3.7.26 \(\int \frac {1}{\sqrt {x} (2+b x)^{5/2}} \, dx\)

Optimal. Leaf size=37 \[ \frac {\sqrt {x}}{3 \sqrt {b x+2}}+\frac {\sqrt {x}}{3 (b x+2)^{3/2}} \]

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Rubi [A]  time = 0.00, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {45, 37} \begin {gather*} \frac {\sqrt {x}}{3 \sqrt {b x+2}}+\frac {\sqrt {x}}{3 (b x+2)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[x]*(2 + b*x)^(5/2)),x]

[Out]

Sqrt[x]/(3*(2 + b*x)^(3/2)) + Sqrt[x]/(3*Sqrt[2 + 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 45

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*} \int \frac {1}{\sqrt {x} (2+b x)^{5/2}} \, dx &=\frac {\sqrt {x}}{3 (2+b x)^{3/2}}+\frac {1}{3} \int \frac {1}{\sqrt {x} (2+b x)^{3/2}} \, dx\\ &=\frac {\sqrt {x}}{3 (2+b x)^{3/2}}+\frac {\sqrt {x}}{3 \sqrt {2+b x}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 23, normalized size = 0.62 \begin {gather*} \frac {\sqrt {x} (b x+3)}{3 (b x+2)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.07, size = 23, normalized size = 0.62 \begin {gather*} \frac {\sqrt {x} (b x+3)}{3 (b x+2)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.81, size = 32, normalized size = 0.86 \begin {gather*} \frac {{\left (b x + 3\right )} \sqrt {b x + 2} \sqrt {x}}{3 \, {\left (b^{2} x^{2} + 4 \, b x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 1.22, size = 79, normalized size = 2.14 \begin {gather*} \frac {8 \, {\left (3 \, {\left (\sqrt {b x + 2} \sqrt {b} - \sqrt {{\left (b x + 2\right )} b - 2 \, b}\right )}^{2} + 2 \, b\right )} b^{\frac {5}{2}}}{3 \, {\left ({\left (\sqrt {b x + 2} \sqrt {b} - \sqrt {{\left (b x + 2\right )} b - 2 \, b}\right )}^{2} + 2 \, b\right )}^{3} {\left | b \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8/3*(3*(sqrt(b*x + 2)*sqrt(b) - sqrt((b*x + 2)*b - 2*b))^2 + 2*b)*b^(5/2)/(((sqrt(b*x + 2)*sqrt(b) - sqrt((b*x
+ 2)*b - 2*b))^2 + 2*b)^3*abs(b))

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maple [A]  time = 0.00, size = 18, normalized size = 0.49 \begin {gather*} \frac {\left (b x +3\right ) \sqrt {x}}{3 \left (b x +2\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+2)^(5/2)/x^(1/2),x)

[Out]

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

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maxima [A]  time = 1.32, size = 24, normalized size = 0.65 \begin {gather*} -\frac {{\left (b - \frac {3 \, {\left (b x + 2\right )}}{x}\right )} x^{\frac {3}{2}}}{6 \, {\left (b x + 2\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(b - 3*(b*x + 2)/x)*x^(3/2)/(b*x + 2)^(3/2)

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mupad [B]  time = 0.36, size = 42, normalized size = 1.14 \begin {gather*} \frac {3\,\sqrt {x}\,\sqrt {b\,x+2}+b\,x^{3/2}\,\sqrt {b\,x+2}}{3\,b^2\,x^2+12\,b\,x+12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(1/2)*(b*x + 2)^(5/2)),x)

[Out]

(3*x^(1/2)*(b*x + 2)^(1/2) + b*x^(3/2)*(b*x + 2)^(1/2))/(12*b*x + 3*b^2*x^2 + 12)

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sympy [B]  time = 1.84, size = 75, normalized size = 2.03 \begin {gather*} \frac {b x}{3 b^{\frac {3}{2}} x \sqrt {1 + \frac {2}{b x}} + 6 \sqrt {b} \sqrt {1 + \frac {2}{b x}}} + \frac {3}{3 b^{\frac {3}{2}} x \sqrt {1 + \frac {2}{b x}} + 6 \sqrt {b} \sqrt {1 + \frac {2}{b x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*x/(3*b**(3/2)*x*sqrt(1 + 2/(b*x)) + 6*sqrt(b)*sqrt(1 + 2/(b*x))) + 3/(3*b**(3/2)*x*sqrt(1 + 2/(b*x)) + 6*sqr
t(b)*sqrt(1 + 2/(b*x)))

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