3.2.30 \(\int \frac {\sqrt {x}}{(a+b x^2)^3} \, dx\) [130]

Optimal. Leaf size=147 \[ x^{3/2} \left (\frac {1}{4 a \left (a+b x^2\right )^2}+\frac {5}{16 a^2 \left (a+b x^2\right )}\right )+\frac {5 \left (\arctan \left (\frac {\sqrt {2} \sqrt [4]{\frac {a}{b}} \sqrt {x}}{\sqrt {\frac {a}{b}}-x}\right )-\log \left (\frac {\sqrt {\frac {a}{b}}+\sqrt {2} \sqrt [4]{\frac {a}{b}} \sqrt {x}+x}{\sqrt {a+b x^2}}\right )\right )}{32 \sqrt {2} a^2 \sqrt [4]{\frac {a}{b}} b} \]

[Out]

(1/4/a/(b*x^2+a)^2+5/16/a^2/(b*x^2+a))*x^(3/2)+5/64/a^2/b/(a/b)^(1/4)*2^(1/2)*(-ln((x+(a/b)^(1/4)*2^(1/2)*x^(1
/2)+(a/b)^(1/2))/(b*x^2+a)^(1/2))+arctan((a/b)^(1/4)*2^(1/2)*x^(1/2)/((a/b)^(1/2)-x)))

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Rubi [A]
time = 0.11, antiderivative size = 239, normalized size of antiderivative = 1.63, number of steps used = 12, number of rules used = 8, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {296, 335, 303, 1176, 631, 210, 1179, 642} \begin {gather*} -\frac {5 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{9/4} b^{3/4}}+\frac {5 \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt {2} a^{9/4} b^{3/4}}+\frac {5 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{64 \sqrt {2} a^{9/4} b^{3/4}}-\frac {5 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{64 \sqrt {2} a^{9/4} b^{3/4}}+\frac {5 x^{3/2}}{16 a^2 \left (a+b x^2\right )}+\frac {x^{3/2}}{4 a \left (a+b x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]/(a + b*x^2)^3,x]

[Out]

x^(3/2)/(4*a*(a + b*x^2)^2) + (5*x^(3/2))/(16*a^2*(a + b*x^2)) - (5*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/
4)])/(32*Sqrt[2]*a^(9/4)*b^(3/4)) + (5*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(32*Sqrt[2]*a^(9/4)*b^(3
/4)) + (5*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(64*Sqrt[2]*a^(9/4)*b^(3/4)) - (5*Log[Sq
rt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(64*Sqrt[2]*a^(9/4)*b^(3/4))

Rule 210

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

Rule 296

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] /; Free
Q[{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 303

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 335

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 631

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 642

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]

Rule 1176

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 1179

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]

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=\frac {x^{3/2}}{4 a \left (a+b x^2\right )^2}+\frac {5 \int \frac {\sqrt {x}}{\left (a+b x^2\right )^2} \, dx}{8 a}\\ &=\frac {x^{3/2}}{4 a \left (a+b x^2\right )^2}+\frac {5 x^{3/2}}{16 a^2 \left (a+b x^2\right )}+\frac {5 \int \frac {\sqrt {x}}{a+b x^2} \, dx}{32 a^2}\\ &=\frac {x^{3/2}}{4 a \left (a+b x^2\right )^2}+\frac {5 x^{3/2}}{16 a^2 \left (a+b x^2\right )}+\frac {5 \text {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{16 a^2}\\ &=\frac {x^{3/2}}{4 a \left (a+b x^2\right )^2}+\frac {5 x^{3/2}}{16 a^2 \left (a+b x^2\right )}-\frac {5 \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{32 a^2 \sqrt {b}}+\frac {5 \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{32 a^2 \sqrt {b}}\\ &=\frac {x^{3/2}}{4 a \left (a+b x^2\right )^2}+\frac {5 x^{3/2}}{16 a^2 \left (a+b x^2\right )}+\frac {5 \text {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 )}{64 a^2 b}+\frac {5 \text {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 )}{64 a^2 b}+\frac {5 \text {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 )}{64 \sqrt {2} a^{9/4} b^{3/4}}+\frac {5 \text {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 )}{64 \sqrt {2} a^{9/4} b^{3/4}}\\ &=\frac {x^{3/2}}{4 a \left (a+b x^2\right )^2}+\frac {5 x^{3/2}}{16 a^2 \left (a+b x^2\right )}+\frac {5 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{9/4} b^{3/4}}-\frac {5 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{9/4} b^{3/4}}+\frac {5 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{9/4} b^{3/4}}-\frac {5 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{9/4} b^{3/4}}\\ &=\frac {x^{3/2}}{4 a \left (a+b x^2\right )^2}+\frac {5 x^{3/2}}{16 a^2 \left (a+b x^2\right )}-\frac {5 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{9/4} b^{3/4}}+\frac {5 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{9/4} b^{3/4}}+\frac {5 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{9/4} b^{3/4}}-\frac {5 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{9/4} b^{3/4}}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.20, size = 138, normalized size = 0.94 \begin {gather*} \frac {\frac {4 \sqrt [4]{a} x^{3/2} \left (9 a+5 b x^2\right )}{\left (a+b x^2\right )^2}-\frac {5 \sqrt {2} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{b^{3/4}}-\frac {5 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{b^{3/4}}}{64 a^{9/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]/(a + b*x^2)^3,x]

