3.6.40 \(\int x^{3/2} (2-b x)^{3/2} \, dx\) [540]

Optimal. Leaf size=109 \[ -\frac {3 \sqrt {x} \sqrt {2-b x}}{8 b^2}-\frac {x^{3/2} \sqrt {2-b x}}{8 b}+\frac {1}{4} x^{5/2} \sqrt {2-b x}+\frac {1}{4} x^{5/2} (2-b x)^{3/2}+\frac {3 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{4 b^{5/2}} \]

[Out]

1/4*x^(5/2)*(-b*x+2)^(3/2)+3/4*arcsin(1/2*b^(1/2)*x^(1/2)*2^(1/2))/b^(5/2)-1/8*x^(3/2)*(-b*x+2)^(1/2)/b+1/4*x^
(5/2)*(-b*x+2)^(1/2)-3/8*x^(1/2)*(-b*x+2)^(1/2)/b^2

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Rubi [A]
time = 0.02, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {52, 56, 222} \begin {gather*} \frac {3 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{4 b^{5/2}}-\frac {3 \sqrt {x} \sqrt {2-b x}}{8 b^2}+\frac {1}{4} x^{5/2} (2-b x)^{3/2}+\frac {1}{4} x^{5/2} \sqrt {2-b x}-\frac {x^{3/2} \sqrt {2-b x}}{8 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(3/2)*(2 - b*x)^(3/2),x]

[Out]

(-3*Sqrt[x]*Sqrt[2 - b*x])/(8*b^2) - (x^(3/2)*Sqrt[2 - b*x])/(8*b) + (x^(5/2)*Sqrt[2 - b*x])/4 + (x^(5/2)*(2 -
b*x)^(3/2))/4 + (3*ArcSin[(Sqrt[b]*Sqrt[x])/Sqrt[2]])/(4*b^(5/2))

Rule 52

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 56

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 222

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 x^{3/2} (2-b x)^{3/2} \, dx &=\frac {1}{4} x^{5/2} (2-b x)^{3/2}+\frac {3}{4} \int x^{3/2} \sqrt {2-b x} \, dx\\ &=\frac {1}{4} x^{5/2} \sqrt {2-b x}+\frac {1}{4} x^{5/2} (2-b x)^{3/2}+\frac {1}{4} \int \frac {x^{3/2}}{\sqrt {2-b x}} \, dx\\ &=-\frac {x^{3/2} \sqrt {2-b x}}{8 b}+\frac {1}{4} x^{5/2} \sqrt {2-b x}+\frac {1}{4} x^{5/2} (2-b x)^{3/2}+\frac {3 \int \frac {\sqrt {x}}{\sqrt {2-b x}} \, dx}{8 b}\\ &=-\frac {3 \sqrt {x} \sqrt {2-b x}}{8 b^2}-\frac {x^{3/2} \sqrt {2-b x}}{8 b}+\frac {1}{4} x^{5/2} \sqrt {2-b x}+\frac {1}{4} x^{5/2} (2-b x)^{3/2}+\frac {3 \int \frac {1}{\sqrt {x} \sqrt {2-b x}} \, dx}{8 b^2}\\ &=-\frac {3 \sqrt {x} \sqrt {2-b x}}{8 b^2}-\frac {x^{3/2} \sqrt {2-b x}}{8 b}+\frac {1}{4} x^{5/2} \sqrt {2-b x}+\frac {1}{4} x^{5/2} (2-b x)^{3/2}+\frac {3 \text {Subst}\left (\int \frac {1}{\sqrt {2-b x^2}} \, dx,x,\sqrt {x}\right )}{4 b^2}\\ &=-\frac {3 \sqrt {x} \sqrt {2-b x}}{8 b^2}-\frac {x^{3/2} \sqrt {2-b x}}{8 b}+\frac {1}{4} x^{5/2} \sqrt {2-b x}+\frac {1}{4} x^{5/2} (2-b x)^{3/2}+\frac {3 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{4 b^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.10, size = 81, normalized size = 0.74 \begin {gather*} -\frac {\sqrt {x} \sqrt {2-b x} \left (3+b x-6 b^2 x^2+2 b^3 x^3\right )}{8 b^2}-\frac {3 \log \left (-\sqrt {-b} \sqrt {x}+\sqrt {2-b x}\right )}{4 (-b)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(3/2)*(2 - b*x)^(3/2),x]

