Optimal. Leaf size=216 \[ \frac{4 c (d+e x)^{7/2} \left (a B e^2-2 A c d e+5 B c d^2\right )}{7 e^6}+\frac{2 (d+e x)^{3/2} \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{3 e^6}-\frac{2 \sqrt{d+e x} \left (a e^2+c d^2\right )^2 (B d-A e)}{e^6}-\frac{4 c (d+e x)^{5/2} \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{5 e^6}-\frac{2 c^2 (d+e x)^{9/2} (5 B d-A e)}{9 e^6}+\frac{2 B c^2 (d+e x)^{11/2}}{11 e^6} \]
[Out]
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Rubi [A] time = 0.263366, antiderivative size = 216, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ \frac{4 c (d+e x)^{7/2} \left (a B e^2-2 A c d e+5 B c d^2\right )}{7 e^6}+\frac{2 (d+e x)^{3/2} \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{3 e^6}-\frac{2 \sqrt{d+e x} \left (a e^2+c d^2\right )^2 (B d-A e)}{e^6}-\frac{4 c (d+e x)^{5/2} \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{5 e^6}-\frac{2 c^2 (d+e x)^{9/2} (5 B d-A e)}{9 e^6}+\frac{2 B c^2 (d+e x)^{11/2}}{11 e^6} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(a + c*x^2)^2)/Sqrt[d + e*x],x]
[Out]
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Rubi in Sympy [A] time = 55.6133, size = 221, normalized size = 1.02 \[ \frac{2 B c^{2} \left (d + e x\right )^{\frac{11}{2}}}{11 e^{6}} + \frac{2 c^{2} \left (d + e x\right )^{\frac{9}{2}} \left (A e - 5 B d\right )}{9 e^{6}} + \frac{4 c \left (d + e x\right )^{\frac{7}{2}} \left (- 2 A c d e + B a e^{2} + 5 B c d^{2}\right )}{7 e^{6}} + \frac{4 c \left (d + e x\right )^{\frac{5}{2}} \left (A a e^{3} + 3 A c d^{2} e - 3 B a d e^{2} - 5 B c d^{3}\right )}{5 e^{6}} + \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (a e^{2} + c d^{2}\right ) \left (- 4 A c d e + B a e^{2} + 5 B c d^{2}\right )}{3 e^{6}} + \frac{2 \sqrt{d + e x} \left (A e - B d\right ) \left (a e^{2} + c d^{2}\right )^{2}}{e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(c*x**2+a)**2/(e*x+d)**(1/2),x)
[Out]
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Mathematica [A] time = 0.306642, size = 213, normalized size = 0.99 \[ \frac{2 \sqrt{d+e x} \left (11 A e \left (315 a^2 e^4+42 a c e^2 \left (8 d^2-4 d e x+3 e^2 x^2\right )+c^2 \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )\right )+B \left (1155 a^2 e^4 (e x-2 d)+198 a c e^2 \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )-5 c^2 \left (256 d^5-128 d^4 e x+96 d^3 e^2 x^2-80 d^2 e^3 x^3+70 d e^4 x^4-63 e^5 x^5\right )\right )\right )}{3465 e^6} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(a + c*x^2)^2)/Sqrt[d + e*x],x]
[Out]
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Maple [A] time = 0.012, size = 259, normalized size = 1.2 \[{\frac{630\,B{c}^{2}{x}^{5}{e}^{5}+770\,A{c}^{2}{e}^{5}{x}^{4}-700\,B{c}^{2}d{e}^{4}{x}^{4}-880\,A{c}^{2}d{e}^{4}{x}^{3}+1980\,Bac{e}^{5}{x}^{3}+800\,B{c}^{2}{d}^{2}{e}^{3}{x}^{3}+2772\,Aac{e}^{5}{x}^{2}+1056\,A{c}^{2}{d}^{2}{e}^{3}{x}^{2}-2376\,Bacd{e}^{4}{x}^{2}-960\,B{c}^{2}{d}^{3}{e}^{2}{x}^{2}-3696\,Aacd{e}^{4}x-1408\,A{c}^{2}{d}^{3}{e}^{2}x+2310\,B{a}^{2}{e}^{5}x+3168\,Bac{d}^{2}{e}^{3}x+1280\,B{c}^{2}{d}^{4}ex+6930\,A{a}^{2}{e}^{5}+7392\,A{d}^{2}ac{e}^{3}+2816\,A{d}^{4}{c}^{2}e-4620\,Bd{a}^{2}{e}^{4}-6336\,aBc{d}^{3}{e}^{2}-2560\,B{c}^{2}{d}^{5}}{3465\,{e}^{6}}\sqrt{ex+d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+a)^2/(e*x+d)^(1/2),x)
[Out]
