Optimal. Leaf size=152 \[ \frac{b \sqrt{a^2+2 a b x^2+b^2 x^4} \left (c+d x^2\right )^{3/2}}{3 d \left (a+b x^2\right )}+\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4} \sqrt{c+d x^2}}{a+b x^2}-\frac{a \sqrt{c} \sqrt{a^2+2 a b x^2+b^2 x^4} \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{a+b x^2} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.305133, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.162 \[ \frac{b \sqrt{a^2+2 a b x^2+b^2 x^4} \left (c+d x^2\right )^{3/2}}{3 d \left (a+b x^2\right )}+\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4} \sqrt{c+d x^2}}{a+b x^2}-\frac{a \sqrt{c} \sqrt{a^2+2 a b x^2+b^2 x^4} \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{a+b x^2} \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[c + d*x^2]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/x,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 20.0972, size = 105, normalized size = 0.69 \[ - \frac{a \sqrt{c} \sqrt{\left (a + b x^{2}\right )^{2}} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{a + b x^{2}} + \frac{a \sqrt{c + d x^{2}} \sqrt{\left (a + b x^{2}\right )^{2}}}{a + b x^{2}} + \frac{b \left (c + d x^{2}\right )^{\frac{3}{2}} \sqrt{\left (a + b x^{2}\right )^{2}}}{3 d \left (a + b x^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x**2+c)**(1/2)*((b*x**2+a)**2)**(1/2)/x,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.113774, size = 96, normalized size = 0.63 \[ \frac{\sqrt{\left (a+b x^2\right )^2} \left (\sqrt{c+d x^2} \left (3 a d+b c+b d x^2\right )-3 a \sqrt{c} d \log \left (\sqrt{c} \sqrt{c+d x^2}+c\right )+3 a \sqrt{c} d \log (x)\right )}{3 d \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
[In] Integrate[(Sqrt[c + d*x^2]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/x,x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.01, size = 79, normalized size = 0.5 \[{\frac{1}{ \left ( 3\,b{x}^{2}+3\,a \right ) d}\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}} \left ( b \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}-3\,\sqrt{c}\ln \left ( 2\,{\frac{\sqrt{c}\sqrt{d{x}^{2}+c}+c}{x}} \right ) ad+3\,\sqrt{d{x}^{2}+c}ad \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x^2+c)^(1/2)*((b*x^2+a)^2)^(1/2)/x,x)
[Out]
_______________________________________________________________________________________
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(sqrt(d*x^2 + c)*sqrt((b*x^2 + a)^2)/x,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.279204, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a \sqrt{c} d \log \left (-\frac{d x^{2} - 2 \, \sqrt{d x^{2} + c} \sqrt{c} + 2 \, c}{x^{2}}\right ) + 2 \,{\left (b d x^{2} + b c + 3 \, a d\right )} \sqrt{d x^{2} + c}}{6 \, d}, -\frac{3 \, a \sqrt{-c} d \arctan \left (\frac{c}{\sqrt{d x^{2} + c} \sqrt{-c}}\right ) -{\left (b d x^{2} + b c + 3 \, a d\right )} \sqrt{d x^{2} + c}}{3 \, d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^2 + c)*sqrt((b*x^2 + a)^2)/x,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x**2+c)**(1/2)*((b*x**2+a)**2)**(1/2)/x,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.275687, size = 113, normalized size = 0.74 \[ \frac{a c \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right ){\rm sign}\left (b x^{2} + a\right )}{\sqrt{-c}} + \frac{{\left (d x^{2} + c\right )}^{\frac{3}{2}} b d^{2}{\rm sign}\left (b x^{2} + a\right ) + 3 \, \sqrt{d x^{2} + c} a d^{3}{\rm sign}\left (b x^{2} + a\right )}{3 \, d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^2 + c)*sqrt((b*x^2 + a)^2)/x,x, algorithm="giac")
[Out]