3.7.20 \(\int \frac {1}{x^{3/2} (2+b x)^{3/2}} \, dx\) [620]

Optimal. Leaf size=32 \[ \frac {1}{\sqrt {x} \sqrt {2+b x}}-\frac {\sqrt {2+b x}}{\sqrt {x}} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 32, 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 = {47, 37} \begin {gather*} \frac {1}{\sqrt {x} \sqrt {b x+2}}-\frac {\sqrt {b x+2}}{\sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^(3/2)*(2 + b*x)^(3/2)),x]

[Out]

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

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Mathematica [A]
time = 0.05, size = 21, normalized size = 0.66 \begin {gather*} \frac {-1-b x}{\sqrt {x} \sqrt {2+b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-1 - b*x)/(Sqrt[x]*Sqrt[2 + b*x])

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Mathics [A]
time = 2.54, size = 31, normalized size = 0.97 \begin {gather*} \frac {\sqrt {b} \left (-1-b x\right ) \sqrt {\frac {2+b x}{b x}}}{2+b x} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Sqrt[b] (-1 - b x) Sqrt[(2 + b x) / (b x)] / (2 + b x)

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Maple [A]
time = 0.14, size = 27, normalized size = 0.84

method	result	size
gosper	\(-\frac {b x +1}{\sqrt {x}\, \sqrt {b x +2}}\)	\(18\)
meijerg	\(-\frac {\sqrt {2}\, \left (b x +1\right )}{2 \sqrt {x}\, \sqrt {\frac {b x}{2}+1}}\)	\(22\)
default	\(-\frac {1}{\sqrt {x}\, \sqrt {b x +2}}-\frac {b \sqrt {x}}{\sqrt {b x +2}}\)	\(27\)
risch	\(-\frac {\sqrt {b x +2}}{2 \sqrt {x}}-\frac {b \sqrt {x}}{2 \sqrt {b x +2}}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.27, size = 26, normalized size = 0.81 \begin {gather*} -\frac {b \sqrt {x}}{2 \, \sqrt {b x + 2}} - \frac {\sqrt {b x + 2}}{2 \, \sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.31, size = 28, normalized size = 0.88 \begin {gather*} -\frac {\sqrt {b x + 2} {\left (b x + 1\right )} \sqrt {x}}{b x^{2} + 2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.86, size = 34, normalized size = 1.06 \begin {gather*} - \frac {\sqrt {b}}{\sqrt {1 + \frac {2}{b x}}} - \frac {1}{\sqrt {b} x \sqrt {1 + \frac {2}{b x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.00, size = 65, normalized size = 2.03 \begin {gather*} 2 \left (-\frac {2 b \sqrt {x} \sqrt {b x+2}}{8 \left (b x+2\right )}+\frac {2 \sqrt {b}}{2 \left (\left (\sqrt {b x+2}-\sqrt {b} \sqrt {x}\right )^{2}-2\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.35, size = 17, normalized size = 0.53 \begin {gather*} -\frac {b\,x+1}{\sqrt {x}\,\sqrt {b\,x+2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(3/2)*(b*x + 2)^(3/2)),x)

[Out]

-(b*x + 1)/(x^(1/2)*(b*x + 2)^(1/2))

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