3.7.15 \(\int \frac {1}{x^{9/2} \sqrt {2+b x}} \, dx\) [615]

Optimal. Leaf size=80 \[ -\frac {\sqrt {2+b x}}{7 x^{7/2}}+\frac {3 b \sqrt {2+b x}}{35 x^{5/2}}-\frac {2 b^2 \sqrt {2+b x}}{35 x^{3/2}}+\frac {2 b^3 \sqrt {2+b x}}{35 \sqrt {x}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 80, 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 b^3 \sqrt {b x+2}}{35 \sqrt {x}}-\frac {2 b^2 \sqrt {b x+2}}{35 x^{3/2}}+\frac {3 b \sqrt {b x+2}}{35 x^{5/2}}-\frac {\sqrt {b x+2}}{7 x^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^(9/2)*Sqrt[2 + b*x]),x]

[Out]

-1/7*Sqrt[2 + b*x]/x^(7/2) + (3*b*Sqrt[2 + b*x])/(35*x^(5/2)) - (2*b^2*Sqrt[2 + b*x])/(35*x^(3/2)) + (2*b^3*Sq
rt[2 + b*x])/(35*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^{9/2} \sqrt {2+b x}} \, dx &=-\frac {\sqrt {2+b x}}{7 x^{7/2}}-\frac {1}{7} (3 b) \int \frac {1}{x^{7/2} \sqrt {2+b x}} \, dx\\ &=-\frac {\sqrt {2+b x}}{7 x^{7/2}}+\frac {3 b \sqrt {2+b x}}{35 x^{5/2}}+\frac {1}{35} \left (6 b^2\right ) \int \frac {1}{x^{5/2} \sqrt {2+b x}} \, dx\\ &=-\frac {\sqrt {2+b x}}{7 x^{7/2}}+\frac {3 b \sqrt {2+b x}}{35 x^{5/2}}-\frac {2 b^2 \sqrt {2+b x}}{35 x^{3/2}}-\frac {1}{35} \left (2 b^3\right ) \int \frac {1}{x^{3/2} \sqrt {2+b x}} \, dx\\ &=-\frac {\sqrt {2+b x}}{7 x^{7/2}}+\frac {3 b \sqrt {2+b x}}{35 x^{5/2}}-\frac {2 b^2 \sqrt {2+b x}}{35 x^{3/2}}+\frac {2 b^3 \sqrt {2+b x}}{35 \sqrt {x}}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 40, normalized size = 0.50 \begin {gather*} \frac {\sqrt {2+b x} \left (-5+3 b x-2 b^2 x^2+2 b^3 x^3\right )}{35 x^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(9/2)*Sqrt[2 + b*x]),x]

[Out]

(Sqrt[2 + b*x]*(-5 + 3*b*x - 2*b^2*x^2 + 2*b^3*x^3))/(35*x^(7/2))

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Mathics [A]
time = 13.57, size = 89, normalized size = 1.11 \begin {gather*} \frac {\sqrt {b} \left (-40-36 b x-10 b^2 x^2+5 b^3 x^3 \left (1+3 b x+2 b^2 x^2\right )+2 b^6 x^6\right ) \sqrt {\frac {2+b x}{b x}}}{35 x^3 \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^(9/2)*Sqrt[2 + b*x]),x]')

[Out]

Sqrt[b] (-40 - 36 b x - 10 b ^ 2 x ^ 2 + 5 b ^ 3 x ^ 3 (1 + 3 b x + 2 b ^ 2 x ^ 2) + 2 b ^ 6 x ^ 6) Sqrt[(2 +
b x) / (b x)] / (35 x ^ 3 (8 + 12 b x + 6 b ^ 2 x ^ 2 + b ^ 3 x ^ 3))

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Maple [A]
time = 0.13, size = 59, normalized size = 0.74

method	result	size
gosper	\(\frac {\sqrt {b x +2}\, \left (2 b^{3} x^{3}-2 x^{2} b^{2}+3 b x -5\right )}{35 x^{\frac {7}{2}}}\)	\(35\)
meijerg	\(-\frac {\sqrt {2}\, \left (-\frac {2}{5} b^{3} x^{3}+\frac {2}{5} x^{2} b^{2}-\frac {3}{5} b x +1\right ) \sqrt {\frac {b x}{2}+1}}{7 x^{\frac {7}{2}}}\)	\(39\)
risch	\(\frac {2 b^{4} x^{4}+2 b^{3} x^{3}-x^{2} b^{2}+b x -10}{35 x^{\frac {7}{2}} \sqrt {b x +2}}\)	\(42\)
default	\(-\frac {\sqrt {b x +2}}{7 x^{\frac {7}{2}}}-\frac {3 b \left (-\frac {\sqrt {b x +2}}{5 x^{\frac {5}{2}}}-\frac {2 b \left (-\frac {\sqrt {b x +2}}{3 x^{\frac {3}{2}}}+\frac {b \sqrt {b x +2}}{3 \sqrt {x}}\right )}{5}\right )}{7}\)	\(59\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7*(b*x+2)^(1/2)/x^(7/2)-3/7*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 = 0.26, size = 56, normalized size = 0.70 \begin {gather*} \frac {\sqrt {b x + 2} b^{3}}{8 \, \sqrt {x}} - \frac {{\left (b x + 2\right )}^{\frac {3}{2}} b^{2}}{8 \, x^{\frac {3}{2}}} + \frac {3 \, {\left (b x + 2\right )}^{\frac {5}{2}} b}{40 \, x^{\frac {5}{2}}} - \frac {{\left (b x + 2\right )}^{\frac {7}{2}}}{56 \, x^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*sqrt(b*x + 2)*b^3/sqrt(x) - 1/8*(b*x + 2)^(3/2)*b^2/x^(3/2) + 3/40*(b*x + 2)^(5/2)*b/x^(5/2) - 1/56*(b*x +
2)^(7/2)/x^(7/2)

