\(\int \frac {x^{5/2}}{(a-b x)^{5/2}} \, dx\) [602]

Optimal result

Integrand size = 16, antiderivative size = 95 \[ \int \frac {x^{5/2}}{(a-b x)^{5/2}} \, dx=\frac {2 x^{5/2}}{3 b (a-b x)^{3/2}}-\frac {10 x^{3/2}}{3 b^2 \sqrt {a-b x}}-\frac {5 \sqrt {x} \sqrt {a-b x}}{b^3}+\frac {5 a \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{b^{7/2}} \]

[Out]

2/3*x^(5/2)/b/(-b*x+a)^(3/2)+5*a*arctan(b^(1/2)*x^(1/2)/(-b*x+a)^(1/2))/b^(7/2)-10/3*x^(3/2)/b^2/(-b*x+a)^(1/2
)-5*x^(1/2)*(-b*x+a)^(1/2)/b^3

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {49, 52, 65, 223, 209} \[ \int \frac {x^{5/2}}{(a-b x)^{5/2}} \, dx=\frac {5 a \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{b^{7/2}}-\frac {5 \sqrt {x} \sqrt {a-b x}}{b^3}-\frac {10 x^{3/2}}{3 b^2 \sqrt {a-b x}}+\frac {2 x^{5/2}}{3 b (a-b x)^{3/2}} \]

[In]

Int[x^(5/2)/(a - b*x)^(5/2),x]

[Out]

(2*x^(5/2))/(3*b*(a - b*x)^(3/2)) - (10*x^(3/2))/(3*b^2*Sqrt[a - b*x]) - (5*Sqrt[x]*Sqrt[a - b*x])/b^3 + (5*a*
ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a - b*x]])/b^(7/2)

Rule 49

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

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 {2 x^{5/2}}{3 b (a-b x)^{3/2}}-\frac {5 \int \frac {x^{3/2}}{(a-b x)^{3/2}} \, dx}{3 b} \\ & = \frac {2 x^{5/2}}{3 b (a-b x)^{3/2}}-\frac {10 x^{3/2}}{3 b^2 \sqrt {a-b x}}+\frac {5 \int \frac {\sqrt {x}}{\sqrt {a-b x}} \, dx}{b^2} \\ & = \frac {2 x^{5/2}}{3 b (a-b x)^{3/2}}-\frac {10 x^{3/2}}{3 b^2 \sqrt {a-b x}}-\frac {5 \sqrt {x} \sqrt {a-b x}}{b^3}+\frac {(5 a) \int \frac {1}{\sqrt {x} \sqrt {a-b x}} \, dx}{2 b^3} \\ & = \frac {2 x^{5/2}}{3 b (a-b x)^{3/2}}-\frac {10 x^{3/2}}{3 b^2 \sqrt {a-b x}}-\frac {5 \sqrt {x} \sqrt {a-b x}}{b^3}+\frac {(5 a) \text {Subst}\left (\int \frac {1}{\sqrt {a-b x^2}} \, dx,x,\sqrt {x}\right )}{b^3} \\ & = \frac {2 x^{5/2}}{3 b (a-b x)^{3/2}}-\frac {10 x^{3/2}}{3 b^2 \sqrt {a-b x}}-\frac {5 \sqrt {x} \sqrt {a-b x}}{b^3}+\frac {(5 a) \text {Subst}\left (\int \frac {1}{1+b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a-b x}}\right )}{b^3} \\ & = \frac {2 x^{5/2}}{3 b (a-b x)^{3/2}}-\frac {10 x^{3/2}}{3 b^2 \sqrt {a-b x}}-\frac {5 \sqrt {x} \sqrt {a-b x}}{b^3}+\frac {5 a \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{b^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.86 \[ \int \frac {x^{5/2}}{(a-b x)^{5/2}} \, dx=-\frac {\sqrt {x} \left (15 a^2-20 a b x+3 b^2 x^2\right )}{3 b^3 (a-b x)^{3/2}}+\frac {10 a \arctan \left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a-b x}}\right )}{b^{7/2}} \]

