Optimal. Leaf size=150 \[ -\frac{5805 \sqrt{1-2 x}}{134456 (3 x+2)}-\frac{1935 \sqrt{1-2 x}}{19208 (3 x+2)^2}-\frac{387 \sqrt{1-2 x}}{1372 (3 x+2)^3}+\frac{387}{686 \sqrt{1-2 x} (3 x+2)^3}+\frac{43}{294 (1-2 x)^{3/2} (3 x+2)^3}+\frac{1}{84 (1-2 x)^{3/2} (3 x+2)^4}-\frac{1935 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{67228} \]
[Out]
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Rubi [A] time = 0.163779, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{5805 \sqrt{1-2 x}}{134456 (3 x+2)}-\frac{1935 \sqrt{1-2 x}}{19208 (3 x+2)^2}-\frac{387 \sqrt{1-2 x}}{1372 (3 x+2)^3}+\frac{387}{686 \sqrt{1-2 x} (3 x+2)^3}+\frac{43}{294 (1-2 x)^{3/2} (3 x+2)^3}+\frac{1}{84 (1-2 x)^{3/2} (3 x+2)^4}-\frac{1935 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{67228} \]
Antiderivative was successfully verified.
[In] Int[(3 + 5*x)/((1 - 2*x)^(5/2)*(2 + 3*x)^5),x]
[Out]
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Rubi in Sympy [A] time = 14.7344, size = 133, normalized size = 0.89 \[ - \frac{5805 \sqrt{- 2 x + 1}}{134456 \left (3 x + 2\right )} - \frac{1935 \sqrt{- 2 x + 1}}{19208 \left (3 x + 2\right )^{2}} - \frac{1935 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{470596} + \frac{129}{686 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}} + \frac{43}{686 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{2}} - \frac{43}{588 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{3}} + \frac{1}{84 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((3+5*x)/(1-2*x)**(5/2)/(2+3*x)**5,x)
[Out]
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Mathematica [A] time = 0.182707, size = 73, normalized size = 0.49 \[ \frac{-\frac{7 \left (1880820 x^5+3343680 x^4+1069281 x^3-1034451 x^2-611202 x-48490\right )}{(1-2 x)^{3/2} (3 x+2)^4}-11610 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{2823576} \]
Antiderivative was successfully verified.
[In] Integrate[(3 + 5*x)/((1 - 2*x)^(5/2)*(2 + 3*x)^5),x]
[Out]
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Maple [A] time = 0.022, size = 84, normalized size = 0.6 \[{\frac{176}{50421} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{2080}{117649}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{3888}{117649\, \left ( -4-6\,x \right ) ^{4}} \left ({\frac{5225}{192} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{119623}{576} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{921935}{1728} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{2378705}{5184}\sqrt{1-2\,x}} \right ) }-{\frac{1935\,\sqrt{21}}{470596}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((3+5*x)/(1-2*x)^(5/2)/(2+3*x)^5,x)
[Out]
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Maxima [A] time = 1.50868, size = 173, normalized size = 1.15 \[ \frac{1935}{941192} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{470205 \,{\left (2 \, x - 1\right )}^{5} + 4022865 \,{\left (2 \, x - 1\right )}^{4} + 12458691 \,{\left (2 \, x - 1\right )}^{3} + 15872031 \,{\left (2 \, x - 1\right )}^{2} + 11327232 \, x - 7353920}{201684 \,{\left (81 \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} - 756 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + 2646 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 4116 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 2401 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)/((3*x + 2)^5*(-2*x + 1)^(5/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.219016, size = 185, normalized size = 1.23 \[ \frac{\sqrt{7}{\left (5805 \, \sqrt{3}{\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} + 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + \sqrt{7}{\left (1880820 \, x^{5} + 3343680 \, x^{4} + 1069281 \, x^{3} - 1034451 \, x^{2} - 611202 \, x - 48490\right )}\right )}}{2823576 \,{\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \sqrt{-2 \, x + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)/((3*x + 2)^5*(-2*x + 1)^(5/2)),x, algorithm="fricas")
[Out]
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3+5*x)/(1-2*x)**(5/2)/(2+3*x)**5,x)
[Out]
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GIAC/XCAS [A] time = 0.230776, size = 163, normalized size = 1.09 \[ \frac{1935}{941192} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{16 \,{\left (780 \, x - 467\right )}}{352947 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} - \frac{3 \,{\left (141075 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 1076607 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 2765805 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 2378705 \, \sqrt{-2 \, x + 1}\right )}}{7529536 \,{\left (3 \, x + 2\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)/((3*x + 2)^5*(-2*x + 1)^(5/2)),x, algorithm="giac")
[Out]