Optimal. Leaf size=212 \[ -\frac{a c^2 \sqrt{a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{8 d^{3/2} (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2} (8 b c-15 a d x)}{60 d^2 (a+b x)}-\frac{a c x \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{c+d x^2}}{8 d (a+b x)}+\frac{b x^2 \sqrt{a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2}}{5 d (a+b x)} \]
[Out]
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Rubi [A] time = 0.393442, antiderivative size = 212, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171 \[ -\frac{a c^2 \sqrt{a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{8 d^{3/2} (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2} (8 b c-15 a d x)}{60 d^2 (a+b x)}-\frac{a c x \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{c+d x^2}}{8 d (a+b x)}+\frac{b x^2 \sqrt{a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2}}{5 d (a+b x)} \]
Antiderivative was successfully verified.
[In] Int[x^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c + d*x^2],x]
[Out]
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Rubi in Sympy [A] time = 40.4385, size = 196, normalized size = 0.92 \[ - \frac{a c^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{8 d^{\frac{3}{2}} \left (a + b x\right )} - \frac{a c x \sqrt{c + d x^{2}} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{8 d \left (a + b x\right )} + \frac{b x^{2} \left (c + d x^{2}\right )^{\frac{3}{2}} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{5 d \left (a + b x\right )} - \frac{\left (c + d x^{2}\right )^{\frac{3}{2}} \left (- 30 a d x + 16 b c\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{120 d^{2} \left (a + b x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2*((b*x+a)**2)**(1/2)*(d*x**2+c)**(1/2),x)
[Out]
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Mathematica [A] time = 0.128794, size = 108, normalized size = 0.51 \[ \frac{\sqrt{(a+b x)^2} \left (\sqrt{c+d x^2} \left (15 a d x \left (c+2 d x^2\right )+8 b \left (-2 c^2+c d x^2+3 d^2 x^4\right )\right )-15 a c^2 \sqrt{d} \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )\right )}{120 d^2 (a+b x)} \]
Antiderivative was successfully verified.
[In] Integrate[x^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c + d*x^2],x]
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Maple [C] time = 0.043, size = 105, normalized size = 0.5 \[{\frac{{\it csgn} \left ( bx+a \right ) }{120} \left ( 24\,{d}^{5/2} \left ( d{x}^{2}+c \right ) ^{3/2}{x}^{2}b+30\,ax \left ( d{x}^{2}+c \right ) ^{3/2}{d}^{5/2}-16\,{d}^{3/2} \left ( d{x}^{2}+c \right ) ^{3/2}bc-15\,acx\sqrt{d{x}^{2}+c}{d}^{5/2}-15\,a{c}^{2}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{2} \right ){d}^{-{\frac{7}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2*((b*x+a)^2)^(1/2)*(d*x^2+c)^(1/2),x)
[Out]
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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^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.323893, size = 1, normalized size = 0. \[ \left [\frac{15 \, a c^{2} d \log \left (2 \, \sqrt{d x^{2} + c} d x -{\left (2 \, d x^{2} + c\right )} \sqrt{d}\right ) + 2 \,{\left (24 \, b d^{2} x^{4} + 30 \, a d^{2} x^{3} + 8 \, b c d x^{2} + 15 \, a c d x - 16 \, b c^{2}\right )} \sqrt{d x^{2} + c} \sqrt{d}}{240 \, d^{\frac{5}{2}}}, -\frac{15 \, a c^{2} d \arctan \left (\frac{\sqrt{-d} x}{\sqrt{d x^{2} + c}}\right ) -{\left (24 \, b d^{2} x^{4} + 30 \, a d^{2} x^{3} + 8 \, b c d x^{2} + 15 \, a c d x - 16 \, b c^{2}\right )} \sqrt{d x^{2} + c} \sqrt{-d}}{120 \, \sqrt{-d} d^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^2 + c)*sqrt((b*x + a)^2)*x^2,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int x^{2} \sqrt{c + d x^{2}} \sqrt{\left (a + b x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2*((b*x+a)**2)**(1/2)*(d*x**2+c)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.269735, size = 158, normalized size = 0.75 \[ \frac{a c^{2}{\rm ln}\left ({\left | -\sqrt{d} x + \sqrt{d x^{2} + c} \right |}\right ){\rm sign}\left (b x + a\right )}{8 \, d^{\frac{3}{2}}} + \frac{1}{120} \, \sqrt{d x^{2} + c}{\left ({\left (2 \,{\left (3 \,{\left (4 \, b x{\rm sign}\left (b x + a\right ) + 5 \, a{\rm sign}\left (b x + a\right )\right )} x + \frac{4 \, b c{\rm sign}\left (b x + a\right )}{d}\right )} x + \frac{15 \, a c{\rm sign}\left (b x + a\right )}{d}\right )} x - \frac{16 \, b c^{2}{\rm sign}\left (b x + a\right )}{d^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^2 + c)*sqrt((b*x + a)^2)*x^2,x, algorithm="giac")
[Out]