3.6.20 \(\int \frac {\sqrt {2-b x}}{x^{9/2}} \, dx\) [520]

Optimal. Leaf size=62 \[ -\frac {(2-b x)^{3/2}}{7 x^{7/2}}-\frac {2 b (2-b x)^{3/2}}{35 x^{5/2}}-\frac {2 b^2 (2-b x)^{3/2}}{105 x^{3/2}} \]

[Out]

-1/7*(-b*x+2)^(3/2)/x^(7/2)-2/35*b*(-b*x+2)^(3/2)/x^(5/2)-2/105*b^2*(-b*x+2)^(3/2)/x^(3/2)

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Rubi [A]
time = 0.01, antiderivative size = 62, 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 b^2 (2-b x)^{3/2}}{105 x^{3/2}}-\frac {2 b (2-b x)^{3/2}}{35 x^{5/2}}-\frac {(2-b x)^{3/2}}{7 x^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[2 - b*x]/x^(9/2),x]

[Out]

-1/7*(2 - b*x)^(3/2)/x^(7/2) - (2*b*(2 - b*x)^(3/2))/(35*x^(5/2)) - (2*b^2*(2 - b*x)^(3/2))/(105*x^(3/2))

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 {\sqrt {2-b x}}{x^{9/2}} \, dx &=-\frac {(2-b x)^{3/2}}{7 x^{7/2}}+\frac {1}{7} (2 b) \int \frac {\sqrt {2-b x}}{x^{7/2}} \, dx\\ &=-\frac {(2-b x)^{3/2}}{7 x^{7/2}}-\frac {2 b (2-b x)^{3/2}}{35 x^{5/2}}+\frac {1}{35} \left (2 b^2\right ) \int \frac {\sqrt {2-b x}}{x^{5/2}} \, dx\\ &=-\frac {(2-b x)^{3/2}}{7 x^{7/2}}-\frac {2 b (2-b x)^{3/2}}{35 x^{5/2}}-\frac {2 b^2 (2-b x)^{3/2}}{105 x^{3/2}}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 41, normalized size = 0.66 \begin {gather*} \frac {\sqrt {2-b x} \left (-30+3 b x+2 b^2 x^2+2 b^3 x^3\right )}{105 x^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[2 - b*x]/x^(9/2),x]

[Out]

(Sqrt[2 - b*x]*(-30 + 3*b*x + 2*b^2*x^2 + 2*b^3*x^3))/(105*x^(7/2))

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in optimal.
time = 11.22, size = 363, normalized size = 5.85 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {\sqrt {b} \left (-120+132 b x-34 b^2 x^2+3 b^3 x^3 \left (1-2 b x\right )+2 b^5 x^5\right ) \sqrt {\frac {2-b x}{b x}}}{105 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 {-120 I b^{\frac {9}{2}} \sqrt {1-\frac {2}{b x}}}{420 b^4 x^3-420 b^5 x^4+105 b^6 x^5}+\frac {I 132 b^{\frac {11}{2}} x \sqrt {1-\frac {2}{b x}}}{420 b^4 x^3-420 b^5 x^4+105 b^6 x^5}-\frac {34 I b^{\frac {13}{2}} x^2 \sqrt {1-\frac {2}{b x}}}{420 b^4 x^3-420 b^5 x^4+105 b^6 x^5}+\frac {I 3 b^{\frac {15}{2}} x^3 \sqrt {1-\frac {2}{b x}}}{420 b^4 x^3-420 b^5 x^4+105 b^6 x^5}-\frac {6 I b^{\frac {17}{2}} x^4 \sqrt {1-\frac {2}{b x}}}{420 b^4 x^3-420 b^5 x^4+105 b^6 x^5}+\frac {I 2 b^{\frac {19}{2}} x^5 \sqrt {1-\frac {2}{b x}}}{420 b^4 x^3-420 b^5 x^4+105 b^6 x^5}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[Sqrt[2 - b*x]/x^(9/2),x]')