[In]

Integrate[x^(5/2)/(a - b*x)^(5/2),x]

[Out]

-1/3*(Sqrt[x]*(15*a^2 - 20*a*b*x + 3*b^2*x^2))/(b^3*(a - b*x)^(3/2)) + (10*a*ArcTan[(Sqrt[b]*Sqrt[x])/(-Sqrt[a
] + Sqrt[a - b*x])])/b^(7/2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(159\) vs. \(2(71)=142\).

Time = 0.11 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.68

method	result	size
risch	\(-\frac {\sqrt {x}\, \sqrt {-b x +a}}{b^{3}}+\frac {\left (\frac {5 a \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-b \,x^{2}+a x}}\right )}{2 b^{\frac {7}{2}}}+\frac {14 a \sqrt {-b \left (-\frac {a}{b}+x \right )^{2}-\left (-\frac {a}{b}+x \right ) a}}{3 b^{4} \left (-\frac {a}{b}+x \right )}+\frac {2 a^{2} \sqrt {-b \left (-\frac {a}{b}+x \right )^{2}-\left (-\frac {a}{b}+x \right ) a}}{3 b^{5} \left (-\frac {a}{b}+x \right )^{2}}\right ) \sqrt {x \left (-b x +a \right )}}{\sqrt {x}\, \sqrt {-b x +a}}\)	\(160\)

[In]

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

[Out]

-x^(1/2)*(-b*x+a)^(1/2)/b^3+(5/2/b^(7/2)*a*arctan(b^(1/2)*(x-1/2*a/b)/(-b*x^2+a*x)^(1/2))+14/3/b^4*a/(-a/b+x)*
(-b*(-a/b+x)^2-(-a/b+x)*a)^(1/2)+2/3/b^5*a^2/(-a/b+x)^2*(-b*(-a/b+x)^2-(-a/b+x)*a)^(1/2))*(x*(-b*x+a))^(1/2)/x
^(1/2)/(-b*x+a)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 215, normalized size of antiderivative = 2.26 \[ \int \frac {x^{5/2}}{(a-b x)^{5/2}} \, dx=\left [-\frac {15 \, {\left (a b^{2} x^{2} - 2 \, a^{2} b x + a^{3}\right )} \sqrt {-b} \log \left (-2 \, b x + 2 \, \sqrt {-b x + a} \sqrt {-b} \sqrt {x} + a\right ) + 2 \, {\left (3 \, b^{3} x^{2} - 20 \, a b^{2} x + 15 \, a^{2} b\right )} \sqrt {-b x + a} \sqrt {x}}{6 \, {\left (b^{6} x^{2} - 2 \, a b^{5} x + a^{2} b^{4}\right )}}, -\frac {15 \, {\left (a b^{2} x^{2} - 2 \, a^{2} b x + a^{3}\right )} \sqrt {b} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right ) + {\left (3 \, b^{3} x^{2} - 20 \, a b^{2} x + 15 \, a^{2} b\right )} \sqrt {-b x + a} \sqrt {x}}{3 \, {\left (b^{6} x^{2} - 2 \, a b^{5} x + a^{2} b^{4}\right )}}\right ] \]

[In]

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

[Out]

[-1/6*(15*(a*b^2*x^2 - 2*a^2*b*x + a^3)*sqrt(-b)*log(-2*b*x + 2*sqrt(-b*x + a)*sqrt(-b)*sqrt(x) + a) + 2*(3*b^
3*x^2 - 20*a*b^2*x + 15*a^2*b)*sqrt(-b*x + a)*sqrt(x))/(b^6*x^2 - 2*a*b^5*x + a^2*b^4), -1/3*(15*(a*b^2*x^2 -
2*a^2*b*x + a^3)*sqrt(b)*arctan(sqrt(-b*x + a)/(sqrt(b)*sqrt(x))) + (3*b^3*x^2 - 20*a*b^2*x + 15*a^2*b)*sqrt(-
b*x + a)*sqrt(x))/(b^6*x^2 - 2*a*b^5*x + a^2*b^4)]

