Integrand size = 16, antiderivative size = 125 \[ \int x^{3/2} (a-b x)^{3/2} \, dx=-\frac {3 a^3 \sqrt {x} \sqrt {a-b x}}{64 b^2}-\frac {a^2 x^{3/2} \sqrt {a-b x}}{32 b}+\frac {3}{8} a x^{5/2} \sqrt {a-b x}-\frac {1}{4} b x^{7/2} \sqrt {a-b x}+\frac {3 a^4 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{64 b^{5/2}} \] Output:
-3/64*a^3*x^(1/2)*(-b*x+a)^(1/2)/b^2-1/32*a^2*x^(3/2)*(-b*x+a)^(1/2)/b+3/8 *a*x^(5/2)*(-b*x+a)^(1/2)-1/4*b*x^(7/2)*(-b*x+a)^(1/2)+3/64*a^4*arctan(b^( 1/2)*x^(1/2)/(-b*x+a)^(1/2))/b^(5/2)
Time = 0.31 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.79 \[ \int x^{3/2} (a-b x)^{3/2} \, dx=\frac {-\sqrt {b} \sqrt {x} \sqrt {a-b x} \left (3 a^3+2 a^2 b x-24 a b^2 x^2+16 b^3 x^3\right )+6 a^4 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a-b x}}\right )}{64 b^{5/2}} \] Input:
Integrate[x^(3/2)*(a - b*x)^(3/2),x]
Output:
(-(Sqrt[b]*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)) + 6*a^4*ArcTan[(Sqrt[b]*Sqrt[x])/(-Sqrt[a] + Sqrt[a - b*x])])/(64* b^(5/2))
Time = 0.21 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.05, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {60, 60, 60, 60, 65, 216}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x^{3/2} (a-b x)^{3/2} \, dx\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {3}{8} a \int x^{3/2} \sqrt {a-b x}dx+\frac {1}{4} x^{5/2} (a-b x)^{3/2}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {3}{8} a \left (\frac {1}{6} a \int \frac {x^{3/2}}{\sqrt {a-b x}}dx+\frac {1}{3} x^{5/2} \sqrt {a-b x}\right )+\frac {1}{4} x^{5/2} (a-b x)^{3/2}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {3}{8} a \left (\frac {1}{6} a \left (\frac {3 a \int \frac {\sqrt {x}}{\sqrt {a-b x}}dx}{4 b}-\frac {x^{3/2} \sqrt {a-b x}}{2 b}\right )+\frac {1}{3} x^{5/2} \sqrt {a-b x}\right )+\frac {1}{4} x^{5/2} (a-b x)^{3/2}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {3}{8} a \left (\frac {1}{6} a \left (\frac {3 a \left (\frac {a \int \frac {1}{\sqrt {x} \sqrt {a-b x}}dx}{2 b}-\frac {\sqrt {x} \sqrt {a-b x}}{b}\right )}{4 b}-\frac {x^{3/2} \sqrt {a-b x}}{2 b}\right )+\frac {1}{3} x^{5/2} \sqrt {a-b x}\right )+\frac {1}{4} x^{5/2} (a-b x)^{3/2}\) |
\(\Big \downarrow \) 65 |
\(\displaystyle \frac {3}{8} a \left (\frac {1}{6} a \left (\frac {3 a \left (\frac {a \int \frac {1}{\frac {b x}{a-b x}+1}d\frac {\sqrt {x}}{\sqrt {a-b x}}}{b}-\frac {\sqrt {x} \sqrt {a-b x}}{b}\right )}{4 b}-\frac {x^{3/2} \sqrt {a-b x}}{2 b}\right )+\frac {1}{3} x^{5/2} \sqrt {a-b x}\right )+\frac {1}{4} x^{5/2} (a-b x)^{3/2}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {3}{8} a \left (\frac {1}{6} a \left (\frac {3 a \left (\frac {a \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{b^{3/2}}-\frac {\sqrt {x} \sqrt {a-b x}}{b}\right )}{4 b}-\frac {x^{3/2} \sqrt {a-b x}}{2 b}\right )+\frac {1}{3} x^{5/2} \sqrt {a-b x}\right )+\frac {1}{4} x^{5/2} (a-b x)^{3/2}\) |
