Optimal. Leaf size=127 \[ -\frac{8 b^3 (d+e x)^{7/2} (b d-a e)}{7 e^5}+\frac{12 b^2 (d+e x)^{5/2} (b d-a e)^2}{5 e^5}-\frac{8 b (d+e x)^{3/2} (b d-a e)^3}{3 e^5}+\frac{2 \sqrt{d+e x} (b d-a e)^4}{e^5}+\frac{2 b^4 (d+e x)^{9/2}}{9 e^5} \]
[Out]
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Rubi [A] time = 0.13037, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ -\frac{8 b^3 (d+e x)^{7/2} (b d-a e)}{7 e^5}+\frac{12 b^2 (d+e x)^{5/2} (b d-a e)^2}{5 e^5}-\frac{8 b (d+e x)^{3/2} (b d-a e)^3}{3 e^5}+\frac{2 \sqrt{d+e x} (b d-a e)^4}{e^5}+\frac{2 b^4 (d+e x)^{9/2}}{9 e^5} \]
Antiderivative was successfully verified.
[In] Int[(a^2 + 2*a*b*x + b^2*x^2)^2/Sqrt[d + e*x],x]
[Out]
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Rubi in Sympy [A] time = 54.5858, size = 117, normalized size = 0.92 \[ \frac{2 b^{4} \left (d + e x\right )^{\frac{9}{2}}}{9 e^{5}} + \frac{8 b^{3} \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )}{7 e^{5}} + \frac{12 b^{2} \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )^{2}}{5 e^{5}} + \frac{8 b \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{3}}{3 e^{5}} + \frac{2 \sqrt{d + e x} \left (a e - b d\right )^{4}}{e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b**2*x**2+2*a*b*x+a**2)**2/(e*x+d)**(1/2),x)
[Out]
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Mathematica [A] time = 0.121315, size = 153, normalized size = 1.2 \[ \frac{2 \sqrt{d+e x} \left (315 a^4 e^4+420 a^3 b e^3 (e x-2 d)+126 a^2 b^2 e^2 \left (8 d^2-4 d e x+3 e^2 x^2\right )+36 a b^3 e \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )+b^4 \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )\right )}{315 e^5} \]
Antiderivative was successfully verified.
[In] Integrate[(a^2 + 2*a*b*x + b^2*x^2)^2/Sqrt[d + e*x],x]
[Out]
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Maple [A] time = 0.011, size = 186, normalized size = 1.5 \[{\frac{70\,{x}^{4}{b}^{4}{e}^{4}+360\,{x}^{3}a{b}^{3}{e}^{4}-80\,{x}^{3}{b}^{4}d{e}^{3}+756\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}-432\,{x}^{2}a{b}^{3}d{e}^{3}+96\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}+840\,x{a}^{3}b{e}^{4}-1008\,x{a}^{2}{b}^{2}d{e}^{3}+576\,xa{b}^{3}{d}^{2}{e}^{2}-128\,x{b}^{4}{d}^{3}e+630\,{a}^{4}{e}^{4}-1680\,{a}^{3}bd{e}^{3}+2016\,{a}^{2}{b}^{2}{d}^{2}{e}^{2}-1152\,a{b}^{3}{d}^{3}e+256\,{b}^{4}{d}^{4}}{315\,{e}^{5}}\sqrt{ex+d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^(1/2),x)
[Out]
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Maxima [A] time = 0.740799, size = 333, normalized size = 2.62 \[ \frac{2 \,{\left (315 \, \sqrt{e x + d} a^{4} + 42 \,{\left (\frac{10 \,{\left ({\left (e x + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{e x + d} d\right )} a b}{e} + \frac{{\left (3 \,{\left (e x + d\right )}^{\frac{5}{2}} - 10 \,{\left (e x + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{e x + d} d^{2}\right )} b^{2}}{e^{2}}\right )} a^{2} + \frac{84 \,{\left (3 \,{\left (e x + d\right )}^{\frac{5}{2}} - 10 \,{\left (e x + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{e x + d} d^{2}\right )} a^{2} b^{2}}{e^{2}} + \frac{36 \,{\left (5 \,{\left (e x + d\right )}^{\frac{7}{2}} - 21 \,{\left (e x + d\right )}^{\frac{5}{2}} d + 35 \,{\left (e x + d\right )}^{\frac{3}{2}} d^{2} - 35 \, \sqrt{e x + d} d^{3}\right )} a b^{3}}{e^{3}} + \frac{{\left (35 \,{\left (e x + d\right )}^{\frac{9}{2}} - 180 \,{\left (e x + d\right )}^{\frac{7}{2}} d + 378 \,{\left (e x + d\right )}^{\frac{5}{2}} d^{2} - 420 \,{\left (e x + d\right )}^{\frac{3}{2}} d^{3} + 