Optimal. Leaf size=240 \[ \frac{2 e (f+g x)^{7/2} \left (a e^2 g^2+c \left (3 d^2 g^2-12 d e f g+10 e^2 f^2\right )\right )}{7 g^6}-\frac{2 (f+g x)^{5/2} (e f-d g) \left (3 a e^2 g^2+c \left (d^2 g^2-8 d e f g+10 e^2 f^2\right )\right )}{5 g^6}-\frac{2 \sqrt{f+g x} \left (a g^2+c f^2\right ) (e f-d g)^3}{g^6}+\frac{2 (f+g x)^{3/2} (e f-d g)^2 \left (3 a e g^2+c f (5 e f-2 d g)\right )}{3 g^6}-\frac{2 c e^2 (f+g x)^{9/2} (5 e f-3 d g)}{9 g^6}+\frac{2 c e^3 (f+g x)^{11/2}}{11 g^6} \]
[Out]
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Rubi [A] time = 0.683848, antiderivative size = 240, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{2 e (f+g x)^{7/2} \left (a e^2 g^2+c \left (3 d^2 g^2-12 d e f g+10 e^2 f^2\right )\right )}{7 g^6}-\frac{2 (f+g x)^{5/2} (e f-d g) \left (3 a e^2 g^2+c \left (d^2 g^2-8 d e f g+10 e^2 f^2\right )\right )}{5 g^6}-\frac{2 \sqrt{f+g x} \left (a g^2+c f^2\right ) (e f-d g)^3}{g^6}+\frac{2 (f+g x)^{3/2} (e f-d g)^2 \left (3 a e g^2+c f (5 e f-2 d g)\right )}{3 g^6}-\frac{2 c e^2 (f+g x)^{9/2} (5 e f-3 d g)}{9 g^6}+\frac{2 c e^3 (f+g x)^{11/2}}{11 g^6} \]
Antiderivative was successfully verified.
[In] Int[((d + e*x)^3*(a + c*x^2))/Sqrt[f + g*x],x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{2 c e^{3} \left (f + g x\right )^{\frac{11}{2}}}{11 g^{6}} + \frac{2 c e^{2} \left (f + g x\right )^{\frac{9}{2}} \left (3 d g - 5 e f\right )}{9 g^{6}} + \frac{2 e \left (f + g x\right )^{\frac{7}{2}} \left (a e^{2} g^{2} + 3 c d^{2} g^{2} - 12 c d e f g + 10 c e^{2} f^{2}\right )}{7 g^{6}} + \frac{2 \left (a g^{2} + c f^{2}\right ) \left (d g - e f\right )^{3} \int ^{\sqrt{f + g x}} \frac{1}{g^{5}}\, dx}{g} + \frac{2 \left (f + g x\right )^{\frac{5}{2}} \left (d g - e f\right ) \left (3 a e^{2} g^{2} + c d^{2} g^{2} - 8 c d e f g + 10 c e^{2} f^{2}\right )}{5 g^{6}} + \frac{2 \left (f + g x\right )^{\frac{3}{2}} \left (d g - e f\right )^{2} \left (3 a e g^{2} - 2 c d f g + 5 c e f^{2}\right )}{3 g^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**3*(c*x**2+a)/(g*x+f)**(1/2),x)
[Out]
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Mathematica [A] time = 0.485726, size = 282, normalized size = 1.18 \[ \frac{2 \sqrt{f+g x} \left (99 a g^2 \left (35 d^3 g^3+35 d^2 e g^2 (g x-2 f)+7 d e^2 g \left (8 f^2-4 f g x+3 g^2 x^2\right )+e^3 \left (-16 f^3+8 f^2 g x-6 f g^2 x^2+5 g^3 x^3\right )\right )+c \left (231 d^3 g^3 \left (8 f^2-4 f g x+3 g^2 x^2\right )+297 d^2 e g^2 \left (-16 f^3+8 f^2 g x-6 f g^2 x^2+5 g^3 x^3\right )+33 d e^2 g \left (128 f^4-64 f^3 g x+48 f^2 g^2 x^2-40 f g^3 x^3+35 g^4 x^4\right )-5 e^3 \left (256 f^5-128 f^4 g x+96 f^3 g^2 x^2-80 f^2 g^3 x^3+70 f g^4 x^4-63 g^5 x^5\right )\right )\right )}{3465 g^6} \]
Antiderivative was successfully verified.