Input:
Int[x^(3/2)*(a - b*x)^(3/2),x]
Output:
(x^(5/2)*(a - b*x)^(3/2))/4 + (3*a*((x^(5/2)*Sqrt[a - b*x])/3 + (a*(-1/2*( x^(3/2)*Sqrt[a - b*x])/b + (3*a*(-((Sqrt[x]*Sqrt[a - b*x])/b) + (a*ArcTan[ (Sqrt[b]*Sqrt[x])/Sqrt[a - b*x]])/b^(3/2)))/(4*b)))/6))/8
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] + Simp[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] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2 Sub st[Int[1/(b - d*x^2), x], x, Sqrt[b*x]/Sqrt[c + d*x]], x] /; FreeQ[{b, c, d }, x] && !GtQ[c, 0]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Time = 0.08 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.82
method | result | size |
risch | \(-\frac {\left (16 b^{3} x^{3}-24 a \,b^{2} x^{2}+2 a^{2} b x +3 a^{3}\right ) \sqrt {x}\, \sqrt {-b x +a}}{64 b^{2}}+\frac {3 a^{4} \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-b \,x^{2}+a x}}\right ) \sqrt {x \left (-b x +a \right )}}{128 b^{\frac {5}{2}} \sqrt {x}\, \sqrt {-b x +a}}\) | \(102\) |
default | \(-\frac {x^{\frac {3}{2}} \left (-b x +a \right )^{\frac {5}{2}}}{4 b}+\frac {3 a \left (-\frac {\sqrt {x}\, \left (-b x +a \right )^{\frac {5}{2}}}{3 b}+\frac {a \left (\frac {\sqrt {x}\, \left (-b x +a \right )^{\frac {3}{2}}}{2}+\frac {3 a \left (\sqrt {x}\, \sqrt {-b x +a}+\frac {a \sqrt {x \left (-b x +a \right )}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-b \,x^{2}+a x}}\right )}{2 \sqrt {-b x +a}\, \sqrt {x}\, \sqrt {b}}\right )}{4}\right )}{6 b}\right )}{8 b}\) | \(129\) |
Input:
int(x^(3/2)*(-b*x+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/64*(16*b^3*x^3-24*a*b^2*x^2+2*a^2*b*x+3*a^3)/b^2*x^(1/2)*(-b*x+a)^(1/2) +3/128*a^4/b^(5/2)*arctan(b^(1/2)*(x-1/2*a/b)/(-b*x^2+a*x)^(1/2))*(x*(-b*x +a))^(1/2)/x^(1/2)/(-b*x+a)^(1/2)
Time = 0.09 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.38 \[ \int x^{3/2} (a-b x)^{3/2} \, dx=\left [-\frac {3 \, a^{4} \sqrt {-b} \log \left (-2 \, b x + 2 \, \sqrt {-b x + a} \sqrt {-b} \sqrt {x} + a\right ) + 2 \, {\left (16 \, b^{4} x^{3} - 24 \, a b^{3} x^{2} + 2 \, a^{2} b^{2} x + 3 \, a^{3} b\right )} \sqrt {-b x + a} \sqrt {x}}{128 \, b^{3}}, -\frac {3 \, a^{4} \sqrt {b} \arctan \left (\frac {\sqrt {-b x + a} \sqrt {b} \sqrt {x}}{b x - a}\right ) + {\left (16 \, b^{4} x^{3} - 24 \, a b^{3} x^{2} + 2 \, a^{2} b^{2} x + 3 \, a^{3} b\right )} \sqrt {-b x + a} \sqrt {x}}{64 \, b^{3}}\right ] \] Input:
integrate(x^(3/2)*(-b*x+a)^(3/2),x, algorithm="fricas")
Output:
[-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)*s qrt(x))/b^3, -1/64*(3*a^4*sqrt(b)*arctan(sqrt(-b*x + a)*sqrt(b)*sqrt(x)/(b *x - a)) + (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]
Result contains complex when optimal does not.
