\(\int \frac {1}{x^{5/2} \sqrt {a+b x}} \, dx\) [574]

Optimal result

Integrand size = 15, antiderivative size = 44 \[ \int \frac {1}{x^{5/2} \sqrt {a+b x}} \, dx=-\frac {2 \sqrt {a+b x}}{3 a x^{3/2}}+\frac {4 b \sqrt {a+b x}}{3 a^2 \sqrt {x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 44, 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} \[ \int \frac {1}{x^{5/2} \sqrt {a+b x}} \, dx=\frac {4 b \sqrt {a+b x}}{3 a^2 \sqrt {x}}-\frac {2 \sqrt {a+b x}}{3 a x^{3/2}} \]

[In]

Int[1/(x^(5/2)*Sqrt[a + b*x]),x]

[Out]

(-2*Sqrt[a + b*x])/(3*a*x^(3/2)) + (4*b*Sqrt[a + b*x])/(3*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 \sqrt {a+b x}}{3 a x^{3/2}}-\frac {(2 b) \int \frac {1}{x^{3/2} \sqrt {a+b x}} \, dx}{3 a} \\ & = -\frac {2 \sqrt {a+b x}}{3 a x^{3/2}}+\frac {4 b \sqrt {a+b x}}{3 a^2 \sqrt {x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.61 \[ \int \frac {1}{x^{5/2} \sqrt {a+b x}} \, dx=-\frac {2 (a-2 b x) \sqrt {a+b x}}{3 a^2 x^{3/2}} \]

[In]

Integrate[1/(x^(5/2)*Sqrt[a + b*x]),x]

[Out]

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

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.50

method	result	size
gosper	\(-\frac {2 \sqrt {b x +a}\, \left (-2 b x +a \right )}{3 x^{\frac {3}{2}} a^{2}}\)	\(22\)
risch	\(-\frac {2 \sqrt {b x +a}\, \left (-2 b x +a \right )}{3 x^{\frac {3}{2}} a^{2}}\)	\(22\)
default	\(-\frac {2 \sqrt {b x +a}}{3 a \,x^{\frac {3}{2}}}+\frac {4 b \sqrt {b x +a}}{3 a^{2} \sqrt {x}}\)	\(33\)

[In]

int(1/x^(5/2)/(b*x+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.52 \[ \int \frac {1}{x^{5/2} \sqrt {a+b x}} \, dx=\frac {2 \, {\left (2 \, b x - a\right )} \sqrt {b x + a}}{3 \, a^{2} x^{\frac {3}{2}}} \]

[In]

integrate(1/x^(5/2)/(b*x+a)^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 1.54 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.95 \[ \int \frac {1}{x^{5/2} \sqrt {a+b x}} \, dx=- \frac {2 \sqrt {b} \sqrt {\frac {a}{b x} + 1}}{3 a x} + \frac {4 b^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}}{3 a^{2}} \]

[In]

integrate(1/x**(5/2)/(b*x+a)**(1/2),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.70 \[ \int \frac {1}{x^{5/2} \sqrt {a+b x}} \, dx=\frac {2 \, {\left (\frac {3 \, \sqrt {b x + a} b}{\sqrt {x}} - \frac {{\left (b x + a\right )}^{\frac {3}{2}}}{x^{\frac {3}{2}}}\right )}}{3 \, a^{2}} \]

[In]

integrate(1/x^(5/2)/(b*x+a)^(1/2),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.14 \[ \int \frac {1}{x^{5/2} \sqrt {a+b x}} \, dx=\frac {2 \, {\left (\frac {2 \, {\left (b x + a\right )} b^{3}}{a^{2}} - \frac {3 \, b^{3}}{a}\right )} \sqrt {b x + a} b}{3 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {3}{2}} {\left | b \right |}} \]

[In]

integrate(1/x^(5/2)/(b*x+a)^(1/2),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.52 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.57 \[ \int \frac {1}{x^{5/2} \sqrt {a+b x}} \, dx=-\frac {\left (\frac {2}{3\,a}-\frac {4\,b\,x}{3\,a^2}\right )\,\sqrt {a+b\,x}}{x^{3/2}} \]

[In]

int(1/(x^(5/2)*(a + b*x)^(1/2)),x)

[Out]

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