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 6.75 (sec) , antiderivative size = 971, normalized size of antiderivative = 10.22 \[ \int \frac {x^{5/2}}{(a-b x)^{5/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Piecewise((-30*I*a**(81/2)*b**22*x**(51/2)*sqrt(-1 + b*x/a)*acosh(sqrt(b)*sqrt(x)/sqrt(a))/(6*a**(79/2)*b**(51
/2)*x**(51/2)*sqrt(-1 + b*x/a) - 6*a**(77/2)*b**(53/2)*x**(53/2)*sqrt(-1 + b*x/a)) + 15*pi*a**(81/2)*b**22*x**
(51/2)*sqrt(-1 + b*x/a)/(6*a**(79/2)*b**(51/2)*x**(51/2)*sqrt(-1 + b*x/a) - 6*a**(77/2)*b**(53/2)*x**(53/2)*sq
rt(-1 + b*x/a)) + 30*I*a**(79/2)*b**23*x**(53/2)*sqrt(-1 + b*x/a)*acosh(sqrt(b)*sqrt(x)/sqrt(a))/(6*a**(79/2)*
b**(51/2)*x**(51/2)*sqrt(-1 + b*x/a) - 6*a**(77/2)*b**(53/2)*x**(53/2)*sqrt(-1 + b*x/a)) - 15*pi*a**(79/2)*b**
23*x**(53/2)*sqrt(-1 + b*x/a)/(6*a**(79/2)*b**(51/2)*x**(51/2)*sqrt(-1 + b*x/a) - 6*a**(77/2)*b**(53/2)*x**(53
/2)*sqrt(-1 + b*x/a)) + 30*I*a**40*b**(45/2)*x**26/(6*a**(79/2)*b**(51/2)*x**(51/2)*sqrt(-1 + b*x/a) - 6*a**(7
7/2)*b**(53/2)*x**(53/2)*sqrt(-1 + b*x/a)) - 40*I*a**39*b**(47/2)*x**27/(6*a**(79/2)*b**(51/2)*x**(51/2)*sqrt(
-1 + b*x/a) - 6*a**(77/2)*b**(53/2)*x**(53/2)*sqrt(-1 + b*x/a)) + 6*I*a**38*b**(49/2)*x**28/(6*a**(79/2)*b**(5
1/2)*x**(51/2)*sqrt(-1 + b*x/a) - 6*a**(77/2)*b**(53/2)*x**(53/2)*sqrt(-1 + b*x/a)), Abs(b*x/a) > 1), (15*a**(
81/2)*b**22*x**(51/2)*sqrt(1 - b*x/a)*asin(sqrt(b)*sqrt(x)/sqrt(a))/(3*a**(79/2)*b**(51/2)*x**(51/2)*sqrt(1 -
b*x/a) - 3*a**(77/2)*b**(53/2)*x**(53/2)*sqrt(1 - b*x/a)) - 15*a**(79/2)*b**23*x**(53/2)*sqrt(1 - b*x/a)*asin(
sqrt(b)*sqrt(x)/sqrt(a))/(3*a**(79/2)*b**(51/2)*x**(51/2)*sqrt(1 - b*x/a) - 3*a**(77/2)*b**(53/2)*x**(53/2)*sq
rt(1 - b*x/a)) - 15*a**40*b**(45/2)*x**26/(3*a**(79/2)*b**(51/2)*x**(51/2)*sqrt(1 - b*x/a) - 3*a**(77/2)*b**(5
3/2)*x**(53/2)*sqrt(1 - b*x/a)) + 20*a**39*b**(47/2)*x**27/(3*a**(79/2)*b**(51/2)*x**(51/2)*sqrt(1 - b*x/a) -
3*a**(77/2)*b**(53/2)*x**(53/2)*sqrt(1 - b*x/a)) - 3*a**38*b**(49/2)*x**28/(3*a**(79/2)*b**(51/2)*x**(51/2)*sq
rt(1 - b*x/a) - 3*a**(77/2)*b**(53/2)*x**(53/2)*sqrt(1 - b*x/a)), True))

