3.603 \(\int \frac{x^{3/2}}{(a-b x)^{5/2}} \, dx\)

Optimal. Leaf size=72 \[ -\frac{2 \sqrt{x}}{b^2 \sqrt{a-b x}}+\frac{2 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{b^{5/2}}+\frac{2 x^{3/2}}{3 b (a-b x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0220513, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {47, 63, 217, 203} \[ -\frac{2 \sqrt{x}}{b^2 \sqrt{a-b x}}+\frac{2 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{b^{5/2}}+\frac{2 x^{3/2}}{3 b (a-b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[x^(3/2)/(a - b*x)^(5/2),x]

[Out]

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

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

Mathematica [A]  time = 0.169204, size = 82, normalized size = 1.14 \[ \frac{2 \left (\sqrt{b} \sqrt{x} (4 b x-3 a)+3 \sqrt{a} (a-b x) \sqrt{1-\frac{b x}{a}} \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )\right )}{3 b^{5/2} (a-b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(3/2)/(a - b*x)^(5/2),x]

[Out]

(2*(Sqrt[b]*Sqrt[x]*(-3*a + 4*b*x) + 3*Sqrt[a]*(a - b*x)*Sqrt[1 - (b*x)/a]*ArcSin[(Sqrt[b]*Sqrt[x])/Sqrt[a]]))
/(3*b^(5/2)*(a - b*x)^(3/2))

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Maple [F]  time = 0.024, size = 0, normalized size = 0. \begin{align*} \int{{x}^{{\frac{3}{2}}} \left ( -bx+a \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(3/2)/(-b*x+a)^(5/2),x)

[Out]

int(x^(3/2)/(-b*x+a)^(5/2),x)

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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(x^(3/2)/(-b*x+a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.90345, size = 456, normalized size = 6.33 \begin{align*} \left [-\frac{3 \,{\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \sqrt{-b} \log \left (-2 \, b x + 2 \, \sqrt{-b x + a} \sqrt{-b} \sqrt{x} + a\right ) - 2 \,{\left (4 \, b^{2} x - 3 \, a b\right )} \sqrt{-b x + a} \sqrt{x}}{3 \,{\left (b^{5} x^{2} - 2 \, a b^{4} x + a^{2} b^{3}\right )}}, -\frac{2 \,{\left (3 \,{\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \sqrt{b} \arctan \left (\frac{\sqrt{-b x + a}}{\sqrt{b} \sqrt{x}}\right ) -{\left (4 \, b^{2} x - 3 \, a b\right )} \sqrt{-b x + a} \sqrt{x}\right )}}{3 \,{\left (b^{5} x^{2} - 2 \, a b^{4} x + a^{2} b^{3}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)/(-b*x+a)^(5/2),x, algorithm="fricas")

[Out]

[-1/3*(3*(b^2*x^2 - 2*a*b*x + a^2)*sqrt(-b)*log(-2*b*x + 2*sqrt(-b*x + a)*sqrt(-b)*sqrt(x) + a) - 2*(4*b^2*x -
3*a*b)*sqrt(-b*x + a)*sqrt(x))/(b^5*x^2 - 2*a*b^4*x + a^2*b^3), -2/3*(3*(b^2*x^2 - 2*a*b*x + a^2)*sqrt(b)*arc
tan(sqrt(-b*x + a)/(sqrt(b)*sqrt(x))) - (4*b^2*x - 3*a*b)*sqrt(-b*x + a)*sqrt(x))/(b^5*x^2 - 2*a*b^4*x + a^2*b
^3)]

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Sympy [B]  time = 6.04487, size = 835, normalized size = 11.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(3/2)/(-b*x+a)**(5/2),x)

[Out]