[Out]

-1/8*(Sqrt[x]*Sqrt[2 - b*x]*(3 + b*x - 6*b^2*x^2 + 2*b^3*x^3))/b^2 - (3*Log[-(Sqrt[-b]*Sqrt[x]) + Sqrt[2 - b*x
]])/(4*(-b)^(5/2))

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Mathics [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[x^(3/2)*(2 - b*x)^(3/2),x]')

[Out]

Timed out

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Maple [A]
time = 0.11, size = 122, normalized size = 1.12

method	result	size
meijerg	\(-\frac {12 \left (-\frac {\sqrt {\pi }\, \sqrt {x}\, \sqrt {2}\, \left (-b \right )^{\frac {5}{2}} \left (10 b^{3} x^{3}-30 x^{2} b^{2}+5 b x +15\right ) \sqrt {-\frac {b x}{2}+1}}{480 b^{2}}+\frac {\sqrt {\pi }\, \left (-b \right )^{\frac {5}{2}} \arcsin \left (\frac {\sqrt {b}\, \sqrt {x}\, \sqrt {2}}{2}\right )}{16 b^{\frac {5}{2}}}\right )}{\left (-b \right )^{\frac {3}{2}} \sqrt {\pi }\, b}\)	\(89\)
risch	\(\frac {\left (2 b^{3} x^{3}-6 x^{2} b^{2}+b x +3\right ) \sqrt {x}\, \left (b x -2\right ) \sqrt {\left (-b x +2\right ) x}}{8 b^{2} \sqrt {-x \left (b x -2\right )}\, \sqrt {-b x +2}}+\frac {3 \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {1}{b}\right )}{\sqrt {-x^{2} b +2 x}}\right ) \sqrt {\left (-b x +2\right ) x}}{8 b^{\frac {5}{2}} \sqrt {x}\, \sqrt {-b x +2}}\)	\(114\)
default	\(-\frac {x^{\frac {3}{2}} \left (-b x +2\right )^{\frac {5}{2}}}{4 b}+\frac {-\frac {\sqrt {x}\, \left (-b x +2\right )^{\frac {5}{2}}}{4 b}+\frac {\frac {\left (-b x +2\right )^{\frac {3}{2}} \sqrt {x}}{2}+\frac {3 \sqrt {x}\, \sqrt {-b x +2}}{2}+\frac {3 \sqrt {\left (-b x +2\right ) x}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {1}{b}\right )}{\sqrt {-x^{2} b +2 x}}\right )}{2 \sqrt {-b x +2}\, \sqrt {x}\, \sqrt {b}}}{4 b}}{b}\)	\(122\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(3/2)*(-b*x+2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/4/b*x^(3/2)*(-b*x+2)^(5/2)+3/4/b*(-1/3/b*x^(1/2)*(-b*x+2)^(5/2)+1/3/b*(1/2*(-b*x+2)^(3/2)*x^(1/2)+3/2*x^(1/
2)*(-b*x+2)^(1/2)+3/2*((-b*x+2)*x)^(1/2)/(-b*x+2)^(1/2)/x^(1/2)/b^(1/2)*arctan(b^(1/2)*(x-1/b)/(-b*x^2+2*x)^(1
/2))))

