Optimal. Leaf size=115 \[ -\frac{b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{a^2 \sqrt{b c-a d}}+\frac{(a d+2 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{2 a^2 c^{3/2}}-\frac{\sqrt{c+d x^2}}{2 a c x^2} \]
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Rubi [A] time = 0.362275, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{a^2 \sqrt{b c-a d}}+\frac{(a d+2 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{2 a^2 c^{3/2}}-\frac{\sqrt{c+d x^2}}{2 a c x^2} \]
Antiderivative was successfully verified.
[In] Int[1/(x^3*(a + b*x^2)*Sqrt[c + d*x^2]),x]
[Out]
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Rubi in Sympy [A] time = 47.8614, size = 97, normalized size = 0.84 \[ - \frac{\sqrt{c + d x^{2}}}{2 a c x^{2}} + \frac{b^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{2}}}{\sqrt{a d - b c}} \right )}}{a^{2} \sqrt{a d - b c}} + \frac{\left (\frac{a d}{2} + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{a^{2} c^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**3/(b*x**2+a)/(d*x**2+c)**(1/2),x)
[Out]
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Mathematica [C] time = 0.668216, size = 292, normalized size = 2.54 \[ \frac{-\frac{\sqrt{c} \left (b^{3/2} c x^2 \log \left (\frac{2 a^2 \left (\sqrt{c+d x^2} \sqrt{b c-a d}-i \sqrt{a} d x+\sqrt{b} c\right )}{b^{3/2} \left (\sqrt{b} x+i \sqrt{a}\right ) \sqrt{b c-a d}}\right )+b^{3/2} c x^2 \log \left (\frac{2 a^2 \left (\sqrt{c+d x^2} \sqrt{b c-a d}+i \sqrt{a} d x+\sqrt{b} c\right )}{b^{3/2} \left (\sqrt{b} x-i \sqrt{a}\right ) \sqrt{b c-a d}}\right )+a \sqrt{c+d x^2} \sqrt{b c-a d}\right )}{x^2 \sqrt{b c-a d}}+(a d+2 b c) \log \left (\sqrt{c} \sqrt{c+d x^2}+c\right )+\log (x) (-(a d+2 b c))}{2 a^2 c^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^3*(a + b*x^2)*Sqrt[c + d*x^2]),x]
[Out]
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Maple [B] time = 0.02, size = 385, normalized size = 3.4 \[ -{\frac{1}{2\,ac{x}^{2}}\sqrt{d{x}^{2}+c}}+{\frac{d}{2\,a}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ){c}^{-{\frac{3}{2}}}}+{\frac{b}{{a}^{2}}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ){\frac{1}{\sqrt{c}}}}-{\frac{b}{2\,{a}^{2}}\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}+2\,{\frac{d\sqrt{-ab}}{b} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d+2\,{\frac{d\sqrt{-ab}}{b} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ( x-{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}}-{\frac{b}{2\,{a}^{2}}\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}-2\,{\frac{d\sqrt{-ab}}{b} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d-2\,{\frac{d\sqrt{-ab}}{b} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^3/(b*x^2+a)/(d*x^2+c)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )} \sqrt{d x^{2} + c} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)*sqrt(d*x^2 + c)*x^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.394214, size = 1, normalized size = 0.01 \[ \left [\frac{b c^{\frac{3}{2}} x^{2} \sqrt{\frac{b}{b c - a d}} \log \left (\frac{b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \,{\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} - 4 \,{\left (2 \, b^{2} c^{2} - 3 \, a b c d + a^{2} d^{2} +{\left (b^{2} c d - a b d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{\frac{b}{b c - a d}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) +{\left (2 \, b c + a d\right )} x^{2} \log \left (-\frac{{\left (d x^{2} + 2 \, c\right )} \sqrt{c} + 2 \, \sqrt{d x^{2} + c} c}{x^{2}}\right ) - 2 \, \sqrt{d x^{2} + c} a \sqrt{c}}{4 \, a^{2} c^{\frac{3}{2}} x^{2}}, \frac{b \sqrt{-c} c x^{2} \sqrt{\frac{b}{b c - a d}} \log \left (\frac{b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \,{\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} - 4 \,{\left (2 \, b^{2} c^{2} - 3 \, a b c d + a^{2} d^{2} +{\left (b^{2} c d - a b d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{\frac{b}{b c - a d}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 2 \,{\left (2 \, b c + a d\right )} x^{2} \arctan \left (\frac{\sqrt{-c}}{\sqrt{d x^{2} + c}}\right ) - 2 \, \sqrt{d x^{2} + c} a \sqrt{-c}}{4 \, a^{2} \sqrt{-c} c x^{2}}, \frac{2 \, b c^{\frac{3}{2}} x^{2} \sqrt{-\frac{b}{b c - a d}} \arctan \left (-\frac{b d x^{2} + 2 \, b c - a d}{2 \, \sqrt{d x^{2} + c}{\left (b c - a d\right )} \sqrt{-\frac{b}{b c - a d}}}\right ) +{\left (2 \, b c + a d\right )} x^{2} \log \left (-\frac{{\left (d x^{2} + 2 \, c\right )} \sqrt{c} + 2 \, \sqrt{d x^{2} + c} c}{x^{2}}\right ) - 2 \, \sqrt{d x^{2} + c} a \sqrt{c}}{4 \, a^{2} c^{\frac{3}{2}} x^{2}}, \frac{b \sqrt{-c} c x^{2} \sqrt{-\frac{b}{b c - a d}} \arctan \left (-\frac{b d x^{2} + 2 \, b c - a d}{2 \, \sqrt{d x^{2} + c}{\left (b c - a d\right )} \sqrt{-\frac{b}{b c - a d}}}\right ) +{\left (2 \, b c + a d\right )} x^{2} \arctan \left (\frac{\sqrt{-c}}{\sqrt{d x^{2} + c}}\right ) - \sqrt{d x^{2} + c} a \sqrt{-c}}{2 \, a^{2} \sqrt{-c} c x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)*sqrt(d*x^2 + c)*x^3),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{3} \left (a + b x^{2}\right ) \sqrt{c + d x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**3/(b*x**2+a)/(d*x**2+c)**(1/2),x)
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GIAC/XCAS [A] time = 0.248291, size = 159, normalized size = 1.38 \[ \frac{1}{2} \, d^{2}{\left (\frac{2 \, b^{2} \arctan \left (\frac{\sqrt{d x^{2} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} a^{2} d^{2}} - \frac{{\left (2 \, b c + a d\right )} \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{a^{2} \sqrt{-c} c d^{2}} - \frac{\sqrt{d x^{2} + c}}{a c d^{2} x^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)*sqrt(d*x^2 + c)*x^3),x, algorithm="giac")
[Out]