Optimal. Leaf size=229 \[ \frac{231 b^{5/2} e^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 (b d-a e)^{13/2}}-\frac{231 b^2 e^3}{8 \sqrt{d+e x} (b d-a e)^6}-\frac{77 b e^3}{8 (d+e x)^{3/2} (b d-a e)^5}-\frac{231 e^3}{40 (d+e x)^{5/2} (b d-a e)^4}-\frac{33 e^2}{8 (a+b x) (d+e x)^{5/2} (b d-a e)^3}+\frac{11 e}{12 (a+b x)^2 (d+e x)^{5/2} (b d-a e)^2}-\frac{1}{3 (a+b x)^3 (d+e x)^{5/2} (b d-a e)} \]
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Rubi [A] time = 0.542479, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{231 b^{5/2} e^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 (b d-a e)^{13/2}}-\frac{231 b^2 e^3}{8 \sqrt{d+e x} (b d-a e)^6}-\frac{77 b e^3}{8 (d+e x)^{3/2} (b d-a e)^5}-\frac{231 e^3}{40 (d+e x)^{5/2} (b d-a e)^4}-\frac{33 e^2}{8 (a+b x) (d+e x)^{5/2} (b d-a e)^3}+\frac{11 e}{12 (a+b x)^2 (d+e x)^{5/2} (b d-a e)^2}-\frac{1}{3 (a+b x)^3 (d+e x)^{5/2} (b d-a e)} \]
Antiderivative was successfully verified.
[In] Int[1/((d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^2),x]
[Out]
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Rubi in Sympy [A] time = 124.054, size = 226, normalized size = 0.99 \[ - \frac{231 b^{\frac{5}{2}} e^{3} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{8 \left (a e - b d\right )^{\frac{13}{2}}} - \frac{231 b^{3} e^{2} \sqrt{d + e x}}{8 \left (a + b x\right ) \left (a e - b d\right )^{6}} - \frac{77 b^{3} e \sqrt{d + e x}}{4 \left (a + b x\right )^{2} \left (a e - b d\right )^{5}} - \frac{77 b^{3} \sqrt{d + e x}}{5 \left (a + b x\right )^{3} \left (a e - b d\right )^{4}} - \frac{66 b^{2}}{5 \left (a + b x\right )^{3} \sqrt{d + e x} \left (a e - b d\right )^{3}} + \frac{22 b}{15 \left (a + b x\right )^{3} \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{2}} - \frac{2}{5 \left (a + b x\right )^{3} \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(e*x+d)**(7/2)/(b**2*x**2+2*a*b*x+a**2)**2,x)
[Out]
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Mathematica [A] time = 0.97706, size = 193, normalized size = 0.84 \[ \frac{231 b^{5/2} e^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 (b d-a e)^{13/2}}-\frac{\sqrt{d+e x} \left (-\frac{230 b^3 e (b d-a e)}{(a+b x)^2}+\frac{40 b^3 (b d-a e)^2}{(a+b x)^3}+\frac{1065 b^3 e^2}{a+b x}+\frac{320 b e^3 (b d-a e)}{(d+e x)^2}+\frac{48 e^3 (b d-a e)^2}{(d+e x)^3}+\frac{2400 b^2 e^3}{d+e x}\right )}{120 (b d-a e)^6} \]
Antiderivative was successfully verified.
[In] Integrate[1/((d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^2),x]
[Out]
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Maple [A] time = 0.035, size = 344, normalized size = 1.5 \[ -{\frac{2\,{e}^{3}}{5\, \left ( ae-bd \right ) ^{4}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}}-20\,{\frac{{e}^{3}{b}^{2}}{ \left ( ae-bd \right ) ^{6}\sqrt{ex+d}}}+{\frac{8\,{e}^{3}b}{3\, \left ( ae-bd \right ) ^{5}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}}-{\frac{71\,{e}^{3}{b}^{5}}{8\, \left ( ae-bd \right ) ^{6} \left ( bex+ae \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{5}{2}}}}-{\frac{59\,{b}^{4}{e}^{4}a}{3\, \left ( ae-bd \right ) ^{6} \left ( bex+ae \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{59\,{e}^{3}{b}^{5}d}{3\, \left ( ae-bd \right ) ^{6} \left ( bex+ae \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{89\,{e}^{5}{b}^{3}{a}^{2}}{8\, \left ( ae-bd \right ) ^{6} \left ( bex+ae \right ) ^{3}}\sqrt{ex+d}}+{\frac{89\,{b}^{4}{e}^{4}ad}{4\, \left ( ae-bd \right ) ^{6} \left ( bex+ae \right ) ^{3}}\sqrt{ex+d}}-{\frac{89\,{e}^{3}{b}^{5}{d}^{2}}{8\, \left ( ae-bd \right ) ^{6} \left ( bex+ae \right ) ^{3}}\sqrt{ex+d}}-{\frac{231\,{b}^{3}{e}^{3}}{8\, \left ( ae-bd \right ) ^{6}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^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(1/((b^2*x^2 + 2*a*b*x + a^2)^2*(e*x + d)^(7/2)),x, algorithm="maxima")
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Fricas [A] time = 0.251486, size = 1, normalized size = 0. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b^2*x^2 + 2*a*b*x + a^2)^2*(e*x + d)^(7/2)),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(1/(e*x+d)**(7/2)/(b**2*x**2+2*a*b*x+a**2)**2,x)
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GIAC/XCAS [A] time = 0.221782, size = 635, normalized size = 2.77 \[ -\frac{231 \, b^{3} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{3}}{8 \,{\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} \sqrt{-b^{2} d + a b e}} - \frac{2 \,{\left (150 \,{\left (x e + d\right )}^{2} b^{2} e^{3} + 20 \,{\left (x e + d\right )} b^{2} d e^{3} + 3 \, b^{2} d^{2} e^{3} - 20 \,{\left (x e + d\right )} a b e^{4} - 6 \, a b d e^{4} + 3 \, a^{2} e^{5}\right )}}{15 \,{\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )}{\left (x e + d\right )}^{\frac{5}{2}}} - \frac{213 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{5} e^{3} - 472 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{5} d e^{3} + 267 \, \sqrt{x e + d} b^{5} d^{2} e^{3} + 472 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{4} e^{4} - 534 \, \sqrt{x e + d} a b^{4} d e^{4} + 267 \, \sqrt{x e + d} a^{2} b^{3} e^{5}}{24 \,{\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b^2*x^2 + 2*a*b*x + a^2)^2*(e*x + d)^(7/2)),x, algorithm="giac")
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