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Maxima [A]
time = 0.35, size = 147, normalized size = 1.35 \begin {gather*} \frac {\frac {3 \, \sqrt {-b x + 2} b^{3}}{\sqrt {x}} + \frac {11 \, {\left (-b x + 2\right )}^{\frac {3}{2}} b^{2}}{x^{\frac {3}{2}}} - \frac {11 \, {\left (-b x + 2\right )}^{\frac {5}{2}} b}{x^{\frac {5}{2}}} - \frac {3 \, {\left (-b x + 2\right )}^{\frac {7}{2}}}{x^{\frac {7}{2}}}}{4 \, {\left (b^{6} - \frac {4 \, {\left (b x - 2\right )} b^{5}}{x} + \frac {6 \, {\left (b x - 2\right )}^{2} b^{4}}{x^{2}} - \frac {4 \, {\left (b x - 2\right )}^{3} b^{3}}{x^{3}} + \frac {{\left (b x - 2\right )}^{4} b^{2}}{x^{4}}\right )}} - \frac {3 \, \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right )}{4 \, b^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)*(-b*x+2)^(3/2),x, algorithm="maxima")

[Out]

1/4*(3*sqrt(-b*x + 2)*b^3/sqrt(x) + 11*(-b*x + 2)^(3/2)*b^2/x^(3/2) - 11*(-b*x + 2)^(5/2)*b/x^(5/2) - 3*(-b*x
+ 2)^(7/2)/x^(7/2))/(b^6 - 4*(b*x - 2)*b^5/x + 6*(b*x - 2)^2*b^4/x^2 - 4*(b*x - 2)^3*b^3/x^3 + (b*x - 2)^4*b^2
/x^4) - 3/4*arctan(sqrt(-b*x + 2)/(sqrt(b)*sqrt(x)))/b^(5/2)

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Fricas [A]
time = 0.32, size = 139, normalized size = 1.28 \begin {gather*} \left [-\frac {{\left (2 \, b^{4} x^{3} - 6 \, b^{3} x^{2} + b^{2} x + 3 \, b\right )} \sqrt {-b x + 2} \sqrt {x} + 3 \, \sqrt {-b} \log \left (-b x + \sqrt {-b x + 2} \sqrt {-b} \sqrt {x} + 1\right )}{8 \, b^{3}}, -\frac {{\left (2 \, b^{4} x^{3} - 6 \, b^{3} x^{2} + b^{2} x + 3 \, b\right )} \sqrt {-b x + 2} \sqrt {x} + 6 \, \sqrt {b} \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right )}{8 \, b^{3}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)*(-b*x+2)^(3/2),x, algorithm="fricas")

[Out]

[-1/8*((2*b^4*x^3 - 6*b^3*x^2 + b^2*x + 3*b)*sqrt(-b*x + 2)*sqrt(x) + 3*sqrt(-b)*log(-b*x + sqrt(-b*x + 2)*sqr
t(-b)*sqrt(x) + 1))/b^3, -1/8*((2*b^4*x^3 - 6*b^3*x^2 + b^2*x + 3*b)*sqrt(-b*x + 2)*sqrt(x) + 6*sqrt(b)*arctan
(sqrt(-b*x + 2)/(sqrt(b)*sqrt(x))))/b^3]

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Sympy [A]
time = 11.51, size = 250, normalized size = 2.29 \begin {gather*} \begin {cases} - \frac {i b^{2} x^{\frac {9}{2}}}{4 \sqrt {b x - 2}} + \frac {5 i b x^{\frac {7}{2}}}{4 \sqrt {b x - 2}} - \frac {13 i x^{\frac {5}{2}}}{8 \sqrt {b x - 2}} - \frac {i x^{\frac {3}{2}}}{8 b \sqrt {b x - 2}} + \frac {3 i \sqrt {x}}{4 b^{2} \sqrt {b x - 2}} - \frac {3 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{4 b^{\frac {5}{2}}} & \text {for}\: \left |{b x}\right | > 2 \\\frac {b^{2} x^{\frac {9}{2}}}{4 \sqrt {- b x + 2}} - \frac {5 b x^{\frac {7}{2}}}{4 \sqrt {- b x + 2}} + \frac {13 x^{\frac {5}{2}}}{8 \sqrt {- b x + 2}} + \frac {x^{\frac {3}{2}}}{8 b \sqrt {- b x + 2}} - \frac {3 \sqrt {x}}{4 b^{2} \sqrt {- b x + 2}} + \frac {3 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{4 b^{\frac {5}{2}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(3/2)*(-b*x+2)**(3/2),x)

