3.2244 \(\int \frac{1}{\sqrt{a+b \sqrt{x}} x^3} \, dx\)

Optimal. Leaf size=136 \[ \frac{35 b^3 \sqrt{a+b \sqrt{x}}}{32 a^4 \sqrt{x}}-\frac{35 b^2 \sqrt{a+b \sqrt{x}}}{48 a^3 x}-\frac{35 b^4 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{x}}}{\sqrt{a}}\right )}{32 a^{9/2}}+\frac{7 b \sqrt{a+b \sqrt{x}}}{12 a^2 x^{3/2}}-\frac{\sqrt{a+b \sqrt{x}}}{2 a x^2} \]

[Out]

-Sqrt[a + b*Sqrt[x]]/(2*a*x^2) + (7*b*Sqrt[a + b*Sqrt[x]])/(12*a^2*x^(3/2)) - (35*b^2*Sqrt[a + b*Sqrt[x]])/(48
*a^3*x) + (35*b^3*Sqrt[a + b*Sqrt[x]])/(32*a^4*Sqrt[x]) - (35*b^4*ArcTanh[Sqrt[a + b*Sqrt[x]]/Sqrt[a]])/(32*a^
(9/2))

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Rubi [A]  time = 0.0589333, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {266, 51, 63, 208} \[ \frac{35 b^3 \sqrt{a+b \sqrt{x}}}{32 a^4 \sqrt{x}}-\frac{35 b^2 \sqrt{a+b \sqrt{x}}}{48 a^3 x}-\frac{35 b^4 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{x}}}{\sqrt{a}}\right )}{32 a^{9/2}}+\frac{7 b \sqrt{a+b \sqrt{x}}}{12 a^2 x^{3/2}}-\frac{\sqrt{a+b \sqrt{x}}}{2 a x^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[a + b*Sqrt[x]]*x^3),x]

[Out]

-Sqrt[a + b*Sqrt[x]]/(2*a*x^2) + (7*b*Sqrt[a + b*Sqrt[x]])/(12*a^2*x^(3/2)) - (35*b^2*Sqrt[a + b*Sqrt[x]])/(48
*a^3*x) + (35*b^3*Sqrt[a + b*Sqrt[x]])/(32*a^4*Sqrt[x]) - (35*b^4*ArcTanh[Sqrt[a + b*Sqrt[x]]/Sqrt[a]])/(32*a^
(9/2))

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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}{\sqrt{a+b \sqrt{x}} x^3} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{x^5 \sqrt{a+b x}} \, dx,x,\sqrt{x}\right )\\ &=-\frac{\sqrt{a+b \sqrt{x}}}{2 a x^2}-\frac{(7 b) \operatorname{Subst}\left (\int \frac{1}{x^4 \sqrt{a+b x}} \, dx,x,\sqrt{x}\right )}{4 a}\\ &=-\frac{\sqrt{a+b \sqrt{x}}}{2 a x^2}+\frac{7 b \sqrt{a+b \sqrt{x}}}{12 a^2 x^{3/2}}+\frac{\left (35 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{a+b x}} \, dx,x,\sqrt{x}\right )}{24 a^2}\\ &=-\frac{\sqrt{a+b \sqrt{x}}}{2 a x^2}+\frac{7 b \sqrt{a+b \sqrt{x}}}{12 a^2 x^{3/2}}-\frac{35 b^2 \sqrt{a+b \sqrt{x}}}{48 a^3 x}-\frac{\left (35 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x}} \, dx,x,\sqrt{x}\right )}{32 a^3}\\ &=-\frac{\sqrt{a+b \sqrt{x}}}{2 a x^2}+\frac{7 b \sqrt{a+b \sqrt{x}}}{12 a^2 x^{3/2}}-\frac{35 b^2 \sqrt{a+b \sqrt{x}}}{48 a^3 x}+\frac{35 b^3 \sqrt{a+b \sqrt{x}}}{32 a^4 \sqrt{x}}+\frac{\left (35 b^4\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\sqrt{x}\right )}{64 a^4}\\ &=-\frac{\sqrt{a+b \sqrt{x}}}{2 a x^2}+\frac{7 b \sqrt{a+b \sqrt{x}}}{12 a^2 x^{3/2}}-\frac{35 b^2 \sqrt{a+b \sqrt{x}}}{48 a^3 x}+\frac{35 b^3 \sqrt{a+b \sqrt{x}}}{32 a^4 \sqrt{x}}+\frac{\left (35 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sqrt{x}}\right )}{32 a^4}\\ &=-\frac{\sqrt{a+b \sqrt{x}}}{2 a x^2}+\frac{7 b \sqrt{a+b \sqrt{x}}}{12 a^2 x^{3/2}}-\frac{35 b^2 \sqrt{a+b \sqrt{x}}}{48 a^3 x}+\frac{35 b^3 \sqrt{a+b \sqrt{x}}}{32 a^4 \sqrt{x}}-\frac{35 b^4 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{x}}}{\sqrt{a}}\right )}{32 a^{9/2}}\\ \end{align*}

