3.7.45 \(\int \frac {1}{x^{3/2} (2-b x)^{5/2}} \, dx\) [645]

Optimal. Leaf size=58 \[ \frac {1}{3 \sqrt {x} (2-b x)^{3/2}}+\frac {2}{3 \sqrt {x} \sqrt {2-b x}}-\frac {2 \sqrt {2-b x}}{3 \sqrt {x}} \]

[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 = 58, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {47, 37} \begin {gather*} -\frac {2 \sqrt {2-b x}}{3 \sqrt {x}}+\frac {2}{3 \sqrt {x} \sqrt {2-b x}}+\frac {1}{3 \sqrt {x} (2-b x)^{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 = 0.08, size = 33, normalized size = 0.57 \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]

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

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in optimal.
time = 4.49, size = 182, normalized size = 3.14 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {\sqrt {b} \left (-3+6 b x-2 b^2 x^2\right ) \sqrt {\frac {2-b x}{b x}}}{3 \left (4-4 b x+b^2 x^2\right )},\frac {1}{\text {Abs}\left [b x\right ]}>\frac {1}{2}\right \}\right \},\frac {-3 I b^{\frac {9}{2}} \sqrt {1-\frac {2}{b x}}}{12 b^4-12 b^5 x+3 b^6 x^2}+\frac {I 6 b^{\frac {11}{2}} x \sqrt {1-\frac {2}{b x}}}{12 b^4-12 b^5 x+3 b^6 x^2}-\frac {2 I b^{\frac {13}{2}} x^2 \sqrt {1-\frac {2}{b x}}}{12 b^4-12 b^5 x+3 b^6 x^2}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Piecewise[{{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)), 1 /
Abs[b x] > 1 / 2}}, -3 I b ^ (9 / 2) Sqrt[1 - 2 / (b x)] / (12 b ^ 4 - 12 b ^ 5 x + 3 b ^ 6 x ^ 2) + I 6 b ^ (
11 / 2) x Sqrt[1 - 2 / (b x)] / (12 b ^ 4 - 12 b ^ 5 x + 3 b ^ 6 x ^ 2) - 2 I b ^ (13 / 2) x ^ 2 Sqrt[1 - 2 /
(b x)] / (12 b ^ 4 - 12 b ^ 5 x + 3 b ^ 6 x ^ 2)]

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Maple [A]
time = 0.13, size = 45, normalized size = 0.78

method	result	size
gosper	\(-\frac {2 x^{2} b^{2}-6 b x +3}{3 \sqrt {x}\, \left (-b x +2\right )^{\frac {3}{2}}}\)	\(28\)
meijerg	\(-\frac {\sqrt {2}\, \left (2 x^{2} b^{2}-6 b x +3\right )}{12 \sqrt {x}\, \left (-\frac {b x}{2}+1\right )^{\frac {3}{2}}}\)	\(31\)
default	\(-\frac {1}{\left (-b x +2\right )^{\frac {3}{2}} \sqrt {x}}+2 b \left (\frac {\sqrt {x}}{3 \left (-b x +2\right )^{\frac {3}{2}}}+\frac {\sqrt {x}}{3 \sqrt {-b x +2}}\right )\)	\(45\)
risch	\(\frac {\left (b x -2\right ) \sqrt {\left (-b x +2\right ) x}}{4 \sqrt {-x \left (b x -2\right )}\, \sqrt {x}\, \sqrt {-b x +2}}+\frac {b \left (5 b x -12\right ) \sqrt {x}\, \sqrt {\left (-b x +2\right ) x}}{12 \sqrt {-x \left (b x -2\right )}\, \left (b x -2\right ) \sqrt {-b x +2}}\)	\(87\)

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 = 42, normalized size = 0.72 \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 = 0.32, size = 46, normalized size = 0.79 \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 [C] Result contains complex when optimal does not.
time = 2.55, size = 245, normalized size = 4.22 \begin {gather*} \begin {cases} - \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}} & \text {for}\: \frac {1}{\left |{b x}\right |} > \frac {1}{2} \\- \frac {2 i 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 i 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 i b^{\frac {9}{2}} \sqrt {1 - \frac {2}{b x}}}{3 b^{6} x^{2} - 12 b^{5} x + 12 b^{4}} & \text {otherwise} \end {cases} \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]

Piecewise((-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), 1/Abs(b*x) > 1/2), (-2*I*b**(13/2)*x**2*sqrt(1 - 2/(b*x))/(3*b**6*x**2 - 12*b**5*x + 12*b**4) + 6*I*b**(11
/2)*x*sqrt(1 - 2/(b*x))/(3*b**6*x**2 - 12*b**5*x + 12*b**4) - 3*I*b**(9/2)*sqrt(1 - 2/(b*x))/(3*b**6*x**2 - 12
*b**5*x + 12*b**4), True))

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Giac [A]
time = 0.01, size = 100, normalized size = 1.72 \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(-b*x + 2)*sqrt(x)/(b*x - 2)^2 + sqrt(-b)/((sqrt(-b)*sqrt(x) - sqrt(-b*x + 2))^2 -
2)

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Mupad [B]
time = 0.37, size = 59, normalized size = 1.02 \begin {gather*} \frac {3\,\sqrt {2-b\,x}-6\,b\,x\,\sqrt {2-b\,x}+2\,b^2\,x^2\,\sqrt {2-b\,x}}{\sqrt {x}\,\left (x\,\left (12\,b-3\,b^2\,x\right )-12\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(3/2)*(2 - b*x)^(5/2)),x)

[Out]

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

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