3.566 \(\int \frac {(2-b x)^{5/2}}{\sqrt {x}} \, dx\)

Optimal. Leaf size=82 \[ \frac {1}{3} \sqrt {x} (2-b x)^{5/2}+\frac {5}{6} \sqrt {x} (2-b x)^{3/2}+\frac {5}{2} \sqrt {x} \sqrt {2-b x}+\frac {5 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{\sqrt {b}} \]

[Out]

5*arcsin(1/2*b^(1/2)*x^(1/2)*2^(1/2))/b^(1/2)+5/6*(-b*x+2)^(3/2)*x^(1/2)+1/3*(-b*x+2)^(5/2)*x^(1/2)+5/2*x^(1/2
)*(-b*x+2)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {50, 54, 216} \[ \frac {1}{3} \sqrt {x} (2-b x)^{5/2}+\frac {5}{6} \sqrt {x} (2-b x)^{3/2}+\frac {5}{2} \sqrt {x} \sqrt {2-b x}+\frac {5 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{\sqrt {b}} \]

Antiderivative was successfully verified.

[In]

Int[(2 - b*x)^(5/2)/Sqrt[x],x]

[Out]

(5*Sqrt[x]*Sqrt[2 - b*x])/2 + (5*Sqrt[x]*(2 - b*x)^(3/2))/6 + (Sqrt[x]*(2 - b*x)^(5/2))/3 + (5*ArcSin[(Sqrt[b]
*Sqrt[x])/Sqrt[2]])/Sqrt[b]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 54

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin {align*} \int \frac {(2-b x)^{5/2}}{\sqrt {x}} \, dx &=\frac {1}{3} \sqrt {x} (2-b x)^{5/2}+\frac {5}{3} \int \frac {(2-b x)^{3/2}}{\sqrt {x}} \, dx\\ &=\frac {5}{6} \sqrt {x} (2-b x)^{3/2}+\frac {1}{3} \sqrt {x} (2-b x)^{5/2}+\frac {5}{2} \int \frac {\sqrt {2-b x}}{\sqrt {x}} \, dx\\ &=\frac {5}{2} \sqrt {x} \sqrt {2-b x}+\frac {5}{6} \sqrt {x} (2-b x)^{3/2}+\frac {1}{3} \sqrt {x} (2-b x)^{5/2}+\frac {5}{2} \int \frac {1}{\sqrt {x} \sqrt {2-b x}} \, dx\\ &=\frac {5}{2} \sqrt {x} \sqrt {2-b x}+\frac {5}{6} \sqrt {x} (2-b x)^{3/2}+\frac {1}{3} \sqrt {x} (2-b x)^{5/2}+5 \operatorname {Subst}\left (\int \frac {1}{\sqrt {2-b x^2}} \, dx,x,\sqrt {x}\right )\\ &=\frac {5}{2} \sqrt {x} \sqrt {2-b x}+\frac {5}{6} \sqrt {x} (2-b x)^{3/2}+\frac {1}{3} \sqrt {x} (2-b x)^{5/2}+\frac {5 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{\sqrt {b}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 58, normalized size = 0.71 \[ \frac {1}{6} \sqrt {x} \sqrt {2-b x} \left (2 b^2 x^2-13 b x+33\right )+\frac {5 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{\sqrt {b}} \]

Antiderivative was successfully verified.

[In]

Integrate[(2 - b*x)^(5/2)/Sqrt[x],x]

[Out]

(Sqrt[x]*Sqrt[2 - b*x]*(33 - 13*b*x + 2*b^2*x^2))/6 + (5*ArcSin[(Sqrt[b]*Sqrt[x])/Sqrt[2]])/Sqrt[b]

________________________________________________________________________________________

fricas [A]  time = 0.45, size = 125, normalized size = 1.52 \[ \left [\frac {{\left (2 \, b^{3} x^{2} - 13 \, b^{2} x + 33 \, b\right )} \sqrt {-b x + 2} \sqrt {x} - 15 \, \sqrt {-b} \log \left (-b x + \sqrt {-b x + 2} \sqrt {-b} \sqrt {x} + 1\right )}{6 \, b}, \frac {{\left (2 \, b^{3} x^{2} - 13 \, b^{2} x + 33 \, b\right )} \sqrt {-b x + 2} \sqrt {x} - 30 \, \sqrt {b} \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right )}{6 \, b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x+2)^(5/2)/x^(1/2),x, algorithm="fricas")

[Out]

[1/6*((2*b^3*x^2 - 13*b^2*x + 33*b)*sqrt(-b*x + 2)*sqrt(x) - 15*sqrt(-b)*log(-b*x + sqrt(-b*x + 2)*sqrt(-b)*sq
rt(x) + 1))/b, 1/6*((2*b^3*x^2 - 13*b^2*x + 33*b)*sqrt(-b*x + 2)*sqrt(x) - 30*sqrt(b)*arctan(sqrt(-b*x + 2)/(s
qrt(b)*sqrt(x))))/b]