[Out]

Piecewise[{{Sqrt[b] (-120 + 132 b x - 34 b ^ 2 x ^ 2 + 3 b ^ 3 x ^ 3 (1 - 2 b x) + 2 b ^ 5 x ^ 5) Sqrt[(2 - b
x) / (b x)] / (105 x ^ 3 (4 - 4 b x + b ^ 2 x ^ 2)), 1 / Abs[b x] > 1 / 2}}, -120 I b ^ (9 / 2) Sqrt[1 - 2 / (
b x)] / (420 b ^ 4 x ^ 3 - 420 b ^ 5 x ^ 4 + 105 b ^ 6 x ^ 5) + I 132 b ^ (11 / 2) x Sqrt[1 - 2 / (b x)] / (42
0 b ^ 4 x ^ 3 - 420 b ^ 5 x ^ 4 + 105 b ^ 6 x ^ 5) - 34 I b ^ (13 / 2) x ^ 2 Sqrt[1 - 2 / (b x)] / (420 b ^ 4
x ^ 3 - 420 b ^ 5 x ^ 4 + 105 b ^ 6 x ^ 5) + I 3 b ^ (15 / 2) x ^ 3 Sqrt[1 - 2 / (b x)] / (420 b ^ 4 x ^ 3 - 4
20 b ^ 5 x ^ 4 + 105 b ^ 6 x ^ 5) - 6 I b ^ (17 / 2) x ^ 4 Sqrt[1 - 2 / (b x)] / (420 b ^ 4 x ^ 3 - 420 b ^ 5
x ^ 4 + 105 b ^ 6 x ^ 5) + I 2 b ^ (19 / 2) x ^ 5 Sqrt[1 - 2 / (b x)] / (420 b ^ 4 x ^ 3 - 420 b ^ 5 x ^ 4 + 1
05 b ^ 6 x ^ 5)]

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Maple [A]
time = 0.13, size = 63, normalized size = 1.02

method	result	size
gosper	\(-\frac {\left (2 x^{2} b^{2}+6 b x +15\right ) \left (-b x +2\right )^{\frac {3}{2}}}{105 x^{\frac {7}{2}}}\)	\(28\)
meijerg	\(-\frac {2 \sqrt {2}\, \left (-\frac {1}{15} b^{3} x^{3}-\frac {1}{15} x^{2} b^{2}-\frac {1}{10} b x +1\right ) \sqrt {-\frac {b x}{2}+1}}{7 x^{\frac {7}{2}}}\)	\(39\)
default	\(-\frac {2 \sqrt {-b x +2}}{7 x^{\frac {7}{2}}}-\frac {b \left (-\frac {\sqrt {-b x +2}}{5 x^{\frac {5}{2}}}+\frac {2 b \left (-\frac {\sqrt {-b x +2}}{3 x^{\frac {3}{2}}}-\frac {b \sqrt {-b x +2}}{3 \sqrt {x}}\right )}{5}\right )}{7}\)	\(63\)
risch	\(-\frac {\sqrt {\left (-b x +2\right ) x}\, \left (2 b^{4} x^{4}-2 b^{3} x^{3}-x^{2} b^{2}-36 b x +60\right )}{105 x^{\frac {7}{2}} \sqrt {-b x +2}\, \sqrt {-x \left (b x -2\right )}}\)	\(64\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-b*x+2)^(1/2)/x^(9/2),x,method=_RETURNVERBOSE)

[Out]

-2/7*(-b*x+2)^(1/2)/x^(7/2)-1/7*b*(-1/5*(-b*x+2)^(1/2)/x^(5/2)+2/5*b*(-1/3*(-b*x+2)^(1/2)/x^(3/2)-1/3*b*(-b*x+
2)^(1/2)/x^(1/2)))