[Out]

((4*a^(1/4)*x^(3/2)*(9*a + 5*b*x^2))/(a + b*x^2)^2 - (5*Sqrt[2]*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*
b^(1/4)*Sqrt[x])])/b^(3/4) - (5*Sqrt[2]*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/b^(3
/4))/(64*a^(9/4))

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Maple [A]
time = 0.04, size = 150, normalized size = 1.02

method	result	size
derivativedivides	\(\frac {x^{\frac {3}{2}}}{4 a \left (x^{2} b +a \right )^{2}}+\frac {\frac {5 x^{\frac {3}{2}}}{16 a \left (x^{2} b +a \right )}+\frac {5 \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{128 a b \left (\frac {a}{b}\right )^{\frac {1}{4}}}}{a}\)	\(150\)
default	\(\frac {x^{\frac {3}{2}}}{4 a \left (x^{2} b +a \right )^{2}}+\frac {\frac {5 x^{\frac {3}{2}}}{16 a \left (x^{2} b +a \right )}+\frac {5 \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{128 a b \left (\frac {a}{b}\right )^{\frac {1}{4}}}}{a}\)	\(150\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x^(3/2)/a/(b*x^2+a)^2+5/4/a*(1/4*x^(3/2)/a/(b*x^2+a)+1/32/a/b/(a/b)^(1/4)*2^(1/2)*(ln((x-(a/b)^(1/4)*2^(1/
2)*x^(1/2)+(a/b)^(1/2))/(x+(a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/2)))+2*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+2
*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)))

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Maxima [A]
time = 0.45, size = 217, normalized size = 1.48 \begin {gather*} \frac {5 \, b x^{\frac {7}{2}} + 9 \, a x^{\frac {3}{2}}}{16 \, {\left (a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + a^{4}\right )}} + \frac {5 \, {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{128 \, a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(5*b*x^(7/2) + 9*a*x^(3/2))/(a^2*b^2*x^4 + 2*a^3*b*x^2 + a^4) + 5/128*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt
(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(sqrt(a)*sqrt(b))*sqrt(b)) + 2*sqrt(2)*ar
ctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(sqrt(a)*sqrt(b))*
sqrt(b)) - sqrt(2)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(1/4)*b^(3/4)) + sqrt(2)*log(
-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(1/4)*b^(3/4)))/a^2

