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

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 = 0.0160369, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {765} \[ 2 \sqrt{x} (A c+b B)-\frac{2 A b}{\sqrt{x}}+\frac{2}{3} B c x^{3/2} \]

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

Rule 765

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand
Integrand[(e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, e, f, g, m}, x] && IntegerQ[p] && (
GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

Mathematica [A]  time = 0.0114129, 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 \begin{align*} -{\frac{-2\,Bc{x}^{2}-6\,Acx-6\,bBx+6\,Ab}{3}{\frac{1}{\sqrt{x}}}} \end{align*}

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 = 1.03003, size = 36, normalized size = 1.03 \begin{align*} \frac{2}{3} \, B c x^{\frac{3}{2}} - \frac{2 \, A b}{\sqrt{x}} + 2 \,{\left (B b + A c\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)/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 = 1.77157, size = 66, normalized size = 1.89 \begin{align*} \frac{2 \,{\left (B c x^{2} - 3 \, A b + 3 \,{\left (B b + A c\right )} x\right )}}{3 \, \sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)/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 = 0.889824, size = 41, normalized size = 1.17 \begin{align*} - \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} \end{align*}

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 [A]  time = 1.16114, size = 39, normalized size = 1.11 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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