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Maxima [A]
time = 0.28, size = 44, normalized size = 0.71 \begin {gather*} -\frac {{\left (-b x + 2\right )}^{\frac {3}{2}} b^{2}}{12 \, x^{\frac {3}{2}}} - \frac {{\left (-b x + 2\right )}^{\frac {5}{2}} b}{10 \, x^{\frac {5}{2}}} - \frac {{\left (-b x + 2\right )}^{\frac {7}{2}}}{28 \, x^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x+2)^(1/2)/x^(9/2),x, algorithm="maxima")

[Out]

-1/12*(-b*x + 2)^(3/2)*b^2/x^(3/2) - 1/10*(-b*x + 2)^(5/2)*b/x^(5/2) - 1/28*(-b*x + 2)^(7/2)/x^(7/2)

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Fricas [A]
time = 0.30, size = 35, normalized size = 0.56 \begin {gather*} \frac {{\left (2 \, b^{3} x^{3} + 2 \, b^{2} x^{2} + 3 \, b x - 30\right )} \sqrt {-b x + 2}}{105 \, x^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x+2)^(1/2)/x^(9/2),x, algorithm="fricas")

[Out]

1/105*(2*b^3*x^3 + 2*b^2*x^2 + 3*b*x - 30)*sqrt(-b*x + 2)/x^(7/2)

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Sympy [C] Result contains complex when optimal does not.
time = 11.45, size = 556, normalized size = 8.97 \begin {gather*} \begin {cases} \frac {2 b^{\frac {19}{2}} x^{5} \sqrt {-1 + \frac {2}{b x}}}{105 b^{6} x^{5} - 420 b^{5} x^{4} + 420 b^{4} x^{3}} - \frac {6 b^{\frac {17}{2}} x^{4} \sqrt {-1 + \frac {2}{b x}}}{105 b^{6} x^{5} - 420 b^{5} x^{4} + 420 b^{4} x^{3}} + \frac {3 b^{\frac {15}{2}} x^{3} \sqrt {-1 + \frac {2}{b x}}}{105 b^{6} x^{5} - 420 b^{5} x^{4} + 420 b^{4} x^{3}} - \frac {34 b^{\frac {13}{2}} x^{2} \sqrt {-1 + \frac {2}{b x}}}{105 b^{6} x^{5} - 420 b^{5} x^{4} + 420 b^{4} x^{3}} + \frac {132 b^{\frac {11}{2}} x \sqrt {-1 + \frac {2}{b x}}}{105 b^{6} x^{5} - 420 b^{5} x^{4} + 420 b^{4} x^{3}} - \frac {120 b^{\frac {9}{2}} \sqrt {-1 + \frac {2}{b x}}}{105 b^{6} x^{5} - 420 b^{5} x^{4} + 420 b^{4} x^{3}} & \text {for}\: \frac {1}{\left |{b x}\right |} > \frac {1}{2} \\\frac {2 i b^{\frac {19}{2}} x^{5} \sqrt {1 - \frac {2}{b x}}}{105 b^{6} x^{5} - 420 b^{5} x^{4} + 420 b^{4} x^{3}} - \frac {6 i b^{\frac {17}{2}} x^{4} \sqrt {1 - \frac {2}{b x}}}{105 b^{6} x^{5} - 420 b^{5} x^{4} + 420 b^{4} x^{3}} + \frac {3 i b^{\frac {15}{2}} x^{3} \sqrt {1 - \frac {2}{b x}}}{105 b^{6} x^{5} - 420 b^{5} x^{4} + 420 b^{4} x^{3}} - \frac {34 i b^{\frac {13}{2}} x^{2} \sqrt {1 - \frac {2}{b x}}}{105 b^{6} x^{5} - 420 b^{5} x^{4} + 420 b^{4} x^{3}} + \frac {132 i b^{\frac {11}{2}} x \sqrt {1 - \frac {2}{b x}}}{105 b^{6} x^{5} - 420 b^{5} x^{4} + 420 b^{4} x^{3}} - \frac {120 i b^{\frac {9}{2}} \sqrt {1 - \frac {2}{b x}}}{105 b^{6} x^{5} - 420 b^{5} x^{4} + 420 b^{4} x^{3}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x+2)**(1/2)/x**(9/2),x)