Piecewise((-6*I*a**(39/2)*b**11*x**(27/2)*sqrt(-1 + b*x/a)*acosh(sqrt(b)*sqrt(x)/sqrt(a))/(3*a**(39/2)*b**(27/
2)*x**(27/2)*sqrt(-1 + b*x/a) - 3*a**(37/2)*b**(29/2)*x**(29/2)*sqrt(-1 + b*x/a)) + 3*pi*a**(39/2)*b**11*x**(2
7/2)*sqrt(-1 + b*x/a)/(3*a**(39/2)*b**(27/2)*x**(27/2)*sqrt(-1 + b*x/a) - 3*a**(37/2)*b**(29/2)*x**(29/2)*sqrt
(-1 + b*x/a)) + 6*I*a**(37/2)*b**12*x**(29/2)*sqrt(-1 + b*x/a)*acosh(sqrt(b)*sqrt(x)/sqrt(a))/(3*a**(39/2)*b**
(27/2)*x**(27/2)*sqrt(-1 + b*x/a) - 3*a**(37/2)*b**(29/2)*x**(29/2)*sqrt(-1 + b*x/a)) - 3*pi*a**(37/2)*b**12*x
**(29/2)*sqrt(-1 + b*x/a)/(3*a**(39/2)*b**(27/2)*x**(27/2)*sqrt(-1 + b*x/a) - 3*a**(37/2)*b**(29/2)*x**(29/2)*
sqrt(-1 + b*x/a)) + 6*I*a**19*b**(23/2)*x**14/(3*a**(39/2)*b**(27/2)*x**(27/2)*sqrt(-1 + b*x/a) - 3*a**(37/2)*
b**(29/2)*x**(29/2)*sqrt(-1 + b*x/a)) - 8*I*a**18*b**(25/2)*x**15/(3*a**(39/2)*b**(27/2)*x**(27/2)*sqrt(-1 + b
*x/a) - 3*a**(37/2)*b**(29/2)*x**(29/2)*sqrt(-1 + b*x/a)), Abs(b*x)/Abs(a) > 1), (6*a**(39/2)*b**11*x**(27/2)*
sqrt(1 - b*x/a)*asin(sqrt(b)*sqrt(x)/sqrt(a))/(3*a**(39/2)*b**(27/2)*x**(27/2)*sqrt(1 - b*x/a) - 3*a**(37/2)*b
**(29/2)*x**(29/2)*sqrt(1 - b*x/a)) - 6*a**(37/2)*b**12*x**(29/2)*sqrt(1 - b*x/a)*asin(sqrt(b)*sqrt(x)/sqrt(a)
)/(3*a**(39/2)*b**(27/2)*x**(27/2)*sqrt(1 - b*x/a) - 3*a**(37/2)*b**(29/2)*x**(29/2)*sqrt(1 - b*x/a)) - 6*a**1
9*b**(23/2)*x**14/(3*a**(39/2)*b**(27/2)*x**(27/2)*sqrt(1 - b*x/a) - 3*a**(37/2)*b**(29/2)*x**(29/2)*sqrt(1 -
b*x/a)) + 8*a**18*b**(25/2)*x**15/(3*a**(39/2)*b**(27/2)*x**(27/2)*sqrt(1 - b*x/a) - 3*a**(37/2)*b**(29/2)*x**
(29/2)*sqrt(1 - b*x/a)), True))

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Giac [B]  time = 59.183, size = 266, normalized size = 3.69 \begin{align*} -\frac{{\left (\frac{3 \, \sqrt{-b} \log \left ({\left (\sqrt{-b x + a} \sqrt{-b} - \sqrt{{\left (b x - a\right )} b + a b}\right )}^{2}\right )}{b} - \frac{8 \,{\left (3 \, a{\left (\sqrt{-b x + a} \sqrt{-b} - \sqrt{{\left (b x - a\right )} b + a b}\right )}^{4} \sqrt{-b} - 3 \, a^{2}{\left (\sqrt{-b x + a} \sqrt{-b} - \sqrt{{\left (b x - a\right )} b + a b}\right )}^{2} \sqrt{-b} b + 2 \, a^{3} \sqrt{-b} 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}}\right )}{\left | b \right |}}{3 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)/(-b*x+a)^(5/2),x, algorithm="giac")

[Out]

-1/3*(3*sqrt(-b)*log((sqrt(-b*x + a)*sqrt(-b) - sqrt((b*x - a)*b + a*b))^2)/b - 8*(3*a*(sqrt(-b*x + a)*sqrt(-b
) - sqrt((b*x - a)*b + a*b))^4*sqrt(-b) - 3*a^2*(sqrt(-b*x + a)*sqrt(-b) - sqrt((b*x - a)*b + a*b))^2*sqrt(-b)
*b + 2*a^3*sqrt(-b)*b^2)/((sqrt(-b*x + a)*sqrt(-b) - sqrt((b*x - a)*b + a*b))^2 - a*b)^3)*abs(b)/b^3