315 \, \sqrt{e x + d} d^{4}\right )} b^{4}}{e^{4}}\right )}}{315 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^2/sqrt(e*x + d),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.204266, size = 246, normalized size = 1.94 \[ \frac{2 \,{\left (35 \, b^{4} e^{4} x^{4} + 128 \, b^{4} d^{4} - 576 \, a b^{3} d^{3} e + 1008 \, a^{2} b^{2} d^{2} e^{2} - 840 \, a^{3} b d e^{3} + 315 \, a^{4} e^{4} - 20 \,{\left (2 \, b^{4} d e^{3} - 9 \, a b^{3} e^{4}\right )} x^{3} + 6 \,{\left (8 \, b^{4} d^{2} e^{2} - 36 \, a b^{3} d e^{3} + 63 \, a^{2} b^{2} e^{4}\right )} x^{2} - 4 \,{\left (16 \, b^{4} d^{3} e - 72 \, a b^{3} d^{2} e^{2} + 126 \, a^{2} b^{2} d e^{3} - 105 \, a^{3} b e^{4}\right )} x\right )} \sqrt{e x + d}}{315 \, e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^2/sqrt(e*x + d),x, algorithm="fricas")
[Out]
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Sympy [A] time = 36.1641, size = 561, normalized size = 4.42 \[ \begin{cases} - \frac{\frac{2 a^{4} d}{\sqrt{d + e x}} + 2 a^{4} \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right ) + \frac{8 a^{3} b d \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right )}{e} + \frac{8 a^{3} b \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e} + \frac{12 a^{2} b^{2} d \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e^{2}} + \frac{12 a^{2} b^{2} \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{2}} + \frac{8 a b^{3} d \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{3}} + \frac{8 a b^{3} \left (\frac{d^{4}}{\sqrt{d + e x}} + 4 d^{3} \sqrt{d + e x} - 2 d^{2} \left (d + e x\right )^{\frac{3}{2}} + \frac{4 d \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{3}} + \frac{2 b^{4} d \left (\frac{d^{4}}{\sqrt{d + e x}} + 4 d^{3} \sqrt{d + e x} - 2 d^{2} \left (d + e x\right )^{\frac{3}{2}} + \frac{4 d \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{4}} + \frac{2 b^{4} \left (- \frac{d^{5}}{\sqrt{d + e x}} - 5 d^{4} \sqrt{d + e x} + \frac{10 d^{3} \left (d + e x\right )^{\frac{3}{2}}}{3} - 2 d^{2} \left (d + e x\right )^{\frac{5}{2}} + \frac{5 d \left (d + e x\right )^{\frac{7}{2}}}{7} - \frac{\left (d + e x\right )^{\frac{9}{2}}}{9}\right )}{e^{4}}}{e} & \text{for}\: e \neq 0 \\\frac{a^{4} x + 2 a^{3} b x^{2} + 2 a^{2} b^{2} x^{3} + a b^{3} x^{4} + \frac{b^{4} x^{5}}{5}}{\sqrt{d}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b**2*x**2+2*a*b*x+a**2)**2/(e*x+d)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.218487, size = 321, normalized size = 2.53 \[ \frac{2}{315} \,{\left (420 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} a^{3} b e^{\left (-1\right )} + 126 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} e^{8} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d e^{8} + 15 \, \sqrt{x e + d} d^{2} e^{8}\right )} a^{2} b^{2} e^{\left (-10\right )} + 36 \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{18} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{18} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{18} - 35 \, \sqrt{x e + d} d^{3} e^{18}\right )} a b^{3} e^{\left (-21\right )} +{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} e^{32} - 180 \,{\left (x e + d\right )}^{\frac{7}{2}} d e^{32} + 378 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} e^{32} - 420 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} e^{32} + 315 \, \sqrt{x e + d} d^{4} e^{32}\right )} b^{4} e^{\left (-36\right )} + 315 \, \sqrt{x e + d} a^{4}\right )} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^2/sqrt(e*x + d),x, algorithm="giac")
[Out]