3.324 \(\int \frac{(a+b x) (A+B x)}{x^{3/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0127751, antiderivative size = 35, 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, Rules used = {76} \[ 2 \sqrt{x} (a B+A b)-\frac{2 a A}{\sqrt{x}}+\frac{2}{3} b B x^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(A + B*x))/x^(3/2),x]

[Out]

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

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps

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

Mathematica [A]  time = 0.0120495, size = 29, normalized size = 0.83 \[ \frac{2 (b x (3 A+B x)-3 a (A-B x))}{3 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(A + B*x))/x^(3/2),x]

[Out]

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

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Maple [A]  time = 0.003, size = 28, normalized size = 0.8 \begin{align*} -{\frac{-2\,bB{x}^{2}-6\,Abx-6\,Bax+6\,Aa}{3}{\frac{1}{\sqrt{x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(B*x+A)/x^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.12629, size = 36, normalized size = 1.03 \begin{align*} \frac{2}{3} \, B b x^{\frac{3}{2}} - \frac{2 \, A a}{\sqrt{x}} + 2 \,{\left (B a + A b\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(B*x+A)/x^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.62197, size = 66, normalized size = 1.89 \begin{align*} \frac{2 \,{\left (B b x^{2} - 3 \, A a + 3 \,{\left (B a + A b\right )} x\right )}}{3 \, \sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(B*x+A)/x^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.581993, size = 41, normalized size = 1.17 \begin{align*} - \frac{2 A a}{\sqrt{x}} + 2 A b \sqrt{x} + 2 B a \sqrt{x} + \frac{2 B b x^{\frac{3}{2}}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(B*x+A)/x**(3/2),x)

[Out]

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

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Giac [A]  time = 1.22355, size = 39, normalized size = 1.11 \begin{align*} \frac{2}{3} \, B b x^{\frac{3}{2}} + 2 \, B a \sqrt{x} + 2 \, A b \sqrt{x} - \frac{2 \, A a}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(B*x+A)/x^(3/2),x, algorithm="giac")

[Out]

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