\(\int \frac {A+B x}{x^{3/2} (a+b x)^{3/2}} \, dx\) [531]

Optimal result

Integrand size = 20, antiderivative size = 49 \[ \int \frac {A+B x}{x^{3/2} (a+b x)^{3/2}} \, dx=-\frac {2 A}{a \sqrt {x} \sqrt {a+b x}}-\frac {2 (2 A b-a B) \sqrt {x}}{a^2 \sqrt {a+b x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {79, 37} \[ \int \frac {A+B x}{x^{3/2} (a+b x)^{3/2}} \, dx=-\frac {2 \sqrt {x} (2 A b-a B)}{a^2 \sqrt {a+b x}}-\frac {2 A}{a \sqrt {x} \sqrt {a+b x}} \]

[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 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]

Rule 79

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] || L
tQ[p, n]))))

Rubi steps \begin{align*} \text {integral}& = -\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] (verified)

Time = 0.13 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.67 \[ \int \frac {A+B x}{x^{3/2} (a+b x)^{3/2}} \, dx=-\frac {2 (a A+2 A b x-a B x)}{a^2 \sqrt {x} \sqrt {a+b x}} \]

[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] (verified)

Time = 0.51 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.61

method	result	size
gosper	\(-\frac {2 \left (2 A b x -B a x +A a \right )}{\sqrt {x}\, \sqrt {b x +a}\, a^{2}}\)	\(30\)
default	\(-\frac {2 \left (2 A b x -B a x +A a \right )}{\sqrt {x}\, \sqrt {b x +a}\, a^{2}}\)	\(30\)
risch	\(-\frac {2 A \sqrt {b x +a}}{a^{2} \sqrt {x}}-\frac {2 \left (A b -B a \right ) \sqrt {x}}{a^{2} \sqrt {b x +a}}\)	\(41\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.88 \[ \int \frac {A+B x}{x^{3/2} (a+b x)^{3/2}} \, dx=-\frac {2 \, {\left (A a - {\left (B a - 2 \, A b\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{a^{2} b x^{2} + a^{3} x} \]

[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] (verification not implemented)

Time = 5.21 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.29 \[ \int \frac {A+B x}{x^{3/2} (a+b x)^{3/2}} \, dx=A \left (- \frac {2}{a \sqrt {b} x \sqrt {\frac {a}{b x} + 1}} - \frac {4 \sqrt {b}}{a^{2} \sqrt {\frac {a}{b x} + 1}}\right ) + \frac {2 B}{a \sqrt {b} \sqrt {\frac {a}{b x} + 1}} \]

[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)
)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.12 \[ \int \frac {A+B x}{x^{3/2} (a+b x)^{3/2}} \, dx=\frac {2 \, B x}{\sqrt {b x^{2} + a x} a} - \frac {4 \, A b x}{\sqrt {b x^{2} + a x} a^{2}} - \frac {2 \, A}{\sqrt {b x^{2} + a x} a} \]

[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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 160 vs. \(2 (40) = 80\).

Time = 0.32 (sec) , antiderivative size = 160, normalized size of antiderivative = 3.27 \[ \int \frac {A+B x}{x^{3/2} (a+b x)^{3/2}} \, dx=-\frac {2 \, \sqrt {b x + a} A b^{2}}{\sqrt {{\left (b x + a\right )} b - a b} a^{2} {\left | b \right |}} + \frac {4 \, {\left (B^{2} a^{2} b^{3} - 2 \, A B a b^{4} + A^{2} b^{5}\right )}}{{\left (B a {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac {3}{2}} + B a^{2} b^{\frac {5}{2}} - A {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac {5}{2}} - A a b^{\frac {7}{2}}\right )} a {\left | b \right |}} \]

[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^2*a^2*b^3 - 2*A*B*a*b^4 + A^2*b^5)/((B*a*(s
qrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2*b^(3/2) + B*a^2*b^(5/2) - A*(sqrt(b*x + a)*sqrt(b) - sqrt((b
*x + a)*b - a*b))^2*b^(5/2) - A*a*b^(7/2))*a*abs(b))

Mupad [B] (verification not implemented)

Time = 0.84 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.02 \[ \int \frac {A+B x}{x^{3/2} (a+b x)^{3/2}} \, dx=-\frac {\left (\frac {2\,A}{a\,b}+\frac {x\,\left (4\,A\,b-2\,B\,a\right )}{a^2\,b}\right )\,\sqrt {a+b\,x}}{x^{3/2}+\frac {a\,\sqrt {x}}{b}} \]

[In]

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

[Out]

-(((2*A)/(a*b) + (x*(4*A*b - 2*B*a))/(a^2*b))*(a + b*x)^(1/2))/(x^(3/2) + (a*x^(1/2))/b)