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

Optimal. Leaf size=41 \[ -\frac{2 a A}{3 x^{3/2}}-\frac{2 a B}{\sqrt{x}}+2 A c \sqrt{x}+\frac{2}{3} B c x^{3/2} \]

[Out]

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

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Rubi [A]  time = 0.0415767, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ -\frac{2 a A}{3 x^{3/2}}-\frac{2 a B}{\sqrt{x}}+2 A c \sqrt{x}+\frac{2}{3} B c x^{3/2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 5.0343, size = 42, normalized size = 1.02 \[ - \frac{2 A a}{3 x^{\frac{3}{2}}} + 2 A c \sqrt{x} - \frac{2 B a}{\sqrt{x}} + \frac{2 B c x^{\frac{3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.01884, size = 32, normalized size = 0.78 \[ \frac{2 c x^2 (3 A+B x)-2 a (A+3 B x)}{3 x^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 29, normalized size = 0.7 \[ -{\frac{-2\,Bc{x}^{3}-6\,Ac{x}^{2}+6\,aBx+2\,aA}{3}{x}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.68074, size = 39, normalized size = 0.95 \[ \frac{2}{3} \, B c x^{\frac{3}{2}} + 2 \, A c \sqrt{x} - \frac{2 \,{\left (3 \, B a x + A a\right )}}{3 \, x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.286121, size = 38, normalized size = 0.93 \[ \frac{2 \,{\left (B c x^{3} + 3 \, A c x^{2} - 3 \, B a x - A a\right )}}{3 \, x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.18323, size = 42, normalized size = 1.02 \[ - \frac{2 A a}{3 x^{\frac{3}{2}}} + 2 A c \sqrt{x} - \frac{2 B a}{\sqrt{x}} + \frac{2 B c x^{\frac{3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.27904, size = 39, normalized size = 0.95 \[ \frac{2}{3} \, B c x^{\frac{3}{2}} + 2 \, A c \sqrt{x} - \frac{2 \,{\left (3 \, B a x + A a\right )}}{3 \, x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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