3.466 \(\int \frac{1}{\sqrt{x} (a+b x)^3} \, dx\)

Optimal. Leaf size=70 \[ \frac{3 \sqrt{x}}{4 a^2 (a+b x)}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{5/2} \sqrt{b}}+\frac{\sqrt{x}}{2 a (a+b x)^2} \]

[Out]

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

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Rubi [A]  time = 0.0191969, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {51, 63, 205} \[ \frac{3 \sqrt{x}}{4 a^2 (a+b x)}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{5/2} \sqrt{b}}+\frac{\sqrt{x}}{2 a (a+b x)^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[x]*(a + b*x)^3),x]

[Out]

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

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

Mathematica [C]  time = 0.0047083, size = 25, normalized size = 0.36 \[ \frac{2 \sqrt{x} \, _2F_1\left (\frac{1}{2},3;\frac{3}{2};-\frac{b x}{a}\right )}{a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[x]*(a + b*x)^3),x]

[Out]

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

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Maple [A]  time = 0.006, size = 53, normalized size = 0.8 \begin{align*}{\frac{1}{2\,a \left ( bx+a \right ) ^{2}}\sqrt{x}}+{\frac{3}{4\,{a}^{2} \left ( bx+a \right ) }\sqrt{x}}+{\frac{3}{4\,{a}^{2}}\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/(b*x+a)^3/x^(1/2),x)

[Out]

1/2*x^(1/2)/a/(b*x+a)^2+3/4*x^(1/2)/a^2/(b*x+a)+3/4/a^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/(b*x+a)^3/x^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.38426, size = 423, normalized size = 6.04 \begin{align*} \left [-\frac{3 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt{-a b} \log \left (\frac{b x - a - 2 \, \sqrt{-a b} \sqrt{x}}{b x + a}\right ) - 2 \,{\left (3 \, a b^{2} x + 5 \, a^{2} b\right )} \sqrt{x}}{8 \,{\left (a^{3} b^{3} x^{2} + 2 \, a^{4} b^{2} x + a^{5} b\right )}}, -\frac{3 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b}}{b \sqrt{x}}\right ) -{\left (3 \, a b^{2} x + 5 \, a^{2} b\right )} \sqrt{x}}{4 \,{\left (a^{3} b^{3} x^{2} + 2 \, a^{4} b^{2} x + a^{5} b\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/8*(3*(b^2*x^2 + 2*a*b*x + a^2)*sqrt(-a*b)*log((b*x - a - 2*sqrt(-a*b)*sqrt(x))/(b*x + a)) - 2*(3*a*b^2*x +
5*a^2*b)*sqrt(x))/(a^3*b^3*x^2 + 2*a^4*b^2*x + a^5*b), -1/4*(3*(b^2*x^2 + 2*a*b*x + a^2)*sqrt(a*b)*arctan(sqr
t(a*b)/(b*sqrt(x))) - (3*a*b^2*x + 5*a^2*b)*sqrt(x))/(a^3*b^3*x^2 + 2*a^4*b^2*x + a^5*b)]

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Sympy [A]  time = 61.4325, size = 712, normalized size = 10.17 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{x^{\frac{5}{2}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{2}{5 b^{3} x^{\frac{5}{2}}} & \text{for}\: a = 0 \\\frac{2 \sqrt{x}}{a^{3}} & \text{for}\: b = 0 \\\frac{10 i a^{\frac{3}{2}} b \sqrt{x} \sqrt{\frac{1}{b}}}{8 i a^{\frac{9}{2}} b \sqrt{\frac{1}{b}} + 16 i a^{\frac{7}{2}} b^{2} x \sqrt{\frac{1}{b}} + 8 i a^{\frac{5}{2}} b^{3} x^{2} \sqrt{\frac{1}{b}}} + \frac{6 i \sqrt{a} b^{2} x^{\frac{3}{2}} \sqrt{\frac{1}{b}}}{8 i a^{\frac{9}{2}} b \sqrt{\frac{1}{b}} + 16 i a^{\frac{7}{2}} b^{2} x \sqrt{\frac{1}{b}} + 8 i a^{\frac{5}{2}} b^{3} x^{2} \sqrt{\frac{1}{b}}} + \frac{3 a^{2} \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{8 i a^{\frac{9}{2}} b \sqrt{\frac{1}{b}} + 16 i a^{\frac{7}{2}} b^{2} x \sqrt{\frac{1}{b}} + 8 i a^{\frac{5}{2}} b^{3} x^{2} \sqrt{\frac{1}{b}}} - \frac{3 a^{2} \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{8 i a^{\frac{9}{2}} b \sqrt{\frac{1}{b}} + 16 i a^{\frac{7}{2}} b^{2} x \sqrt{\frac{1}{b}} + 8 i a^{\frac{5}{2}} b^{3} x^{2} \sqrt{\frac{1}{b}}} + \frac{6 a b x \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{8 i a^{\frac{9}{2}} b \sqrt{\frac{1}{b}} + 16 i a^{\frac{7}{2}} b^{2} x \sqrt{\frac{1}{b}} + 8 i a^{\frac{5}{2}} b^{3} x^{2} \sqrt{\frac{1}{b}}} - \frac{6 a b x \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{8 i a^{\frac{9}{2}} b \sqrt{\frac{1}{b}} + 16 i a^{\frac{7}{2}} b^{2} x \sqrt{\frac{1}{b}} + 8 i a^{\frac{5}{2}} b^{3} x^{2} \sqrt{\frac{1}{b}}} + \frac{3 b^{2} x^{2} \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{8 i a^{\frac{9}{2}} b \sqrt{\frac{1}{b}} + 16 i a^{\frac{7}{2}} b^{2} x \sqrt{\frac{1}{b}} + 8 i a^{\frac{5}{2}} b^{3} x^{2} \sqrt{\frac{1}{b}}} - \frac{3 b^{2} x^{2} \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{8 i a^{\frac{9}{2}} b \sqrt{\frac{1}{b}} + 16 i a^{\frac{7}{2}} b^{2} x \sqrt{\frac{1}{b}} + 8 i a^{\frac{5}{2}} b^{3} x^{2} \sqrt{\frac{1}{b}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)**3/x**(1/2),x)

