3.299 \(\int \frac{\sqrt{x}}{(a+b x^2)^2} \, dx\)

Optimal. Leaf size=218 \[ \frac{\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{5/4} b^{3/4}}-\frac{\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{5/4} b^{3/4}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{5/4} b^{3/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{5/4} b^{3/4}}+\frac{x^{3/2}}{2 a \left (a+b x^2\right )} \]

[Out]

x^(3/2)/(2*a*(a + b*x^2)) - ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)]/(4*Sqrt[2]*a^(5/4)*b^(3/4)) + ArcTan
[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)]/(4*Sqrt[2]*a^(5/4)*b^(3/4)) + Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sq
rt[x] + Sqrt[b]*x]/(8*Sqrt[2]*a^(5/4)*b^(3/4)) - Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]/(8
*Sqrt[2]*a^(5/4)*b^(3/4))

________________________________________________________________________________________

Rubi [A]  time = 0.147091, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.533, Rules used = {290, 329, 297, 1162, 617, 204, 1165, 628} \[ \frac{\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{5/4} b^{3/4}}-\frac{\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{5/4} b^{3/4}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{5/4} b^{3/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{5/4} b^{3/4}}+\frac{x^{3/2}}{2 a \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]/(a + b*x^2)^2,x]

[Out]

x^(3/2)/(2*a*(a + b*x^2)) - ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)]/(4*Sqrt[2]*a^(5/4)*b^(3/4)) + ArcTan
[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)]/(4*Sqrt[2]*a^(5/4)*b^(3/4)) + Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sq
rt[x] + Sqrt[b]*x]/(8*Sqrt[2]*a^(5/4)*b^(3/4)) - Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]/(8
*Sqrt[2]*a^(5/4)*b^(3/4))

Rule 290

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(
a*c*n*(p + 1)), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[
{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 297

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

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

Mathematica [C]  time = 0.0048495, size = 29, normalized size = 0.13 \[ \frac{2 x^{3/2} \, _2F_1\left (\frac{3}{4},2;\frac{7}{4};-\frac{b x^2}{a}\right )}{3 a^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]/(a + b*x^2)^2,x]

[Out]

(2*x^(3/2)*Hypergeometric2F1[3/4, 2, 7/4, -((b*x^2)/a)])/(3*a^2)

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 158, normalized size = 0.7 \begin{align*}{\frac{1}{2\,a \left ( b{x}^{2}+a \right ) }{x}^{{\frac{3}{2}}}}+{\frac{\sqrt{2}}{16\,ab}\ln \left ({ \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{\sqrt{2}}{8\,ab}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{\sqrt{2}}{8\,ab}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^(3/2)/a/(b*x^2+a)+1/16/a/b/(1/b*a)^(1/4)*2^(1/2)*ln((x-(1/b*a)^(1/4)*x^(1/2)*2^(1/2)+(1/b*a)^(1/2))/(x+(
1/b*a)^(1/4)*x^(1/2)*2^(1/2)+(1/b*a)^(1/2)))+1/8/a/b/(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x^(1/2
)+1)+1/8/a/b/(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x^(1/2)-1)

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.39898, size = 467, normalized size = 2.14 \begin{align*} -\frac{4 \,{\left (a b x^{2} + a^{2}\right )} \left (-\frac{1}{a^{5} b^{3}}\right )^{\frac{1}{4}} \arctan \left (\sqrt{-a^{3} b \sqrt{-\frac{1}{a^{5} b^{3}}} + x} a b \left (-\frac{1}{a^{5} b^{3}}\right )^{\frac{1}{4}} - a b \sqrt{x} \left (-\frac{1}{a^{5} b^{3}}\right )^{\frac{1}{4}}\right ) -{\left (a b x^{2} + a^{2}\right )} \left (-\frac{1}{a^{5} b^{3}}\right )^{\frac{1}{4}} \log \left (a^{4} b^{2} \left (-\frac{1}{a^{5} b^{3}}\right )^{\frac{3}{4}} + \sqrt{x}\right ) +{\left (a b x^{2} + a^{2}\right )} \left (-\frac{1}{a^{5} b^{3}}\right )^{\frac{1}{4}} \log \left (-a^{4} b^{2} \left (-\frac{1}{a^{5} b^{3}}\right )^{\frac{3}{4}} + \sqrt{x}\right ) - 4 \, x^{\frac{3}{2}}}{8 \,{\left (a b x^{2} + a^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(4*(a*b*x^2 + a^2)*(-1/(a^5*b^3))^(1/4)*arctan(sqrt(-a^3*b*sqrt(-1/(a^5*b^3)) + x)*a*b*(-1/(a^5*b^3))^(1/
4) - a*b*sqrt(x)*(-1/(a^5*b^3))^(1/4)) - (a*b*x^2 + a^2)*(-1/(a^5*b^3))^(1/4)*log(a^4*b^2*(-1/(a^5*b^3))^(3/4)
+ sqrt(x)) + (a*b*x^2 + a^2)*(-1/(a^5*b^3))^(1/4)*log(-a^4*b^2*(-1/(a^5*b^3))^(3/4) + sqrt(x)) - 4*x^(3/2))/(
a*b*x^2 + a^2)

