Optimal. Leaf size=265 \[ -\frac{2 (d+e x)^{7/2} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{7 e^6}+\frac{2 (d+e x)^{5/2} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{5 e^6}-\frac{2 d^2 \sqrt{d+e x} (B d-A e) (c d-b e)^2}{e^6}-\frac{2 c (d+e x)^{9/2} (-A c e-2 b B e+5 B c d)}{9 e^6}+\frac{2 d (d+e x)^{3/2} (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{3 e^6}+\frac{2 B c^2 (d+e x)^{11/2}}{11 e^6} \]
[Out]
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Rubi [A] time = 0.448912, antiderivative size = 265, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{2 (d+e x)^{7/2} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{7 e^6}+\frac{2 (d+e x)^{5/2} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{5 e^6}-\frac{2 d^2 \sqrt{d+e x} (B d-A e) (c d-b e)^2}{e^6}-\frac{2 c (d+e x)^{9/2} (-A c e-2 b B e+5 B c d)}{9 e^6}+\frac{2 d (d+e x)^{3/2} (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{3 e^6}+\frac{2 B c^2 (d+e x)^{11/2}}{11 e^6} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(b*x + c*x^2)^2)/Sqrt[d + e*x],x]
[Out]
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Rubi in Sympy [A] time = 101.697, size = 291, normalized size = 1.1 \[ \frac{2 B c^{2} \left (d + e x\right )^{\frac{11}{2}}}{11 e^{6}} + \frac{2 c \left (d + e x\right )^{\frac{9}{2}} \left (A c e + 2 B b e - 5 B c d\right )}{9 e^{6}} + \frac{2 d^{2} \sqrt{d + e x} \left (A e - B d\right ) \left (b e - c d\right )^{2}}{e^{6}} - \frac{2 d \left (d + e x\right )^{\frac{3}{2}} \left (b e - c d\right ) \left (2 A b e^{2} - 4 A c d e - 3 B b d e + 5 B c d^{2}\right )}{3 e^{6}} + \frac{2 \left (d + e x\right )^{\frac{7}{2}} \left (2 A b c e^{2} - 4 A c^{2} d e + B b^{2} e^{2} - 8 B b c d e + 10 B c^{2} d^{2}\right )}{7 e^{6}} + \frac{2 \left (d + e x\right )^{\frac{5}{2}} \left (A b^{2} e^{3} - 6 A b c d e^{2} + 6 A c^{2} d^{2} e - 3 B b^{2} d e^{2} + 12 B b c d^{2} e - 10 B c^{2} d^{3}\right )}{5 e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(c*x**2+b*x)**2/(e*x+d)**(1/2),x)
[Out]
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Mathematica [A] time = 0.393456, size = 273, normalized size = 1.03 \[ \frac{2 \sqrt{d+e x} \left (11 A e \left (21 b^2 e^2 \left (8 d^2-4 d e x+3 e^2 x^2\right )+18 b c e \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )+c^2 \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )\right )+B \left (99 b^2 e^2 \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )+22 b c e \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )-5 c^2 \left (256 d^5-128 d^4 e x+96 d^3 e^2 x^2-80 d^2 e^3 x^3+70 d e^4 x^4-63 e^5 x^5\right )\right )\right )}{3465 e^6} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(b*x + c*x^2)^2)/Sqrt[d + e*x],x]
[Out]
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Maple [A] time = 0.01, size = 341, normalized size = 1.3 \[{\frac{630\,B{c}^{2}{x}^{5}{e}^{5}+770\,A{c}^{2}{e}^{5}{x}^{4}+1540\,Bbc{e}^{5}{x}^{4}-700\,B{c}^{2}d{e}^{4}{x}^{4}+1980\,Abc{e}^{5}{x}^{3}-880\,A{c}^{2}d{e}^{4}{x}^{3}+990\,B{b}^{2}{e}^{5}{x}^{3}-1760\,Bbcd{e}^{4}{x}^{3}+800\,B{c}^{2}{d}^{2}{e}^{3}{x}^{3}+1386\,A{b}^{2}{e}^{5}{x}^{2}-2376\,Abcd{e}^{4}{x}^{2}+1056\,A{c}^{2}{d}^{2}{e}^{3}{x}^{2}-1188\,B{b}^{2}d{e}^{4}{x}^{2}+2112\,Bbc{d}^{2}{e}^{3}{x}^{2}-960\,B{c}^{2}{d}^{3}{e}^{2}{x}^{2}-1848\,A{b}^{2}d{e}^{4}x+3168\,Abc{d}^{2}{e}^{3}x-1408\,A{c}^{2}{d}^{3}{e}^{2}x+1584\,B{b}^{2}{d}^{2}{e}^{3}x-2816\,Bbc{d}^{3}{e}^{2}x+1280\,B{c}^{2}{d}^{4}ex+3696\,A{b}^{2}{d}^{2}{e}^{3}-6336\,Abc{d}^{3}{e}^{2}+2816\,A{c}^{2}{d}^{4}e-3168\,B{b}^{2}{d}^{3}{e}^{2}+5632\,Bbc{d}^{4}e-2560\,B{c}^{2}{d}^{5}}{3465\,{e}^{6}}\sqrt{ex+d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+b*x)^2/(e*x+d)^(1/2),x)
[Out]
