Optimal. Leaf size=191 \[ \frac{(b c-a d) \left (5 a^2 d^2+2 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 b^{7/2} d^{5/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (5 a^2 d^2+2 a b c d+b^2 c^2\right )}{8 b^3 d^2}-\frac{\sqrt{a+b x} (c+d x)^{3/2} (5 a d+3 b c)}{12 b^2 d^2}+\frac{x \sqrt{a+b x} (c+d x)^{3/2}}{3 b d} \]
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Rubi [A] time = 0.410091, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{(b c-a d) \left (5 a^2 d^2+2 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 b^{7/2} d^{5/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (5 a^2 d^2+2 a b c d+b^2 c^2\right )}{8 b^3 d^2}-\frac{\sqrt{a+b x} (c+d x)^{3/2} (5 a d+3 b c)}{12 b^2 d^2}+\frac{x \sqrt{a+b x} (c+d x)^{3/2}}{3 b d} \]
Antiderivative was successfully verified.
[In] Int[(x^2*Sqrt[c + d*x])/Sqrt[a + b*x],x]
[Out]
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Rubi in Sympy [A] time = 27.0081, size = 178, normalized size = 0.93 \[ \frac{x \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}}}{3 b d} - \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (5 a d + 3 b c\right )}{12 b^{2} d^{2}} + \frac{\sqrt{a + b x} \sqrt{c + d x} \left (5 a^{2} d^{2} + 2 a b c d + b^{2} c^{2}\right )}{8 b^{3} d^{2}} - \frac{\left (a d - b c\right ) \left (5 a^{2} d^{2} + 2 a b c d + b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{8 b^{\frac{7}{2}} d^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2*(d*x+c)**(1/2)/(b*x+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.132147, size = 161, normalized size = 0.84 \[ \frac{(b c-a d) \left (5 a^2 d^2+2 a b c d+b^2 c^2\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{16 b^{7/2} d^{5/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (15 a^2 d^2-2 a b d (2 c+5 d x)+b^2 \left (-3 c^2+2 c d x+8 d^2 x^2\right )\right )}{24 b^3 d^2} \]
Antiderivative was successfully verified.
[In] Integrate[(x^2*Sqrt[c + d*x])/Sqrt[a + b*x],x]
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Maple [B] time = 0.032, size = 395, normalized size = 2.1 \[ -{\frac{1}{48\,{b}^{3}{d}^{2}}\sqrt{bx+a}\sqrt{dx+c} \left ( -16\,{x}^{2}{b}^{2}{d}^{2}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+15\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{3}{d}^{3}-9\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{2}bc{d}^{2}-3\,{c}^{2}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) a{b}^{2}d-3\,{c}^{3}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){b}^{3}+20\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }xab{d}^{2}-4\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }x{b}^{2}cd-30\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{a}^{2}{d}^{2}+8\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }abcd+6\,{c}^{2}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{b}^{2}\sqrt{bd} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2*(d*x+c)^(1/2)/(b*x+a)^(1/2),x)
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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 + c)*x^2/sqrt(b*x + a),x, algorithm="maxima")
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Fricas [A] time = 0.259889, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (8 \, b^{2} d^{2} x^{2} - 3 \, b^{2} c^{2} - 4 \, a b c d + 15 \, a^{2} d^{2} + 2 \,{\left (b^{2} c d - 5 \, a b d^{2}\right )} x\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} - 3 \,{\left (b^{3} c^{3} + a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} \log \left (-4 \,{\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d}\right )}{96 \, \sqrt{b d} b^{3} d^{2}}, \frac{2 \,{\left (8 \, b^{2} d^{2} x^{2} - 3 \, b^{2} c^{2} - 4 \, a b c d + 15 \, a^{2} d^{2} + 2 \,{\left (b^{2} c d - 5 \, a b d^{2}\right )} x\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c} + 3 \,{\left (b^{3} c^{3} + a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x + a} \sqrt{d x + c} b d}\right )}{48 \, \sqrt{-b d} b^{3} d^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x + c)*x^2/sqrt(b*x + a),x, algorithm="fricas")
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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(x**2*(d*x+c)**(1/2)/(b*x+a)**(1/2),x)
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GIAC/XCAS [A] time = 0.248564, size = 279, normalized size = 1.46 \[ \frac{{\left (\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \sqrt{b x + a}{\left (2 \,{\left (b x + a\right )}{\left (\frac{4 \,{\left (b x + a\right )}}{b^{2}} + \frac{b^{6} c d^{3} - 13 \, a b^{5} d^{4}}{b^{7} d^{4}}\right )} - \frac{3 \,{\left (b^{7} c^{2} d^{2} + 2 \, a b^{6} c d^{3} - 11 \, a^{2} b^{5} d^{4}\right )}}{b^{7} d^{4}}\right )} - \frac{3 \,{\left (b^{3} c^{3} + a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} b d^{2}}\right )}{\left | b \right |}}{24 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x + c)*x^2/sqrt(b*x + a),x, algorithm="giac")
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