3.6.28 \(\int x^{3/2} (a-b x)^{3/2} \, dx\) [528]

Optimal. Leaf size=124 \[ -\frac {3 a^3 \sqrt {x} \sqrt {a-b x}}{64 b^2}-\frac {a^2 x^{3/2} \sqrt {a-b x}}{32 b}+\frac {1}{8} a x^{5/2} \sqrt {a-b x}+\frac {1}{4} x^{5/2} (a-b x)^{3/2}+\frac {3 a^4 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{64 b^{5/2}} \]

[Out]

1/4*x^(5/2)*(-b*x+a)^(3/2)+3/64*a^4*arctan(b^(1/2)*x^(1/2)/(-b*x+a)^(1/2))/b^(5/2)-1/32*a^2*x^(3/2)*(-b*x+a)^(
1/2)/b+1/8*a*x^(5/2)*(-b*x+a)^(1/2)-3/64*a^3*x^(1/2)*(-b*x+a)^(1/2)/b^2

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Rubi [A]
time = 0.03, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {52, 65, 223, 209} \begin {gather*} \frac {3 a^4 \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{64 b^{5/2}}-\frac {3 a^3 \sqrt {x} \sqrt {a-b x}}{64 b^2}-\frac {a^2 x^{3/2} \sqrt {a-b x}}{32 b}+\frac {1}{8} a x^{5/2} \sqrt {a-b x}+\frac {1}{4} x^{5/2} (a-b x)^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(3/2)*(a - b*x)^(3/2),x]

[Out]

(-3*a^3*Sqrt[x]*Sqrt[a - b*x])/(64*b^2) - (a^2*x^(3/2)*Sqrt[a - b*x])/(32*b) + (a*x^(5/2)*Sqrt[a - b*x])/8 + (
x^(5/2)*(a - b*x)^(3/2))/4 + (3*a^4*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a - b*x]])/(64*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 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 209

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

Rule 223

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

Rubi steps

\begin {align*} \int x^{3/2} (a-b x)^{3/2} \, dx &=\frac {1}{4} x^{5/2} (a-b x)^{3/2}+\frac {1}{8} (3 a) \int x^{3/2} \sqrt {a-b x} \, dx\\ &=\frac {1}{8} a x^{5/2} \sqrt {a-b x}+\frac {1}{4} x^{5/2} (a-b x)^{3/2}+\frac {1}{16} a^2 \int \frac {x^{3/2}}{\sqrt {a-b x}} \, dx\\ &=-\frac {a^2 x^{3/2} \sqrt {a-b x}}{32 b}+\frac {1}{8} a x^{5/2} \sqrt {a-b x}+\frac {1}{4} x^{5/2} (a-b x)^{3/2}+\frac {\left (3 a^3\right ) \int \frac {\sqrt {x}}{\sqrt {a-b x}} \, dx}{64 b}\\ &=-\frac {3 a^3 \sqrt {x} \sqrt {a-b x}}{64 b^2}-\frac {a^2 x^{3/2} \sqrt {a-b x}}{32 b}+\frac {1}{8} a x^{5/2} \sqrt {a-b x}+\frac {1}{4} x^{5/2} (a-b x)^{3/2}+\frac {\left (3 a^4\right ) \int \frac {1}{\sqrt {x} \sqrt {a-b x}} \, dx}{128 b^2}\\ &=-\frac {3 a^3 \sqrt {x} \sqrt {a-b x}}{64 b^2}-\frac {a^2 x^{3/2} \sqrt {a-b x}}{32 b}+\frac {1}{8} a x^{5/2} \sqrt {a-b x}+\frac {1}{4} x^{5/2} (a-b x)^{3/2}+\frac {\left (3 a^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-b x^2}} \, dx,x,\sqrt {x}\right )}{64 b^2}\\ &=-\frac {3 a^3 \sqrt {x} \sqrt {a-b x}}{64 b^2}-\frac {a^2 x^{3/2} \sqrt {a-b x}}{32 b}+\frac {1}{8} a x^{5/2} \sqrt {a-b x}+\frac {1}{4} x^{5/2} (a-b x)^{3/2}+\frac {\left (3 a^4\right ) \text {Subst}\left (\int \frac {1}{1+b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a-b x}}\right )}{64 b^2}\\ &=-\frac {3 a^3 \sqrt {x} \sqrt {a-b x}}{64 b^2}-\frac {a^2 x^{3/2} \sqrt {a-b x}}{32 b}+\frac {1}{8} a x^{5/2} \sqrt {a-b x}+\frac {1}{4} x^{5/2} (a-b x)^{3/2}+\frac {3 a^4 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{64 b^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.14, size = 93, normalized size = 0.75 \begin {gather*} \frac {1}{64} \left (-\frac {\sqrt {x} \sqrt {a-b x} \left (3 a^3+2 a^2 b x-24 a b^2 x^2+16 b^3 x^3\right )}{b^2}-\frac {3 a^4 \log \left (-\sqrt {-b} \sqrt {x}+\sqrt {a-b x}\right )}{(-b)^{5/2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(3/2)*(a - b*x)^(3/2),x]

