Optimal. Leaf size=228 \[ \frac{5 (a+c) (a+b x)^{3/2} (b x+c)^{3/2}}{12 b^3 (a-c)^2}+\frac{\left (4 a c-5 (a+c)^2\right ) (a+b x)^{3/2} \sqrt{b x+c}}{16 b^3 (a-c)^2}-\frac{\left (4 a c-5 (a+c)^2\right ) \sqrt{a+b x} \sqrt{b x+c}}{32 b^3 (a-c)}-\frac{\left (4 a c-5 (a+c)^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{b x+c}}\right )}{32 b^3}-\frac{x (a+b x)^{3/2} (b x+c)^{3/2}}{2 b^2 (a-c)^2}+\frac{b x^4}{2 (a-c)^2}+\frac{x^3 (a+c)}{3 (a-c)^2} \]
[Out]
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Rubi [A] time = 0.751523, antiderivative size = 228, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28 \[ \frac{5 (a+c) (a+b x)^{3/2} (b x+c)^{3/2}}{12 b^3 (a-c)^2}+\frac{\left (4 a c-5 (a+c)^2\right ) (a+b x)^{3/2} \sqrt{b x+c}}{16 b^3 (a-c)^2}-\frac{\left (4 a c-5 (a+c)^2\right ) \sqrt{a+b x} \sqrt{b x+c}}{32 b^3 (a-c)}-\frac{\left (4 a c-5 (a+c)^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{b x+c}}\right )}{32 b^3}-\frac{x (a+b x)^{3/2} (b x+c)^{3/2}}{2 b^2 (a-c)^2}+\frac{b x^4}{2 (a-c)^2}+\frac{x^3 (a+c)}{3 (a-c)^2} \]
Antiderivative was successfully verified.
[In] Int[x^2/(Sqrt[a + b*x] + Sqrt[c + b*x])^2,x]
[Out]
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Rubi in Sympy [A] time = 61.8664, size = 197, normalized size = 0.86 \[ \frac{b x^{4}}{2 \left (a - c\right )^{2}} + \frac{x^{3} \left (a + c\right )}{3 \left (a - c\right )^{2}} - \frac{x \left (a + b x\right )^{\frac{3}{2}} \left (b x + c\right )^{\frac{3}{2}}}{2 b^{2} \left (a - c\right )^{2}} - \frac{\left (a c - \frac{5 \left (a + c\right )^{2}}{4}\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{b x + c}} \right )}}{8 b^{3}} + \frac{\sqrt{a + b x} \left (4 a c - 5 \left (a + c\right )^{2}\right ) \sqrt{b x + c}}{32 b^{3} \left (a - c\right )} + \frac{5 \left (a + c\right ) \left (a + b x\right )^{\frac{3}{2}} \left (b x + c\right )^{\frac{3}{2}}}{12 b^{3} \left (a - c\right )^{2}} + \frac{\sqrt{a + b x} \left (a c - \frac{5 \left (a + c\right )^{2}}{4}\right ) \left (b x + c\right )^{\frac{3}{2}}}{4 b^{3} \left (a - c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/((b*x+a)**(1/2)+(b*x+c)**(1/2))**2,x)
[Out]
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Mathematica [A] time = 0.184766, size = 167, normalized size = 0.73 \[ \frac{3 (a-c)^2 \left (5 a^2+6 a c+5 c^2\right ) \log \left (2 \sqrt{a+b x} \sqrt{b x+c}+a+2 b x+c\right )-2 \sqrt{a+b x} \sqrt{b x+c} \left (15 a^3-2 b x \left (5 a^2-2 a c+5 c^2\right )-7 a^2 c+8 b^2 x^2 (a+c)-7 a c^2+48 b^3 x^3+15 c^3\right )+64 b^3 x^3 (a+c)+96 b^4 x^4}{192 b^3 (a-c)^2} \]
Antiderivative was successfully verified.
[In] Integrate[x^2/(Sqrt[a + b*x] + Sqrt[c + b*x])^2,x]
[Out]
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Maple [C] time = 0.027, size = 604, normalized size = 2.7 \[{\frac{a{x}^{3}}{3\, \left ( a-c \right ) ^{2}}}+{\frac{c{x}^{3}}{3\, \left ( a-c \right ) ^{2}}}+{\frac{b{x}^{4}}{2\, \left ( a-c \right ) ^{2}}}-{\frac{{\it csgn} \left ( b \right ) }{192\, \left ( a-c \right ) ^{2}{b}^{3}}\sqrt{bx+a}\sqrt{bx+c} \left ( 96\,{\it csgn} \left ( b \right ){x}^{3}{b}^{3}\sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}+16\,{\it csgn} \left ( b \right ){x}^{2}a{b}^{2}\sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}+16\,{\it csgn} \left ( b \right ){x}^{2}{b}^{2}c\sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}-20\,{\it csgn} \left ( b \right ) \sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}x{a}^{2}b+8\,{\it csgn} \left ( b \right ) \sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}xabc-20\,{\it csgn} \left ( b \right ) \sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}xb{c}^{2}+30\,{\it csgn} \left ( b \right ) \sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}{a}^{3}-14\,{\it csgn} \left ( b \right ) \sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}{a}^{2}c-14\,{\it csgn} \left ( b \right ) \sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}a{c}^{2}+30\,{\it csgn} \left ( b \right ) \sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}{c}^{3}-15\,\ln \left ( 1/2\, \left ( 2\,{\it csgn} \left ( b \right ) \sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}+2\,bx+a+c \right ){\it csgn} \left ( b \right ) \right ){a}^{4}+12\,\ln \left ( 1/2\, \left ( 2\,{\it csgn} \left ( b \right ) \sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}+2\,bx+a+c \right ){\it csgn} \left ( b \right ) \right ){a}^{3}c+6\,\ln \left ( 1/2\, \left ( 2\,{\it csgn} \left ( b \right ) \sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}+2\,bx+a+c \right ){\it csgn} \left ( b \right ) \right ){a}^{2}{c}^{2}+12\,\ln \left ( 1/2\, \left ( 2\,{\it csgn} \left ( b \right ) \sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}+2\,bx+a+c \right ){\it csgn} \left ( b \right ) \right ) a{c}^{3}-15\,\ln \left ( 1/2\, \left ( 2\,{\it csgn} \left ( b \right ) \sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}+2\,bx+a+c \right ){\it csgn} \left ( b \right ) \right ){c}^{4} \right ){\frac{1}{\sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/((b*x+a)^(1/2)+(b*x+c)^(1/2))^2,x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{{\left (\sqrt{b x + a} + \sqrt{b x + c}\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(sqrt(b*x + a) + sqrt(b*x + c))^2,x, algorithm="maxima")
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Fricas [A] time = 0.309121, size = 2361, normalized size = 10.36 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(sqrt(b*x + a) + sqrt(b*x + c))^2,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\left (\sqrt{a + b x} + \sqrt{b x + c}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/((b*x+a)**(1/2)+(b*x+c)**(1/2))**2,x)
[Out]
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GIAC/XCAS [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(sqrt(b*x + a) + sqrt(b*x + c))^2,x, algorithm="giac")
[Out]