\(\int x^{5/2} (a-b x)^{5/2} \, dx\) [551]

Optimal result

Integrand size = 16, antiderivative size = 171 \[ \int x^{5/2} (a-b x)^{5/2} \, dx=-\frac {5 a^5 \sqrt {x} \sqrt {a-b x}}{512 b^3}-\frac {5 a^4 x^{3/2} \sqrt {a-b x}}{768 b^2}-\frac {a^3 x^{5/2} \sqrt {a-b x}}{192 b}+\frac {1}{32} a^2 x^{7/2} \sqrt {a-b x}+\frac {1}{12} a x^{7/2} (a-b x)^{3/2}+\frac {1}{6} x^{7/2} (a-b x)^{5/2}+\frac {5 a^6 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{512 b^{7/2}} \]

[Out]

1/12*a*x^(7/2)*(-b*x+a)^(3/2)+1/6*x^(7/2)*(-b*x+a)^(5/2)+5/512*a^6*arctan(b^(1/2)*x^(1/2)/(-b*x+a)^(1/2))/b^(7
/2)-5/768*a^4*x^(3/2)*(-b*x+a)^(1/2)/b^2-1/192*a^3*x^(5/2)*(-b*x+a)^(1/2)/b+1/32*a^2*x^(7/2)*(-b*x+a)^(1/2)-5/
512*a^5*x^(1/2)*(-b*x+a)^(1/2)/b^3

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {52, 65, 223, 209} \[ \int x^{5/2} (a-b x)^{5/2} \, dx=\frac {5 a^6 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{512 b^{7/2}}-\frac {5 a^5 \sqrt {x} \sqrt {a-b x}}{512 b^3}-\frac {5 a^4 x^{3/2} \sqrt {a-b x}}{768 b^2}-\frac {a^3 x^{5/2} \sqrt {a-b x}}{192 b}+\frac {1}{32} a^2 x^{7/2} \sqrt {a-b x}+\frac {1}{12} a x^{7/2} (a-b x)^{3/2}+\frac {1}{6} x^{7/2} (a-b x)^{5/2} \]

[In]

Int[x^(5/2)*(a - b*x)^(5/2),x]

[Out]

(-5*a^5*Sqrt[x]*Sqrt[a - b*x])/(512*b^3) - (5*a^4*x^(3/2)*Sqrt[a - b*x])/(768*b^2) - (a^3*x^(5/2)*Sqrt[a - b*x
])/(192*b) + (a^2*x^(7/2)*Sqrt[a - b*x])/32 + (a*x^(7/2)*(a - b*x)^(3/2))/12 + (x^(7/2)*(a - b*x)^(5/2))/6 + (
5*a^6*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a - b*x]])/(512*b^(7/2))

