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

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 = 0.012395, 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, Rules used = {766} \[ -\frac{2 a A}{3 x^{3/2}}-\frac{2 a B}{\sqrt{x}}+2 A c \sqrt{x}+\frac{2}{3} B c x^{3/2} \]

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

Rule 766

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(e*x
)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{(A+B x) \left (a+c x^2\right )}{x^{5/2}} \, dx &=\int \left (\frac{a A}{x^{5/2}}+\frac{a B}{x^{3/2}}+\frac{A c}{\sqrt{x}}+B c \sqrt{x}\right ) \, dx\\ &=-\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}\\ \end{align*}

Mathematica [A]  time = 0.0113768, 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 = 0.004, size = 29, normalized size = 0.7 \begin{align*} -{\frac{-2\,Bc{x}^{3}-6\,Ac{x}^{2}+6\,aBx+2\,aA}{3}{x}^{-{\frac{3}{2}}}} \end{align*}

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 = 1.00301, size = 39, normalized size = 0.95 \begin{align*} \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}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+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 = 1.33047, size = 69, normalized size = 1.68 \begin{align*} \frac{2 \,{\left (B c x^{3} + 3 \, A c x^{2} - 3 \, B a x - A a\right )}}{3 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+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 = 1.02441, size = 42, normalized size = 1.02 \begin{align*} - \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} \end{align*}

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 [A]  time = 1.221, size = 39, normalized size = 0.95 \begin{align*} \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}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+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)