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

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

[Out]

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

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Rubi [A]  time = 0.0427225, 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 \[ -\frac{2 (a B+A b)}{\sqrt{x}}-\frac{2 a A}{3 x^{3/2}}+2 b B \sqrt{x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 5.21823, size = 36, normalized size = 1.03 \[ - \frac{2 A a}{3 x^{\frac{3}{2}}} + 2 B b \sqrt{x} - \frac{2 A b + 2 B a}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0150795, size = 28, normalized size = 0.8 \[ -\frac{2 (a (A+3 B x)+3 b x (A-B x))}{3 x^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 27, normalized size = 0.8 \[ -{\frac{-6\,bB{x}^{2}+6\,Abx+6\,Bax+2\,Aa}{3}{x}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(B*x+A)/x^(5/2),x)

[Out]

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

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Maxima [A]  time = 1.3415, size = 36, normalized size = 1.03 \[ 2 \, B b \sqrt{x} - \frac{2 \,{\left (A a + 3 \,{\left (B a + A b\right )} x\right )}}{3 \, x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.206176, size = 36, normalized size = 1.03 \[ \frac{2 \,{\left (3 \, B b x^{2} - A a - 3 \,{\left (B a + A b\right )} x\right )}}{3 \, x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.64516, size = 41, normalized size = 1.17 \[ - \frac{2 A a}{3 x^{\frac{3}{2}}} - \frac{2 A b}{\sqrt{x}} - \frac{2 B a}{\sqrt{x}} + 2 B b \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.257555, size = 36, normalized size = 1.03 \[ 2 \, B b \sqrt{x} - \frac{2 \,{\left (3 \, B a x + 3 \, A b x + A a\right )}}{3 \, x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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