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

Optimal. Leaf size=35 \[ 2 \sqrt{x} (a B+A b)-\frac{2 a A}{\sqrt{x}}+\frac{2}{3} b B x^{3/2} \]

[Out]

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

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Rubi [A]  time = 0.0438658, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ 2 \sqrt{x} (a B+A b)-\frac{2 a A}{\sqrt{x}}+\frac{2}{3} b B x^{3/2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 5.38978, size = 36, normalized size = 1.03 \[ - \frac{2 A a}{\sqrt{x}} + \frac{2 B b x^{\frac{3}{2}}}{3} + \sqrt{x} \left (2 A b + 2 B a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0175277, size = 29, normalized size = 0.83 \[ \frac{2 (b x (3 A+B x)-3 a (A-B x))}{3 \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 28, normalized size = 0.8 \[ -{\frac{-2\,bB{x}^{2}-6\,Abx-6\,Bax+6\,Aa}{3}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(B*x+A)/x^(3/2),x)

[Out]

-2/3*(-B*b*x^2-3*A*b*x-3*B*a*x+3*A*a)/x^(1/2)

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Maxima [A]  time = 1.344, size = 36, normalized size = 1.03 \[ \frac{2}{3} \, B b x^{\frac{3}{2}} - \frac{2 \, A a}{\sqrt{x}} + 2 \,{\left (B a + A b\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.208627, size = 35, normalized size = 1. \[ \frac{2 \,{\left (B b x^{2} - 3 \, A a + 3 \,{\left (B a + A b\right )} x\right )}}{3 \, \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 5.1316, size = 41, normalized size = 1.17 \[ - \frac{2 A a}{\sqrt{x}} + 2 A b \sqrt{x} + 2 B a \sqrt{x} + \frac{2 B b x^{\frac{3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.253459, size = 39, normalized size = 1.11 \[ \frac{2}{3} \, B b x^{\frac{3}{2}} + 2 \, B a \sqrt{x} + 2 \, A b \sqrt{x} - \frac{2 \, A a}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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