\(\int \frac {1}{x^{3/2} (a+b x)^{3/2}} \, dx\) [581]

Optimal result

Integrand size = 15, antiderivative size = 39 \[ \int \frac {1}{x^{3/2} (a+b x)^{3/2}} \, dx=\frac {2}{a \sqrt {x} \sqrt {a+b x}}-\frac {4 \sqrt {a+b x}}{a^2 \sqrt {x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {47, 37} \[ \int \frac {1}{x^{3/2} (a+b x)^{3/2}} \, dx=\frac {2}{a \sqrt {x} \sqrt {a+b x}}-\frac {4 \sqrt {a+b x}}{a^2 \sqrt {x}} \]

[In]

Int[1/(x^(3/2)*(a + b*x)^(3/2)),x]

[Out]

2/(a*Sqrt[x]*Sqrt[a + b*x]) - (4*Sqrt[a + b*x])/(a^2*Sqrt[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 47

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] - Dist[d*(Simplify[m + n + 2]/((b*c - a*d)*(m + 1))), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rubi steps \begin{align*} \text {integral}& = \frac {2}{a \sqrt {x} \sqrt {a+b x}}+\frac {2 \int \frac {1}{x^{3/2} \sqrt {a+b x}} \, dx}{a} \\ & = \frac {2}{a \sqrt {x} \sqrt {a+b x}}-\frac {4 \sqrt {a+b x}}{a^2 \sqrt {x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.64 \[ \int \frac {1}{x^{3/2} (a+b x)^{3/2}} \, dx=-\frac {2 (a+2 b x)}{a^2 \sqrt {x} \sqrt {a+b x}} \]

[In]

Integrate[1/(x^(3/2)*(a + b*x)^(3/2)),x]

[Out]

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

Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.56

method	result	size
gosper	\(-\frac {2 \left (2 b x +a \right )}{\sqrt {x}\, \sqrt {b x +a}\, a^{2}}\)	\(22\)
default	\(-\frac {2}{a \sqrt {x}\, \sqrt {b x +a}}-\frac {4 b \sqrt {x}}{a^{2} \sqrt {b x +a}}\)	\(33\)
risch	\(-\frac {2 \sqrt {b x +a}}{a^{2} \sqrt {x}}-\frac {2 b \sqrt {x}}{a^{2} \sqrt {b x +a}}\)	\(33\)

[In]

int(1/x^(3/2)/(b*x+a)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.87 \[ \int \frac {1}{x^{3/2} (a+b x)^{3/2}} \, dx=-\frac {2 \, {\left (2 \, b x + a\right )} \sqrt {b x + a} \sqrt {x}}{a^{2} b x^{2} + a^{3} x} \]

[In]

integrate(1/x^(3/2)/(b*x+a)^(3/2),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.81 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.05 \[ \int \frac {1}{x^{3/2} (a+b x)^{3/2}} \, dx=- \frac {2}{a \sqrt {b} x \sqrt {\frac {a}{b x} + 1}} - \frac {4 \sqrt {b}}{a^{2} \sqrt {\frac {a}{b x} + 1}} \]

[In]

integrate(1/x**(3/2)/(b*x+a)**(3/2),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.82 \[ \int \frac {1}{x^{3/2} (a+b x)^{3/2}} \, dx=-\frac {2 \, b \sqrt {x}}{\sqrt {b x + a} a^{2}} - \frac {2 \, \sqrt {b x + a}}{a^{2} \sqrt {x}} \]

[In]

integrate(1/x^(3/2)/(b*x+a)^(3/2),x, algorithm="maxima")

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (31) = 62\).

Time = 0.31 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.10 \[ \int \frac {1}{x^{3/2} (a+b x)^{3/2}} \, dx=-\frac {4 \, b^{\frac {5}{2}}}{{\left ({\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )} a {\left | b \right |}} - \frac {2 \, \sqrt {b x + a} b^{2}}{\sqrt {{\left (b x + a\right )} b - a b} a^{2} {\left | b \right |}} \]

[In]

integrate(1/x^(3/2)/(b*x+a)^(3/2),x, algorithm="giac")

[Out]

-4*b^(5/2)/(((sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2 + a*b)*a*abs(b)) - 2*sqrt(b*x + a)*b^2/(sqrt(
(b*x + a)*b - a*b)*a^2*abs(b))

Mupad [B] (verification not implemented)

Time = 0.47 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^{3/2} (a+b x)^{3/2}} \, dx=-\frac {2\,a\,\sqrt {a+b\,x}+4\,b\,x\,\sqrt {a+b\,x}}{\sqrt {x}\,\left (a^3+b\,x\,a^2\right )} \]

[In]

int(1/(x^(3/2)*(a + b*x)^(3/2)),x)

[Out]

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