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.99 \[ \int \frac {x^{5/2}}{(a-b x)^{5/2}} \, dx=\frac {2 \, a b^{2} + \frac {10 \, {\left (b x - a\right )} a b}{x} - \frac {15 \, {\left (b x - a\right )}^{2} a}{x^{2}}}{3 \, {\left (\frac {{\left (-b x + a\right )}^{\frac {3}{2}} b^{4}}{x^{\frac {3}{2}}} + \frac {{\left (-b x + a\right )}^{\frac {5}{2}} b^{3}}{x^{\frac {5}{2}}}\right )}} - \frac {5 \, a \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right )}{b^{\frac {7}{2}}} \]

[In]

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

[Out]

1/3*(2*a*b^2 + 10*(b*x - a)*a*b/x - 15*(b*x - a)^2*a/x^2)/((-b*x + a)^(3/2)*b^4/x^(3/2) + (-b*x + a)^(5/2)*b^3
/x^(5/2)) - 5*a*arctan(sqrt(-b*x + a)/(sqrt(b)*sqrt(x)))/b^(7/2)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 221 vs. \(2 (71) = 142\).

Time = 15.71 (sec) , antiderivative size = 221, normalized size of antiderivative = 2.33 \[ \int \frac {x^{5/2}}{(a-b x)^{5/2}} \, dx=\frac {{\left (\frac {15 \, a \log \left ({\left (\sqrt {-b x + a} \sqrt {-b} - \sqrt {{\left (b x - a\right )} b + a b}\right )}^{2}\right )}{\sqrt {-b} b^{2}} - \frac {6 \, \sqrt {{\left (b x - a\right )} b + a b} \sqrt {-b x + a}}{b^{3}} - \frac {8 \, {\left (9 \, a^{2} {\left (\sqrt {-b x + a} \sqrt {-b} - \sqrt {{\left (b x - a\right )} b + a b}\right )}^{4} - 12 \, a^{3} {\left (\sqrt {-b x + a} \sqrt {-b} - \sqrt {{\left (b x - a\right )} b + a b}\right )}^{2} b + 7 \, a^{4} b^{2}\right )}}{{\left ({\left (\sqrt {-b x + a} \sqrt {-b} - \sqrt {{\left (b x - a\right )} b + a b}\right )}^{2} - a b\right )}^{3} \sqrt {-b} b}\right )} {\left | b \right |}}{6 \, b^{2}} \]

[In]

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

[Out]

1/6*(15*a*log((sqrt(-b*x + a)*sqrt(-b) - sqrt((b*x - a)*b + a*b))^2)/(sqrt(-b)*b^2) - 6*sqrt((b*x - a)*b + a*b
)*sqrt(-b*x + a)/b^3 - 8*(9*a^2*(sqrt(-b*x + a)*sqrt(-b) - sqrt((b*x - a)*b + a*b))^4 - 12*a^3*(sqrt(-b*x + a)
*sqrt(-b) - sqrt((b*x - a)*b + a*b))^2*b + 7*a^4*b^2)/(((sqrt(-b*x + a)*sqrt(-b) - sqrt((b*x - a)*b + a*b))^2
- a*b)^3*sqrt(-b)*b))*abs(b)/b^2

Mupad [F(-1)]

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

[In]

int(x^(5/2)/(a - b*x)^(5/2),x)

[Out]

int(x^(5/2)/(a - b*x)^(5/2), x)