3.7.28 \(\int \frac {1}{x^{5/2} (2+b x)^{5/2}} \, dx\) [628]

Optimal. Leaf size=71 \[ \frac {1}{3 x^{3/2} (2+b x)^{3/2}}+\frac {1}{x^{3/2} \sqrt {2+b x}}-\frac {2 \sqrt {2+b x}}{3 x^{3/2}}+\frac {2 b \sqrt {2+b x}}{3 \sqrt {x}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 4, 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 x^{3/2}}+\frac {1}{x^{3/2} \sqrt {b x+2}}+\frac {1}{3 x^{3/2} (b x+2)^{3/2}}+\frac {2 b \sqrt {b x+2}}{3 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^(5/2)*(2 + b*x)^(5/2)),x]

[Out]

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

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Mathematica [A]
time = 0.07, size = 40, normalized size = 0.56 \begin {gather*} \frac {-1+3 b x+6 b^2 x^2+2 b^3 x^3}{3 x^{3/2} (2+b x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathics [A]
time = 7.24, size = 76, normalized size = 1.07 \begin {gather*} \frac {\sqrt {b} \left (-2+b x \left (5+10 b^2 x^2+2 b^3 x^3\right )+15 b^2 x^2\right ) \sqrt {\frac {2+b x}{b x}}}{3 x \left (8+12 b x+6 b^2 x^2+b^3 x^3\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Sqrt[b] (-2 + b x (5 + 10 b ^ 2 x ^ 2 + 2 b ^ 3 x ^ 3) + 15 b ^ 2 x ^ 2) Sqrt[(2 + b x) / (b x)] / (3 x (8 + 1
2 b x + 6 b ^ 2 x ^ 2 + b ^ 3 x ^ 3))

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Maple [A]
time = 0.13, size = 58, normalized size = 0.82

method	result	size
gosper	\(\frac {2 b^{3} x^{3}+6 x^{2} b^{2}+3 b x -1}{3 x^{\frac {3}{2}} \left (b x +2\right )^{\frac {3}{2}}}\)	\(35\)
meijerg	\(-\frac {\sqrt {2}\, \left (-2 b^{3} x^{3}-6 x^{2} b^{2}-3 b x +1\right )}{12 x^{\frac {3}{2}} \left (\frac {b x}{2}+1\right )^{\frac {3}{2}}}\)	\(39\)
risch	\(\frac {4 x^{2} b^{2}+7 b x -2}{12 x^{\frac {3}{2}} \sqrt {b x +2}}+\frac {b^{2} \left (4 b x +9\right ) \sqrt {x}}{12 \left (b x +2\right )^{\frac {3}{2}}}\)	\(49\)
default	\(-\frac {1}{3 x^{\frac {3}{2}} \left (b x +2\right )^{\frac {3}{2}}}-b \left (-\frac {1}{\left (b x +2\right )^{\frac {3}{2}} \sqrt {x}}-2 b \left (\frac {\sqrt {x}}{3 \left (b x +2\right )^{\frac {3}{2}}}+\frac {\sqrt {x}}{3 \sqrt {b x +2}}\right )\right )\)	\(58\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3/x^(3/2)/(b*x+2)^(3/2)-b*(-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 = 0.29, size = 55, normalized size = 0.77 \begin {gather*} \frac {3 \, \sqrt {b x + 2} b}{8 \, \sqrt {x}} - \frac {{\left (b^{3} - \frac {9 \, {\left (b x + 2\right )} b^{2}}{x}\right )} x^{\frac {3}{2}}}{24 \, {\left (b x + 2\right )}^{\frac {3}{2}}} - \frac {{\left (b x + 2\right )}^{\frac {3}{2}}}{24 \, x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/8*sqrt(b*x + 2)*b/sqrt(x) - 1/24*(b^3 - 9*(b*x + 2)*b^2/x)*x^(3/2)/(b*x + 2)^(3/2) - 1/24*(b*x + 2)^(3/2)/x^
(3/2)