[Out]

(-((Sqrt[x]*Sqrt[a - b*x]*(3*a^3 + 2*a^2*b*x - 24*a*b^2*x^2 + 16*b^3*x^3))/b^2) - (3*a^4*Log[-(Sqrt[-b]*Sqrt[x
]) + Sqrt[a - b*x]])/(-b)^(5/2))/64

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Maple [A]
time = 0.11, size = 129, normalized size = 1.04

method	result	size
risch	\(-\frac {\left (16 b^{3} x^{3}-24 a \,b^{2} x^{2}+2 a^{2} b x +3 a^{3}\right ) \sqrt {x}\, \sqrt {-b x +a}}{64 b^{2}}+\frac {3 a^{4} \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-x^{2} b +a x}}\right ) \sqrt {x \left (-b x +a \right )}}{128 b^{\frac {5}{2}} \sqrt {x}\, \sqrt {-b x +a}}\)	\(102\)
default	\(-\frac {x^{\frac {3}{2}} \left (-b x +a \right )^{\frac {5}{2}}}{4 b}+\frac {3 a \left (-\frac {\sqrt {x}\, \left (-b x +a \right )^{\frac {5}{2}}}{3 b}+\frac {a \left (\frac {\left (-b x +a \right )^{\frac {3}{2}} \sqrt {x}}{2}+\frac {3 a \left (\sqrt {x}\, \sqrt {-b x +a}+\frac {a \sqrt {x \left (-b x +a \right )}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-x^{2} b +a x}}\right )}{2 \sqrt {-b x +a}\, \sqrt {x}\, \sqrt {b}}\right )}{4}\right )}{6 b}\right )}{8 b}\)	\(129\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4/b*x^(3/2)*(-b*x+a)^(5/2)+3/8*a/b*(-1/3/b*x^(1/2)*(-b*x+a)^(5/2)+1/6*a/b*(1/2*(-b*x+a)^(3/2)*x^(1/2)+3/4*a
*(x^(1/2)*(-b*x+a)^(1/2)+1/2*a*(x*(-b*x+a))^(1/2)/(-b*x+a)^(1/2)/x^(1/2)/b^(1/2)*arctan(b^(1/2)*(x-1/2*a/b)/(-
b*x^2+a*x)^(1/2)))))

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Maxima [A]
time = 0.52, size = 170, normalized size = 1.37 \begin {gather*} -\frac {3 \, a^{4} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right )}{64 \, b^{\frac {5}{2}}} + \frac {\frac {3 \, \sqrt {-b x + a} a^{4} b^{3}}{\sqrt {x}} + \frac {11 \, {\left (-b x + a\right )}^{\frac {3}{2}} a^{4} b^{2}}{x^{\frac {3}{2}}} - \frac {11 \, {\left (-b x + a\right )}^{\frac {5}{2}} a^{4} b}{x^{\frac {5}{2}}} - \frac {3 \, {\left (-b x + a\right )}^{\frac {7}{2}} a^{4}}{x^{\frac {7}{2}}}}{64 \, {\left (b^{6} - \frac {4 \, {\left (b x - a\right )} b^{5}}{x} + \frac {6 \, {\left (b x - a\right )}^{2} b^{4}}{x^{2}} - \frac {4 \, {\left (b x - a\right )}^{3} b^{3}}{x^{3}} + \frac {{\left (b x - a\right )}^{4} b^{2}}{x^{4}}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/64*a^4*arctan(sqrt(-b*x + a)/(sqrt(b)*sqrt(x)))/b^(5/2) + 1/64*(3*sqrt(-b*x + a)*a^4*b^3/sqrt(x) + 11*(-b*x
+ a)^(3/2)*a^4*b^2/x^(3/2) - 11*(-b*x + a)^(5/2)*a^4*b/x^(5/2) - 3*(-b*x + a)^(7/2)*a^4/x^(7/2))/(b^6 - 4*(b*
x - a)*b^5/x + 6*(b*x - a)^2*b^4/x^2 - 4*(b*x - a)^3*b^3/x^3 + (b*x - a)^4*b^2/x^4)

