Optimal. Leaf size=148 \[ \frac{b \sqrt{a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2}}{3 d (a+b x)}+\frac{a x \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{c+d x^2}}{2 (a+b x)}+\frac{a c \sqrt{a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{2 \sqrt{d} (a+b x)} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.149308, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156 \[ \frac{b \sqrt{a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2}}{3 d (a+b x)}+\frac{a x \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{c+d x^2}}{2 (a+b x)}+\frac{a c \sqrt{a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{2 \sqrt{d} (a+b x)} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c + d*x^2],x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 26.1897, size = 136, normalized size = 0.92 \[ \frac{a c \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{2 \sqrt{d} \left (a + b x\right )} + \frac{a x \sqrt{c + d x^{2}} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{2 \left (a + b x\right )} + \frac{b \left (c + d x^{2}\right )^{\frac{3}{2}} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{3 d \left (a + b x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(((b*x+a)**2)**(1/2)*(d*x**2+c)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0746258, size = 85, normalized size = 0.57 \[ \frac{\sqrt{(a+b x)^2} \left (\sqrt{c+d x^2} \left (3 a d x+2 b \left (c+d x^2\right )\right )+3 a c \sqrt{d} \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )\right )}{6 d (a+b x)} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c + d*x^2],x]
[Out]
_______________________________________________________________________________________
Maple [C] time = 0.013, size = 65, normalized size = 0.4 \[{\frac{{\it csgn} \left ( bx+a \right ) }{6} \left ( 2\,b \left ( d{x}^{2}+c \right ) ^{3/2}\sqrt{d}+3\,ax\sqrt{d{x}^{2}+c}{d}^{3/2}+3\,ac\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ) d \right ){d}^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(((b*x+a)^2)^(1/2)*(d*x^2+c)^(1/2),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 + a)^2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.339508, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a c d \log \left (-2 \, \sqrt{d x^{2} + c} d x -{\left (2 \, d x^{2} + c\right )} \sqrt{d}\right ) + 2 \,{\left (2 \, b d x^{2} + 3 \, a d x + 2 \, b c\right )} \sqrt{d x^{2} + c} \sqrt{d}}{12 \, d^{\frac{3}{2}}}, \frac{3 \, a c d \arctan \left (\frac{\sqrt{-d} x}{\sqrt{d x^{2} + c}}\right ) +{\left (2 \, b d x^{2} + 3 \, a d x + 2 \, b c\right )} \sqrt{d x^{2} + c} \sqrt{-d}}{6 \, \sqrt{-d} d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^2 + c)*sqrt((b*x + a)^2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{c + d x^{2}} \sqrt{\left (a + b x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((b*x+a)**2)**(1/2)*(d*x**2+c)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.271617, size = 107, normalized size = 0.72 \[ -\frac{a c{\rm ln}\left ({\left | -\sqrt{d} x + \sqrt{d x^{2} + c} \right |}\right ){\rm sign}\left (b x + a\right )}{2 \, \sqrt{d}} + \frac{1}{6} \, \sqrt{d x^{2} + c}{\left ({\left (2 \, b x{\rm sign}\left (b x + a\right ) + 3 \, a{\rm sign}\left (b x + a\right )\right )} x + \frac{2 \, b c{\rm sign}\left (b x + a\right )}{d}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^2 + c)*sqrt((b*x + a)^2),x, algorithm="giac")
[Out]