[Out]

Piecewise((zoo/x**(5/2), Eq(a, 0) & Eq(b, 0)), (-2/(5*b**3*x**(5/2)), Eq(a, 0)), (2*sqrt(x)/a**3, Eq(b, 0)), (
10*I*a**(3/2)*b*sqrt(x)*sqrt(1/b)/(8*I*a**(9/2)*b*sqrt(1/b) + 16*I*a**(7/2)*b**2*x*sqrt(1/b) + 8*I*a**(5/2)*b*
*3*x**2*sqrt(1/b)) + 6*I*sqrt(a)*b**2*x**(3/2)*sqrt(1/b)/(8*I*a**(9/2)*b*sqrt(1/b) + 16*I*a**(7/2)*b**2*x*sqrt
(1/b) + 8*I*a**(5/2)*b**3*x**2*sqrt(1/b)) + 3*a**2*log(-I*sqrt(a)*sqrt(1/b) + sqrt(x))/(8*I*a**(9/2)*b*sqrt(1/
b) + 16*I*a**(7/2)*b**2*x*sqrt(1/b) + 8*I*a**(5/2)*b**3*x**2*sqrt(1/b)) - 3*a**2*log(I*sqrt(a)*sqrt(1/b) + sqr
t(x))/(8*I*a**(9/2)*b*sqrt(1/b) + 16*I*a**(7/2)*b**2*x*sqrt(1/b) + 8*I*a**(5/2)*b**3*x**2*sqrt(1/b)) + 6*a*b*x
*log(-I*sqrt(a)*sqrt(1/b) + sqrt(x))/(8*I*a**(9/2)*b*sqrt(1/b) + 16*I*a**(7/2)*b**2*x*sqrt(1/b) + 8*I*a**(5/2)
*b**3*x**2*sqrt(1/b)) - 6*a*b*x*log(I*sqrt(a)*sqrt(1/b) + sqrt(x))/(8*I*a**(9/2)*b*sqrt(1/b) + 16*I*a**(7/2)*b
**2*x*sqrt(1/b) + 8*I*a**(5/2)*b**3*x**2*sqrt(1/b)) + 3*b**2*x**2*log(-I*sqrt(a)*sqrt(1/b) + sqrt(x))/(8*I*a**
(9/2)*b*sqrt(1/b) + 16*I*a**(7/2)*b**2*x*sqrt(1/b) + 8*I*a**(5/2)*b**3*x**2*sqrt(1/b)) - 3*b**2*x**2*log(I*sqr
t(a)*sqrt(1/b) + sqrt(x))/(8*I*a**(9/2)*b*sqrt(1/b) + 16*I*a**(7/2)*b**2*x*sqrt(1/b) + 8*I*a**(5/2)*b**3*x**2*
sqrt(1/b)), True))

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Giac [A]  time = 1.22448, size = 63, normalized size = 0.9 \begin{align*} \frac{3 \, \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{4 \, \sqrt{a b} a^{2}} + \frac{3 \, b x^{\frac{3}{2}} + 5 \, a \sqrt{x}}{4 \,{\left (b x + a\right )}^{2} a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/4*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^2) + 1/4*(3*b*x^(3/2) + 5*a*sqrt(x))/((b*x + a)^2*a^2)