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Maxima [A] time = 0.681488, size = 335, normalized size = 1.55 \[ \frac{2 \,{\left (315 \,{\left (e x + d\right )}^{\frac{11}{2}} B c^{2} - 385 \,{\left (5 \, B c^{2} d - A c^{2} e\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 990 \,{\left (5 \, B c^{2} d^{2} - 2 \, A c^{2} d e + B a c e^{2}\right )}{\left (e x + d\right )}^{\frac{7}{2}} - 1386 \,{\left (5 \, B c^{2} d^{3} - 3 \, A c^{2} d^{2} e + 3 \, B a c d e^{2} - A a c e^{3}\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 1155 \,{\left (5 \, B c^{2} d^{4} - 4 \, A c^{2} d^{3} e + 6 \, B a c d^{2} e^{2} - 4 \, A a c d e^{3} + B a^{2} e^{4}\right )}{\left (e x + d\right )}^{\frac{3}{2}} - 3465 \,{\left (B c^{2} d^{5} - A c^{2} d^{4} e + 2 \, B a c d^{3} e^{2} - 2 \, A a c d^{2} e^{3} + B a^{2} d e^{4} - A a^{2} e^{5}\right )} \sqrt{e x + d}\right )}}{3465 \, e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2*(B*x + A)/sqrt(e*x + d),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.264657, size = 333, normalized size = 1.54 \[ \frac{2 \,{\left (315 \, B c^{2} e^{5} x^{5} - 1280 \, B c^{2} d^{5} + 1408 \, A c^{2} d^{4} e - 3168 \, B a c d^{3} e^{2} + 3696 \, A a c d^{2} e^{3} - 2310 \, B a^{2} d e^{4} + 3465 \, A a^{2} e^{5} - 35 \,{\left (10 \, B c^{2} d e^{4} - 11 \, A c^{2} e^{5}\right )} x^{4} + 10 \,{\left (40 \, B c^{2} d^{2} e^{3} - 44 \, A c^{2} d e^{4} + 99 \, B a c e^{5}\right )} x^{3} - 6 \,{\left (80 \, B c^{2} d^{3} e^{2} - 88 \, A c^{2} d^{2} e^{3} + 198 \, B a c d e^{4} - 231 \, A a c e^{5}\right )} x^{2} +{\left (640 \, B c^{2} d^{4} e - 704 \, A c^{2} d^{3} e^{2} + 1584 \, B a c d^{2} e^{3} - 1848 \, A a c d e^{4} + 1155 \, B a^{2} e^{5}\right )} x\right )} \sqrt{e x + d}}{3465 \, e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2*(B*x + A)/sqrt(e*x + d),x, algorithm="fricas")
[Out]
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Sympy [A] time = 54.7771, size = 772, normalized size = 3.57 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(c*x**2+a)**2/(e*x+d)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.284517, size = 447, normalized size = 2.07 \[ \frac{2}{3465} \,{\left (1155 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} B a^{2} e^{\left (-1\right )} + 462 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} e^{8} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d e^{8} + 15 \, \sqrt{x e + d} d^{2} e^{8}\right )} A a c e^{\left (-10\right )} + 198 \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{18} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{18} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{18} - 35 \, \sqrt{x e + d} d^{3} e^{18}\right )} B a c e^{\left (-21\right )} + 11 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} e^{32} - 180 \,{\left (x e + d\right )}^{\frac{7}{2}} d e^{32} + 378 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} e^{32} - 420 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} e^{32} + 315 \, \sqrt{x e + d} d^{4} e^{32}\right )} A c^{2} e^{\left (-36\right )} + 5 \,{\left (63 \,{\left (x e + d\right )}^{\frac{11}{2}} e^{50} - 385 \,{\left (x e + d\right )}^{\frac{9}{2}} d e^{50} + 990 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{2} e^{50} - 1386 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{3} e^{50} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{4} e^{50} - 693 \, \sqrt{x e + d} d^{5} e^{50}\right )} B c^{2} e^{\left (-55\right )} + 3465 \, \sqrt{x e + d} A a^{2}\right )} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2*(B*x + A)/sqrt(e*x + d),x, algorithm="giac")
[Out]