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

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

[Out]

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

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Rubi [A]  time = 0.0409498, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ 2 \sqrt{x} (A c+b B)-\frac{2 A b}{\sqrt{x}}+\frac{2}{3} B c x^{3/2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 6.12318, size = 36, normalized size = 1.03 \[ - \frac{2 A b}{\sqrt{x}} + \frac{2 B c x^{\frac{3}{2}}}{3} + \sqrt{x} \left (2 A c + 2 B b\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0211512, size = 29, normalized size = 0.83 \[ \frac{2 (B x (3 b+c x)-3 A (b-c x))}{3 \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 28, normalized size = 0.8 \[ -{\frac{-2\,Bc{x}^{2}-6\,Acx-6\,xBb+6\,Ab}{3}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.682003, size = 36, normalized size = 1.03 \[ \frac{2}{3} \, B c x^{\frac{3}{2}} - \frac{2 \, A b}{\sqrt{x}} + 2 \,{\left (B b + A c\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.299777, size = 35, normalized size = 1. \[ \frac{2 \,{\left (B c x^{2} - 3 \, A b + 3 \,{\left (B b + A c\right )} x\right )}}{3 \, \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.31027, size = 41, normalized size = 1.17 \[ - \frac{2 A b}{\sqrt{x}} + 2 A c \sqrt{x} + 2 B b \sqrt{x} + \frac{2 B c x^{\frac{3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.269387, size = 39, normalized size = 1.11 \[ \frac{2}{3} \, B c x^{\frac{3}{2}} + 2 \, B b \sqrt{x} + 2 \, A c \sqrt{x} - \frac{2 \, A b}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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