Optimal. Leaf size=194 \[ \frac{e \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{a^{3/2} d^2}-\frac{2 c x}{a^2 d \sqrt{a+c x^2}}+\frac{e^2 (a e+c d x)}{a d^2 \sqrt{a+c x^2} \left (a e^2+c d^2\right )}-\frac{e^4 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{d^2 \left (a e^2+c d^2\right )^{3/2}}-\frac{e}{a d^2 \sqrt{a+c x^2}}-\frac{1}{a d x \sqrt{a+c x^2}} \]
[Out]
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Rubi [A] time = 0.401405, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 11, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ \frac{e \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{a^{3/2} d^2}-\frac{2 c x}{a^2 d \sqrt{a+c x^2}}+\frac{e^2 (a e+c d x)}{a d^2 \sqrt{a+c x^2} \left (a e^2+c d^2\right )}-\frac{e^4 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{d^2 \left (a e^2+c d^2\right )^{3/2}}-\frac{e}{a d^2 \sqrt{a+c x^2}}-\frac{1}{a d x \sqrt{a+c x^2}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^2*(d + e*x)*(a + c*x^2)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 46.8053, size = 172, normalized size = 0.89 \[ - \frac{e^{4} \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{d^{2} \left (a e^{2} + c d^{2}\right )^{\frac{3}{2}}} - \frac{1}{a d x \sqrt{a + c x^{2}}} + \frac{e^{2} \left (a e + c d x\right )}{a d^{2} \sqrt{a + c x^{2}} \left (a e^{2} + c d^{2}\right )} - \frac{e}{a d^{2} \sqrt{a + c x^{2}}} - \frac{2 c x}{a^{2} d \sqrt{a + c x^{2}}} + \frac{e \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{2}}}{\sqrt{a}} \right )}}{a^{\frac{3}{2}} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**2/(e*x+d)/(c*x**2+a)**(3/2),x)
[Out]
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Mathematica [A] time = 0.589887, size = 188, normalized size = 0.97 \[ \frac{e \log \left (\sqrt{a} \sqrt{a+c x^2}+a\right )}{a^{3/2} d^2}-\frac{e \log (x)}{a^{3/2} d^2}-\frac{\sqrt{a+c x^2} \left (\frac{c (a e+c d x)}{\left (a+c x^2\right ) \left (a e^2+c d^2\right )}+\frac{1}{d x}\right )}{a^2}-\frac{e^4 \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{d^2 \left (a e^2+c d^2\right )^{3/2}}+\frac{e^4 \log (d+e x)}{d^2 \left (a e^2+c d^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^2*(d + e*x)*(a + c*x^2)^(3/2)),x]
[Out]
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Maple [B] time = 0.017, size = 363, normalized size = 1.9 \[ -{\frac{1}{adx}{\frac{1}{\sqrt{c{x}^{2}+a}}}}-2\,{\frac{cx}{{a}^{2}d\sqrt{c{x}^{2}+a}}}+{\frac{{e}^{3}}{{d}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) }{\frac{1}{\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}+{\frac{{e}^{2}cx}{d \left ( a{e}^{2}+c{d}^{2} \right ) a}{\frac{1}{\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}-{\frac{{e}^{3}}{{d}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) }\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ( x+{\frac{d}{e}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}-{\frac{e}{a{d}^{2}}{\frac{1}{\sqrt{c{x}^{2}+a}}}}+{\frac{e}{{d}^{2}}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^2/(e*x+d)/(c*x^2+a)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} + a\right )}^{\frac{3}{2}}{\left (e x + d\right )} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^2 + a)^(3/2)*(e*x + d)*x^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.534645, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^2 + a)^(3/2)*(e*x + d)*x^2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{2} \left (a + c x^{2}\right )^{\frac{3}{2}} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**2/(e*x+d)/(c*x**2+a)**(3/2),x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^2 + a)^(3/2)*(e*x + d)*x^2),x, algorithm="giac")
[Out]