[In] Integrate[((d + e*x)^3*(a + c*x^2))/Sqrt[f + g*x],x]
[Out]
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Maple [A] time = 0.01, size = 365, normalized size = 1.5 \[{\frac{630\,{e}^{3}c{x}^{5}{g}^{5}+2310\,cd{e}^{2}{g}^{5}{x}^{4}-700\,c{e}^{3}f{g}^{4}{x}^{4}+990\,a{e}^{3}{g}^{5}{x}^{3}+2970\,c{d}^{2}e{g}^{5}{x}^{3}-2640\,cd{e}^{2}f{g}^{4}{x}^{3}+800\,c{e}^{3}{f}^{2}{g}^{3}{x}^{3}+4158\,ad{e}^{2}{g}^{5}{x}^{2}-1188\,a{e}^{3}f{g}^{4}{x}^{2}+1386\,c{d}^{3}{g}^{5}{x}^{2}-3564\,c{d}^{2}ef{g}^{4}{x}^{2}+3168\,cd{e}^{2}{f}^{2}{g}^{3}{x}^{2}-960\,c{e}^{3}{f}^{3}{g}^{2}{x}^{2}+6930\,a{d}^{2}e{g}^{5}x-5544\,ad{e}^{2}f{g}^{4}x+1584\,a{e}^{3}{f}^{2}{g}^{3}x-1848\,c{d}^{3}f{g}^{4}x+4752\,c{d}^{2}e{f}^{2}{g}^{3}x-4224\,cd{e}^{2}{f}^{3}{g}^{2}x+1280\,c{e}^{3}{f}^{4}gx+6930\,{d}^{3}a{g}^{5}-13860\,a{d}^{2}ef{g}^{4}+11088\,ad{e}^{2}{f}^{2}{g}^{3}-3168\,a{e}^{3}{f}^{3}{g}^{2}+3696\,c{d}^{3}{f}^{2}{g}^{3}-9504\,c{d}^{2}e{f}^{3}{g}^{2}+8448\,cd{e}^{2}{f}^{4}g-2560\,c{e}^{3}{f}^{5}}{3465\,{g}^{6}}\sqrt{gx+f}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^3*(c*x^2+a)/(g*x+f)^(1/2),x)
[Out]
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Maxima [A] time = 0.700273, size = 440, normalized size = 1.83 \[ \frac{2 \,{\left (315 \,{\left (g x + f\right )}^{\frac{11}{2}} c e^{3} - 385 \,{\left (5 \, c e^{3} f - 3 \, c d e^{2} g\right )}{\left (g x + f\right )}^{\frac{9}{2}} + 495 \,{\left (10 \, c e^{3} f^{2} - 12 \, c d e^{2} f g +{\left (3 \, c d^{2} e + a e^{3}\right )} g^{2}\right )}{\left (g x + f\right )}^{\frac{7}{2}} - 693 \,{\left (10 \, c e^{3} f^{3} - 18 \, c d e^{2} f^{2} g + 3 \,{\left (3 \, c d^{2} e + a e^{3}\right )} f g^{2} -{\left (c d^{3} + 3 \, a d e^{2}\right )} g^{3}\right )}{\left (g x + f\right )}^{\frac{5}{2}} + 1155 \,{\left (5 \, c e^{3} f^{4} - 12 \, c d e^{2} f^{3} g + 3 \, a d^{2} e g^{4} + 3 \,{\left (3 \, c d^{2} e + a e^{3}\right )} f^{2} g^{2} - 2 \,{\left (c d^{3} + 3 \, a d e^{2}\right )} f g^{3}\right )}{\left (g x + f\right )}^{\frac{3}{2}} - 3465 \,{\left (c e^{3} f^{5} - 3 \, c d e^{2} f^{4} g + 3 \, a d^{2} e f g^{4} - a d^{3} g^{5} +{\left (3 \, c d^{2} e + a e^{3}\right )} f^{3} g^{2} -{\left (c d^{3} + 3 \, a d e^{2}\right )} f^{2} g^{3}\right )} \sqrt{g x + f}\right )}}{3465 \, g^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)*(e*x + d)^3/sqrt(g*x + f),x, algorithm="maxima")
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Fricas [A] time = 0.276816, size = 437, normalized size = 1.82 \[ \frac{2 \,{\left (315 \, c e^{3} g^{5} x^{5} - 1280 \, c e^{3} f^{5} + 4224 \, c d e^{2} f^{4} g - 6930 \, a d^{2} e f g^{4} + 3465 \, a d^{3} g^{5} - 1584 \,{\left (3 \, c d^{2} e + a e^{3}\right )} f^{3} g^{2} + 1848 \,{\left (c d^{3} + 3 \, a d e^{2}\right )} f^{2} g^{3} - 35 \,{\left (10 \, c e^{3} f g^{4} - 33 \, c d e^{2} g^{5}\right )} x^{4} + 