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x+2)^(5/2)/x^(1/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: Warning, choosing root of [1,0,%%%{4,[1,
1]%%%}+%%%{4,[1,0]%%%}+%%%{-4,[0,1]%%%}+%%%{-8,[0,0]%%%},0,%%%{6,[2,2]%%%}+%%%{4,[2,1]%%%}+%%%{6,[2,0]%%%}+%%%
{-4,[1,2]%%%}+%%%{-28,[1,1]%%%}+%%%{-8,[1,0]%%%}+%%%{6,[0,2]%%%}+%%%{8,[0,1]%%%}+%%%{24,[0,0]%%%},0,%%%{4,[3,3
]%%%}+%%%{-4,[3,2]%%%}+%%%{-4,[3,1]%%%}+%%%{4,[3,0]%%%}+%%%{4,[2,3]%%%}+%%%{-64,[2,2]%%%}+%%%{20,[2,1]%%%}+%%%
{8,[2,0]%%%}+%%%{-4,[1,3]%%%}+%%%{-20,[1,2]%%%}+%%%{128,[1,1]%%%}+%%%{-16,[1,0]%%%}+%%%{-4,[0,3]%%%}+%%%{8,[0,
2]%%%}+%%%{16,[0,1]%%%}+%%%{-32,[0,0]%%%},0,%%%{1,[4,4]%%%}+%%%{-4,[4,3]%%%}+%%%{6,[4,2]%%%}+%%%{-4,[4,1]%%%}+
%%%{1,[4,0]%%%}+%%%{4,[3,4]%%%}+%%%{-12,[3,3]%%%}+%%%{20,[3,2]%%%}+%%%{-20,[3,1]%%%}+%%%{8,[3,0]%%%}+%%%{6,[2,
4]%%%}+%%%{-20,[2,3]%%%}+%%%{46,[2,2]%%%}+%%%{-40,[2,1]%%%}+%%%{24,[2,0]%%%}+%%%{4,[1,4]%%%}+%%%{-20,[1,3]%%%}
+%%%{40,[1,2]%%%}+%%%{-48,[1,1]%%%}+%%%{32,[1,0]%%%}+%%%{1,[0,4]%%%}+%%%{-8,[0,3]%%%}+%%%{24,[0,2]%%%}+%%%{-32
,[0,1]%%%}+%%%{16,[0,0]%%%}] at parameters values [-15.6438432182,61.7937478349]Warning, choosing root of [1,0
,%%%{4,[1,1]%%%}+%%%{4,[1,0]%%%}+%%%{-4,[0,1]%%%}+%%%{-8,[0,0]%%%},0,%%%{6,[2,2]%%%}+%%%{4,[2,1]%%%}+%%%{6,[2,
0]%%%}+%%%{-4,[1,2]%%%}+%%%{-28,[1,1]%%%}+%%%{-8,[1,0]%%%}+%%%{6,[0,2]%%%}+%%%{8,[0,1]%%%}+%%%{24,[0,0]%%%},0,
%%%{4,[3,3]%%%}+%%%{-4,[3,2]%%%}+%%%{-4,[3,1]%%%}+%%%{4,[3,0]%%%}+%%%{4,[2,3]%%%}+%%%{-64,[2,2]%%%}+%%%{20,[2,
1]%%%}+%%%{8,[2,0]%%%}+%%%{-4,[1,3]%%%}+%%%{-20,[1,2]%%%}+%%%{128,[1,1]%%%}+%%%{-16,[1,0]%%%}+%%%{-4,[0,3]%%%}
+%%%{8,[0,2]%%%}+%%%{16,[0,1]%%%}+%%%{-32,[0,0]%%%},0,%%%{1,[4,4]%%%}+%%%{-4,[4,3]%%%}+%%%{6,[4,2]%%%}+%%%{-4,
[4,1]%%%}+%%%{1,[4,0]%%%}+%%%{4,[3,4]%%%}+%%%{-12,[3,3]%%%}+%%%{20,[3,2]%%%}+%%%{-20,[3,1]%%%}+%%%{8,[3,0]%%%}
+%%%{6,[2,4]%%%}+%%%{-20,[2,3]%%%}+%%%{46,[2,2]%%%}+%%%{-40,[2,1]%%%}+%%%{24,[2,0]%%%}+%%%{4,[1,4]%%%}+%%%{-20
,[1,3]%%%}+%%%{40,[1,2]%%%}+%%%{-48,[1,1]%%%}+%%%{32,[1,0]%%%}+%%%{1,[0,4]%%%}+%%%{-8,[0,3]%%%}+%%%{24,[0,2]%%
%}+%%%{-32,[0,1]%%%}+%%%{16,[0,0]%%%}] at parameters values [-29.292030761,78.6493344628]1/abs(b)*b^2/b*(2*((1
/6/b*sqrt(-b*x+2)*sqrt(-b*x+2)+5/12/b)*sqrt(-b*x+2)*sqrt(-b*x+2)+5/4/b)*sqrt(-b*x+2)*sqrt(-b*(-b*x+2)+2*b)+5/s
qrt(-b)*ln(abs(sqrt(-b*(-b*x+2)+2*b)-sqrt(-b)*sqrt(-b*x+2))))

