Optimal. Leaf size=125 \[ -\frac{8 b^3 (d+e x)^{3/2} (b d-a e)}{3 e^5}+\frac{12 b^2 \sqrt{d+e x} (b d-a e)^2}{e^5}+\frac{8 b (b d-a e)^3}{e^5 \sqrt{d+e x}}-\frac{2 (b d-a e)^4}{3 e^5 (d+e x)^{3/2}}+\frac{2 b^4 (d+e x)^{5/2}}{5 e^5} \]
[Out]
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Rubi [A] time = 0.128688, antiderivative size = 125, 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 \[ -\frac{8 b^3 (d+e x)^{3/2} (b d-a e)}{3 e^5}+\frac{12 b^2 \sqrt{d+e x} (b d-a e)^2}{e^5}+\frac{8 b (b d-a e)^3}{e^5 \sqrt{d+e x}}-\frac{2 (b d-a e)^4}{3 e^5 (d+e x)^{3/2}}+\frac{2 b^4 (d+e x)^{5/2}}{5 e^5} \]
Antiderivative was successfully verified.
[In] Int[(a^2 + 2*a*b*x + b^2*x^2)^2/(d + e*x)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 53.615, size = 116, normalized size = 0.93 \[ \frac{2 b^{4} \left (d + e x\right )^{\frac{5}{2}}}{5 e^{5}} + \frac{8 b^{3} \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )}{3 e^{5}} + \frac{12 b^{2} \sqrt{d + e x} \left (a e - b d\right )^{2}}{e^{5}} - \frac{8 b \left (a e - b d\right )^{3}}{e^{5} \sqrt{d + e x}} - \frac{2 \left (a e - b d\right )^{4}}{3 e^{5} \left (d + e x\right )^{\frac{3}{2}}} \]
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)**(5/2),x)
[Out]
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Mathematica [A] time = 0.20862, size = 110, normalized size = 0.88 \[ \frac{2 \sqrt{d+e x} \left (b^2 \left (90 a^2 e^2-160 a b d e+73 b^2 d^2\right )-2 b^3 e x (7 b d-10 a e)+\frac{60 b (b d-a e)^3}{d+e x}-\frac{5 (b d-a e)^4}{(d+e x)^2}+3 b^4 e^2 x^2\right )}{15 e^5} \]
Antiderivative was successfully verified.
[In] Integrate[(a^2 + 2*a*b*x + b^2*x^2)^2/(d + e*x)^(5/2),x]
[Out]
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Maple [A] time = 0.011, size = 186, normalized size = 1.5 \[ -{\frac{-6\,{x}^{4}{b}^{4}{e}^{4}-40\,{x}^{3}a{b}^{3}{e}^{4}+16\,{x}^{3}{b}^{4}d{e}^{3}-180\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}+240\,{x}^{2}a{b}^{3}d{e}^{3}-96\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}+120\,x{a}^{3}b{e}^{4}-720\,x{a}^{2}{b}^{2}d{e}^{3}+960\,xa{b}^{3}{d}^{2}{e}^{2}-384\,x{b}^{4}{d}^{3}e+10\,{a}^{4}{e}^{4}+80\,{a}^{3}bd{e}^{3}-480\,{a}^{2}{b}^{2}{d}^{2}{e}^{2}+640\,a{b}^{3}{d}^{3}e-256\,{b}^{4}{d}^{4}}{15\,{e}^{5}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \]
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)^(5/2),x)
[Out]
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Maxima [A] time = 0.739769, size = 252, normalized size = 2.02 \[ \frac{2 \,{\left (\frac{3 \,{\left (e x + d\right )}^{\frac{5}{2}} b^{4} - 20 \,{\left (b^{4} d - a b^{3} e\right )}{\left (e x + d\right )}^{\frac{3}{2}} + 90 \,{\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )} \sqrt{e x + d}}{e^{4}} - \frac{5 \,{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4} - 12 \,{\left (b^{4} d^{3} - 3 \, a b^{3} d^{2} e + 3 \, a^{2} b^{2} d e^{2} - a^{3} b e^{3}\right )}{\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac{3}{2}} e^{4}}\right )}}{15 \, e} \]
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)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.205044, size = 259, normalized size = 2.07 \[ \frac{2 \,{\left (3 \, b^{4} e^{4} x^{4} + 128 \, b^{4} d^{4} - 320 \, a b^{3} d^{3} e + 240 \, a^{2} b^{2} d^{2} e^{2} - 40 \, a^{3} b d e^{3} - 5 \, a^{4} e^{4} - 4 \,{\left (2 \, b^{4} d e^{3} - 5 \, a b^{3} e^{4}\right )} x^{3} + 6 \,{\left (8 \, b^{4} d^{2} e^{2} - 20 \, a b^{3} d e^{3} + 15 \, a^{2} b^{2} e^{4}\right )} x^{2} + 12 \,{\left (16 \, b^{4} d^{3} e - 40 \, a b^{3} d^{2} e^{2} + 30 \, a^{2} b^{2} d e^{3} - 5 \, a^{3} b e^{4}\right )} x\right )}}{15 \,{\left (e^{6} x + d e^{5}\right )} \sqrt{e x + d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^2/(e*x + d)^(5/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x\right )^{4}}{\left (d + e x\right )^{\frac{5}{2}}}\, dx \]
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)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.213063, size = 309, normalized size = 2.47 \[ \frac{2}{15} \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{4} e^{20} - 20 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{4} d e^{20} + 90 \, \sqrt{x e + d} b^{4} d^{2} e^{20} + 20 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{3} e^{21} - 180 \, \sqrt{x e + d} a b^{3} d e^{21} + 90 \, \sqrt{x e + d} a^{2} b^{2} e^{22}\right )} e^{\left (-25\right )} + \frac{2 \,{\left (12 \,{\left (x e + d\right )} b^{4} d^{3} - b^{4} d^{4} - 36 \,{\left (x e + d\right )} a b^{3} d^{2} e + 4 \, a b^{3} d^{3} e + 36 \,{\left (x e + d\right )} a^{2} b^{2} d e^{2} - 6 \, a^{2} b^{2} d^{2} e^{2} - 12 \,{\left (x e + d\right )} a^{3} b e^{3} + 4 \, a^{3} b d e^{3} - a^{4} e^{4}\right )} e^{\left (-5\right )}}{3 \,{\left (x e + d\right )}^{\frac{3}{2}}} \]
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)^(5/2),x, algorithm="giac")
[Out]