3.7.27 \(\int \frac {1}{x^{3/2} (2+b x)^{5/2}} \, dx\) [627]

Optimal. Leaf size=55 \[ \frac {1}{3 \sqrt {x} (2+b x)^{3/2}}+\frac {2}{3 \sqrt {x} \sqrt {2+b x}}-\frac {2 \sqrt {2+b x}}{3 \sqrt {x}} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, 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 {2 \sqrt {b x+2}}{3 \sqrt {x}}+\frac {2}{3 \sqrt {x} \sqrt {b x+2}}+\frac {1}{3 \sqrt {x} (b x+2)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^(3/2)*(2 + b*x)^(5/2)),x]

[Out]

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

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Mathematica [A]
time = 0.07, size = 32, normalized size = 0.58 \begin {gather*} \frac {-3-6 b x-2 b^2 x^2}{3 \sqrt {x} (2+b x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(3/2)*(2 + b*x)^(5/2)),x]

[Out]

(-3 - 6*b*x - 2*b^2*x^2)/(3*Sqrt[x]*(2 + b*x)^(3/2))

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Mathics [A]
time = 4.44, size = 48, normalized size = 0.87 \begin {gather*} \frac {\sqrt {b} \left (-3-6 b x-2 b^2 x^2\right ) \sqrt {\frac {2+b x}{b x}}}{12+12 b x+3 b^2 x^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[1/(x^(3/2)*(2 + b*x)^(5/2)),x]')

[Out]

Sqrt[b] (-3 - 6 b x - 2 b ^ 2 x ^ 2) Sqrt[(2 + b x) / (b x)] / (3 (4 + 4 b x + b ^ 2 x ^ 2))

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Maple [A]
time = 0.13, size = 42, normalized size = 0.76

method	result	size
gosper	\(-\frac {2 x^{2} b^{2}+6 b x +3}{3 \sqrt {x}\, \left (b x +2\right )^{\frac {3}{2}}}\)	\(27\)
meijerg	\(-\frac {\sqrt {2}\, \left (2 x^{2} b^{2}+6 b x +3\right )}{12 \sqrt {x}\, \left (\frac {b x}{2}+1\right )^{\frac {3}{2}}}\)	\(31\)
risch	\(-\frac {\sqrt {b x +2}}{4 \sqrt {x}}-\frac {b \left (5 b x +12\right ) \sqrt {x}}{12 \left (b x +2\right )^{\frac {3}{2}}}\)	\(33\)
default	\(-\frac {1}{\left (b x +2\right )^{\frac {3}{2}} \sqrt {x}}-2 b \left (\frac {\sqrt {x}}{3 \left (b x +2\right )^{\frac {3}{2}}}+\frac {\sqrt {x}}{3 \sqrt {b x +2}}\right )\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(3/2)/(b*x+2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 40, normalized size = 0.73 \begin {gather*} \frac {{\left (b^{2} - \frac {6 \, {\left (b x + 2\right )} b}{x}\right )} x^{\frac {3}{2}}}{12 \, {\left (b x + 2\right )}^{\frac {3}{2}}} - \frac {\sqrt {b x + 2}}{4 \, \sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(3/2)/(b*x+2)^(5/2),x, algorithm="maxima")

[Out]

1/12*(b^2 - 6*(b*x + 2)*b/x)*x^(3/2)/(b*x + 2)^(3/2) - 1/4*sqrt(b*x + 2)/sqrt(x)

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Fricas [A]
time = 0.31, size = 45, normalized size = 0.82 \begin {gather*} -\frac {{\left (2 \, b^{2} x^{2} + 6 \, b x + 3\right )} \sqrt {b x + 2} \sqrt {x}}{3 \, {\left (b^{2} x^{3} + 4 \, b x^{2} + 4 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(3/2)/(b*x+2)^(5/2),x, algorithm="fricas")

[Out]

-1/3*(2*b^2*x^2 + 6*b*x + 3)*sqrt(b*x + 2)*sqrt(x)/(b^2*x^3 + 4*b*x^2 + 4*x)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 117 vs. \(2 (49) = 98\).
time = 2.43, size = 117, normalized size = 2.13 \begin {gather*} - \frac {2 b^{\frac {13}{2}} x^{2} \sqrt {1 + \frac {2}{b x}}}{3 b^{6} x^{2} + 12 b^{5} x + 12 b^{4}} - \frac {6 b^{\frac {11}{2}} x \sqrt {1 + \frac {2}{b x}}}{3 b^{6} x^{2} + 12 b^{5} x + 12 b^{4}} - \frac {3 b^{\frac {9}{2}} \sqrt {1 + \frac {2}{b x}}}{3 b^{6} x^{2} + 12 b^{5} x + 12 b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**(3/2)/(b*x+2)**(5/2),x)

[Out]

-2*b**(13/2)*x**2*sqrt(1 + 2/(b*x))/(3*b**6*x**2 + 12*b**5*x + 12*b**4) - 6*b**(11/2)*x*sqrt(1 + 2/(b*x))/(3*b
**6*x**2 + 12*b**5*x + 12*b**4) - 3*b**(9/2)*sqrt(1 + 2/(b*x))/(3*b**6*x**2 + 12*b**5*x + 12*b**4)

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Giac [A]
time = 0.01, size = 96, normalized size = 1.75 \begin {gather*} 2 \left (\frac {2 \left (-\frac {\frac {1}{576}\cdot 60 b^{3} \sqrt {x} \sqrt {x}}{b}-\frac {\frac {1}{576}\cdot 144 b^{2}}{b}\right ) \sqrt {x} \sqrt {b x+2}}{\left (b x+2\right )^{2}}+\frac {2 \sqrt {b}}{4 \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)^(5/2),x)

[Out]

-1/12*(5*b^2*x + 12*b)*sqrt(x)/(b*x + 2)^(3/2) + sqrt(b)/((sqrt(b)*sqrt(x) - sqrt(b*x + 2))^2 - 2)

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Mupad [B]
time = 0.38, size = 57, normalized size = 1.04 \begin {gather*} -\frac {3\,\sqrt {b\,x+2}+6\,b\,x\,\sqrt {b\,x+2}+2\,b^2\,x^2\,\sqrt {b\,x+2}}{\sqrt {x}\,\left (x\,\left (3\,x\,b^2+12\,b\right )+12\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(3/2)*(b*x + 2)^(5/2)),x)

[Out]

-(3*(b*x + 2)^(1/2) + 6*b*x*(b*x + 2)^(1/2) + 2*b^2*x^2*(b*x + 2)^(1/2))/(x^(1/2)*(x*(12*b + 3*b^2*x) + 12))

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