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*} \text {integral}& = \frac {1}{6} x^{7/2} (a-b x)^{5/2}+\frac {1}{12} (5 a) \int x^{5/2} (a-b x)^{3/2} \, dx \\ & = \frac {1}{12} a x^{7/2} (a-b x)^{3/2}+\frac {1}{6} x^{7/2} (a-b x)^{5/2}+\frac {1}{8} a^2 \int x^{5/2} \sqrt {a-b x} \, dx \\ & = \frac {1}{32} a^2 x^{7/2} \sqrt {a-b x}+\frac {1}{12} a x^{7/2} (a-b x)^{3/2}+\frac {1}{6} x^{7/2} (a-b x)^{5/2}+\frac {1}{64} a^3 \int \frac {x^{5/2}}{\sqrt {a-b x}} \, dx \\ & = -\frac {a^3 x^{5/2} \sqrt {a-b x}}{192 b}+\frac {1}{32} a^2 x^{7/2} \sqrt {a-b x}+\frac {1}{12} a x^{7/2} (a-b x)^{3/2}+\frac {1}{6} x^{7/2} (a-b x)^{5/2}+\frac {\left (5 a^4\right ) \int \frac {x^{3/2}}{\sqrt {a-b x}} \, dx}{384 b} \\ & = -\frac {5 a^4 x^{3/2} \sqrt {a-b x}}{768 b^2}-\frac {a^3 x^{5/2} \sqrt {a-b x}}{192 b}+\frac {1}{32} a^2 x^{7/2} \sqrt {a-b x}+\frac {1}{12} a x^{7/2} (a-b x)^{3/2}+\frac {1}{6} x^{7/2} (a-b x)^{5/2}+\frac {\left (5 a^5\right ) \int \frac {\sqrt {x}}{\sqrt {a-b x}} \, dx}{512 b^2} \\ & = -\frac {5 a^5 \sqrt {x} \sqrt {a-b x}}{512 b^3}-\frac {5 a^4 x^{3/2} \sqrt {a-b x}}{768 b^2}-\frac {a^3 x^{5/2} \sqrt {a-b x}}{192 b}+\frac {1}{32} a^2 x^{7/2} \sqrt {a-b x}+\frac {1}{12} a x^{7/2} (a-b x)^{3/2}+\frac {1}{6} x^{7/2} (a-b x)^{5/2}+\frac {\left (5 a^6\right ) \int \frac {1}{\sqrt {x} \sqrt {a-b x}} \, dx}{1024 b^3} \\ & = -\frac {5 a^5 \sqrt {x} \sqrt {a-b x}}{512 b^3}-\frac {5 a^4 x^{3/2} \sqrt {a-b x}}{768 b^2}-\frac {a^3 x^{5/2} \sqrt {a-b x}}{192 b}+\frac {1}{32} a^2 x^{7/2} \sqrt {a-b x}+\frac {1}{12} a x^{7/2} (a-b x)^{3/2}+\frac {1}{6} x^{7/2} (a-b x)^{5/2}+\frac {\left (5 a^6\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-b x^2}} \, dx,x,\sqrt {x}\right )}{512 b^3} \\ & = -\frac {5 a^5 \sqrt {x} \sqrt {a-b x}}{512 b^3}-\frac {5 a^4 x^{3/2} \sqrt {a-b x}}{768 b^2}-\frac {a^3 x^{5/2} \sqrt {a-b x}}{192 b}+\frac {1}{32} a^2 x^{7/2} \sqrt {a-b x}+\frac {1}{12} a x^{7/2} (a-b x)^{3/2}+\frac {1}{6} x^{7/2} (a-b x)^{5/2}+\frac {\left (5 a^6\right ) \text {Subst}\left (\int \frac {1}{1+b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a-b x}}\right )}{512 b^3} \\ & = -\frac {5 a^5 \sqrt {x} \sqrt {a-b x}}{512 b^3}-\frac {5 a^4 x^{3/2} \sqrt {a-b x}}{768 b^2}-\frac {a^3 x^{5/2} \sqrt {a-b x}}{192 b}+\frac {1}{32} a^2 x^{7/2} \sqrt {a-b x}+\frac {1}{12} a x^{7/2} (a-b x)^{3/2}+\frac {1}{6} x^{7/2} (a-b x)^{5/2}+\frac {5 a^6 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{512 b^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.52 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.70 \[ \int x^{5/2} (a-b x)^{5/2} \, dx=\frac {\sqrt {b} \sqrt {x} \sqrt {a-b x} \left (-15 a^5-10 a^4 b x-8 a^3 b^2 x^2+432 a^2 b^3 x^3-640 a b^4 x^4+256 b^5 x^5\right )+30 a^6 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a-b x}}\right )}{1536 b^{7/2}} \]

[In]

Integrate[x^(5/2)*(a - b*x)^(5/2),x]

[Out]

(Sqrt[b]*Sqrt[x]*Sqrt[a - b*x]*(-15*a^5 - 10*a^4*b*x - 8*a^3*b^2*x^2 + 432*a^2*b^3*x^3 - 640*a*b^4*x^4 + 256*b
^5*x^5) + 30*a^6*ArcTan[(Sqrt[b]*Sqrt[x])/(-Sqrt[a] + Sqrt[a - b*x])])/(1536*b^(7/2))

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.73

method	result	size
risch	\(-\frac {\left (-256 b^{5} x^{5}+640 a \,b^{4} x^{4}-432 a^{2} b^{3} x^{3}+8 a^{3} b^{2} x^{2}+10 a^{4} b x +15 a^{5}\right ) \sqrt {x}\, \sqrt {-b x +a}}{1536 b^{3}}+\frac {5 a^{6} \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-b \,x^{2}+a x}}\right ) \sqrt {x \left (-b x +a \right )}}{1024 b^{\frac {7}{2}} \sqrt {x}\, \sqrt {-b x +a}}\)	\(124\)
default	\(-\frac {x^{\frac {5}{2}} \left (-b x +a \right )^{\frac {7}{2}}}{6 b}+\frac {5 a \left (-\frac {x^{\frac {3}{2}} \left (-b x +a \right )^{\frac {7}{2}}}{5 b}+\frac {3 a \left (-\frac {\sqrt {x}\, \left (-b x +a \right )^{\frac {7}{2}}}{4 b}+\frac {a \left (\frac {\left (-b x +a \right )^{\frac {5}{2}} \sqrt {x}}{3}+\frac {5 a \left (\frac {\left (-b x +a \right )^{\frac {3}{2}} \sqrt {x}}{2}+\frac {3 a \left (\sqrt {x}\, \sqrt {-b x +a}+\frac {a \sqrt {x \left (-b x +a \right )}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-b \,x^{2}+a x}}\right )}{2 \sqrt {-b x +a}\, \sqrt {x}\, \sqrt {b}}\right )}{4}\right )}{6}\right )}{8 b}\right )}{10 b}\right )}{12 b}\)	\(169\)