Time = 10.08 (sec) , antiderivative size = 323, normalized size of antiderivative = 2.58 \[ \int x^{3/2} (a-b x)^{3/2} \, dx=\begin {cases} \frac {3 i a^{\frac {7}{2}} \sqrt {x}}{64 b^{2} \sqrt {-1 + \frac {b x}{a}}} - \frac {i a^{\frac {5}{2}} x^{\frac {3}{2}}}{64 b \sqrt {-1 + \frac {b x}{a}}} - \frac {13 i a^{\frac {3}{2}} x^{\frac {5}{2}}}{32 \sqrt {-1 + \frac {b x}{a}}} + \frac {5 i \sqrt {a} b x^{\frac {7}{2}}}{8 \sqrt {-1 + \frac {b x}{a}}} - \frac {3 i a^{4} \operatorname {acosh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{64 b^{\frac {5}{2}}} - \frac {i b^{2} x^{\frac {9}{2}}}{4 \sqrt {a} \sqrt {-1 + \frac {b x}{a}}} & \text {for}\: \left |{\frac {b x}{a}}\right | > 1 \\- \frac {3 a^{\frac {7}{2}} \sqrt {x}}{64 b^{2} \sqrt {1 - \frac {b x}{a}}} + \frac {a^{\frac {5}{2}} x^{\frac {3}{2}}}{64 b \sqrt {1 - \frac {b x}{a}}} + \frac {13 a^{\frac {3}{2}} x^{\frac {5}{2}}}{32 \sqrt {1 - \frac {b x}{a}}} - \frac {5 \sqrt {a} b x^{\frac {7}{2}}}{8 \sqrt {1 - \frac {b x}{a}}} + \frac {3 a^{4} \operatorname {asin}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{64 b^{\frac {5}{2}}} + \frac {b^{2} x^{\frac {9}{2}}}{4 \sqrt {a} \sqrt {1 - \frac {b x}{a}}} & \text {otherwise} \end {cases} \] Input:
integrate(x**(3/2)*(-b*x+a)**(3/2),x)
Output:
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))
Time = 0.11 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.36 \[ \int x^{3/2} (a-b x)^{3/2} \, dx=-\frac {3 \, a^{4} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right )}{64 \, b^{\frac {5}{2}}} + \frac {\frac {3 \, \sqrt {-b x + a} a^{4} b^{3}}{\sqrt {x}} + \frac {11 \, {\left (-b x + a\right )}^{\frac {3}{2}} a^{4} b^{2}}{x^{\frac {3}{2}}} - \frac {11 \, {\left (-b x + a\right )}^{\frac {5}{2}} a^{4} b}{x^{\frac {5}{2}}} - \frac {3 \, {\left (-b x + a\right )}^{\frac {7}{2}} a^{4}}{x^{\frac {7}{2}}}}{64 \, {\left (b^{6} - \frac {4 \, {\left (b x - a\right )} b^{5}}{x} + \frac {6 \, {\left (b x - a\right )}^{2} b^{4}}{x^{2}} - \frac {4 \, {\left (b x - a\right )}^{3} b^{3}}{x^{3}} + \frac {{\left (b x - a\right )}^{4} b^{2}}{x^{4}}\right )}} \] Input:
integrate(x^(3/2)*(-b*x+a)^(3/2),x, algorithm="maxima")
Output:
-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)
Timed out. \[ \int x^{3/2} (a-b x)^{3/2} \, dx=\text {Timed out} \] Input:
integrate(x^(3/2)*(-b*x+a)^(3/2),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int x^{3/2} (a-b x)^{3/2} \, dx=\int x^{3/2}\,{\left (a-b\,x\right )}^{3/2} \,d x \] Input:
int(x^(3/2)*(a - b*x)^(3/2),x)
Output:
int(x^(3/2)*(a - b*x)^(3/2), x)
Time = 0.16 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.82 \[ \int x^{3/2} (a-b x)^{3/2} \, dx=\frac {-3 \sqrt {x}\, \sqrt {-b x +a}\, a^{3} b -2 \sqrt {x}\, \sqrt {-b x +a}\, a^{2} b^{2} x +24 \sqrt {x}\, \sqrt {-b x +a}\, a \,b^{3} x^{2}-16 \sqrt {x}\, \sqrt {-b x +a}\, b^{4} x^{3}-3 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {-b x +a}+\sqrt {x}\, \sqrt {b}\, i}{\sqrt {a}}\right ) a^{4} i}{64 b^{3}} \] Input:
int(x^(3/2)*(-b*x+a)^(3/2),x)
Output:
( - 3*sqrt(x)*sqrt(a - b*x)*a**3*b - 2*sqrt(x)*sqrt(a - b*x)*a**2*b**2*x + 24*sqrt(x)*sqrt(a - b*x)*a*b**3*x**2 - 16*sqrt(x)*sqrt(a - b*x)*b**4*x**3 - 3*sqrt(b)*log((sqrt(a - b*x) + sqrt(x)*sqrt(b)*i)/sqrt(a))*a**4*i)/(64* b**3)