3.735 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{x^{5/2}} \, dx\)

Optimal. Leaf size=59 \[ -\frac{2 a^2 A}{3 x^{3/2}}-\frac{2 a (a B+2 A b)}{\sqrt{x}}+2 b \sqrt{x} (2 a B+A b)+\frac{2}{3} b^2 B x^{3/2} \]

[Out]

(-2*a^2*A)/(3*x^(3/2)) - (2*a*(2*A*b + a*B))/Sqrt[x] + 2*b*(A*b + 2*a*B)*Sqrt[x]
+ (2*b^2*B*x^(3/2))/3

_______________________________________________________________________________________

Rubi [A]  time = 0.0781606, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074 \[ -\frac{2 a^2 A}{3 x^{3/2}}-\frac{2 a (a B+2 A b)}{\sqrt{x}}+2 b \sqrt{x} (2 a B+A b)+\frac{2}{3} b^2 B x^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2))/x^(5/2),x]

[Out]

(-2*a^2*A)/(3*x^(3/2)) - (2*a*(2*A*b + a*B))/Sqrt[x] + 2*b*(A*b + 2*a*B)*Sqrt[x]
+ (2*b^2*B*x^(3/2))/3

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 19.4871, size = 60, normalized size = 1.02 \[ - \frac{2 A a^{2}}{3 x^{\frac{3}{2}}} + \frac{2 B b^{2} x^{\frac{3}{2}}}{3} - \frac{2 a \left (2 A b + B a\right )}{\sqrt{x}} + 2 b \sqrt{x} \left (A b + 2 B a\right ) \]

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)/x**(5/2),x)

[Out]

-2*A*a**2/(3*x**(3/2)) + 2*B*b**2*x**(3/2)/3 - 2*a*(2*A*b + B*a)/sqrt(x) + 2*b*s
qrt(x)*(A*b + 2*B*a)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0289188, size = 47, normalized size = 0.8 \[ \frac{2 \left (a^2 (-(A+3 B x))+6 a b x (B x-A)+b^2 x^2 (3 A+B x)\right )}{3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2))/x^(5/2),x]

[Out]

(2*(6*a*b*x*(-A + B*x) + b^2*x^2*(3*A + B*x) - a^2*(A + 3*B*x)))/(3*x^(3/2))

_______________________________________________________________________________________

Maple [A]  time = 0.01, size = 51, normalized size = 0.9 \[ -{\frac{-2\,B{b}^{2}{x}^{3}-6\,A{b}^{2}{x}^{2}-12\,B{x}^{2}ab+12\,aAbx+6\,{a}^{2}Bx+2\,A{a}^{2}}{3}{x}^{-{\frac{3}{2}}}} \]

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)/x^(5/2),x)

[Out]

-2/3*(-B*b^2*x^3-3*A*b^2*x^2-6*B*a*b*x^2+6*A*a*b*x+3*B*a^2*x+A*a^2)/x^(3/2)

_______________________________________________________________________________________

Maxima [A]  time = 0.683079, size = 69, normalized size = 1.17 \[ \frac{2}{3} \, B b^{2} x^{\frac{3}{2}} + 2 \,{\left (2 \, B a b + A b^{2}\right )} \sqrt{x} - \frac{2 \,{\left (A a^{2} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} x\right )}}{3 \, x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)*(B*x + A)/x^(5/2),x, algorithm="maxima")

[Out]

2/3*B*b^2*x^(3/2) + 2*(2*B*a*b + A*b^2)*sqrt(x) - 2/3*(A*a^2 + 3*(B*a^2 + 2*A*a*
b)*x)/x^(3/2)

_______________________________________________________________________________________

Fricas [A]  time = 0.298781, size = 68, normalized size = 1.15 \[ \frac{2 \,{\left (B b^{2} x^{3} - A a^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} x^{2} - 3 \,{\left (B a^{2} + 2 \, A a b\right )} x\right )}}{3 \, x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)*(B*x + A)/x^(5/2),x, algorithm="fricas")

[Out]

2/3*(B*b^2*x^3 - A*a^2 + 3*(2*B*a*b + A*b^2)*x^2 - 3*(B*a^2 + 2*A*a*b)*x)/x^(3/2
)

_______________________________________________________________________________________

Sympy [A]  time = 2.57098, size = 73, normalized size = 1.24 \[ - \frac{2 A a^{2}}{3 x^{\frac{3}{2}}} - \frac{4 A a b}{\sqrt{x}} + 2 A b^{2} \sqrt{x} - \frac{2 B a^{2}}{\sqrt{x}} + 4 B a b \sqrt{x} + \frac{2 B b^{2} x^{\frac{3}{2}}}{3} \]

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)/x**(5/2),x)

[Out]

-2*A*a**2/(3*x**(3/2)) - 4*A*a*b/sqrt(x) + 2*A*b**2*sqrt(x) - 2*B*a**2/sqrt(x) +
4*B*a*b*sqrt(x) + 2*B*b**2*x**(3/2)/3

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.269531, size = 69, normalized size = 1.17 \[ \frac{2}{3} \, B b^{2} x^{\frac{3}{2}} + 4 \, B a b \sqrt{x} + 2 \, A b^{2} \sqrt{x} - \frac{2 \,{\left (3 \, B a^{2} x + 6 \, A a b x + A a^{2}\right )}}{3 \, x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)*(B*x + A)/x^(5/2),x, algorithm="giac")

[Out]

2/3*B*b^2*x^(3/2) + 4*B*a*b*sqrt(x) + 2*A*b^2*sqrt(x) - 2/3*(3*B*a^2*x + 6*A*a*b
*x + A*a^2)/x^(3/2)