Optimal. Leaf size=208 \[ \frac{6 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^4 (a+b x) (d+e x)^{5/2}}-\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{7 e^4 (a+b x) (d+e x)^{7/2}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{9 e^4 (a+b x) (d+e x)^{9/2}}-\frac{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^4 (a+b x) (d+e x)^{3/2}} \]
[Out]
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Rubi [A] time = 0.204643, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{6 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^4 (a+b x) (d+e x)^{5/2}}-\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{7 e^4 (a+b x) (d+e x)^{7/2}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{9 e^4 (a+b x) (d+e x)^{9/2}}-\frac{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^4 (a+b x) (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/(d + e*x)^(11/2),x]
[Out]
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Rubi in Sympy [A] time = 22.3622, size = 168, normalized size = 0.81 \[ - \frac{16 b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{63 e^{3} \left (d + e x\right )^{\frac{5}{2}}} + \frac{32 b^{2} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{315 e^{4} \left (a + b x\right ) \left (d + e x\right )^{\frac{5}{2}}} - \frac{4 b \left (3 a + 3 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{63 e^{2} \left (d + e x\right )^{\frac{7}{2}}} - \frac{2 \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{9 e \left (d + e x\right )^{\frac{9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b**2*x**2+2*a*b*x+a**2)**(3/2)/(e*x+d)**(11/2),x)
[Out]
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Mathematica [A] time = 0.208277, size = 97, normalized size = 0.47 \[ \frac{2 \left ((a+b x)^2\right )^{3/2} \left (189 b^2 (d+e x)^2 (b d-a e)-135 b (d+e x) (b d-a e)^2+35 (b d-a e)^3-105 b^3 (d+e x)^3\right )}{315 e^4 (a+b x)^3 (d+e x)^{9/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/(d + e*x)^(11/2),x]
[Out]
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Maple [A] time = 0.01, size = 132, normalized size = 0.6 \[ -{\frac{210\,{x}^{3}{b}^{3}{e}^{3}+378\,{x}^{2}a{b}^{2}{e}^{3}+252\,{x}^{2}{b}^{3}d{e}^{2}+270\,x{a}^{2}b{e}^{3}+216\,xa{b}^{2}d{e}^{2}+144\,x{b}^{3}{d}^{2}e+70\,{a}^{3}{e}^{3}+60\,{a}^{2}bd{e}^{2}+48\,a{b}^{2}{d}^{2}e+32\,{b}^{3}{d}^{3}}{315\, \left ( bx+a \right ) ^{3}{e}^{4}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{9}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^(11/2),x)
[Out]
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Maxima [A] time = 0.743124, size = 215, normalized size = 1.03 \[ -\frac{2 \,{\left (105 \, b^{3} e^{3} x^{3} + 16 \, b^{3} d^{3} + 24 \, a b^{2} d^{2} e + 30 \, a^{2} b d e^{2} + 35 \, a^{3} e^{3} + 63 \,{\left (2 \, b^{3} d e^{2} + 3 \, a b^{2} e^{3}\right )} x^{2} + 9 \,{\left (8 \, b^{3} d^{2} e + 12 \, a b^{2} d e^{2} + 15 \, a^{2} b e^{3}\right )} x\right )}}{315 \,{\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )} \sqrt{e x + d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)/(e*x + d)^(11/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.20792, size = 215, normalized size = 1.03 \[ -\frac{2 \,{\left (105 \, b^{3} e^{3} x^{3} + 16 \, b^{3} d^{3} + 24 \, a b^{2} d^{2} e + 30 \, a^{2} b d e^{2} + 35 \, a^{3} e^{3} + 63 \,{\left (2 \, b^{3} d e^{2} + 3 \, a b^{2} e^{3}\right )} x^{2} + 9 \,{\left (8 \, b^{3} d^{2} e + 12 \, a b^{2} d e^{2} + 15 \, a^{2} b e^{3}\right )} x\right )}}{315 \,{\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )} \sqrt{e x + d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)/(e*x + d)^(11/2),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((b**2*x**2+2*a*b*x+a**2)**(3/2)/(e*x+d)**(11/2),x)
[Out]
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GIAC/XCAS [A] time = 0.220727, size = 262, normalized size = 1.26 \[ -\frac{2 \,{\left (105 \,{\left (x e + d\right )}^{3} b^{3}{\rm sign}\left (b x + a\right ) - 189 \,{\left (x e + d\right )}^{2} b^{3} d{\rm sign}\left (b x + a\right ) + 135 \,{\left (x e + d\right )} b^{3} d^{2}{\rm sign}\left (b x + a\right ) - 35 \, b^{3} d^{3}{\rm sign}\left (b x + a\right ) + 189 \,{\left (x e + d\right )}^{2} a b^{2} e{\rm sign}\left (b x + a\right ) - 270 \,{\left (x e + d\right )} a b^{2} d e{\rm sign}\left (b x + a\right ) + 105 \, a b^{2} d^{2} e{\rm sign}\left (b x + a\right ) + 135 \,{\left (x e + d\right )} a^{2} b e^{2}{\rm sign}\left (b x + a\right ) - 105 \, a^{2} b d e^{2}{\rm sign}\left (b x + a\right ) + 35 \, a^{3} e^{3}{\rm sign}\left (b x + a\right )\right )} e^{\left (-4\right )}}{315 \,{\left (x e + d\right )}^{\frac{9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)/(e*x + d)^(11/2),x, algorithm="giac")
[Out]