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

Optimal. Leaf size=95 \[ -\frac{10 x^{3/2}}{3 b^2 \sqrt{a-b x}}-\frac{5 \sqrt{x} \sqrt{a-b x}}{b^3}+\frac{5 a \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{b^{7/2}}+\frac{2 x^{5/2}}{3 b (a-b x)^{3/2}} \]

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.0285071, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {47, 50, 63, 217, 203} \[ -\frac{10 x^{3/2}}{3 b^2 \sqrt{a-b x}}-\frac{5 \sqrt{x} \sqrt{a-b x}}{b^3}+\frac{5 a \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{b^{7/2}}+\frac{2 x^{5/2}}{3 b (a-b x)^{3/2}} \]

Antiderivative was successfully verified.

[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 47

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 50

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 63

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 217

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]

Rule 203

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

Rubi steps

\begin{align*} \int \frac{x^{5/2}}{(a-b x)^{5/2}} \, dx &=\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) \operatorname{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) \operatorname{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 [C]  time = 0.0115855, size = 51, normalized size = 0.54 \[ \frac{2 x^{7/2} \sqrt{1-\frac{b x}{a}} \, _2F_1\left (\frac{5}{2},\frac{7}{2};\frac{9}{2};\frac{b x}{a}\right )}{7 a^2 \sqrt{a-b x}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(5/2)/(a - b*x)^(5/2),x]

[Out]

(2*x^(7/2)*Sqrt[1 - (b*x)/a]*Hypergeometric2F1[5/2, 7/2, 9/2, (b*x)/a])/(7*a^2*Sqrt[a - b*x])

________________________________________________________________________________________

Maple [B]  time = 0.032, size = 160, normalized size = 1.7 \begin{align*} -{\frac{1}{{b}^{3}}\sqrt{x}\sqrt{-bx+a}}+{ \left ({\frac{5\,a}{2}\arctan \left ({\sqrt{b} \left ( x-{\frac{a}{2\,b}} \right ){\frac{1}{\sqrt{-b{x}^{2}+ax}}}} \right ){b}^{-{\frac{7}{2}}}}+{\frac{2\,{a}^{2}}{3\,{b}^{5}}\sqrt{-b \left ( x-{\frac{a}{b}} \right ) ^{2}-a \left ( x-{\frac{a}{b}} \right ) } \left ( x-{\frac{a}{b}} \right ) ^{-2}}+{\frac{14\,a}{3\,{b}^{4}}\sqrt{-b \left ( x-{\frac{a}{b}} \right ) ^{2}-a \left ( x-{\frac{a}{b}} \right ) } \left ( x-{\frac{a}{b}} \right ) ^{-1}} \right ) \sqrt{x \left ( -bx+a \right ) }{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{-bx+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.84299, size = 518, normalized size = 5.45 \begin{align*} \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 ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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 [B]  time = 15.4067, size = 972, normalized size = 10.23 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)/Abs(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)*sqrt(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**(53/2)*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)*sqrt(1 - b*x/a) - 3*a**(77/2)*b**(53/2)*x**(53/2)*sqrt(1 - b*x/a)), True))

________________________________________________________________________________________

Giac [B]  time = 59.3158, size = 298, normalized size = 3.14 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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