3.342 \(\int \frac{1}{x^4 \sqrt{a+b x}} \, dx\)

Optimal. Leaf size=90 \[ -\frac{5 b^2 \sqrt{a+b x}}{8 a^3 x}+\frac{5 b^3 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{8 a^{7/2}}+\frac{5 b \sqrt{a+b x}}{12 a^2 x^2}-\frac{\sqrt{a+b x}}{3 a x^3} \]

[Out]

-Sqrt[a + b*x]/(3*a*x^3) + (5*b*Sqrt[a + b*x])/(12*a^2*x^2) - (5*b^2*Sqrt[a + b*x])/(8*a^3*x) + (5*b^3*ArcTanh
[Sqrt[a + b*x]/Sqrt[a]])/(8*a^(7/2))

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Rubi [A]  time = 0.0253039, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {51, 63, 208} \[ -\frac{5 b^2 \sqrt{a+b x}}{8 a^3 x}+\frac{5 b^3 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{8 a^{7/2}}+\frac{5 b \sqrt{a+b x}}{12 a^2 x^2}-\frac{\sqrt{a+b x}}{3 a x^3} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*Sqrt[a + b*x]),x]

[Out]

-Sqrt[a + b*x]/(3*a*x^3) + (5*b*Sqrt[a + b*x])/(12*a^2*x^2) - (5*b^2*Sqrt[a + b*x])/(8*a^3*x) + (5*b^3*ArcTanh
[Sqrt[a + b*x]/Sqrt[a]])/(8*a^(7/2))

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^4 \sqrt{a+b x}} \, dx &=-\frac{\sqrt{a+b x}}{3 a x^3}-\frac{(5 b) \int \frac{1}{x^3 \sqrt{a+b x}} \, dx}{6 a}\\ &=-\frac{\sqrt{a+b x}}{3 a x^3}+\frac{5 b \sqrt{a+b x}}{12 a^2 x^2}+\frac{\left (5 b^2\right ) \int \frac{1}{x^2 \sqrt{a+b x}} \, dx}{8 a^2}\\ &=-\frac{\sqrt{a+b x}}{3 a x^3}+\frac{5 b \sqrt{a+b x}}{12 a^2 x^2}-\frac{5 b^2 \sqrt{a+b x}}{8 a^3 x}-\frac{\left (5 b^3\right ) \int \frac{1}{x \sqrt{a+b x}} \, dx}{16 a^3}\\ &=-\frac{\sqrt{a+b x}}{3 a x^3}+\frac{5 b \sqrt{a+b x}}{12 a^2 x^2}-\frac{5 b^2 \sqrt{a+b x}}{8 a^3 x}-\frac{\left (5 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x}\right )}{8 a^3}\\ &=-\frac{\sqrt{a+b x}}{3 a x^3}+\frac{5 b \sqrt{a+b x}}{12 a^2 x^2}-\frac{5 b^2 \sqrt{a+b x}}{8 a^3 x}+\frac{5 b^3 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{8 a^{7/2}}\\ \end{align*}

Mathematica [C]  time = 0.0084621, size = 33, normalized size = 0.37 \[ \frac{2 b^3 \sqrt{a+b x} \, _2F_1\left (\frac{1}{2},4;\frac{3}{2};\frac{b x}{a}+1\right )}{a^4} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*Sqrt[a + b*x]),x]

[Out]

(2*b^3*Sqrt[a + b*x]*Hypergeometric2F1[1/2, 4, 3/2, 1 + (b*x)/a])/a^4

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Maple [A]  time = 0.006, size = 90, normalized size = 1. \begin{align*} 2\,{b}^{3} \left ( -1/6\,{\frac{\sqrt{bx+a}}{a{b}^{3}{x}^{3}}}-5/6\,{\frac{1}{a} \left ( -1/4\,{\frac{\sqrt{bx+a}}{a{b}^{2}{x}^{2}}}-3/4\,{\frac{1}{a} \left ( -1/2\,{\frac{\sqrt{bx+a}}{abx}}+1/2\,{\frac{1}{{a}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) } \right ) } \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(b*x+a)^(1/2),x)

[Out]

2*b^3*(-1/6*(b*x+a)^(1/2)/a/b^3/x^3-5/6/a*(-1/4*(b*x+a)^(1/2)/a/b^2/x^2-3/4/a*(-1/2*(b*x+a)^(1/2)/a/b/x+1/2*ar
ctanh((b*x+a)^(1/2)/a^(1/2))/a^(3/2))))

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.49112, size = 356, normalized size = 3.96 \begin{align*} \left [\frac{15 \, \sqrt{a} b^{3} x^{3} \log \left (\frac{b x + 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) - 2 \,{\left (15 \, a b^{2} x^{2} - 10 \, a^{2} b x + 8 \, a^{3}\right )} \sqrt{b x + a}}{48 \, a^{4} x^{3}}, -\frac{15 \, \sqrt{-a} b^{3} x^{3} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) +{\left (15 \, a b^{2} x^{2} - 10 \, a^{2} b x + 8 \, a^{3}\right )} \sqrt{b x + a}}{24 \, a^{4} x^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/48*(15*sqrt(a)*b^3*x^3*log((b*x + 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) - 2*(15*a*b^2*x^2 - 10*a^2*b*x + 8*a^3)
*sqrt(b*x + a))/(a^4*x^3), -1/24*(15*sqrt(-a)*b^3*x^3*arctan(sqrt(b*x + a)*sqrt(-a)/a) + (15*a*b^2*x^2 - 10*a^
2*b*x + 8*a^3)*sqrt(b*x + a))/(a^4*x^3)]

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Sympy [A]  time = 9.4177, size = 129, normalized size = 1.43 \begin{align*} - \frac{1}{3 \sqrt{b} x^{\frac{7}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{\sqrt{b}}{12 a x^{\frac{5}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{5 b^{\frac{3}{2}}}{24 a^{2} x^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{5 b^{\frac{5}{2}}}{8 a^{3} \sqrt{x} \sqrt{\frac{a}{b x} + 1}} + \frac{5 b^{3} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{8 a^{\frac{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**4/(b*x+a)**(1/2),x)

[Out]

-1/(3*sqrt(b)*x**(7/2)*sqrt(a/(b*x) + 1)) + sqrt(b)/(12*a*x**(5/2)*sqrt(a/(b*x) + 1)) - 5*b**(3/2)/(24*a**2*x*
*(3/2)*sqrt(a/(b*x) + 1)) - 5*b**(5/2)/(8*a**3*sqrt(x)*sqrt(a/(b*x) + 1)) + 5*b**3*asinh(sqrt(a)/(sqrt(b)*sqrt
(x)))/(8*a**(7/2))

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Giac [A]  time = 1.20976, size = 113, normalized size = 1.26 \begin{align*} -\frac{\frac{15 \, b^{4} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{3}} + \frac{15 \,{\left (b x + a\right )}^{\frac{5}{2}} b^{4} - 40 \,{\left (b x + a\right )}^{\frac{3}{2}} a b^{4} + 33 \, \sqrt{b x + a} a^{2} b^{4}}{a^{3} b^{3} x^{3}}}{24 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(15*b^4*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*a^3) + (15*(b*x + a)^(5/2)*b^4 - 40*(b*x + a)^(3/2)*a*b
^4 + 33*sqrt(b*x + a)*a^2*b^4)/(a^3*b^3*x^3))/b