Optimal. Leaf size=77 \[ \frac{a \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{\sqrt{b} (b c-a d)^{3/2}}-\frac{c}{d \sqrt{c+d x^2} (b c-a d)} \]
[Out]
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Rubi [A] time = 0.20655, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{a \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{\sqrt{b} (b c-a d)^{3/2}}-\frac{c}{d \sqrt{c+d x^2} (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[x^3/((a + b*x^2)*(c + d*x^2)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 24.3372, size = 63, normalized size = 0.82 \[ \frac{a \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{2}}}{\sqrt{a d - b c}} \right )}}{\sqrt{b} \left (a d - b c\right )^{\frac{3}{2}}} + \frac{c}{d \sqrt{c + d x^{2}} \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3/(b*x**2+a)/(d*x**2+c)**(3/2),x)
[Out]
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Mathematica [A] time = 0.128109, size = 78, normalized size = 1.01 \[ \frac{\frac{c}{d \sqrt{c+d x^2}}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{\sqrt{b} \sqrt{b c-a d}}}{a d-b c} \]
Antiderivative was successfully verified.
[In] Integrate[x^3/((a + b*x^2)*(c + d*x^2)^(3/2)),x]
[Out]
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Maple [B] time = 0.018, size = 653, normalized size = 8.5 \[ -{\frac{1}{bd}{\frac{1}{\sqrt{d{x}^{2}+c}}}}+{\frac{a}{2\, \left ( ad-bc \right ) b}{\frac{1}{\sqrt{ \left ( x-{\frac{1}{b}\sqrt{-ab}} \right ) ^{2}d+2\,{\frac{d\sqrt{-ab}}{b} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}}}}}-{\frac{axd}{2\,{b}^{2} \left ( ad-bc \right ) c}\sqrt{-ab}{\frac{1}{\sqrt{ \left ( x-{\frac{1}{b}\sqrt{-ab}} \right ) ^{2}d+2\,{\frac{d\sqrt{-ab}}{b} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}}}}}-{\frac{a}{2\, \left ( ad-bc \right ) b}\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}+2\,{\frac{d\sqrt{-ab}}{b} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d+2\,{\frac{d\sqrt{-ab}}{b} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ( x-{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}}+{\frac{a}{2\, \left ( ad-bc \right ) b}{\frac{1}{\sqrt{ \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ) ^{2}d-2\,{\frac{d\sqrt{-ab}}{b} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}}}}}+{\frac{axd}{2\,{b}^{2} \left ( ad-bc \right ) c}\sqrt{-ab}{\frac{1}{\sqrt{ \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ) ^{2}d-2\,{\frac{d\sqrt{-ab}}{b} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}}}}}-{\frac{a}{2\, \left ( ad-bc \right ) b}\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}-2\,{\frac{d\sqrt{-ab}}{b} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d-2\,{\frac{d\sqrt{-ab}}{b} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3/(b*x^2+a)/(d*x^2+c)^(3/2),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(x^3/((b*x^2 + a)*(d*x^2 + c)^(3/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.254401, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \, \sqrt{b^{2} c - a b d} \sqrt{d x^{2} + c} c +{\left (a d^{2} x^{2} + a c d\right )} \log \left (\frac{{\left (b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \,{\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2}\right )} \sqrt{b^{2} c - a b d} - 4 \,{\left (2 \, b^{3} c^{2} - 3 \, a b^{2} c d + a^{2} b d^{2} +{\left (b^{3} c d - a b^{2} d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right )}{4 \,{\left (b c^{2} d - a c d^{2} +{\left (b c d^{2} - a d^{3}\right )} x^{2}\right )} \sqrt{b^{2} c - a b d}}, -\frac{2 \, \sqrt{-b^{2} c + a b d} \sqrt{d x^{2} + c} c +{\left (a d^{2} x^{2} + a c d\right )} \arctan \left (-\frac{{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt{-b^{2} c + a b d}}{2 \,{\left (b^{2} c - a b d\right )} \sqrt{d x^{2} + c}}\right )}{2 \,{\left (b c^{2} d - a c d^{2} +{\left (b c d^{2} - a d^{3}\right )} x^{2}\right )} \sqrt{-b^{2} c + a b d}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/((b*x^2 + a)*(d*x^2 + c)^(3/2)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{\left (a + b x^{2}\right ) \left (c + d x^{2}\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3/(b*x**2+a)/(d*x**2+c)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.228596, size = 105, normalized size = 1.36 \[ -\frac{\frac{a d \arctan \left (\frac{\sqrt{d x^{2} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d}{\left (b c - a d\right )}} + \frac{c}{\sqrt{d x^{2} + c}{\left (b c - a d\right )}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/((b*x^2 + a)*(d*x^2 + c)^(3/2)),x, algorithm="giac")
[Out]