3.734 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{x^{3/2}} \, dx\)

Optimal. Leaf size=59 \[ -\frac{2 a^2 A}{\sqrt{x}}+\frac{2}{3} b x^{3/2} (2 a B+A b)+2 a \sqrt{x} (a B+2 A b)+\frac{2}{5} b^2 B x^{5/2} \]

[Out]

(-2*a^2*A)/Sqrt[x] + 2*a*(2*A*b + a*B)*Sqrt[x] + (2*b*(A*b + 2*a*B)*x^(3/2))/3 +
(2*b^2*B*x^(5/2))/5

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Rubi [A]  time = 0.0780771, 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 \[ -\frac{2 a^2 A}{\sqrt{x}}+\frac{2}{3} b x^{3/2} (2 a B+A b)+2 a \sqrt{x} (a B+2 A b)+\frac{2}{5} b^2 B x^{5/2} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2))/x^(3/2),x]

[Out]

(-2*a^2*A)/Sqrt[x] + 2*a*(2*A*b + a*B)*Sqrt[x] + (2*b*(A*b + 2*a*B)*x^(3/2))/3 +
(2*b^2*B*x^(5/2))/5

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Rubi in Sympy [A]  time = 19.5386, size = 60, normalized size = 1.02 \[ - \frac{2 A a^{2}}{\sqrt{x}} + \frac{2 B b^{2} x^{\frac{5}{2}}}{5} + 2 a \sqrt{x} \left (2 A b + B a\right ) + \frac{2 b x^{\frac{3}{2}} \left (A b + 2 B a\right )}{3} \]

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**(3/2),x)

[Out]

-2*A*a**2/sqrt(x) + 2*B*b**2*x**(5/2)/5 + 2*a*sqrt(x)*(2*A*b + B*a) + 2*b*x**(3/
2)*(A*b + 2*B*a)/3

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Mathematica [A]  time = 0.0267141, size = 49, normalized size = 0.83 \[ \frac{-30 a^2 (A-B x)+20 a b x (3 A+B x)+2 b^2 x^2 (5 A+3 B x)}{15 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2))/x^(3/2),x]

[Out]

(-30*a^2*(A - B*x) + 20*a*b*x*(3*A + B*x) + 2*b^2*x^2*(5*A + 3*B*x))/(15*Sqrt[x]
)

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Maple [A]  time = 0.01, size = 52, normalized size = 0.9 \[ -{\frac{-6\,B{b}^{2}{x}^{3}-10\,A{b}^{2}{x}^{2}-20\,B{x}^{2}ab-60\,aAbx-30\,{a}^{2}Bx+30\,A{a}^{2}}{15}{\frac{1}{\sqrt{x}}}} \]

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)/x^(3/2),x)

[Out]

-2/15*(-3*B*b^2*x^3-5*A*b^2*x^2-10*B*a*b*x^2-30*A*a*b*x-15*B*a^2*x+15*A*a^2)/x^(
1/2)

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Maxima [A]  time = 0.683815, size = 69, normalized size = 1.17 \[ \frac{2}{5} \, B b^{2} x^{\frac{5}{2}} - \frac{2 \, A a^{2}}{\sqrt{x}} + \frac{2}{3} \,{\left (2 \, B a b + A b^{2}\right )} x^{\frac{3}{2}} + 2 \,{\left (B a^{2} + 2 \, A a b\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)*(B*x + A)/x^(3/2),x, algorithm="maxima")

[Out]

2/5*B*b^2*x^(5/2) - 2*A*a^2/sqrt(x) + 2/3*(2*B*a*b + A*b^2)*x^(3/2) + 2*(B*a^2 +
2*A*a*b)*sqrt(x)

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Fricas [A]  time = 0.29751, size = 69, normalized size = 1.17 \[ \frac{2 \,{\left (3 \, B b^{2} x^{3} - 15 \, A a^{2} + 5 \,{\left (2 \, B a b + A b^{2}\right )} x^{2} + 15 \,{\left (B a^{2} + 2 \, A a b\right )} x\right )}}{15 \, \sqrt{x}} \]

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^(3/2),x, algorithm="fricas")

[Out]

2/15*(3*B*b^2*x^3 - 15*A*a^2 + 5*(2*B*a*b + A*b^2)*x^2 + 15*(B*a^2 + 2*A*a*b)*x)
/sqrt(x)

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Sympy [A]  time = 5.21499, size = 75, normalized size = 1.27 \[ - \frac{2 A a^{2}}{\sqrt{x}} + 4 A a b \sqrt{x} + \frac{2 A b^{2} x^{\frac{3}{2}}}{3} + 2 B a^{2} \sqrt{x} + \frac{4 B a b x^{\frac{3}{2}}}{3} + \frac{2 B b^{2} x^{\frac{5}{2}}}{5} \]

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**(3/2),x)

[Out]

-2*A*a**2/sqrt(x) + 4*A*a*b*sqrt(x) + 2*A*b**2*x**(3/2)/3 + 2*B*a**2*sqrt(x) + 4
*B*a*b*x**(3/2)/3 + 2*B*b**2*x**(5/2)/5

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GIAC/XCAS [A]  time = 0.269161, size = 72, normalized size = 1.22 \[ \frac{2}{5} \, B b^{2} x^{\frac{5}{2}} + \frac{4}{3} \, B a b x^{\frac{3}{2}} + \frac{2}{3} \, A b^{2} x^{\frac{3}{2}} + 2 \, B a^{2} \sqrt{x} + 4 \, A a b \sqrt{x} - \frac{2 \, A a^{2}}{\sqrt{x}} \]

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^(3/2),x, algorithm="giac")

[Out]

2/5*B*b^2*x^(5/2) + 4/3*B*a*b*x^(3/2) + 2/3*A*b^2*x^(3/2) + 2*B*a^2*sqrt(x) + 4*
A*a*b*sqrt(x) - 2*A*a^2/sqrt(x)