Mathematica [C]  time = 0.0069564, size = 41, normalized size = 0.3 \[ -\frac{4 b^4 \sqrt{a+b \sqrt{x}} \, _2F_1\left (\frac{1}{2},5;\frac{3}{2};\frac{\sqrt{x} b}{a}+1\right )}{a^5} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[a + b*Sqrt[x]]*x^3),x]

[Out]

(-4*b^4*Sqrt[a + b*Sqrt[x]]*Hypergeometric2F1[1/2, 5, 3/2, 1 + (b*Sqrt[x])/a])/a^5

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Maple [A]  time = 0.005, size = 124, normalized size = 0.9 \begin{align*} 4\,{b}^{4} \left ( -1/8\,{\frac{\sqrt{a+b\sqrt{x}}}{a{b}^{4}{x}^{2}}}-{\frac{7}{8\,a} \left ( -1/6\,{\frac{\sqrt{a+b\sqrt{x}}}{a{b}^{3}{x}^{3/2}}}-5/6\,{\frac{1}{a} \left ( -1/4\,{\frac{\sqrt{a+b\sqrt{x}}}{xa{b}^{2}}}-3/4\,{\frac{1}{a} \left ( -1/2\,{\frac{\sqrt{a+b\sqrt{x}}}{ab\sqrt{x}}}+1/2\,{\frac{1}{{a}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{a+b\sqrt{x}}}{\sqrt{a}}} \right ) } \right ) } \right ) } \right ) } \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*b^4*(-1/8*(a+b*x^(1/2))^(1/2)/a/b^4/x^2-7/8/a*(-1/6*(a+b*x^(1/2))^(1/2)/a/b^3/x^(3/2)-5/6/a*(-1/4*(a+b*x^(1/
2))^(1/2)/a/b^2/x-3/4/a*(-1/2*(a+b*x^(1/2))^(1/2)/a/b/x^(1/2)+1/2/a^(3/2)*arctanh((a+b*x^(1/2))^(1/2)/a^(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^3/(a+b*x^(1/2))^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.33949, size = 475, normalized size = 3.49 \begin{align*} \left [\frac{105 \, \sqrt{a} b^{4} x^{2} \log \left (\frac{b x - 2 \, \sqrt{b \sqrt{x} + a} \sqrt{a} \sqrt{x} + 2 \, a \sqrt{x}}{x}\right ) - 2 \,{\left (70 \, a^{2} b^{2} x + 48 \, a^{4} - 7 \,{\left (15 \, a b^{3} x + 8 \, a^{3} b\right )} \sqrt{x}\right )} \sqrt{b \sqrt{x} + a}}{192 \, a^{5} x^{2}}, \frac{105 \, \sqrt{-a} b^{4} x^{2} \arctan \left (\frac{\sqrt{b \sqrt{x} + a} \sqrt{-a}}{a}\right ) -{\left (70 \, a^{2} b^{2} x + 48 \, a^{4} - 7 \,{\left (15 \, a b^{3} x + 8 \, a^{3} b\right )} \sqrt{x}\right )} \sqrt{b \sqrt{x} + a}}{96 \, a^{5} x^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/192*(105*sqrt(a)*b^4*x^2*log((b*x - 2*sqrt(b*sqrt(x) + a)*sqrt(a)*sqrt(x) + 2*a*sqrt(x))/x) - 2*(70*a^2*b^2
*x + 48*a^4 - 7*(15*a*b^3*x + 8*a^3*b)*sqrt(x))*sqrt(b*sqrt(x) + a))/(a^5*x^2), 1/96*(105*sqrt(-a)*b^4*x^2*arc
tan(sqrt(b*sqrt(x) + a)*sqrt(-a)/a) - (70*a^2*b^2*x + 48*a^4 - 7*(15*a*b^3*x + 8*a^3*b)*sqrt(x))*sqrt(b*sqrt(x
) + a))/(a^5*x^2)]

