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