[In]

int(x^(5/2)*(-b*x+a)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/1536*(-256*b^5*x^5+640*a*b^4*x^4-432*a^2*b^3*x^3+8*a^3*b^2*x^2+10*a^4*b*x+15*a^5)/b^3*x^(1/2)*(-b*x+a)^(1/2
)+5/1024*a^6/b^(7/2)*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)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.22 \[ \int x^{5/2} (a-b x)^{5/2} \, dx=\left [-\frac {15 \, a^{6} \sqrt {-b} \log \left (-2 \, b x + 2 \, \sqrt {-b x + a} \sqrt {-b} \sqrt {x} + a\right ) - 2 \, {\left (256 \, b^{6} x^{5} - 640 \, a b^{5} x^{4} + 432 \, a^{2} b^{4} x^{3} - 8 \, a^{3} b^{3} x^{2} - 10 \, a^{4} b^{2} x - 15 \, a^{5} b\right )} \sqrt {-b x + a} \sqrt {x}}{3072 \, b^{4}}, -\frac {15 \, a^{6} \sqrt {b} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right ) - {\left (256 \, b^{6} x^{5} - 640 \, a b^{5} x^{4} + 432 \, a^{2} b^{4} x^{3} - 8 \, a^{3} b^{3} x^{2} - 10 \, a^{4} b^{2} x - 15 \, a^{5} b\right )} \sqrt {-b x + a} \sqrt {x}}{1536 \, b^{4}}\right ] \]

[In]

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

[Out]

[-1/3072*(15*a^6*sqrt(-b)*log(-2*b*x + 2*sqrt(-b*x + a)*sqrt(-b)*sqrt(x) + a) - 2*(256*b^6*x^5 - 640*a*b^5*x^4
+ 432*a^2*b^4*x^3 - 8*a^3*b^3*x^2 - 10*a^4*b^2*x - 15*a^5*b)*sqrt(-b*x + a)*sqrt(x))/b^4, -1/1536*(15*a^6*sqr
t(b)*arctan(sqrt(-b*x + a)/(sqrt(b)*sqrt(x))) - (256*b^6*x^5 - 640*a*b^5*x^4 + 432*a^2*b^4*x^3 - 8*a^3*b^3*x^2
- 10*a^4*b^2*x - 15*a^5*b)*sqrt(-b*x + a)*sqrt(x))/b^4]

Sympy [F(-1)]

Timed out. \[ \int x^{5/2} (a-b x)^{5/2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.42 \[ \int x^{5/2} (a-b x)^{5/2} \, dx=-\frac {5 \, a^{6} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right )}{512 \, b^{\frac {7}{2}}} + \frac {\frac {15 \, \sqrt {-b x + a} a^{6} b^{5}}{\sqrt {x}} + \frac {85 \, {\left (-b x + a\right )}^{\frac {3}{2}} a^{6} b^{4}}{x^{\frac {3}{2}}} + \frac {198 \, {\left (-b x + a\right )}^{\frac {5}{2}} a^{6} b^{3}}{x^{\frac {5}{2}}} - \frac {198 \, {\left (-b x + a\right )}^{\frac {7}{2}} a^{6} b^{2}}{x^{\frac {7}{2}}} - \frac {85 \, {\left (-b x + a\right )}^{\frac {9}{2}} a^{6} b}{x^{\frac {9}{2}}} - \frac {15 \, {\left (-b x + a\right )}^{\frac {11}{2}} a^{6}}{x^{\frac {11}{2}}}}{1536 \, {\left (b^{9} - \frac {6 \, {\left (b x - a\right )} b^{8}}{x} + \frac {15 \, {\left (b x - a\right )}^{2} b^{7}}{x^{2}} - \frac {20 \, {\left (b x - a\right )}^{3} b^{6}}{x^{3}} + \frac {15 \, {\left (b x - a\right )}^{4} b^{5}}{x^{4}} - \frac {6 \, {\left (b x - a\right )}^{5} b^{4}}{x^{5}} + \frac {{\left (b x - a\right )}^{6} b^{3}}{x^{6}}\right )}} \]

[In]

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

[Out]

-5/512*a^6*arctan(sqrt(-b*x + a)/(sqrt(b)*sqrt(x)))/b^(7/2) + 1/1536*(15*sqrt(-b*x + a)*a^6*b^5/sqrt(x) + 85*(
-b*x + a)^(3/2)*a^6*b^4/x^(3/2) + 198*(-b*x + a)^(5/2)*a^6*b^3/x^(5/2) - 198*(-b*x + a)^(7/2)*a^6*b^2/x^(7/2)
- 85*(-b*x + a)^(9/2)*a^6*b/x^(9/2) - 15*(-b*x + a)^(11/2)*a^6/x^(11/2))/(b^9 - 6*(b*x - a)*b^8/x + 15*(b*x -
a)^2*b^7/x^2 - 20*(b*x - a)^3*b^6/x^3 + 15*(b*x - a)^4*b^5/x^4 - 6*(b*x - a)^5*b^4/x^5 + (b*x - a)^6*b^3/x^6)

Giac [F(-1)]

Timed out. \[ \int x^{5/2} (a-b x)^{5/2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int x^{5/2} (a-b x)^{5/2} \, dx=\int x^{5/2}\,{\left (a-b\,x\right )}^{5/2} \,d x \]

[In]

int(x^(5/2)*(a - b*x)^(5/2),x)

[Out]

int(x^(5/2)*(a - b*x)^(5/2), x)