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Sympy [A]  time = 11.2865, size = 173, normalized size = 1.27 \begin{align*} - \frac{1}{2 \sqrt{b} x^{\frac{9}{4}} \sqrt{\frac{a}{b \sqrt{x}} + 1}} + \frac{\sqrt{b}}{12 a x^{\frac{7}{4}} \sqrt{\frac{a}{b \sqrt{x}} + 1}} - \frac{7 b^{\frac{3}{2}}}{48 a^{2} x^{\frac{5}{4}} \sqrt{\frac{a}{b \sqrt{x}} + 1}} + \frac{35 b^{\frac{5}{2}}}{96 a^{3} x^{\frac{3}{4}} \sqrt{\frac{a}{b \sqrt{x}} + 1}} + \frac{35 b^{\frac{7}{2}}}{32 a^{4} \sqrt [4]{x} \sqrt{\frac{a}{b \sqrt{x}} + 1}} - \frac{35 b^{4} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt [4]{x}} \right )}}{32 a^{\frac{9}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(2*sqrt(b)*x**(9/4)*sqrt(a/(b*sqrt(x)) + 1)) + sqrt(b)/(12*a*x**(7/4)*sqrt(a/(b*sqrt(x)) + 1)) - 7*b**(3/2)
/(48*a**2*x**(5/4)*sqrt(a/(b*sqrt(x)) + 1)) + 35*b**(5/2)/(96*a**3*x**(3/4)*sqrt(a/(b*sqrt(x)) + 1)) + 35*b**(
7/2)/(32*a**4*x**(1/4)*sqrt(a/(b*sqrt(x)) + 1)) - 35*b**4*asinh(sqrt(a)/(sqrt(b)*x**(1/4)))/(32*a**(9/2))

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Giac [A]  time = 1.11156, size = 127, normalized size = 0.93 \begin{align*} \frac{1}{96} \, b^{4}{\left (\frac{105 \, \arctan \left (\frac{\sqrt{b \sqrt{x} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{4}} + \frac{105 \,{\left (b \sqrt{x} + a\right )}^{\frac{7}{2}} - 385 \,{\left (b \sqrt{x} + a\right )}^{\frac{5}{2}} a + 511 \,{\left (b \sqrt{x} + a\right )}^{\frac{3}{2}} a^{2} - 279 \, \sqrt{b \sqrt{x} + a} a^{3}}{a^{4} b^{4} x^{2}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/96*b^4*(105*arctan(sqrt(b*sqrt(x) + a)/sqrt(-a))/(sqrt(-a)*a^4) + (105*(b*sqrt(x) + a)^(7/2) - 385*(b*sqrt(x
) + a)^(5/2)*a + 511*(b*sqrt(x) + a)^(3/2)*a^2 - 279*sqrt(b*sqrt(x) + a)*a^3)/(a^4*b^4*x^2))