[Out]

Piecewise((2*b**(19/2)*x**5*sqrt(-1 + 2/(b*x))/(105*b**6*x**5 - 420*b**5*x**4 + 420*b**4*x**3) - 6*b**(17/2)*x
**4*sqrt(-1 + 2/(b*x))/(105*b**6*x**5 - 420*b**5*x**4 + 420*b**4*x**3) + 3*b**(15/2)*x**3*sqrt(-1 + 2/(b*x))/(
105*b**6*x**5 - 420*b**5*x**4 + 420*b**4*x**3) - 34*b**(13/2)*x**2*sqrt(-1 + 2/(b*x))/(105*b**6*x**5 - 420*b**
5*x**4 + 420*b**4*x**3) + 132*b**(11/2)*x*sqrt(-1 + 2/(b*x))/(105*b**6*x**5 - 420*b**5*x**4 + 420*b**4*x**3) -
120*b**(9/2)*sqrt(-1 + 2/(b*x))/(105*b**6*x**5 - 420*b**5*x**4 + 420*b**4*x**3), 1/Abs(b*x) > 1/2), (2*I*b**(
19/2)*x**5*sqrt(1 - 2/(b*x))/(105*b**6*x**5 - 420*b**5*x**4 + 420*b**4*x**3) - 6*I*b**(17/2)*x**4*sqrt(1 - 2/(
b*x))/(105*b**6*x**5 - 420*b**5*x**4 + 420*b**4*x**3) + 3*I*b**(15/2)*x**3*sqrt(1 - 2/(b*x))/(105*b**6*x**5 -
420*b**5*x**4 + 420*b**4*x**3) - 34*I*b**(13/2)*x**2*sqrt(1 - 2/(b*x))/(105*b**6*x**5 - 420*b**5*x**4 + 420*b*
*4*x**3) + 132*I*b**(11/2)*x*sqrt(1 - 2/(b*x))/(105*b**6*x**5 - 420*b**5*x**4 + 420*b**4*x**3) - 120*I*b**(9/2
)*sqrt(1 - 2/(b*x))/(105*b**6*x**5 - 420*b**5*x**4 + 420*b**4*x**3), True))

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Giac [A]
time = 0.01, size = 139, normalized size = 2.24 \begin {gather*} \frac {2 b^{2} \left (\left (-\frac {70}{7350} b^{7} \sqrt {-b x+2} \sqrt {-b x+2}+\frac {490}{7350} b^{7}\right ) \sqrt {-b x+2} \sqrt {-b x+2}-\frac {1225}{7350} b^{7}\right ) \sqrt {-b x+2} \sqrt {-b x+2} \sqrt {-b x+2} \sqrt {-b \left (-b x+2\right )+2 b}}{\left |b\right | b \left (-b \left (-b x+2\right )+2 b\right )^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x+2)^(1/2)/x^(9/2),x)

[Out]

1/105*(35*b^7 + 2*((b*x - 2)*b^7 + 7*b^7)*(b*x - 2))*(b*x - 2)*sqrt(-b*x + 2)*b/(((b*x - 2)*b + 2*b)^(7/2)*abs
(b))

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Mupad [B]
time = 0.22, size = 34, normalized size = 0.55 \begin {gather*} \frac {\sqrt {2-b\,x}\,\left (\frac {2\,b^3\,x^3}{105}+\frac {2\,b^2\,x^2}{105}+\frac {b\,x}{35}-\frac {2}{7}\right )}{x^{7/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2 - b*x)^(1/2)/x^(9/2),x)

[Out]

((2 - b*x)^(1/2)*((b*x)/35 + (2*b^2*x^2)/105 + (2*b^3*x^3)/105 - 2/7))/x^(7/2)

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