3.488 \(\int \frac{1}{x^{5/2} (-a+b x)^3} \, dx\)

Optimal. Leaf size=97 \[ -\frac{35 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{9/2}}-\frac{7}{4 a^2 x^{3/2} (a-b x)}+\frac{35 b}{4 a^4 \sqrt{x}}+\frac{35}{12 a^3 x^{3/2}}-\frac{1}{2 a x^{3/2} (a-b x)^2} \]

[Out]

35/(12*a^3*x^(3/2)) + (35*b)/(4*a^4*Sqrt[x]) - 1/(2*a*x^(3/2)*(a - b*x)^2) - 7/(4*a^2*x^(3/2)*(a - b*x)) - (35
*b^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(4*a^(9/2))

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Rubi [A]  time = 0.0303405, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {51, 63, 208} \[ -\frac{35 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{9/2}}-\frac{7}{4 a^2 x^{3/2} (a-b x)}+\frac{35 b}{4 a^4 \sqrt{x}}+\frac{35}{12 a^3 x^{3/2}}-\frac{1}{2 a x^{3/2} (a-b x)^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^(5/2)*(-a + b*x)^3),x]

[Out]

35/(12*a^3*x^(3/2)) + (35*b)/(4*a^4*Sqrt[x]) - 1/(2*a*x^(3/2)*(a - b*x)^2) - 7/(4*a^2*x^(3/2)*(a - b*x)) - (35
*b^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(4*a^(9/2))

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && 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 208

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

Rubi steps

\begin{align*} \int \frac{1}{x^{5/2} (-a+b x)^3} \, dx &=-\frac{1}{2 a x^{3/2} (a-b x)^2}-\frac{7 \int \frac{1}{x^{5/2} (-a+b x)^2} \, dx}{4 a}\\ &=-\frac{1}{2 a x^{3/2} (a-b x)^2}-\frac{7}{4 a^2 x^{3/2} (a-b x)}+\frac{35 \int \frac{1}{x^{5/2} (-a+b x)} \, dx}{8 a^2}\\ &=\frac{35}{12 a^3 x^{3/2}}-\frac{1}{2 a x^{3/2} (a-b x)^2}-\frac{7}{4 a^2 x^{3/2} (a-b x)}+\frac{(35 b) \int \frac{1}{x^{3/2} (-a+b x)} \, dx}{8 a^3}\\ &=\frac{35}{12 a^3 x^{3/2}}+\frac{35 b}{4 a^4 \sqrt{x}}-\frac{1}{2 a x^{3/2} (a-b x)^2}-\frac{7}{4 a^2 x^{3/2} (a-b x)}+\frac{\left (35 b^2\right ) \int \frac{1}{\sqrt{x} (-a+b x)} \, dx}{8 a^4}\\ &=\frac{35}{12 a^3 x^{3/2}}+\frac{35 b}{4 a^4 \sqrt{x}}-\frac{1}{2 a x^{3/2} (a-b x)^2}-\frac{7}{4 a^2 x^{3/2} (a-b x)}+\frac{\left (35 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{-a+b x^2} \, dx,x,\sqrt{x}\right )}{4 a^4}\\ &=\frac{35}{12 a^3 x^{3/2}}+\frac{35 b}{4 a^4 \sqrt{x}}-\frac{1}{2 a x^{3/2} (a-b x)^2}-\frac{7}{4 a^2 x^{3/2} (a-b x)}-\frac{35 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{9/2}}\\ \end{align*}

Mathematica [C]  time = 0.005165, size = 26, normalized size = 0.27 \[ \frac{2 \, _2F_1\left (-\frac{3}{2},3;-\frac{1}{2};\frac{b x}{a}\right )}{3 a^3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(5/2)*(-a + b*x)^3),x]

[Out]