5 \,{\left (80 \, c e^{3} f^{2} g^{3} - 264 \, c d e^{2} f g^{4} + 99 \,{\left (3 \, c d^{2} e + a e^{3}\right )} g^{5}\right )} x^{3} - 3 \,{\left (160 \, c e^{3} f^{3} g^{2} - 528 \, c d e^{2} f^{2} g^{3} + 198 \,{\left (3 \, c d^{2} e + a e^{3}\right )} f g^{4} - 231 \,{\left (c d^{3} + 3 \, a d e^{2}\right )} g^{5}\right )} x^{2} +{\left (640 \, c e^{3} f^{4} g - 2112 \, c d e^{2} f^{3} g^{2} + 3465 \, a d^{2} e g^{5} + 792 \,{\left (3 \, c d^{2} e + a e^{3}\right )} f^{2} g^{3} - 924 \,{\left (c d^{3} + 3 \, a d e^{2}\right )} f g^{4}\right )} x\right )} \sqrt{g x + f}}{3465 \, g^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)*(e*x + d)^3/sqrt(g*x + f),x, algorithm="fricas")
[Out]
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Sympy [A] time = 76.1024, size = 1040, normalized size = 4.33 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**3*(c*x**2+a)/(g*x+f)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.270841, size = 612, normalized size = 2.55 \[ \frac{2 \,{\left (3465 \, \sqrt{g x + f} a d^{3} + \frac{3465 \,{\left ({\left (g x + f\right )}^{\frac{3}{2}} - 3 \, \sqrt{g x + f} f\right )} a d^{2} e}{g} + \frac{231 \,{\left (3 \,{\left (g x + f\right )}^{\frac{5}{2}} g^{8} - 10 \,{\left (g x + f\right )}^{\frac{3}{2}} f g^{8} + 15 \, \sqrt{g x + f} f^{2} g^{8}\right )} c d^{3}}{g^{10}} + \frac{693 \,{\left (3 \,{\left (g x + f\right )}^{\frac{5}{2}} g^{8} - 10 \,{\left (g x + f\right )}^{\frac{3}{2}} f g^{8} + 15 \, \sqrt{g x + f} f^{2} g^{8}\right )} a d e^{2}}{g^{10}} + \frac{297 \,{\left (5 \,{\left (g x + f\right )}^{\frac{7}{2}} g^{18} - 21 \,{\left (g x + f\right )}^{\frac{5}{2}} f g^{18} + 35 \,{\left (g x + f\right )}^{\frac{3}{2}} f^{2} g^{18} - 35 \, \sqrt{g x + f} f^{3} g^{18}\right )} c d^{2} e}{g^{21}} + \frac{99 \,{\left (5 \,{\left (g x + f\right )}^{\frac{7}{2}} g^{18} - 21 \,{\left (g x + f\right )}^{\frac{5}{2}} f g^{18} + 35 \,{\left (g x + f\right )}^{\frac{3}{2}} f^{2} g^{18} - 35 \, \sqrt{g x + f} f^{3} g^{18}\right )} a e^{3}}{g^{21}} + \frac{33 \,{\left (35 \,{\left (g x + f\right )}^{\frac{9}{2}} g^{32} - 180 \,{\left (g x + f\right )}^{\frac{7}{2}} f g^{32} + 378 \,{\left (g x + f\right )}^{\frac{5}{2}} f^{2} g^{32} - 420 \,{\left (g x + f\right )}^{\frac{3}{2}} f^{3} g^{32} + 315 \, \sqrt{g x + f} f^{4} g^{32}\right )} c d e^{2}}{g^{36}} + \frac{5 \,{\left (63 \,{\left (g x + f\right )}^{\frac{11}{2}} g^{50} - 385 \,{\left (g x + f\right )}^{\frac{9}{2}} f g^{50} + 990 \,{\left (g x + f\right )}^{\frac{7}{2}} f^{2} g^{50} - 1386 \,{\left (g x + f\right )}^{\frac{5}{2}} f^{3} g^{50} + 1155 \,{\left (g x + f\right )}^{\frac{3}{2}} f^{4} g^{50} - 693 \, \sqrt{g x + f} f^{5} g^{50}\right )} c e^{3}}{g^{55}}\right )}}{3465 \, g} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)*(e*x + d)^3/sqrt(g*x + f),x, algorithm="giac")
[Out]