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Maxima [A] time = 0.700892, size = 393, normalized size = 1.48 \[ \frac{2 \,{\left (315 \,{\left (e x + d\right )}^{\frac{11}{2}} B c^{2} - 385 \,{\left (5 \, B c^{2} d -{\left (2 \, B b c + A c^{2}\right )} e\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 495 \,{\left (10 \, B c^{2} d^{2} - 4 \,{\left (2 \, B b c + A c^{2}\right )} d e +{\left (B b^{2} + 2 \, A b c\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{7}{2}} - 693 \,{\left (10 \, B c^{2} d^{3} - A b^{2} e^{3} - 6 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e + 3 \,{\left (B b^{2} + 2 \, A b c\right )} d e^{2}\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 1155 \,{\left (5 \, B c^{2} d^{4} - 2 \, A b^{2} d e^{3} - 4 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e + 3 \,{\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{3}{2}} - 3465 \,{\left (B c^{2} d^{5} - A b^{2} d^{2} e^{3} -{\left (2 \, B b c + A c^{2}\right )} d^{4} e +{\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2}\right )} \sqrt{e x + d}\right )}}{3465 \, e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^2*(B*x + A)/sqrt(e*x + d),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.273866, size = 392, normalized size = 1.48 \[ \frac{2 \,{\left (315 \, B c^{2} e^{5} x^{5} - 1280 \, B c^{2} d^{5} + 1848 \, A b^{2} d^{2} e^{3} + 1408 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e - 1584 \,{\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} - 35 \,{\left (10 \, B c^{2} d e^{4} - 11 \,{\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 5 \,{\left (80 \, B c^{2} d^{2} e^{3} - 88 \,{\left (2 \, B b c + A c^{2}\right )} d e^{4} + 99 \,{\left (B b^{2} + 2 \, A b c\right )} e^{5}\right )} x^{3} - 3 \,{\left (160 \, B c^{2} d^{3} e^{2} - 231 \, A b^{2} e^{5} - 176 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 198 \,{\left (B b^{2} + 2 \, A b c\right )} d e^{4}\right )} x^{2} + 4 \,{\left (160 \, B c^{2} d^{4} e - 231 \, A b^{2} d e^{4} - 176 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 198 \,{\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x\right )} \sqrt{e x + d}}{3465 \, e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^2*(B*x + A)/sqrt(e*x + d),x, algorithm="fricas")
[Out]
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Sympy [A] time = 74.4374, size = 944, normalized size = 3.56 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(c*x**2+b*x)**2/(e*x+d)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.295763, size = 583, normalized size = 2.2 \[ \frac{2}{3465} \,{\left (231 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} e^{8} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d e^{8} + 15 \, \sqrt{x e + d} d^{2} e^{8}\right )} A b^{2} e^{\left (-10\right )} + 99 \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{18} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{18} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{18} - 35 \, \sqrt{x e + d} d^{3} e^{18}\right )} B b^{2} e^{\left (-21\right )} + 198 \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{18} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{18} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{18} - 35 \, \sqrt{x e + d} d^{3} e^{18}\right )} A b c e^{\left (-21\right )} + 22 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} e^{32} - 180 \,{\left (x e + d\right )}^{\frac{7}{2}} d e^{32} + 378 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} e^{32} - 420 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} e^{32} + 315 \, \sqrt{x e + d} d^{4} e^{32}\right )} B b c e^{\left (-36\right )} + 11 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} e^{32} - 180 \,{\left (x e + d\right )}^{\frac{7}{2}} d e^{32} + 378 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} e^{32} - 420 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} e^{32} + 315 \, \sqrt{x e + d} d^{4} e^{32}\right )} A c^{2} e^{\left (-36\right )} + 5 \,{\left (63 \,{\left (x e + d\right )}^{\frac{11}{2}} e^{50} - 385 \,{\left (x e + d\right )}^{\frac{9}{2}} d e^{50} + 990 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{2} e^{50} - 1386 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{3} e^{50} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{4} e^{50} - 693 \, \sqrt{x e + d} d^{5} e^{50}\right )} B c^{2} e^{\left (-55\right )}\right )} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^2*(B*x + A)/sqrt(e*x + d),x, algorithm="giac")
[Out]