}{d}^{3}}{2\,{b}^{4}}}+{\frac{3\,\ln \left ( b{x}^{2}+a \right ){a}^{2}c{d}^{2}}{2\,{b}^{3}}}-{\frac{3\,\ln \left ( b{x}^{2}+a \right ) a{c}^{2}d}{2\,{b}^{2}}}+{\frac{\ln \left ( b{x}^{2}+a \right ){c}^{3}}{2\,b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(d*x^2+c)^3/(b*x^2+a),x)
[Out]
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Maxima [A] time = 1.32954, size = 161, normalized size = 1.85 \[ \frac{2 \, b^{2} d^{3} x^{6} + 3 \,{\left (3 \, b^{2} c d^{2} - a b d^{3}\right )} x^{4} + 6 \,{\left (3 \, b^{2} c^{2} d - 3 \, a b c d^{2} + a^{2} d^{3}\right )} x^{2}}{12 \, b^{3}} + \frac{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x^{2} + a\right )}{2 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^3*x/(b*x^2 + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.226952, size = 162, normalized size = 1.86 \[ \frac{2 \, b^{3} d^{3} x^{6} + 3 \,{\left (3 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{4} + 6 \,{\left (3 \, b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} + 6 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x^{2} + a\right )}{12 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^3*x/(b*x^2 + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.4438, size = 88, normalized size = 1.01 \[ \frac{d^{3} x^{6}}{6 b} - \frac{x^{4} \left (a d^{3} - 3 b c d^{2}\right )}{4 b^{2}} + \frac{x^{2} \left (a^{2} d^{3} - 3 a b c d^{2} + 3 b^{2} c^{2} d\right )}{2 b^{3}} - \frac{\left (a d - b c\right )^{3} \log{\left (a + b x^{2} \right )}}{2 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(d*x**2+c)**3/(b*x**2+a),x)
[Out]
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GIAC/XCAS [A] time = 0.222008, size = 167, normalized size = 1.92 \[ \frac{2 \, b^{2} d^{3} x^{6} + 9 \, b^{2} c d^{2} x^{4} - 3 \, a b d^{3} x^{4} + 18 \, b^{2} c^{2} d x^{2} - 18 \, a b c d^{2} x^{2} + 6 \, a^{2} d^{3} x^{2}}{12 \, b^{3}} + \frac{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^3*x/(b*x^2 + a),x, algorithm="giac")
[Out]