3.296 \(\int \frac{(a+b x) (A+B x)}{x^{7/2}} \, dx\)

Optimal. Leaf size=37 \[ -\frac{2 (a B+A b)}{3 x^{3/2}}-\frac{2 a A}{5 x^{5/2}}-\frac{2 b B}{\sqrt{x}} \]

[Out]

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

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Rubi [A]  time = 0.0501797, antiderivative size = 37, 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 \[ -\frac{2 (a B+A b)}{3 x^{3/2}}-\frac{2 a A}{5 x^{5/2}}-\frac{2 b B}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)*(A + B*x))/x^(7/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 5.29976, size = 41, normalized size = 1.11 \[ - \frac{2 A a}{5 x^{\frac{5}{2}}} - \frac{2 B b}{\sqrt{x}} - \frac{\frac{2 A b}{3} + \frac{2 B a}{3}}{x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(B*x+A)/x**(7/2),x)

[Out]

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

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Mathematica [A]  time = 0.0157182, size = 30, normalized size = 0.81 \[ -\frac{2 (a (3 A+5 B x)+5 b x (A+3 B x))}{15 x^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)*(A + B*x))/x^(7/2),x]

[Out]

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

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Maple [A]  time = 0.006, size = 28, normalized size = 0.8 \[ -{\frac{30\,bB{x}^{2}+10\,Abx+10\,Bax+6\,Aa}{15}{x}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(B*x+A)/x^(7/2),x)

[Out]

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

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Maxima [A]  time = 1.33285, size = 36, normalized size = 0.97 \[ -\frac{2 \,{\left (15 \, B b x^{2} + 3 \, A a + 5 \,{\left (B a + A b\right )} x\right )}}{15 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)/x^(7/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.20682, size = 36, normalized size = 0.97 \[ -\frac{2 \,{\left (15 \, B b x^{2} + 3 \, A a + 5 \,{\left (B a + A b\right )} x\right )}}{15 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)/x^(7/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 7.60234, size = 46, normalized size = 1.24 \[ - \frac{2 A a}{5 x^{\frac{5}{2}}} - \frac{2 A b}{3 x^{\frac{3}{2}}} - \frac{2 B a}{3 x^{\frac{3}{2}}} - \frac{2 B b}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)*(B*x+A)/x**(7/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.262276, size = 36, normalized size = 0.97 \[ -\frac{2 \,{\left (15 \, B b x^{2} + 5 \, B a x + 5 \, A b x + 3 \, A a\right )}}{15 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)/x^(7/2),x, algorithm="giac")

[Out]

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