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

Optimal. Leaf size=95 \[ \frac{35 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{9/2}}+\frac{7}{4 a^2 x^{3/2} (a+b x)}+\frac{35 b}{4 a^4 \sqrt{x}}-\frac{35}{12 a^3 x^{3/2}}+\frac{1}{2 a x^{3/2} (a+b x)^2} \]

[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)) + (3
5*b^(3/2)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(4*a^(9/2))

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Rubi [A]  time = 0.0289663, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {51, 63, 205} \[ \frac{35 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{9/2}}+\frac{7}{4 a^2 x^{3/2} (a+b x)}+\frac{35 b}{4 a^4 \sqrt{x}}-\frac{35}{12 a^3 x^{3/2}}+\frac{1}{2 a x^{3/2} (a+b x)^2} \]

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)) + (3
5*b^(3/2)*ArcTan[(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 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[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} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{9/2}}\\ \end{align*}

Mathematica [C]  time = 0.0051931, size = 27, normalized size = 0.28 \[ -\frac{2 \, _2F_1\left (-\frac{3}{2},3;-\frac{1}{2};-\frac{b x}{a}\right )}{3 a^3 x^{3/2}} \]

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 = 0.015, size = 79, normalized size = 0.8 \begin{align*} -{\frac{2}{3\,{a}^{3}}{x}^{-{\frac{3}{2}}}}+6\,{\frac{b}{{a}^{4}\sqrt{x}}}+{\frac{11\,{b}^{3}}{4\,{a}^{4} \left ( bx+a \right ) ^{2}}{x}^{{\frac{3}{2}}}}+{\frac{13\,{b}^{2}}{4\,{a}^{3} \left ( bx+a \right ) ^{2}}\sqrt{x}}+{\frac{35\,{b}^{2}}{4\,{a}^{4}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \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)+11/4/a^4*b^3/(b*x+a)^2*x^(3/2)+13/4/a^3*b^2/(b*x+a)^2*x^(1/2)+35/4/a^4*b^2/(a
*b)^(1/2)*arctan(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 = 1.40856, size = 545, normalized size = 5.74 \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*(10
5*(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 = 1.17401, size = 96, normalized size = 1.01 \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) + 1
3*a*b^2*sqrt(x))/((b*x + a)^2*a^4)