Optimal. Leaf size=151 \[ -\frac{b d \left (3 a c^2-d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 a^{3/2} \left (a c^2-d^2\right )^2}-\frac{a c-d \sqrt{a+b x^2}}{2 a x^2 \left (a c^2-d^2\right )}+\frac{b c^3 \log \left (c \sqrt{a+b x^2}+d\right )}{\left (a c^2-d^2\right )^2}-\frac{b c^3 \log (x)}{\left (a c^2-d^2\right )^2} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.637981, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207 \[ -\frac{b d \left (3 a c^2-d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 a^{3/2} \left (a c^2-d^2\right )^2}-\frac{a c-d \sqrt{a+b x^2}}{2 a x^2 \left (a c^2-d^2\right )}+\frac{b c^3 \log \left (c \sqrt{a+b x^2}+d\right )}{\left (a c^2-d^2\right )^2}-\frac{b c^3 \log (x)}{\left (a c^2-d^2\right )^2} \]
Antiderivative was successfully verified.
[In] Int[1/(x^3*(a*c + b*c*x^2 + d*Sqrt[a + b*x^2])),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 42.6553, size = 133, normalized size = 0.88 \[ - \frac{b c^{3} \log{\left (- b x^{2} \right )}}{2 \left (- a c^{2} + d^{2}\right )^{2}} + \frac{b c^{3} \log{\left (c \sqrt{a + b x^{2}} + d \right )}}{\left (- a c^{2} + d^{2}\right )^{2}} + \frac{a c - d \sqrt{a + b x^{2}}}{2 a x^{2} \left (- a c^{2} + d^{2}\right )} + \frac{b d \left (- 3 a c^{2} + d^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{2 a^{\frac{3}{2}} \left (- a c^{2} + d^{2}\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**3/(a*c+b*c*x**2+d*(b*x**2+a)**(1/2)),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.94616, size = 430, normalized size = 2.85 \[ \frac{a^{3/2} b c^3 x^2 \log \left (-\frac{2 \left (d^2-a c^2\right )^2 \left (-i \sqrt{b} x \sqrt{a c^2-d^2}+d \sqrt{a+b x^2}+a c\right )}{b^{3/2} c^3 d^2 \left (\sqrt{b} c x+i \sqrt{a c^2-d^2}\right )}\right )+a^{3/2} b c^3 x^2 \log \left (-\frac{2 \left (d^2-a c^2\right )^2 \left (i \sqrt{b} x \sqrt{a c^2-d^2}+d \sqrt{a+b x^2}+a c\right )}{b^{3/2} c^3 d^2 \left (\sqrt{b} c x-i \sqrt{a c^2-d^2}\right )}\right )+a^{3/2} c^2 d \sqrt{a+b x^2}-b x^2 \log (x) \left (2 a^{3/2} c^3-3 a c^2 d+d^3\right )+a^{3/2} b c^3 x^2 \log \left (a c^2+b c^2 x^2-d^2\right )-a^{5/2} c^3+a^{3/2} c d^2-3 a b c^2 d x^2 \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )-\sqrt{a} d^3 \sqrt{a+b x^2}+b d^3 x^2 \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )}{2 a^{3/2} x^2 \left (d^2-a c^2\right )^2} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^3*(a*c + b*c*x^2 + d*Sqrt[a + b*x^2])),x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.067, size = 2459, normalized size = 16.3 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^3/(a*c+b*c*x^2+d*(b*x^2+a)^(1/2)),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b c x^{2} + a c + \sqrt{b x^{2} + a} d\right )} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*c*x^2 + a*c + sqrt(b*x^2 + a)*d)*x^3),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.9211, size = 1, normalized size = 0.01 \[ \left [\frac{a^{\frac{3}{2}} b c^{3} x^{2} \log \left (-\frac{b c^{2} x^{2} + a c^{2} + 2 \, \sqrt{b x^{2} + a} c d + d^{2}}{x^{2}}\right ) - a^{\frac{3}{2}} b c^{3} x^{2} \log \left (-\frac{b c^{2} x^{2} + a c^{2} - 2 \, \sqrt{b x^{2} + a} c d + d^{2}}{x^{2}}\right ) -{\left (3 \, a b c^{2} d - b d^{3}\right )} x^{2} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right ) + 2 \,{\left (a c^{2} d - d^{3}\right )} \sqrt{b x^{2} + a} \sqrt{a} + 2 \,{\left (a b c^{3} x^{2} \log \left (b c^{2} x^{2} + a c^{2} - d^{2}\right ) - 2 \, a b c^{3} x^{2} \log \left (x\right ) - a^{2} c^{3} + a c d^{2}\right )} \sqrt{a}}{4 \,{\left (a^{3} c^{4} - 2 \, a^{2} c^{2} d^{2} + a d^{4}\right )} \sqrt{a} x^{2}}, \frac{\sqrt{-a} a b c^{3} x^{2} \log \left (-\frac{b c^{2} x^{2} + a c^{2} + 2 \, \sqrt{b x^{2} + a} c d + d^{2}}{x^{2}}\right ) - \sqrt{-a} a b c^{3} x^{2} \log \left (-\frac{b c^{2} x^{2} + a c^{2} - 2 \, \sqrt{b x^{2} + a} c d + d^{2}}{x^{2}}\right ) - 2 \,{\left (3 \, a b c^{2} d - b d^{3}\right )} x^{2} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) + 2 \,{\left (a c^{2} d - d^{3}\right )} \sqrt{b x^{2} + a} \sqrt{-a} + 2 \,{\left (a b c^{3} x^{2} \log \left (b c^{2} x^{2} + a c^{2} - d^{2}\right ) - 2 \, a b c^{3} x^{2} \log \left (x\right ) - a^{2} c^{3} + a c d^{2}\right )} \sqrt{-a}}{4 \,{\left (a^{3} c^{4} - 2 \, a^{2} c^{2} d^{2} + a d^{4}\right )} \sqrt{-a} x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*c*x^2 + a*c + sqrt(b*x^2 + a)*d)*x^3),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{3} \left (a c + b c x^{2} + d \sqrt{a + b x^{2}}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**3/(a*c+b*c*x**2+d*(b*x**2+a)**(1/2)),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.278815, size = 275, normalized size = 1.82 \[ \frac{1}{2} \,{\left (\frac{2 \, c^{4}{\rm ln}\left ({\left | \sqrt{b x^{2} + a} c + d \right |}\right )}{a^{2} c^{5} - 2 \, a c^{3} d^{2} + c d^{4}} - \frac{c^{3}{\rm ln}\left (b x^{2}\right )}{a^{2} c^{4} - 2 \, a c^{2} d^{2} + d^{4}} + \frac{{\left (3 \, a c^{2} d - d^{3}\right )} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{{\left (a^{3} c^{4} - 2 \, a^{2} c^{2} d^{2} + a d^{4}\right )} \sqrt{-a}} - \frac{a^{2} c^{3} - a c d^{2} -{\left (a c^{2} d - d^{3}\right )} \sqrt{b x^{2} + a}}{{\left (a c^{2} - d^{2}\right )}^{2} a b x^{2}}\right )} b \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*c*x^2 + a*c + sqrt(b*x^2 + a)*d)*x^3),x, algorithm="giac")
[Out]