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Fricas [A]
time = 0.48, size = 163, normalized size = 1.31 \begin {gather*} \left [-\frac {3 \, a^{4} \sqrt {-b} \log \left (-2 \, b x + 2 \, \sqrt {-b x + a} \sqrt {-b} \sqrt {x} + a\right ) + 2 \, {\left (16 \, b^{4} x^{3} - 24 \, a b^{3} x^{2} + 2 \, a^{2} b^{2} x + 3 \, a^{3} b\right )} \sqrt {-b x + a} \sqrt {x}}{128 \, b^{3}}, -\frac {3 \, a^{4} \sqrt {b} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right ) + {\left (16 \, b^{4} x^{3} - 24 \, a b^{3} x^{2} + 2 \, a^{2} b^{2} x + 3 \, a^{3} b\right )} \sqrt {-b x + a} \sqrt {x}}{64 \, b^{3}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/128*(3*a^4*sqrt(-b)*log(-2*b*x + 2*sqrt(-b*x + a)*sqrt(-b)*sqrt(x) + a) + 2*(16*b^4*x^3 - 24*a*b^3*x^2 + 2
*a^2*b^2*x + 3*a^3*b)*sqrt(-b*x + a)*sqrt(x))/b^3, -1/64*(3*a^4*sqrt(b)*arctan(sqrt(-b*x + a)/(sqrt(b)*sqrt(x)
)) + (16*b^4*x^3 - 24*a*b^3*x^2 + 2*a^2*b^2*x + 3*a^3*b)*sqrt(-b*x + a)*sqrt(x))/b^3]

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Sympy [C] Result contains complex when optimal does not.
time = 12.59, size = 323, normalized size = 2.60 \begin {gather*} \begin {cases} \frac {3 i a^{\frac {7}{2}} \sqrt {x}}{64 b^{2} \sqrt {-1 + \frac {b x}{a}}} - \frac {i a^{\frac {5}{2}} x^{\frac {3}{2}}}{64 b \sqrt {-1 + \frac {b x}{a}}} - \frac {13 i a^{\frac {3}{2}} x^{\frac {5}{2}}}{32 \sqrt {-1 + \frac {b x}{a}}} + \frac {5 i \sqrt {a} b x^{\frac {7}{2}}}{8 \sqrt {-1 + \frac {b x}{a}}} - \frac {3 i a^{4} \operatorname {acosh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{64 b^{\frac {5}{2}}} - \frac {i b^{2} x^{\frac {9}{2}}}{4 \sqrt {a} \sqrt {-1 + \frac {b x}{a}}} & \text {for}\: \left |{\frac {b x}{a}}\right | > 1 \\- \frac {3 a^{\frac {7}{2}} \sqrt {x}}{64 b^{2} \sqrt {1 - \frac {b x}{a}}} + \frac {a^{\frac {5}{2}} x^{\frac {3}{2}}}{64 b \sqrt {1 - \frac {b x}{a}}} + \frac {13 a^{\frac {3}{2}} x^{\frac {5}{2}}}{32 \sqrt {1 - \frac {b x}{a}}} - \frac {5 \sqrt {a} b x^{\frac {7}{2}}}{8 \sqrt {1 - \frac {b x}{a}}} + \frac {3 a^{4} \operatorname {asin}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{64 b^{\frac {5}{2}}} + \frac {b^{2} x^{\frac {9}{2}}}{4 \sqrt {a} \sqrt {1 - \frac {b x}{a}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((3*I*a**(7/2)*sqrt(x)/(64*b**2*sqrt(-1 + b*x/a)) - I*a**(5/2)*x**(3/2)/(64*b*sqrt(-1 + b*x/a)) - 13*
I*a**(3/2)*x**(5/2)/(32*sqrt(-1 + b*x/a)) + 5*I*sqrt(a)*b*x**(7/2)/(8*sqrt(-1 + b*x/a)) - 3*I*a**4*acosh(sqrt(
b)*sqrt(x)/sqrt(a))/(64*b**(5/2)) - I*b**2*x**(9/2)/(4*sqrt(a)*sqrt(-1 + b*x/a)), Abs(b*x/a) > 1), (-3*a**(7/2
)*sqrt(x)/(64*b**2*sqrt(1 - b*x/a)) + a**(5/2)*x**(3/2)/(64*b*sqrt(1 - b*x/a)) + 13*a**(3/2)*x**(5/2)/(32*sqrt
(1 - b*x/a)) - 5*sqrt(a)*b*x**(7/2)/(8*sqrt(1 - b*x/a)) + 3*a**4*asin(sqrt(b)*sqrt(x)/sqrt(a))/(64*b**(5/2)) +
b**2*x**(9/2)/(4*sqrt(a)*sqrt(1 - b*x/a)), True))

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: Warning, choosing root of [1,0,%%%{-4,[1
,0,0]%%%}+%%%{4,[0,1,1]%%%}+%%%{4,[0,1,0]%%%}+%%%{-4,[0,0,1]%%%},0,%%%{6,[2,0,0]%%%}+%%%{-12,[1,1,1]%%%}+%%%{-
4,[1,1,0]%%%}+%%%{4,[

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(3/2)*(a - b*x)^(3/2),x)

[Out]

int(x^(3/2)*(a - b*x)^(3/2), x)

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