Optimal. Leaf size=109 \[ 2 a^4 A \sqrt{x}+\frac{2}{3} a^3 x^{3/2} (a B+4 A b)+\frac{4}{5} a^2 b x^{5/2} (2 a B+3 A b)+\frac{2}{9} b^3 x^{9/2} (4 a B+A b)+\frac{4}{7} a b^2 x^{7/2} (3 a B+2 A b)+\frac{2}{11} b^4 B x^{11/2} \]
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Rubi [A] time = 0.132234, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ 2 a^4 A \sqrt{x}+\frac{2}{3} a^3 x^{3/2} (a B+4 A b)+\frac{4}{5} a^2 b x^{5/2} (2 a B+3 A b)+\frac{2}{9} b^3 x^{9/2} (4 a B+A b)+\frac{4}{7} a b^2 x^{7/2} (3 a B+2 A b)+\frac{2}{11} b^4 B x^{11/2} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/Sqrt[x],x]
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Rubi in Sympy [A] time = 31.8236, size = 112, normalized size = 1.03 \[ 2 A a^{4} \sqrt{x} + \frac{2 B b^{4} x^{\frac{11}{2}}}{11} + \frac{2 a^{3} x^{\frac{3}{2}} \left (4 A b + B a\right )}{3} + \frac{4 a^{2} b x^{\frac{5}{2}} \left (3 A b + 2 B a\right )}{5} + \frac{4 a b^{2} x^{\frac{7}{2}} \left (2 A b + 3 B a\right )}{7} + \frac{2 b^{3} x^{\frac{9}{2}} \left (A b + 4 B a\right )}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**2/x**(1/2),x)
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Mathematica [A] time = 0.0498812, size = 89, normalized size = 0.82 \[ \frac{2 \sqrt{x} \left (1155 a^4 (3 A+B x)+924 a^3 b x (5 A+3 B x)+594 a^2 b^2 x^2 (7 A+5 B x)+220 a b^3 x^3 (9 A+7 B x)+35 b^4 x^4 (11 A+9 B x)\right )}{3465} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/Sqrt[x],x]
[Out]
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Maple [A] time = 0.01, size = 100, normalized size = 0.9 \[{\frac{630\,{b}^{4}B{x}^{5}+770\,A{b}^{4}{x}^{4}+3080\,B{x}^{4}a{b}^{3}+3960\,aA{b}^{3}{x}^{3}+5940\,B{x}^{3}{a}^{2}{b}^{2}+8316\,{a}^{2}A{b}^{2}{x}^{2}+5544\,B{x}^{2}{a}^{3}b+9240\,{a}^{3}Abx+2310\,{a}^{4}Bx+6930\,A{a}^{4}}{3465}\sqrt{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^2/x^(1/2),x)
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Maxima [A] time = 0.682923, size = 134, normalized size = 1.23 \[ \frac{2}{11} \, B b^{4} x^{\frac{11}{2}} + 2 \, A a^{4} \sqrt{x} + \frac{2}{9} \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{\frac{9}{2}} + \frac{4}{7} \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{\frac{7}{2}} + \frac{4}{5} \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{\frac{5}{2}} + \frac{2}{3} \,{\left (B a^{4} + 4 \, A a^{3} b\right )} x^{\frac{3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^2*(B*x + A)/sqrt(x),x, algorithm="maxima")
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Fricas [A] time = 0.289646, size = 134, normalized size = 1.23 \[ \frac{2}{3465} \,{\left (315 \, B b^{4} x^{5} + 3465 \, A a^{4} + 385 \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} + 990 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} + 1386 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} + 1155 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} x\right )} \sqrt{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^2*(B*x + A)/sqrt(x),x, algorithm="fricas")
[Out]
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Sympy [A] time = 10.807, size = 146, normalized size = 1.34 \[ 2 A a^{4} \sqrt{x} + \frac{8 A a^{3} b x^{\frac{3}{2}}}{3} + \frac{12 A a^{2} b^{2} x^{\frac{5}{2}}}{5} + \frac{8 A a b^{3} x^{\frac{7}{2}}}{7} + \frac{2 A b^{4} x^{\frac{9}{2}}}{9} + \frac{2 B a^{4} x^{\frac{3}{2}}}{3} + \frac{8 B a^{3} b x^{\frac{5}{2}}}{5} + \frac{12 B a^{2} b^{2} x^{\frac{7}{2}}}{7} + \frac{8 B a b^{3} x^{\frac{9}{2}}}{9} + \frac{2 B b^{4} x^{\frac{11}{2}}}{11} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**2/x**(1/2),x)
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GIAC/XCAS [A] time = 0.269203, size = 136, normalized size = 1.25 \[ \frac{2}{11} \, B b^{4} x^{\frac{11}{2}} + \frac{8}{9} \, B a b^{3} x^{\frac{9}{2}} + \frac{2}{9} \, A b^{4} x^{\frac{9}{2}} + \frac{12}{7} \, B a^{2} b^{2} x^{\frac{7}{2}} + \frac{8}{7} \, A a b^{3} x^{\frac{7}{2}} + \frac{8}{5} \, B a^{3} b x^{\frac{5}{2}} + \frac{12}{5} \, A a^{2} b^{2} x^{\frac{5}{2}} + \frac{2}{3} \, B a^{4} x^{\frac{3}{2}} + \frac{8}{3} \, A a^{3} b x^{\frac{3}{2}} + 2 \, A a^{4} \sqrt{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^2*(B*x + A)/sqrt(x),x, algorithm="giac")
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