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

Optimal. Leaf size=90 \[ 5 b^2 \sqrt{x} \sqrt{a-b x}+5 a b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )-\frac{2 (a-b x)^{5/2}}{3 x^{3/2}}+\frac{10 b (a-b x)^{3/2}}{3 \sqrt{x}} \]

[Out]

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

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Rubi [A]  time = 0.0296783, antiderivative size = 90, 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} \[ 5 b^2 \sqrt{x} \sqrt{a-b x}+5 a b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )-\frac{2 (a-b x)^{5/2}}{3 x^{3/2}}+\frac{10 b (a-b x)^{3/2}}{3 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Int[(a - b*x)^(5/2)/x^(5/2),x]

[Out]

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

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

Mathematica [C]  time = 0.0107629, size = 51, normalized size = 0.57 \[ -\frac{2 a^2 \sqrt{a-b x} \, _2F_1\left (-\frac{5}{2},-\frac{3}{2};-\frac{1}{2};\frac{b x}{a}\right )}{3 x^{3/2} \sqrt{1-\frac{b x}{a}}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a - b*x)^(5/2)/x^(5/2),x]

[Out]

(-2*a^2*Sqrt[a - b*x]*Hypergeometric2F1[-5/2, -3/2, -1/2, (b*x)/a])/(3*x^(3/2)*Sqrt[1 - (b*x)/a])

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Maple [A]  time = 0.017, size = 86, normalized size = 1. \begin{align*} -{\frac{-3\,{b}^{2}{x}^{2}-14\,abx+2\,{a}^{2}}{3}\sqrt{-bx+a}{x}^{-{\frac{3}{2}}}}+{\frac{5\,a}{2}{b}^{{\frac{3}{2}}}\arctan \left ({\sqrt{b} \left ( x-{\frac{a}{2\,b}} \right ){\frac{1}{\sqrt{-b{x}^{2}+ax}}}} \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((-b*x+a)^(5/2)/x^(5/2),x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.82039, size = 366, normalized size = 4.07 \begin{align*} \left [\frac{15 \, a \sqrt{-b} b x^{2} \log \left (-2 \, b x - 2 \, \sqrt{-b x + a} \sqrt{-b} \sqrt{x} + a\right ) + 2 \,{\left (3 \, b^{2} x^{2} + 14 \, a b x - 2 \, a^{2}\right )} \sqrt{-b x + a} \sqrt{x}}{6 \, x^{2}}, -\frac{15 \, a b^{\frac{3}{2}} x^{2} \arctan \left (\frac{\sqrt{-b x + a}}{\sqrt{b} \sqrt{x}}\right ) -{\left (3 \, b^{2} x^{2} + 14 \, a b x - 2 \, a^{2}\right )} \sqrt{-b x + a} \sqrt{x}}{3 \, x^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/6*(15*a*sqrt(-b)*b*x^2*log(-2*b*x - 2*sqrt(-b*x + a)*sqrt(-b)*sqrt(x) + a) + 2*(3*b^2*x^2 + 14*a*b*x - 2*a^
2)*sqrt(-b*x + a)*sqrt(x))/x^2, -1/3*(15*a*b^(3/2)*x^2*arctan(sqrt(-b*x + a)/(sqrt(b)*sqrt(x))) - (3*b^2*x^2 +
14*a*b*x - 2*a^2)*sqrt(-b*x + a)*sqrt(x))/x^2]

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Sympy [C]  time = 12.2156, size = 248, normalized size = 2.76 \begin{align*} \begin{cases} - \frac{2 a^{2} \sqrt{b} \sqrt{\frac{a}{b x} - 1}}{3 x} + \frac{14 a b^{\frac{3}{2}} \sqrt{\frac{a}{b x} - 1}}{3} - 5 i a b^{\frac{3}{2}} \log{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )} + \frac{5 i a b^{\frac{3}{2}} \log{\left (\frac{a}{b x} \right )}}{2} + 5 a b^{\frac{3}{2}} \operatorname{asin}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )} + b^{\frac{5}{2}} x \sqrt{\frac{a}{b x} - 1} & \text{for}\: \frac{\left |{a}\right |}{\left |{b}\right | \left |{x}\right |} > 1 \\- \frac{2 i a^{2} \sqrt{b} \sqrt{- \frac{a}{b x} + 1}}{3 x} + \frac{14 i a b^{\frac{3}{2}} \sqrt{- \frac{a}{b x} + 1}}{3} + \frac{5 i a b^{\frac{3}{2}} \log{\left (\frac{a}{b x} \right )}}{2} - 5 i a b^{\frac{3}{2}} \log{\left (\sqrt{- \frac{a}{b x} + 1} + 1 \right )} + i b^{\frac{5}{2}} x \sqrt{- \frac{a}{b x} + 1} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-2*a**2*sqrt(b)*sqrt(a/(b*x) - 1)/(3*x) + 14*a*b**(3/2)*sqrt(a/(b*x) - 1)/3 - 5*I*a*b**(3/2)*log(sq
rt(a)/(sqrt(b)*sqrt(x))) + 5*I*a*b**(3/2)*log(a/(b*x))/2 + 5*a*b**(3/2)*asin(sqrt(b)*sqrt(x)/sqrt(a)) + b**(5/
2)*x*sqrt(a/(b*x) - 1), Abs(a)/(Abs(b)*Abs(x)) > 1), (-2*I*a**2*sqrt(b)*sqrt(-a/(b*x) + 1)/(3*x) + 14*I*a*b**(
3/2)*sqrt(-a/(b*x) + 1)/3 + 5*I*a*b**(3/2)*log(a/(b*x))/2 - 5*I*a*b**(3/2)*log(sqrt(-a/(b*x) + 1) + 1) + I*b**
(5/2)*x*sqrt(-a/(b*x) + 1), True))

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out