________________________________________________________________________________________

Sympy [A]  time = 125.801, size = 619, normalized size = 2.84 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo/x**(5/2), Eq(a, 0) & Eq(b, 0)), (2*x**(3/2)/(3*a**2), Eq(b, 0)), (-2/(5*b**2*x**(5/2)), Eq(a, 0
)), (-4*(-1)**(1/4)*a**(85/4)*b**22*x**(3/2)*(1/b)**(85/4)/(-8*(-1)**(1/4)*a**(93/4)*b**22*(1/b)**(85/4) - 8*(
-1)**(1/4)*a**(89/4)*b**23*x**2*(1/b)**(85/4)) - a**22*log(-(-1)**(1/4)*a**(1/4)*(1/b)**(1/4) + sqrt(x))/(-8*(
-1)**(1/4)*a**(93/4)*b**22*(1/b)**(85/4) - 8*(-1)**(1/4)*a**(89/4)*b**23*x**2*(1/b)**(85/4)) + a**22*log((-1)*
*(1/4)*a**(1/4)*(1/b)**(1/4) + sqrt(x))/(-8*(-1)**(1/4)*a**(93/4)*b**22*(1/b)**(85/4) - 8*(-1)**(1/4)*a**(89/4
)*b**23*x**2*(1/b)**(85/4)) + 2*a**22*atan((-1)**(3/4)*sqrt(x)/(a**(1/4)*(1/b)**(1/4)))/(-8*(-1)**(1/4)*a**(93
/4)*b**22*(1/b)**(85/4) - 8*(-1)**(1/4)*a**(89/4)*b**23*x**2*(1/b)**(85/4)) - a**21*b*x**2*log(-(-1)**(1/4)*a*
*(1/4)*(1/b)**(1/4) + sqrt(x))/(-8*(-1)**(1/4)*a**(93/4)*b**22*(1/b)**(85/4) - 8*(-1)**(1/4)*a**(89/4)*b**23*x
**2*(1/b)**(85/4)) + a**21*b*x**2*log((-1)**(1/4)*a**(1/4)*(1/b)**(1/4) + sqrt(x))/(-8*(-1)**(1/4)*a**(93/4)*b
**22*(1/b)**(85/4) - 8*(-1)**(1/4)*a**(89/4)*b**23*x**2*(1/b)**(85/4)) + 2*a**21*b*x**2*atan((-1)**(3/4)*sqrt(
x)/(a**(1/4)*(1/b)**(1/4)))/(-8*(-1)**(1/4)*a**(93/4)*b**22*(1/b)**(85/4) - 8*(-1)**(1/4)*a**(89/4)*b**23*x**2
*(1/b)**(85/4)), True))

________________________________________________________________________________________

Giac [A]  time = 2.16401, size = 269, normalized size = 1.23 \begin{align*} \frac{x^{\frac{3}{2}}}{2 \,{\left (b x^{2} + a\right )} a} + \frac{\sqrt{2} \left (a b^{3}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{8 \, a^{2} b^{3}} + \frac{\sqrt{2} \left (a b^{3}\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{8 \, a^{2} b^{3}} - \frac{\sqrt{2} \left (a b^{3}\right )^{\frac{3}{4}} \log \left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{16 \, a^{2} b^{3}} + \frac{\sqrt{2} \left (a b^{3}\right )^{\frac{3}{4}} \log \left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{16 \, a^{2} b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^(3/2)/((b*x^2 + a)*a) + 1/8*sqrt(2)*(a*b^3)^(3/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(
a/b)^(1/4))/(a^2*b^3) + 1/8*sqrt(2)*(a*b^3)^(3/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^
(1/4))/(a^2*b^3) - 1/16*sqrt(2)*(a*b^3)^(3/4)*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^2*b^3) + 1/1
6*sqrt(2)*(a*b^3)^(3/4)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^2*b^3)