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

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Maple [A]  time = 0.014, size = 69, normalized size = 0.7 \begin{align*}{\frac{2}{3\,{a}^{3}}{x}^{-{\frac{3}{2}}}}+6\,{\frac{b}{{a}^{4}\sqrt{x}}}+2\,{\frac{{b}^{2}}{{a}^{4}} \left ({\frac{1}{ \left ( bx-a \right ) ^{2}} \left ({\frac{11\,b{x}^{3/2}}{8}}-{\frac{13\,a\sqrt{x}}{8}} \right ) }-{\frac{35}{8\,\sqrt{ab}}{\it Artanh} \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(5/2)/(b*x-a)^3,x)

[Out]

2/3/a^3/x^(3/2)+6*b/a^4/x^(1/2)+2/a^4*b^2*((11/8*b*x^(3/2)-13/8*a*x^(1/2))/(b*x-a)^2-35/8/(a*b)^(1/2)*arctanh(
b*x^(1/2)/(a*b)^(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(1/x^(5/2)/(b*x-a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.56587, size = 544, normalized size = 5.61 \begin{align*} \left [\frac{105 \,{\left (b^{3} x^{4} - 2 \, a b^{2} x^{3} + a^{2} b x^{2}\right )} \sqrt{\frac{b}{a}} \log \left (\frac{b x - 2 \, a \sqrt{x} \sqrt{\frac{b}{a}} + a}{b x - a}\right ) + 2 \,{\left (105 \, b^{3} x^{3} - 175 \, a b^{2} x^{2} + 56 \, a^{2} b x + 8 \, a^{3}\right )} \sqrt{x}}{24 \,{\left (a^{4} b^{2} x^{4} - 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )}}, \frac{105 \,{\left (b^{3} x^{4} - 2 \, a b^{2} x^{3} + a^{2} b x^{2}\right )} \sqrt{-\frac{b}{a}} \arctan \left (\frac{a \sqrt{-\frac{b}{a}}}{b \sqrt{x}}\right ) +{\left (105 \, b^{3} x^{3} - 175 \, a b^{2} x^{2} + 56 \, a^{2} b x + 8 \, a^{3}\right )} \sqrt{x}}{12 \,{\left (a^{4} b^{2} x^{4} - 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(5/2)/(b*x-a)^3,x, algorithm="fricas")

[Out]

[1/24*(105*(b^3*x^4 - 2*a*b^2*x^3 + a^2*b*x^2)*sqrt(b/a)*log((b*x - 2*a*sqrt(x)*sqrt(b/a) + a)/(b*x - a)) + 2*
(105*b^3*x^3 - 175*a*b^2*x^2 + 56*a^2*b*x + 8*a^3)*sqrt(x))/(a^4*b^2*x^4 - 2*a^5*b*x^3 + a^6*x^2), 1/12*(105*(
b^3*x^4 - 2*a*b^2*x^3 + a^2*b*x^2)*sqrt(-b/a)*arctan(a*sqrt(-b/a)/(b*sqrt(x))) + (105*b^3*x^3 - 175*a*b^2*x^2
+ 56*a^2*b*x + 8*a^3)*sqrt(x))/(a^4*b^2*x^4 - 2*a^5*b*x^3 + a^6*x^2)]

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Sympy [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(1/x**(5/2)/(b*x-a)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.25547, size = 99, normalized size = 1.02 \begin{align*} \frac{35 \, b^{2} \arctan \left (\frac{b \sqrt{x}}{\sqrt{-a b}}\right )}{4 \, \sqrt{-a b} a^{4}} + \frac{2 \,{\left (9 \, b x + a\right )}}{3 \, a^{4} x^{\frac{3}{2}}} + \frac{11 \, b^{3} x^{\frac{3}{2}} - 13 \, a b^{2} \sqrt{x}}{4 \,{\left (b x - a\right )}^{2} a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(5/2)/(b*x-a)^3,x, algorithm="giac")

[Out]

35/4*b^2*arctan(b*sqrt(x)/sqrt(-a*b))/(sqrt(-a*b)*a^4) + 2/3*(9*b*x + a)/(a^4*x^(3/2)) + 1/4*(11*b^3*x^(3/2) -
13*a*b^2*sqrt(x))/((b*x - a)^2*a^4)