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

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

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Rubi [A]  time = 0.01, 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 = {45, 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 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}{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 = 0.01, size = 33, normalized size = 0.57 \begin {gather*} -\frac {2 b^2 x^2-6 b x+3}{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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IntegrateAlgebraic [A]  time = 0.12, size = 40, normalized size = 0.69 \begin {gather*} \frac {\sqrt {2-b x} \left (-2 b^2 x^2+6 b x-3\right )}{3 \sqrt {x} (b x-2)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(x^(3/2)*(2 - b*x)^(5/2)),x]

[Out]

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

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fricas [A]  time = 1.18, 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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giac [B]  time = 1.12, size = 170, normalized size = 2.93 \begin {gather*} -\frac {\sqrt {-b x + 2} b^{2}}{4 \, \sqrt {{\left (b x - 2\right )} b + 2 \, b} {\left | b \right |}} - \frac {3 \, {\left (\sqrt {-b x + 2} \sqrt {-b} - \sqrt {{\left (b x - 2\right )} b + 2 \, b}\right )}^{4} \sqrt {-b} b^{2} - 24 \, {\left (\sqrt {-b x + 2} \sqrt {-b} - \sqrt {{\left (b x - 2\right )} b + 2 \, b}\right )}^{2} \sqrt {-b} b^{3} + 20 \, \sqrt {-b} b^{4}}{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/x^(3/2)/(-b*x+2)^(5/2),x, algorithm="giac")

[Out]

-1/4*sqrt(-b*x + 2)*b^2/(sqrt((b*x - 2)*b + 2*b)*abs(b)) - 1/3*(3*(sqrt(-b*x + 2)*sqrt(-b) - sqrt((b*x - 2)*b
+ 2*b))^4*sqrt(-b)*b^2 - 24*(sqrt(-b*x + 2)*sqrt(-b) - sqrt((b*x - 2)*b + 2*b))^2*sqrt(-b)*b^3 + 20*sqrt(-b)*b
^4)/(((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 = 28, normalized size = 0.48 \begin {gather*} -\frac {2 b^{2} x^{2}-6 b x +3}{3 \left (-b x +2\right )^{\frac {3}{2}} \sqrt {x}} \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]

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

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maxima [A]  time = 1.35, 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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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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sympy [B]  time = 4.00, size = 243, normalized size = 4.19 \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 {2}{\left |{b x}\right |} > 1 \\- \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), 2/Abs(b*x) > 1), (-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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