________________________________________________________________________________________

maple [A]  time = 0.00, size = 91, normalized size = 1.11 \[ \frac {\left (-b x +2\right )^{\frac {5}{2}} \sqrt {x}}{3}+\frac {5 \left (-b x +2\right )^{\frac {3}{2}} \sqrt {x}}{6}+\frac {5 \sqrt {-b x +2}\, \sqrt {x}}{2}+\frac {5 \sqrt {\left (-b x +2\right ) x}\, \arctan \left (\frac {\left (x -\frac {1}{b}\right ) \sqrt {b}}{\sqrt {-b \,x^{2}+2 x}}\right )}{2 \sqrt {-b x +2}\, \sqrt {b}\, \sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-b*x+2)^(5/2)/x^(1/2),x)

[Out]

1/3*(-b*x+2)^(5/2)*x^(1/2)+5/6*(-b*x+2)^(3/2)*x^(1/2)+5/2*(-b*x+2)^(1/2)*x^(1/2)+5/2*((-b*x+2)*x)^(1/2)/(-b*x+
2)^(1/2)/b^(1/2)/x^(1/2)*arctan((x-1/b)/(-b*x^2+2*x)^(1/2)*b^(1/2))

________________________________________________________________________________________

maxima [A]  time = 2.98, size = 112, normalized size = 1.37 \[ -\frac {5 \, \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right )}{\sqrt {b}} + \frac {\frac {15 \, \sqrt {-b x + 2} b^{2}}{\sqrt {x}} + \frac {40 \, {\left (-b x + 2\right )}^{\frac {3}{2}} b}{x^{\frac {3}{2}}} + \frac {33 \, {\left (-b x + 2\right )}^{\frac {5}{2}}}{x^{\frac {5}{2}}}}{3 \, {\left (b^{3} - \frac {3 \, {\left (b x - 2\right )} b^{2}}{x} + \frac {3 \, {\left (b x - 2\right )}^{2} b}{x^{2}} - \frac {{\left (b x - 2\right )}^{3}}{x^{3}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x+2)^(5/2)/x^(1/2),x, algorithm="maxima")

[Out]

-5*arctan(sqrt(-b*x + 2)/(sqrt(b)*sqrt(x)))/sqrt(b) + 1/3*(15*sqrt(-b*x + 2)*b^2/sqrt(x) + 40*(-b*x + 2)^(3/2)
*b/x^(3/2) + 33*(-b*x + 2)^(5/2)/x^(5/2))/(b^3 - 3*(b*x - 2)*b^2/x + 3*(b*x - 2)^2*b/x^2 - (b*x - 2)^3/x^3)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (2-b\,x\right )}^{5/2}}{\sqrt {x}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2 - b*x)^(5/2)/x^(1/2),x)

[Out]

int((2 - b*x)^(5/2)/x^(1/2), x)

________________________________________________________________________________________

sympy [A]  time = 5.52, size = 209, normalized size = 2.55 \[ \begin {cases} \frac {i b^{3} x^{\frac {7}{2}}}{3 \sqrt {b x - 2}} - \frac {17 i b^{2} x^{\frac {5}{2}}}{6 \sqrt {b x - 2}} + \frac {59 i b x^{\frac {3}{2}}}{6 \sqrt {b x - 2}} - \frac {11 i \sqrt {x}}{\sqrt {b x - 2}} - \frac {5 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{\sqrt {b}} & \text {for}\: \frac {\left |{b x}\right |}{2} > 1 \\- \frac {b^{3} x^{\frac {7}{2}}}{3 \sqrt {- b x + 2}} + \frac {17 b^{2} x^{\frac {5}{2}}}{6 \sqrt {- b x + 2}} - \frac {59 b x^{\frac {3}{2}}}{6 \sqrt {- b x + 2}} + \frac {11 \sqrt {x}}{\sqrt {- b x + 2}} + \frac {5 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{\sqrt {b}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x+2)**(5/2)/x**(1/2),x)

[Out]

Piecewise((I*b**3*x**(7/2)/(3*sqrt(b*x - 2)) - 17*I*b**2*x**(5/2)/(6*sqrt(b*x - 2)) + 59*I*b*x**(3/2)/(6*sqrt(
b*x - 2)) - 11*I*sqrt(x)/sqrt(b*x - 2) - 5*I*acosh(sqrt(2)*sqrt(b)*sqrt(x)/2)/sqrt(b), Abs(b*x)/2 > 1), (-b**3
*x**(7/2)/(3*sqrt(-b*x + 2)) + 17*b**2*x**(5/2)/(6*sqrt(-b*x + 2)) - 59*b*x**(3/2)/(6*sqrt(-b*x + 2)) + 11*sqr
t(x)/sqrt(-b*x + 2) + 5*asin(sqrt(2)*sqrt(b)*sqrt(x)/2)/sqrt(b), True))

________________________________________________________________________________________