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Fricas [A]
time = 0.31, size = 55, normalized size = 0.77 \begin {gather*} \frac {{\left (2 \, b^{3} x^{3} + 6 \, b^{2} x^{2} + 3 \, b x - 1\right )} \sqrt {b x + 2} \sqrt {x}}{3 \, {\left (b^{2} x^{4} + 4 \, b x^{3} + 4 \, x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 257 vs. \(2 (66) = 132\).
time = 4.58, size = 257, normalized size = 3.62 \begin {gather*} \frac {2 b^{\frac {27}{2}} x^{4} \sqrt {1 + \frac {2}{b x}}}{3 b^{12} x^{4} + 18 b^{11} x^{3} + 36 b^{10} x^{2} + 24 b^{9} x} + \frac {10 b^{\frac {25}{2}} x^{3} \sqrt {1 + \frac {2}{b x}}}{3 b^{12} x^{4} + 18 b^{11} x^{3} + 36 b^{10} x^{2} + 24 b^{9} x} + \frac {15 b^{\frac {23}{2}} x^{2} \sqrt {1 + \frac {2}{b x}}}{3 b^{12} x^{4} + 18 b^{11} x^{3} + 36 b^{10} x^{2} + 24 b^{9} x} + \frac {5 b^{\frac {21}{2}} x \sqrt {1 + \frac {2}{b x}}}{3 b^{12} x^{4} + 18 b^{11} x^{3} + 36 b^{10} x^{2} + 24 b^{9} x} - \frac {2 b^{\frac {19}{2}} \sqrt {1 + \frac {2}{b x}}}{3 b^{12} x^{4} + 18 b^{11} x^{3} + 36 b^{10} x^{2} + 24 b^{9} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b**(27/2)*x**4*sqrt(1 + 2/(b*x))/(3*b**12*x**4 + 18*b**11*x**3 + 36*b**10*x**2 + 24*b**9*x) + 10*b**(25/2)*x
**3*sqrt(1 + 2/(b*x))/(3*b**12*x**4 + 18*b**11*x**3 + 36*b**10*x**2 + 24*b**9*x) + 15*b**(23/2)*x**2*sqrt(1 +
2/(b*x))/(3*b**12*x**4 + 18*b**11*x**3 + 36*b**10*x**2 + 24*b**9*x) + 5*b**(21/2)*x*sqrt(1 + 2/(b*x))/(3*b**12
*x**4 + 18*b**11*x**3 + 36*b**10*x**2 + 24*b**9*x) - 2*b**(19/2)*sqrt(1 + 2/(b*x))/(3*b**12*x**4 + 18*b**11*x*
*3 + 36*b**10*x**2 + 24*b**9*x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (49) = 98\).
time = 0.01, size = 166, normalized size = 2.34 \begin {gather*} 2 \left (\frac {2 \left (\frac {\frac {1}{2304}\cdot 192 b^{4} \sqrt {x} \sqrt {x}}{b}+\frac {\frac {1}{2304}\cdot 432 b^{3}}{b}\right ) \sqrt {x} \sqrt {b x+2}}{\left (b x+2\right )^{2}}+\frac {2 \left (-3 b \sqrt {b} \left (\sqrt {b x+2}-\sqrt {b} \sqrt {x}\right )^{4}+18 b \sqrt {b} \left (\sqrt {b x+2}-\sqrt {b} \sqrt {x}\right )^{2}-16 b \sqrt {b}\right )}{12 \left (\left (\sqrt {b x+2}-\sqrt {b} \sqrt {x}\right )^{2}-2\right )^{3}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(4*b^3*x + 9*b^2)*sqrt(x)/(b*x + 2)^(3/2) - 1/3*(3*b^(3/2)*(sqrt(b)*sqrt(x) - sqrt(b*x + 2))^4 - 18*b^(3/
2)*(sqrt(b)*sqrt(x) - sqrt(b*x + 2))^2 + 16*b^(3/2))/((sqrt(b)*sqrt(x) - sqrt(b*x + 2))^2 - 2)^3

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Mupad [B]
time = 0.42, size = 71, normalized size = 1.00 \begin {gather*} \frac {3\,b\,x\,\sqrt {b\,x+2}-\sqrt {b\,x+2}+6\,b^2\,x^2\,\sqrt {b\,x+2}+2\,b^3\,x^3\,\sqrt {b\,x+2}}{x^{3/2}\,\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^(5/2)*(b*x + 2)^(5/2)),x)

[Out]

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

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