Optimal. Leaf size=201 \[ -\frac{231 e^3 (b d-a e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{13/2}}+\frac{231 e^3 \sqrt{d+e x} (b d-a e)^2}{8 b^6}+\frac{77 e^3 (d+e x)^{3/2} (b d-a e)}{8 b^5}-\frac{33 e^2 (d+e x)^{7/2}}{8 b^3 (a+b x)}-\frac{11 e (d+e x)^{9/2}}{12 b^2 (a+b x)^2}-\frac{(d+e x)^{11/2}}{3 b (a+b x)^3}+\frac{231 e^3 (d+e x)^{5/2}}{40 b^4} \]
[Out]
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Rubi [A] time = 0.358634, antiderivative size = 201, 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 \[ -\frac{231 e^3 (b d-a e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{13/2}}+\frac{231 e^3 \sqrt{d+e x} (b d-a e)^2}{8 b^6}+\frac{77 e^3 (d+e x)^{3/2} (b d-a e)}{8 b^5}-\frac{33 e^2 (d+e x)^{7/2}}{8 b^3 (a+b x)}-\frac{11 e (d+e x)^{9/2}}{12 b^2 (a+b x)^2}-\frac{(d+e x)^{11/2}}{3 b (a+b x)^3}+\frac{231 e^3 (d+e x)^{5/2}}{40 b^4} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^(11/2)/(a^2 + 2*a*b*x + b^2*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 82.1152, size = 184, normalized size = 0.92 \[ - \frac{\left (d + e x\right )^{\frac{11}{2}}}{3 b \left (a + b x\right )^{3}} - \frac{11 e \left (d + e x\right )^{\frac{9}{2}}}{12 b^{2} \left (a + b x\right )^{2}} - \frac{33 e^{2} \left (d + e x\right )^{\frac{7}{2}}}{8 b^{3} \left (a + b x\right )} + \frac{231 e^{3} \left (d + e x\right )^{\frac{5}{2}}}{40 b^{4}} - \frac{77 e^{3} \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )}{8 b^{5}} + \frac{231 e^{3} \sqrt{d + e x} \left (a e - b d\right )^{2}}{8 b^{6}} - \frac{231 e^{3} \left (a e - b d\right )^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{8 b^{\frac{13}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**(11/2)/(b**2*x**2+2*a*b*x+a**2)**2,x)
[Out]
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Mathematica [A] time = 0.599167, size = 186, normalized size = 0.93 \[ \frac{\sqrt{d+e x} \left (16 e^3 \left (150 a^2 e^2-320 a b d e+173 b^2 d^2\right )+32 b e^4 x (13 b d-10 a e)+\frac{1335 e^2 (a e-b d)^3}{a+b x}-\frac{310 e (b d-a e)^4}{(a+b x)^2}-\frac{40 (b d-a e)^5}{(a+b x)^3}+48 b^2 e^5 x^2\right )}{120 b^6}-\frac{231 e^3 (b d-a e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{13/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^(11/2)/(a^2 + 2*a*b*x + b^2*x^2)^2,x]
[Out]
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Maple [B] time = 0.031, size = 719, normalized size = 3.6 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(11/2)/(b^2*x^2+2*a*b*x+a^2)^2,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)^(11/2)/(b^2*x^2 + 2*a*b*x + a^2)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.227794, 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)^(11/2)/(b^2*x^2 + 2*a*b*x + a^2)^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((e*x+d)**(11/2)/(b**2*x**2+2*a*b*x+a**2)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.232573, size = 663, normalized size = 3.3 \[ \frac{231 \,{\left (b^{3} d^{3} e^{3} - 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} b d e^{5} - a^{3} e^{6}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{8 \, \sqrt{-b^{2} d + a b e} b^{6}} - \frac{267 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{5} d^{3} e^{3} - 472 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{5} d^{4} e^{3} + 213 \, \sqrt{x e + d} b^{5} d^{5} e^{3} - 801 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{4} d^{2} e^{4} + 1888 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{4} d^{3} e^{4} - 1065 \, \sqrt{x e + d} a b^{4} d^{4} e^{4} + 801 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{2} b^{3} d e^{5} - 2832 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{3} d^{2} e^{5} + 2130 \, \sqrt{x e + d} a^{2} b^{3} d^{3} e^{5} - 267 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{3} b^{2} e^{6} + 1888 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{3} b^{2} d e^{6} - 2130 \, \sqrt{x e + d} a^{3} b^{2} d^{2} e^{6} - 472 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{4} b e^{7} + 1065 \, \sqrt{x e + d} a^{4} b d e^{7} - 213 \, \sqrt{x e + d} a^{5} e^{8}}{24 \,{\left ({\left (x e + d\right )} b - b d + a e\right )}^{3} b^{6}} + \frac{2 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{16} e^{3} + 20 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{16} d e^{3} + 150 \, \sqrt{x e + d} b^{16} d^{2} e^{3} - 20 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{15} e^{4} - 300 \, \sqrt{x e + d} a b^{15} d e^{4} + 150 \, \sqrt{x e + d} a^{2} b^{14} e^{5}\right )}}{15 \, b^{20}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(11/2)/(b^2*x^2 + 2*a*b*x + a^2)^2,x, algorithm="giac")
[Out]