Optimal. Leaf size=120 \[ \frac{2 x^{9/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{9 (a+b x)}+\frac{2 a A x^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac{2 b B x^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}{11 (a+b x)} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.15482, 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 \[ \frac{2 x^{9/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{9 (a+b x)}+\frac{2 a A x^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac{2 b B x^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}{11 (a+b x)} \]
Antiderivative was successfully verified.
[In] Int[x^(5/2)*(A + B*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 19.1039, size = 126, normalized size = 1.05 \[ \frac{B x^{\frac{7}{2}} \left (2 a + 2 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{11 b} + \frac{4 a x^{\frac{7}{2}} \left (11 A b - 7 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{693 b \left (a + b x\right )} + \frac{2 x^{\frac{7}{2}} \left (11 A b - 7 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{99 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(5/2)*(B*x+A)*((b*x+a)**2)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0366185, size = 51, normalized size = 0.42 \[ \frac{2 x^{7/2} \sqrt{(a+b x)^2} (11 a (9 A+7 B x)+7 b x (11 A+9 B x))}{693 (a+b x)} \]
Antiderivative was successfully verified.
[In] Integrate[x^(5/2)*(A + B*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.006, size = 44, normalized size = 0.4 \[{\frac{126\,Bb{x}^{2}+154\,Abx+154\,aBx+198\,aA}{693\,bx+693\,a}{x}^{{\frac{7}{2}}}\sqrt{ \left ( bx+a \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(5/2)*(B*x+A)*((b*x+a)^2)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.702047, size = 47, normalized size = 0.39 \[ \frac{2}{99} \,{\left (9 \, b x^{2} + 11 \, a x\right )} B x^{\frac{7}{2}} + \frac{2}{63} \,{\left (7 \, b x^{2} + 9 \, a x\right )} A x^{\frac{5}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)*(B*x + A)*x^(5/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.304764, size = 43, normalized size = 0.36 \[ \frac{2}{693} \,{\left (63 \, B b x^{5} + 99 \, A a x^{3} + 77 \,{\left (B a + A b\right )} x^{4}\right )} \sqrt{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)*(B*x + A)*x^(5/2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
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**(5/2)*(B*x+A)*((b*x+a)**2)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.274577, size = 72, normalized size = 0.6 \[ \frac{2}{11} \, B b x^{\frac{11}{2}}{\rm sign}\left (b x + a\right ) + \frac{2}{9} \, B a x^{\frac{9}{2}}{\rm sign}\left (b x + a\right ) + \frac{2}{9} \, A b x^{\frac{9}{2}}{\rm sign}\left (b x + a\right ) + \frac{2}{7} \, A a x^{\frac{7}{2}}{\rm sign}\left (b x + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)*(B*x + A)*x^(5/2),x, algorithm="giac")
[Out]