[Out]

Piecewise((-I*b**2*x**(9/2)/(4*sqrt(b*x - 2)) + 5*I*b*x**(7/2)/(4*sqrt(b*x - 2)) - 13*I*x**(5/2)/(8*sqrt(b*x -
2)) - I*x**(3/2)/(8*b*sqrt(b*x - 2)) + 3*I*sqrt(x)/(4*b**2*sqrt(b*x - 2)) - 3*I*acosh(sqrt(2)*sqrt(b)*sqrt(x)
/2)/(4*b**(5/2)), Abs(b*x) > 2), (b**2*x**(9/2)/(4*sqrt(-b*x + 2)) - 5*b*x**(7/2)/(4*sqrt(-b*x + 2)) + 13*x**(
5/2)/(8*sqrt(-b*x + 2)) + x**(3/2)/(8*b*sqrt(-b*x + 2)) - 3*sqrt(x)/(4*b**2*sqrt(-b*x + 2)) + 3*asin(sqrt(2)*s
qrt(b)*sqrt(x)/2)/(4*b**(5/2)), True))

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Giac [A]
time = 0.01, size = 271, normalized size = 2.49 \begin {gather*} -2 b \left (2 \left (\left (\left (\frac {\frac {1}{2880}\cdot 180 b^{6} \sqrt {x} \sqrt {x}}{b^{6}}-\frac {\frac {1}{2880}\cdot 60 b^{5}}{b^{6}}\right ) \sqrt {x} \sqrt {x}-\frac {\frac {1}{2880}\cdot 150 b^{4}}{b^{6}}\right ) \sqrt {x} \sqrt {x}-\frac {\frac {1}{2880}\cdot 450 b^{3}}{b^{6}}\right ) \sqrt {x} \sqrt {-b x+2}-\frac {5 \ln \left (\sqrt {-b x+2}-\sqrt {-b} \sqrt {x}\right )}{8 b^{3} \sqrt {-b}}\right )+4 \left (2 \left (\left (\frac {\frac {1}{144}\cdot 12 b^{4} \sqrt {x} \sqrt {x}}{b^{4}}-\frac {\frac {1}{144}\cdot 6 b^{3}}{b^{4}}\right ) \sqrt {x} \sqrt {x}-\frac {\frac {1}{144}\cdot 18 b^{2}}{b^{4}}\right ) \sqrt {x} \sqrt {-b x+2}-\frac {\ln \left (\sqrt {-b x+2}-\sqrt {-b} \sqrt {x}\right )}{2 b^{2} \sqrt {-b}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)*(-b*x+2)^(3/2),x)

[Out]

-1/24*(((2*(3*x - 1/b)*x - 5/b^2)*x - 15/b^3)*sqrt(-b*x + 2)*sqrt(x) - 30*log(-sqrt(-b)*sqrt(x) + sqrt(-b*x +
2))/(sqrt(-b)*b^3))*b + 1/3*sqrt(-b*x + 2)*((2*x - 1/b)*x - 3/b^2)*sqrt(x) - 2*log(-sqrt(-b)*sqrt(x) + sqrt(-b
*x + 2))/(sqrt(-b)*b^2)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^{3/2}\,{\left (2-b\,x\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(3/2)*(2 - b*x)^(3/2),x)

[Out]

int(x^(3/2)*(2 - b*x)^(3/2), x)

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