\(\int \frac {\sqrt {2+b x}}{x^{9/2}} \, dx\) [512]

Optimal result

Integrand size = 15, antiderivative size = 59 \[ \int \frac {\sqrt {2+b x}}{x^{9/2}} \, dx=-\frac {(2+b x)^{3/2}}{7 x^{7/2}}+\frac {2 b (2+b x)^{3/2}}{35 x^{5/2}}-\frac {2 b^2 (2+b x)^{3/2}}{105 x^{3/2}} \]

[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)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {47, 37} \[ \int \frac {\sqrt {2+b x}}{x^{9/2}} \, dx=-\frac {2 b^2 (b x+2)^{3/2}}{105 x^{3/2}}+\frac {2 b (b x+2)^{3/2}}{35 x^{5/2}}-\frac {(b x+2)^{3/2}}{7 x^{7/2}} \]

[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*} \text {integral}& = -\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*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.68 \[ \int \frac {\sqrt {2+b x}}{x^{9/2}} \, dx=\frac {\sqrt {2+b x} \left (-30-3 b x+2 b^2 x^2-2 b^3 x^3\right )}{105 x^{7/2}} \]

[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))

Maple [A] (verified)

Time = 0.19 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.46

method	result	size
gosper	\(-\frac {\left (b x +2\right )^{\frac {3}{2}} \left (2 b^{2} x^{2}-6 b x +15\right )}{105 x^{\frac {7}{2}}}\)	\(27\)
meijerg	\(-\frac {2 \sqrt {2}\, \left (\frac {1}{15} b^{3} x^{3}-\frac {1}{15} b^{2} x^{2}+\frac {1}{10} b x +1\right ) \sqrt {\frac {b x}{2}+1}}{7 x^{\frac {7}{2}}}\)	\(39\)
risch	\(-\frac {2 b^{4} x^{4}+2 b^{3} x^{3}-b^{2} x^{2}+36 b x +60}{105 x^{\frac {7}{2}} \sqrt {b x +2}}\)	\(43\)
default	\(-\frac {2 \sqrt {b x +2}}{7 x^{\frac {7}{2}}}+\frac {b \left (-\frac {\sqrt {b x +2}}{5 x^{\frac {5}{2}}}-\frac {2 b \left (-\frac {\sqrt {b x +2}}{3 x^{\frac {3}{2}}}+\frac {b \sqrt {b x +2}}{3 \sqrt {x}}\right )}{5}\right )}{7}\)	\(59\)

[In]

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

[Out]

-1/105*(b*x+2)^(3/2)*(2*b^2*x^2-6*b*x+15)/x^(7/2)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.58 \[ \int \frac {\sqrt {2+b x}}{x^{9/2}} \, dx=-\frac {{\left (2 \, b^{3} x^{3} - 2 \, b^{2} x^{2} + 3 \, b x + 30\right )} \sqrt {b x + 2}}{105 \, x^{\frac {7}{2}}} \]

[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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 270 vs. \(2 (53) = 106\).

Time = 10.51 (sec) , antiderivative size = 270, normalized size of antiderivative = 4.58 \[ \int \frac {\sqrt {2+b x}}{x^{9/2}} \, dx=- \frac {2 b^{\frac {19}{2}} x^{5} \sqrt {1 + \frac {2}{b x}}}{105 b^{6} x^{5} + 420 b^{5} x^{4} + 420 b^{4} x^{3}} - \frac {6 b^{\frac {17}{2}} x^{4} \sqrt {1 + \frac {2}{b x}}}{105 b^{6} x^{5} + 420 b^{5} x^{4} + 420 b^{4} x^{3}} - \frac {3 b^{\frac {15}{2}} x^{3} \sqrt {1 + \frac {2}{b x}}}{105 b^{6} x^{5} + 420 b^{5} x^{4} + 420 b^{4} x^{3}} - \frac {34 b^{\frac {13}{2}} x^{2} \sqrt {1 + \frac {2}{b x}}}{105 b^{6} x^{5} + 420 b^{5} x^{4} + 420 b^{4} x^{3}} - \frac {132 b^{\frac {11}{2}} x \sqrt {1 + \frac {2}{b x}}}{105 b^{6} x^{5} + 420 b^{5} x^{4} + 420 b^{4} x^{3}} - \frac {120 b^{\frac {9}{2}} \sqrt {1 + \frac {2}{b x}}}{105 b^{6} x^{5} + 420 b^{5} x^{4} + 420 b^{4} x^{3}} \]

[In]

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

[Out]

-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)*s
qrt(1 + 2/(b*x))/(105*b**6*x**5 + 420*b**5*x**4 + 420*b**4*x**3)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.69 \[ \int \frac {\sqrt {2+b x}}{x^{9/2}} \, dx=-\frac {{\left (b x + 2\right )}^{\frac {3}{2}} b^{2}}{12 \, x^{\frac {3}{2}}} + \frac {{\left (b x + 2\right )}^{\frac {5}{2}} b}{10 \, x^{\frac {5}{2}}} - \frac {{\left (b x + 2\right )}^{\frac {7}{2}}}{28 \, x^{\frac {7}{2}}} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {2+b x}}{x^{9/2}} \, dx=-\frac {{\left (35 \, b^{7} + 2 \, {\left ({\left (b x + 2\right )} b^{7} - 7 \, b^{7}\right )} {\left (b x + 2\right )}\right )} {\left (b x + 2\right )}^{\frac {3}{2}} b}{105 \, {\left ({\left (b x + 2\right )} b - 2 \, b\right )}^{\frac {7}{2}} {\left | b \right |}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.58 \[ \int \frac {\sqrt {2+b x}}{x^{9/2}} \, dx=-\frac {\sqrt {b\,x+2}\,\left (\frac {2\,b^3\,x^3}{105}-\frac {2\,b^2\,x^2}{105}+\frac {b\,x}{35}+\frac {2}{7}\right )}{x^{7/2}} \]

[In]

int((b*x + 2)^(1/2)/x^(9/2),x)

[Out]

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