Optimal. Leaf size=164 \[ \frac{2 (d+e x)^{5/2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{5 e^5}-\frac{4 (d+e x)^{3/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^5}+\frac{2 \sqrt{d+e x} \left (a e^2-b d e+c d^2\right )^2}{e^5}-\frac{4 c (d+e x)^{7/2} (2 c d-b e)}{7 e^5}+\frac{2 c^2 (d+e x)^{9/2}}{9 e^5} \]
[Out]
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Rubi [A] time = 0.200553, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{2 (d+e x)^{5/2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{5 e^5}-\frac{4 (d+e x)^{3/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^5}+\frac{2 \sqrt{d+e x} \left (a e^2-b d e+c d^2\right )^2}{e^5}-\frac{4 c (d+e x)^{7/2} (2 c d-b e)}{7 e^5}+\frac{2 c^2 (d+e x)^{9/2}}{9 e^5} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x + c*x^2)^2/Sqrt[d + e*x],x]
[Out]
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Rubi in Sympy [A] time = 40.6301, size = 160, normalized size = 0.98 \[ \frac{2 c^{2} \left (d + e x\right )^{\frac{9}{2}}}{9 e^{5}} + \frac{4 c \left (d + e x\right )^{\frac{7}{2}} \left (b e - 2 c d\right )}{7 e^{5}} + \frac{2 \left (d + e x\right )^{\frac{5}{2}} \left (2 a c e^{2} + b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right )}{5 e^{5}} + \frac{4 \left (d + e x\right )^{\frac{3}{2}} \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )}{3 e^{5}} + \frac{2 \sqrt{d + e x} \left (a e^{2} - b d e + c d^{2}\right )^{2}}{e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x+a)**2/(e*x+d)**(1/2),x)
[Out]
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Mathematica [A] time = 0.160187, size = 172, normalized size = 1.05 \[ \frac{2 \sqrt{d+e x} \left (21 e^2 \left (15 a^2 e^2+10 a b e (e x-2 d)+b^2 \left (8 d^2-4 d e x+3 e^2 x^2\right )\right )-6 c e \left (3 b \left (16 d^3-8 d^2 e x+6 d e^2 x^2-5 e^3 x^3\right )-7 a e \left (8 d^2-4 d e x+3 e^2 x^2\right )\right )+c^2 \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )\right )}{315 e^5} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x + c*x^2)^2/Sqrt[d + e*x],x]
[Out]
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Maple [A] time = 0.009, size = 194, normalized size = 1.2 \[{\frac{70\,{x}^{4}{c}^{2}{e}^{4}+180\,bc{e}^{4}{x}^{3}-80\,{x}^{3}{c}^{2}d{e}^{3}+252\,{x}^{2}ac{e}^{4}+126\,{b}^{2}{e}^{4}{x}^{2}-216\,bcd{e}^{3}{x}^{2}+96\,{x}^{2}{c}^{2}{d}^{2}{e}^{2}+420\,ab{e}^{4}x-336\,xacd{e}^{3}-168\,{b}^{2}d{e}^{3}x+288\,bc{d}^{2}{e}^{2}x-128\,x{c}^{2}{d}^{3}e+630\,{a}^{2}{e}^{4}-840\,abd{e}^{3}+672\,ac{d}^{2}{e}^{2}+336\,{b}^{2}{d}^{2}{e}^{2}-576\,bc{d}^{3}e+256\,{c}^{2}{d}^{4}}{315\,{e}^{5}}\sqrt{ex+d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x+a)^2/(e*x+d)^(1/2),x)
[Out]
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Maxima [A] time = 0.70578, size = 320, normalized size = 1.95 \[ \frac{2 \,{\left (315 \, \sqrt{e x + d} a^{2} + 42 \, a{\left (\frac{5 \,{\left ({\left (e x + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{e x + d} d\right )} b}{e} + \frac{{\left (3 \,{\left (e x + d\right )}^{\frac{5}{2}} - 10 \,{\left (e x + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{e x + d} d^{2}\right )} c}{e^{2}}\right )} + \frac{21 \,{\left (3 \,{\left (e x + d\right )}^{\frac{5}{2}} - 10 \,{\left (e x + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{e x + d} d^{2}\right )} b^{2}}{e^{2}} + \frac{18 \,{\left (5 \,{\left (e x + d\right )}^{\frac{7}{2}} - 21 \,{\left (e x + d\right )}^{\frac{5}{2}} d + 35 \,{\left (e x + d\right )}^{\frac{3}{2}} d^{2} - 35 \, \sqrt{e x + d} d^{3}\right )} b c}{e^{3}} + \frac{{\left (35 \,{\left (e x + d\right )}^{\frac{9}{2}} - 180 \,{\left (e x + d\right )}^{\frac{7}{2}} d + 378 \,{\left (e x + d\right )}^{\frac{5}{2}} d^{2} - 420 \,{\left (e x + d\right )}^{\frac{3}{2}} d^{3} + 315 \, \sqrt{e x + d} d^{4}\right )} c^{2}}{e^{4}}\right )}}{315 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2/sqrt(e*x + d),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.213025, size = 238, normalized size = 1.45 \[ \frac{2 \,{\left (35 \, c^{2} e^{4} x^{4} + 128 \, c^{2} d^{4} - 288 \, b c d^{3} e - 420 \, a b d e^{3} + 315 \, a^{2} e^{4} + 168 \,{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} - 10 \,{\left (4 \, c^{2} d e^{3} - 9 \, b c e^{4}\right )} x^{3} + 3 \,{\left (16 \, c^{2} d^{2} e^{2} - 36 \, b c d e^{3} + 21 \,{\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} - 2 \,{\left (32 \, c^{2} d^{3} e - 72 \, b c d^{2} e^{2} - 105 \, a b e^{4} + 42 \,{\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x\right )} \sqrt{e x + d}}{315 \, e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2/sqrt(e*x + d),x, algorithm="fricas")
[Out]
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Sympy [A] time = 41.9127, size = 644, normalized size = 3.93 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x+a)**2/(e*x+d)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.206972, size = 375, normalized size = 2.29 \[ \frac{2}{315} \,{\left (210 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} a b e^{\left (-1\right )} + 21 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} e^{8} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d e^{8} + 15 \, \sqrt{x e + d} d^{2} e^{8}\right )} b^{2} e^{\left (-10\right )} + 42 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} e^{8} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d e^{8} + 15 \, \sqrt{x e + d} d^{2} e^{8}\right )} a c e^{\left (-10\right )} + 18 \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{18} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{18} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{18} - 35 \, \sqrt{x e + d} d^{3} e^{18}\right )} b c e^{\left (-21\right )} +{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} e^{32} - 180 \,{\left (x e + d\right )}^{\frac{7}{2}} d e^{32} + 378 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} e^{32} - 420 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} e^{32} + 315 \, \sqrt{x e + d} d^{4} e^{32}\right )} c^{2} e^{\left (-36\right )} + 315 \, \sqrt{x e + d} a^{2}\right )} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2/sqrt(e*x + d),x, algorithm="giac")
[Out]