Optimal. Leaf size=178 \[ -\frac{63 e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{11/2} \sqrt{b d-a e}}-\frac{63 e^4 \sqrt{d+e x}}{128 b^5 (a+b x)}-\frac{21 e^3 (d+e x)^{3/2}}{64 b^4 (a+b x)^2}-\frac{21 e^2 (d+e x)^{5/2}}{80 b^3 (a+b x)^3}-\frac{9 e (d+e x)^{7/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{9/2}}{5 b (a+b x)^5} \]
[Out]
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Rubi [A] time = 0.271623, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{63 e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{11/2} \sqrt{b d-a e}}-\frac{63 e^4 \sqrt{d+e x}}{128 b^5 (a+b x)}-\frac{21 e^3 (d+e x)^{3/2}}{64 b^4 (a+b x)^2}-\frac{21 e^2 (d+e x)^{5/2}}{80 b^3 (a+b x)^3}-\frac{9 e (d+e x)^{7/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{9/2}}{5 b (a+b x)^5} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^(9/2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]
[Out]
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Rubi in Sympy [A] time = 67.6612, size = 163, normalized size = 0.92 \[ - \frac{\left (d + e x\right )^{\frac{9}{2}}}{5 b \left (a + b x\right )^{5}} - \frac{9 e \left (d + e x\right )^{\frac{7}{2}}}{40 b^{2} \left (a + b x\right )^{4}} - \frac{21 e^{2} \left (d + e x\right )^{\frac{5}{2}}}{80 b^{3} \left (a + b x\right )^{3}} - \frac{21 e^{3} \left (d + e x\right )^{\frac{3}{2}}}{64 b^{4} \left (a + b x\right )^{2}} - \frac{63 e^{4} \sqrt{d + e x}}{128 b^{5} \left (a + b x\right )} + \frac{63 e^{5} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{128 b^{\frac{11}{2}} \sqrt{a e - b d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**(9/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)
[Out]
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Mathematica [A] time = 0.399184, size = 161, normalized size = 0.9 \[ -\frac{63 e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{11/2} \sqrt{b d-a e}}-\frac{\sqrt{d+e x} \left (1490 e^3 (a+b x)^3 (b d-a e)+1368 e^2 (a+b x)^2 (b d-a e)^2+656 e (a+b x) (b d-a e)^3+128 (b d-a e)^4+965 e^4 (a+b x)^4\right )}{640 b^5 (a+b x)^5} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^(9/2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]
[Out]
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Maple [B] time = 0.029, size = 463, normalized size = 2.6 \[ -{\frac{193\,{e}^{5}}{128\, \left ( bex+ae \right ) ^{5}b} \left ( ex+d \right ) ^{{\frac{9}{2}}}}-{\frac{237\,{e}^{6}a}{64\, \left ( bex+ae \right ) ^{5}{b}^{2}} \left ( ex+d \right ) ^{{\frac{7}{2}}}}+{\frac{237\,{e}^{5}d}{64\, \left ( bex+ae \right ) ^{5}b} \left ( ex+d \right ) ^{{\frac{7}{2}}}}-{\frac{21\,{a}^{2}{e}^{7}}{5\, \left ( bex+ae \right ) ^{5}{b}^{3}} \left ( ex+d \right ) ^{{\frac{5}{2}}}}+{\frac{42\,{e}^{6}ad}{5\, \left ( bex+ae \right ) ^{5}{b}^{2}} \left ( ex+d \right ) ^{{\frac{5}{2}}}}-{\frac{21\,{e}^{5}{d}^{2}}{5\, \left ( bex+ae \right ) ^{5}b} \left ( ex+d \right ) ^{{\frac{5}{2}}}}-{\frac{147\,{e}^{8}{a}^{3}}{64\, \left ( bex+ae \right ) ^{5}{b}^{4}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{441\,{a}^{2}{e}^{7}d}{64\, \left ( bex+ae \right ) ^{5}{b}^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{441\,{e}^{6}a{d}^{2}}{64\, \left ( bex+ae \right ) ^{5}{b}^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{147\,{e}^{5}{d}^{3}}{64\, \left ( bex+ae \right ) ^{5}b} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{63\,{e}^{9}{a}^{4}}{128\, \left ( bex+ae \right ) ^{5}{b}^{5}}\sqrt{ex+d}}+{\frac{63\,{e}^{8}{a}^{3}d}{32\, \left ( bex+ae \right ) ^{5}{b}^{4}}\sqrt{ex+d}}-{\frac{189\,{a}^{2}{e}^{7}{d}^{2}}{64\, \left ( bex+ae \right ) ^{5}{b}^{3}}\sqrt{ex+d}}+{\frac{63\,{e}^{6}a{d}^{3}}{32\, \left ( bex+ae \right ) ^{5}{b}^{2}}\sqrt{ex+d}}-{\frac{63\,{e}^{5}{d}^{4}}{128\, \left ( bex+ae \right ) ^{5}b}\sqrt{ex+d}}+{\frac{63\,{e}^{5}}{128\,{b}^{5}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(9/2)/(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)^(9/2)/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.226903, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (965 \, b^{4} e^{4} x^{4} + 128 \, b^{4} d^{4} + 144 \, a b^{3} d^{3} e + 168 \, a^{2} b^{2} d^{2} e^{2} + 210 \, a^{3} b d e^{3} + 315 \, a^{4} e^{4} + 10 \,{\left (149 \, b^{4} d e^{3} + 237 \, a b^{3} e^{4}\right )} x^{3} + 6 \,{\left (228 \, b^{4} d^{2} e^{2} + 289 \, a b^{3} d e^{3} + 448 \, a^{2} b^{2} e^{4}\right )} x^{2} + 2 \,{\left (328 \, b^{4} d^{3} e + 384 \, a b^{3} d^{2} e^{2} + 483 \, a^{2} b^{2} d e^{3} + 735 \, a^{3} b e^{4}\right )} x\right )} \sqrt{b^{2} d - a b e} \sqrt{e x + d} - 315 \,{\left (b^{5} e^{5} x^{5} + 5 \, a b^{4} e^{5} x^{4} + 10 \, a^{2} b^{3} e^{5} x^{3} + 10 \, a^{3} b^{2} e^{5} x^{2} + 5 \, a^{4} b e^{5} x + a^{5} e^{5}\right )} \log \left (\frac{\sqrt{b^{2} d - a b e}{\left (b e x + 2 \, b d - a e\right )} - 2 \,{\left (b^{2} d - a b e\right )} \sqrt{e x + d}}{b x + a}\right )}{1280 \,{\left (b^{10} x^{5} + 5 \, a b^{9} x^{4} + 10 \, a^{2} b^{8} x^{3} + 10 \, a^{3} b^{7} x^{2} + 5 \, a^{4} b^{6} x + a^{5} b^{5}\right )} \sqrt{b^{2} d - a b e}}, -\frac{{\left (965 \, b^{4} e^{4} x^{4} + 128 \, b^{4} d^{4} + 144 \, a b^{3} d^{3} e + 168 \, a^{2} b^{2} d^{2} e^{2} + 210 \, a^{3} b d e^{3} + 315 \, a^{4} e^{4} + 10 \,{\left (149 \, b^{4} d e^{3} + 237 \, a b^{3} e^{4}\right )} x^{3} + 6 \,{\left (228 \, b^{4} d^{2} e^{2} + 289 \, a b^{3} d e^{3} + 448 \, a^{2} b^{2} e^{4}\right )} x^{2} + 2 \,{\left (328 \, b^{4} d^{3} e + 384 \, a b^{3} d^{2} e^{2} + 483 \, a^{2} b^{2} d e^{3} + 735 \, a^{3} b e^{4}\right )} x\right )} \sqrt{-b^{2} d + a b e} \sqrt{e x + d} + 315 \,{\left (b^{5} e^{5} x^{5} + 5 \, a b^{4} e^{5} x^{4} + 10 \, a^{2} b^{3} e^{5} x^{3} + 10 \, a^{3} b^{2} e^{5} x^{2} + 5 \, a^{4} b e^{5} x + a^{5} e^{5}\right )} \arctan \left (-\frac{b d - a e}{\sqrt{-b^{2} d + a b e} \sqrt{e x + d}}\right )}{640 \,{\left (b^{10} x^{5} + 5 \, a b^{9} x^{4} + 10 \, a^{2} b^{8} x^{3} + 10 \, a^{3} b^{7} x^{2} + 5 \, a^{4} b^{6} x + a^{5} b^{5}\right )} \sqrt{-b^{2} d + a b e}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(9/2)/(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)**(9/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.236398, size = 451, normalized size = 2.53 \[ \frac{63 \, \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{5}}{128 \, \sqrt{-b^{2} d + a b e} b^{5}} - \frac{965 \,{\left (x e + d\right )}^{\frac{9}{2}} b^{4} e^{5} - 2370 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{4} d e^{5} + 2688 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{4} d^{2} e^{5} - 1470 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{4} d^{3} e^{5} + 315 \, \sqrt{x e + d} b^{4} d^{4} e^{5} + 2370 \,{\left (x e + d\right )}^{\frac{7}{2}} a b^{3} e^{6} - 5376 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{3} d e^{6} + 4410 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{3} d^{2} e^{6} - 1260 \, \sqrt{x e + d} a b^{3} d^{3} e^{6} + 2688 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{2} b^{2} e^{7} - 4410 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{2} d e^{7} + 1890 \, \sqrt{x e + d} a^{2} b^{2} d^{2} e^{7} + 1470 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{3} b e^{8} - 1260 \, \sqrt{x e + d} a^{3} b d e^{8} + 315 \, \sqrt{x e + d} a^{4} e^{9}}{640 \,{\left ({\left (x e + d\right )} b - b d + a e\right )}^{5} b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(9/2)/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="giac")
[Out]