Optimal. Leaf size=208 \[ \frac{3 e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{5/2} (b d-a e)^{7/2}}-\frac{3 e^4 \sqrt{d+e x}}{128 b^2 (a+b x) (b d-a e)^3}+\frac{e^3 \sqrt{d+e x}}{64 b^2 (a+b x)^2 (b d-a e)^2}-\frac{e^2 \sqrt{d+e x}}{80 b^2 (a+b x)^3 (b d-a e)}-\frac{3 e \sqrt{d+e x}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{3/2}}{5 b (a+b x)^5} \]
[Out]
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Rubi [A] time = 0.344784, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{3 e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{5/2} (b d-a e)^{7/2}}-\frac{3 e^4 \sqrt{d+e x}}{128 b^2 (a+b x) (b d-a e)^3}+\frac{e^3 \sqrt{d+e x}}{64 b^2 (a+b x)^2 (b d-a e)^2}-\frac{e^2 \sqrt{d+e x}}{80 b^2 (a+b x)^3 (b d-a e)}-\frac{3 e \sqrt{d+e x}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{3/2}}{5 b (a+b x)^5} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^(3/2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]
[Out]
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Rubi in Sympy [A] time = 95.4016, size = 184, normalized size = 0.88 \[ - \frac{\left (d + e x\right )^{\frac{3}{2}}}{5 b \left (a + b x\right )^{5}} + \frac{3 e^{4} \sqrt{d + e x}}{128 b^{2} \left (a + b x\right ) \left (a e - b d\right )^{3}} + \frac{e^{3} \sqrt{d + e x}}{64 b^{2} \left (a + b x\right )^{2} \left (a e - b d\right )^{2}} + \frac{e^{2} \sqrt{d + e x}}{80 b^{2} \left (a + b x\right )^{3} \left (a e - b d\right )} - \frac{3 e \sqrt{d + e x}}{40 b^{2} \left (a + b x\right )^{4}} + \frac{3 e^{5} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{128 b^{\frac{5}{2}} \left (a e - b d\right )^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**(3/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)
[Out]
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Mathematica [A] time = 0.493525, size = 171, normalized size = 0.82 \[ \frac{3 e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{5/2} (b d-a e)^{7/2}}-\frac{\sqrt{d+e x} \left (10 e^3 (a+b x)^3 (a e-b d)+8 e^2 (a+b x)^2 (b d-a e)^2+176 e (a+b x) (b d-a e)^3+128 (b d-a e)^4+15 e^4 (a+b x)^4\right )}{640 b^2 (a+b x)^5 (b d-a e)^3} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^(3/2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]
[Out]
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Maple [A] time = 0.027, size = 300, normalized size = 1.4 \[{\frac{3\,{e}^{5}{b}^{2}}{128\, \left ( bex+ae \right ) ^{5} \left ({a}^{3}{e}^{3}-3\,{a}^{2}bd{e}^{2}+3\,a{b}^{2}{d}^{2}e-{b}^{3}{d}^{3} \right ) } \left ( ex+d \right ) ^{{\frac{9}{2}}}}+{\frac{7\,{e}^{5}b}{64\, \left ( bex+ae \right ) ^{5} \left ({a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2} \right ) } \left ( ex+d \right ) ^{{\frac{7}{2}}}}+{\frac{{e}^{5}}{5\, \left ( bex+ae \right ) ^{5} \left ( ae-bd \right ) } \left ( ex+d \right ) ^{{\frac{5}{2}}}}-{\frac{7\,{e}^{5}}{64\, \left ( bex+ae \right ) ^{5}b} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{e}^{6}a}{128\, \left ( bex+ae \right ) ^{5}{b}^{2}}\sqrt{ex+d}}+{\frac{3\,{e}^{5}d}{128\, \left ( bex+ae \right ) ^{5}b}\sqrt{ex+d}}+{\frac{3\,{e}^{5}}{ \left ( 128\,{a}^{3}{e}^{3}-384\,{a}^{2}bd{e}^{2}+384\,a{b}^{2}{d}^{2}e-128\,{b}^{3}{d}^{3} \right ){b}^{2}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(3/2)/(b^2*x^2+2*a*b*x+a^2)^3,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((e*x + d)^(3/2)/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.231169, size = 1, normalized size = 0. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(3/2)/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="fricas")
[Out]
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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((e*x+d)**(3/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.227594, size = 556, normalized size = 2.67 \[ -\frac{3 \, \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{5}}{128 \,{\left (b^{5} d^{3} - 3 \, a b^{4} d^{2} e + 3 \, a^{2} b^{3} d e^{2} - a^{3} b^{2} e^{3}\right )} \sqrt{-b^{2} d + a b e}} - \frac{15 \,{\left (x e + d\right )}^{\frac{9}{2}} b^{4} e^{5} - 70 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{4} d e^{5} + 128 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{4} d^{2} e^{5} + 70 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{4} d^{3} e^{5} - 15 \, \sqrt{x e + d} b^{4} d^{4} e^{5} + 70 \,{\left (x e + d\right )}^{\frac{7}{2}} a b^{3} e^{6} - 256 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{3} d e^{6} - 210 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{3} d^{2} e^{6} + 60 \, \sqrt{x e + d} a b^{3} d^{3} e^{6} + 128 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{2} b^{2} e^{7} + 210 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{2} d e^{7} - 90 \, \sqrt{x e + d} a^{2} b^{2} d^{2} e^{7} - 70 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{3} b e^{8} + 60 \, \sqrt{x e + d} a^{3} b d e^{8} - 15 \, \sqrt{x e + d} a^{4} e^{9}}{640 \,{\left (b^{5} d^{3} - 3 \, a b^{4} d^{2} e + 3 \, a^{2} b^{3} d e^{2} - a^{3} b^{2} e^{3}\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(3/2)/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="giac")
[Out]