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

Optimal. Leaf size=49 \[ -\frac{2 \sqrt{x} (2 A b-a B)}{a^2 \sqrt{a+b x}}-\frac{2 A}{a \sqrt{x} \sqrt{a+b x}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0141708, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {78, 37} \[ -\frac{2 \sqrt{x} (2 A b-a B)}{a^2 \sqrt{a+b x}}-\frac{2 A}{a \sqrt{x} \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{A+B x}{x^{3/2} (a+b x)^{3/2}} \, dx &=-\frac{2 A}{a \sqrt{x} \sqrt{a+b x}}+\frac{\left (2 \left (-A b+\frac{a B}{2}\right )\right ) \int \frac{1}{\sqrt{x} (a+b x)^{3/2}} \, dx}{a}\\ &=-\frac{2 A}{a \sqrt{x} \sqrt{a+b x}}-\frac{2 (2 A b-a B) \sqrt{x}}{a^2 \sqrt{a+b x}}\\ \end{align*}

Mathematica [A]  time = 0.0131705, size = 33, normalized size = 0.67 \[ \frac{2 (-a A+a B x-2 A b x)}{a^2 \sqrt{x} \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 30, normalized size = 0.6 \begin{align*} -2\,{\frac{2\,Abx-Bax+Aa}{{a}^{2}\sqrt{x}\sqrt{bx+a}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.59114, size = 74, normalized size = 1.51 \begin{align*} \frac{2 \, B x}{\sqrt{b x^{2} + a x} a} - \frac{4 \, A b x}{\sqrt{b x^{2} + a x} a^{2}} - \frac{2 \, A}{\sqrt{b x^{2} + a x} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*B*x/(sqrt(b*x^2 + a*x)*a) - 4*A*b*x/(sqrt(b*x^2 + a*x)*a^2) - 2*A/(sqrt(b*x^2 + a*x)*a)

________________________________________________________________________________________

Fricas [A]  time = 2.76873, size = 95, normalized size = 1.94 \begin{align*} -\frac{2 \,{\left (A a -{\left (B a - 2 \, A b\right )} x\right )} \sqrt{b x + a} \sqrt{x}}{a^{2} b x^{2} + a^{3} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 41.183, size = 63, normalized size = 1.29 \begin{align*} A \left (- \frac{2}{a \sqrt{b} x \sqrt{\frac{a}{b x} + 1}} - \frac{4 \sqrt{b}}{a^{2} \sqrt{\frac{a}{b x} + 1}}\right ) + \frac{2 B}{a \sqrt{b} \sqrt{\frac{a}{b x} + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*(-2/(a*sqrt(b)*x*sqrt(a/(b*x) + 1)) - 4*sqrt(b)/(a**2*sqrt(a/(b*x) + 1))) + 2*B/(a*sqrt(b)*sqrt(a/(b*x) + 1)
)

________________________________________________________________________________________

Giac [B]  time = 1.23397, size = 126, normalized size = 2.57 \begin{align*} -\frac{2 \, \sqrt{b x + a} A b^{2}}{\sqrt{{\left (b x + a\right )} b - a b} a^{2}{\left | b \right |}} + \frac{4 \,{\left (B a b^{\frac{3}{2}} - A b^{\frac{5}{2}}\right )}}{{\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )} a{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(b*x + a)*A*b^2/(sqrt((b*x + a)*b - a*b)*a^2*abs(b)) + 4*(B*a*b^(3/2) - A*b^(5/2))/(((sqrt(b*x + a)*sqr
t(b) - sqrt((b*x + a)*b - a*b))^2 + a*b)*a*abs(b))