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Fricas [A]
time = 0.32, size = 34, normalized size = 0.42 \begin {gather*} \frac {{\left (2 \, b^{3} x^{3} - 2 \, b^{2} x^{2} + 3 \, b x - 5\right )} \sqrt {b x + 2}}{35 \, x^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 374 vs. \(2 (73) = 146\).
time = 11.82, size = 374, normalized size = 4.68 \begin {gather*} \frac {2 b^{\frac {31}{2}} x^{6} \sqrt {1 + \frac {2}{b x}}}{35 b^{12} x^{6} + 210 b^{11} x^{5} + 420 b^{10} x^{4} + 280 b^{9} x^{3}} + \frac {10 b^{\frac {29}{2}} x^{5} \sqrt {1 + \frac {2}{b x}}}{35 b^{12} x^{6} + 210 b^{11} x^{5} + 420 b^{10} x^{4} + 280 b^{9} x^{3}} + \frac {15 b^{\frac {27}{2}} x^{4} \sqrt {1 + \frac {2}{b x}}}{35 b^{12} x^{6} + 210 b^{11} x^{5} + 420 b^{10} x^{4} + 280 b^{9} x^{3}} + \frac {5 b^{\frac {25}{2}} x^{3} \sqrt {1 + \frac {2}{b x}}}{35 b^{12} x^{6} + 210 b^{11} x^{5} + 420 b^{10} x^{4} + 280 b^{9} x^{3}} - \frac {10 b^{\frac {23}{2}} x^{2} \sqrt {1 + \frac {2}{b x}}}{35 b^{12} x^{6} + 210 b^{11} x^{5} + 420 b^{10} x^{4} + 280 b^{9} x^{3}} - \frac {36 b^{\frac {21}{2}} x \sqrt {1 + \frac {2}{b x}}}{35 b^{12} x^{6} + 210 b^{11} x^{5} + 420 b^{10} x^{4} + 280 b^{9} x^{3}} - \frac {40 b^{\frac {19}{2}} \sqrt {1 + \frac {2}{b x}}}{35 b^{12} x^{6} + 210 b^{11} x^{5} + 420 b^{10} x^{4} + 280 b^{9} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b**(31/2)*x**6*sqrt(1 + 2/(b*x))/(35*b**12*x**6 + 210*b**11*x**5 + 420*b**10*x**4 + 280*b**9*x**3) + 10*b**(
29/2)*x**5*sqrt(1 + 2/(b*x))/(35*b**12*x**6 + 210*b**11*x**5 + 420*b**10*x**4 + 280*b**9*x**3) + 15*b**(27/2)*
x**4*sqrt(1 + 2/(b*x))/(35*b**12*x**6 + 210*b**11*x**5 + 420*b**10*x**4 + 280*b**9*x**3) + 5*b**(25/2)*x**3*sq
rt(1 + 2/(b*x))/(35*b**12*x**6 + 210*b**11*x**5 + 420*b**10*x**4 + 280*b**9*x**3) - 10*b**(23/2)*x**2*sqrt(1 +
2/(b*x))/(35*b**12*x**6 + 210*b**11*x**5 + 420*b**10*x**4 + 280*b**9*x**3) - 36*b**(21/2)*x*sqrt(1 + 2/(b*x))
/(35*b**12*x**6 + 210*b**11*x**5 + 420*b**10*x**4 + 280*b**9*x**3) - 40*b**(19/2)*sqrt(1 + 2/(b*x))/(35*b**12*
x**6 + 210*b**11*x**5 + 420*b**10*x**4 + 280*b**9*x**3)

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Giac [A]
time = 0.01, size = 124, normalized size = 1.55 \begin {gather*} -\frac {512 \sqrt {b} b^{3} \left (-35 \left (\sqrt {b x+2}-\sqrt {b} \sqrt {x}\right )^{6}+42 \left (\sqrt {b x+2}-\sqrt {b} \sqrt {x}\right )^{4}-28 \left (\sqrt {b x+2}-\sqrt {b} \sqrt {x}\right )^{2}+8\right )}{280 \left (\left (\sqrt {b x+2}-\sqrt {b} \sqrt {x}\right )^{2}-2\right )^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

64/35*(35*(sqrt(b)*sqrt(x) - sqrt(b*x + 2))^6 - 42*(sqrt(b)*sqrt(x) - sqrt(b*x + 2))^4 + 28*(sqrt(b)*sqrt(x) -
sqrt(b*x + 2))^2 - 8)*b^(7/2)/((sqrt(b)*sqrt(x) - sqrt(b*x + 2))^2 - 2)^7

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Mupad [B]
time = 0.33, size = 33, normalized size = 0.41 \begin {gather*} \frac {\sqrt {b\,x+2}\,\left (\frac {2\,b^3\,x^3}{35}-\frac {2\,b^2\,x^2}{35}+\frac {3\,b\,x}{35}-\frac {1}{7}\right )}{x^{7/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(9/2)*(b*x + 2)^(1/2)),x)

[Out]

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

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