Optimal. Leaf size=120 \[ \frac{2 x^{7/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{7 (a+b x)}+\frac{2 a A x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac{2 b B x^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}{9 (a+b x)} \]
[Out]
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Rubi [A] time = 0.154175, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ \frac{2 x^{7/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{7 (a+b x)}+\frac{2 a A x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac{2 b B x^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}{9 (a+b x)} \]
Antiderivative was successfully verified.
[In] Int[x^(3/2)*(A + B*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]
[Out]
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Rubi in Sympy [A] time = 18.9906, size = 126, normalized size = 1.05 \[ \frac{B x^{\frac{5}{2}} \left (2 a + 2 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{9 b} + \frac{4 a x^{\frac{5}{2}} \left (9 A b - 5 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{315 b \left (a + b x\right )} + \frac{2 x^{\frac{5}{2}} \left (9 A b - 5 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{63 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(3/2)*(B*x+A)*((b*x+a)**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0424896, size = 51, normalized size = 0.42 \[ \frac{2 x^{5/2} \sqrt{(a+b x)^2} (9 a (7 A+5 B x)+5 b x (9 A+7 B x))}{315 (a+b x)} \]
Antiderivative was successfully verified.
[In] Integrate[x^(3/2)*(A + B*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]
[Out]
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Maple [A] time = 0.006, size = 44, normalized size = 0.4 \[{\frac{70\,Bb{x}^{2}+90\,Abx+90\,aBx+126\,aA}{315\,bx+315\,a}{x}^{{\frac{5}{2}}}\sqrt{ \left ( bx+a \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(3/2)*(B*x+A)*((b*x+a)^2)^(1/2),x)
[Out]
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Maxima [A] time = 0.695614, size = 47, normalized size = 0.39 \[ \frac{2}{63} \,{\left (7 \, b x^{2} + 9 \, a x\right )} B x^{\frac{5}{2}} + \frac{2}{35} \,{\left (5 \, b x^{2} + 7 \, a x\right )} A x^{\frac{3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)*(B*x + A)*x^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.30707, size = 43, normalized size = 0.36 \[ \frac{2}{315} \,{\left (35 \, B b x^{4} + 63 \, A a x^{2} + 45 \,{\left (B a + A b\right )} x^{3}\right )} \sqrt{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)*(B*x + A)*x^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(3/2)*(B*x+A)*((b*x+a)**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.283577, size = 72, normalized size = 0.6 \[ \frac{2}{9} \, B b x^{\frac{9}{2}}{\rm sign}\left (b x + a\right ) + \frac{2}{7} \, B a x^{\frac{7}{2}}{\rm sign}\left (b x + a\right ) + \frac{2}{7} \, A b x^{\frac{7}{2}}{\rm sign}\left (b x + a\right ) + \frac{2}{5} \, A a x^{\frac{5}{2}}{\rm sign}\left (b x + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)*(B*x + A)*x^(3/2),x, algorithm="giac")
[Out]