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 250 vs. \(2 (119) = 238\).
time = 0.63, size = 250, normalized size = 1.70 \begin {gather*} -\frac {20 \, {\left (a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + a^{4}\right )} \left (-\frac {1}{a^{9} b^{3}}\right )^{\frac {1}{4}} \arctan \left (\sqrt {-a^{5} b \sqrt {-\frac {1}{a^{9} b^{3}}} + x} a^{2} b \left (-\frac {1}{a^{9} b^{3}}\right )^{\frac {1}{4}} - a^{2} b \sqrt {x} \left (-\frac {1}{a^{9} b^{3}}\right )^{\frac {1}{4}}\right ) - 5 \, {\left (a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + a^{4}\right )} \left (-\frac {1}{a^{9} b^{3}}\right )^{\frac {1}{4}} \log \left (a^{7} b^{2} \left (-\frac {1}{a^{9} b^{3}}\right )^{\frac {3}{4}} + \sqrt {x}\right ) + 5 \, {\left (a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + a^{4}\right )} \left (-\frac {1}{a^{9} b^{3}}\right )^{\frac {1}{4}} \log \left (-a^{7} b^{2} \left (-\frac {1}{a^{9} b^{3}}\right )^{\frac {3}{4}} + \sqrt {x}\right ) - 4 \, {\left (5 \, b x^{3} + 9 \, a x\right )} \sqrt {x}}{64 \, {\left (a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + a^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/64*(20*(a^2*b^2*x^4 + 2*a^3*b*x^2 + a^4)*(-1/(a^9*b^3))^(1/4)*arctan(sqrt(-a^5*b*sqrt(-1/(a^9*b^3)) + x)*a^
2*b*(-1/(a^9*b^3))^(1/4) - a^2*b*sqrt(x)*(-1/(a^9*b^3))^(1/4)) - 5*(a^2*b^2*x^4 + 2*a^3*b*x^2 + a^4)*(-1/(a^9*
b^3))^(1/4)*log(a^7*b^2*(-1/(a^9*b^3))^(3/4) + sqrt(x)) + 5*(a^2*b^2*x^4 + 2*a^3*b*x^2 + a^4)*(-1/(a^9*b^3))^(
1/4)*log(-a^7*b^2*(-1/(a^9*b^3))^(3/4) + sqrt(x)) - 4*(5*b*x^3 + 9*a*x)*sqrt(x))/(a^2*b^2*x^4 + 2*a^3*b*x^2 +
a^4)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 887 vs. \(2 (116) = 232\).
time = 93.90, size = 887, normalized size = 6.03 \begin {gather*} \begin {cases} \frac {\tilde {\infty }}{x^{\frac {9}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 x^{\frac {3}{2}}}{3 a^{3}} & \text {for}\: b = 0 \\- \frac {2}{9 b^{3} x^{\frac {9}{2}}} & \text {for}\: a = 0 \\\frac {5 a^{2} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{4} b \sqrt [4]{- \frac {a}{b}} + 128 a^{3} b^{2} x^{2} \sqrt [4]{- \frac {a}{b}} + 64 a^{2} b^{3} x^{4} \sqrt [4]{- \frac {a}{b}}} - \frac {5 a^{2} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{4} b \sqrt [4]{- \frac {a}{b}} + 128 a^{3} b^{2} x^{2} \sqrt [4]{- \frac {a}{b}} + 64 a^{2} b^{3} x^{4} \sqrt [4]{- \frac {a}{b}}} + \frac {10 a^{2} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{64 a^{4} b \sqrt [4]{- \frac {a}{b}} + 128 a^{3} b^{2} x^{2} \sqrt [4]{- \frac {a}{b}} + 64 a^{2} b^{3} x^{4} \sqrt [4]{- \frac {a}{b}}} + \frac {36 a b x^{\frac {3}{2}} \sqrt [4]{- \frac {a}{b}}}{64 a^{4} b \sqrt [4]{- \frac {a}{b}} + 128 a^{3} b^{2} x^{2} \sqrt [4]{- \frac {a}{b}} + 64 a^{2} b^{3} x^{4} \sqrt [4]{- \frac {a}{b}}} + \frac {10 a b x^{2} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{4} b \sqrt [4]{- \frac {a}{b}} + 128 a^{3} b^{2} x^{2} \sqrt [4]{- \frac {a}{b}} + 64 a^{2} b^{3} x^{4} \sqrt [4]{- \frac {a}{b}}} - \frac {10 a b x^{2} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{4} b \sqrt [4]{- \frac {a}{b}} + 128 a^{3} b^{2} x^{2} \sqrt [4]{- \frac {a}{b}} + 64 a^{2} b^{3} x^{4} \sqrt [4]{- \frac {a}{b}}} + \frac {20 a b x^{2} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{64 a^{4} b \sqrt [4]{- \frac {a}{b}} + 128 a^{3} b^{2} x^{2} \sqrt [4]{- \frac {a}{b}} + 64 a^{2} b^{3} x^{4} \sqrt [4]{- \frac {a}{b}}} + \frac {20 b^{2} x^{\frac {7}{2}} \sqrt [4]{- \frac {a}{b}}}{64 a^{4} b \sqrt [4]{- \frac {a}{b}} + 128 a^{3} b^{2} x^{2} \sqrt [4]{- \frac {a}{b}} + 64 a^{2} b^{3} x^{4} \sqrt [4]{- \frac {a}{b}}} + \frac {5 b^{2} x^{4} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{4} b \sqrt [4]{- \frac {a}{b}} + 128 a^{3} b^{2} x^{2} \sqrt [4]{- \frac {a}{b}} + 64 a^{2} b^{3} x^{4} \sqrt [4]{- \frac {a}{b}}} - \frac {5 b^{2} x^{4} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{4} b \sqrt [4]{- \frac {a}{b}} + 128 a^{3} b^{2} x^{2} \sqrt [4]{- \frac {a}{b}} + 64 a^{2} b^{3} x^{4} \sqrt [4]{- \frac {a}{b}}} + \frac {10 b^{2} x^{4} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{64 a^{4} b \sqrt [4]{- \frac {a}{b}} + 128 a^{3} b^{2} x^{2} \sqrt [4]{- \frac {a}{b}} + 64 a^{2} b^{3} x^{4} \sqrt [4]{- \frac {a}{b}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo/x**(9/2), Eq(a, 0) & Eq(b, 0)), (2*x**(3/2)/(3*a**3), Eq(b, 0)), (-2/(9*b**3*x**(9/2)), Eq(a, 0
)), (5*a**2*log(sqrt(x) - (-a/b)**(1/4))/(64*a**4*b*(-a/b)**(1/4) + 128*a**3*b**2*x**2*(-a/b)**(1/4) + 64*a**2
*b**3*x**4*(-a/b)**(1/4)) - 5*a**2*log(sqrt(x) + (-a/b)**(1/4))/(64*a**4*b*(-a/b)**(1/4) + 128*a**3*b**2*x**2*
(-a/b)**(1/4) + 64*a**2*b**3*x**4*(-a/b)**(1/4)) + 10*a**2*atan(sqrt(x)/(-a/b)**(1/4))/(64*a**4*b*(-a/b)**(1/4
) + 128*a**3*b**2*x**2*(-a/b)**(1/4) + 64*a**2*b**3*x**4*(-a/b)**(1/4)) + 36*a*b*x**(3/2)*(-a/b)**(1/4)/(64*a*
*4*b*(-a/b)**(1/4) + 128*a**3*b**2*x**2*(-a/b)**(1/4) + 64*a**2*b**3*x**4*(-a/b)**(1/4)) + 10*a*b*x**2*log(sqr
t(x) - (-a/b)**(1/4))/(64*a**4*b*(-a/b)**(1/4) + 128*a**3*b**2*x**2*(-a/b)**(1/4) + 64*a**2*b**3*x**4*(-a/b)**
(1/4)) - 10*a*b*x**2*log(sqrt(x) + (-a/b)**(1/4))/(64*a**4*b*(-a/b)**(1/4) + 128*a**3*b**2*x**2*(-a/b)**(1/4)
+ 64*a**2*b**3*x**4*(-a/b)**(1/4)) + 20*a*b*x**2*atan(sqrt(x)/(-a/b)**(1/4))/(64*a**4*b*(-a/b)**(1/4) + 128*a*
*3*b**2*x**2*(-a/b)**(1/4) + 64*a**2*b**3*x**4*(-a/b)**(1/4)) + 20*b**2*x**(7/2)*(-a/b)**(1/4)/(64*a**4*b*(-a/
b)**(1/4) + 128*a**3*b**2*x**2*(-a/b)**(1/4) + 64*a**2*b**3*x**4*(-a/b)**(1/4)) + 5*b**2*x**4*log(sqrt(x) - (-
a/b)**(1/4))/(64*a**4*b*(-a/b)**(1/4) + 128*a**3*b**2*x**2*(-a/b)**(1/4) + 64*a**2*b**3*x**4*(-a/b)**(1/4)) -
5*b**2*x**4*log(sqrt(x) + (-a/b)**(1/4))/(64*a**4*b*(-a/b)**(1/4) + 128*a**3*b**2*x**2*(-a/b)**(1/4) + 64*a**2
*b**3*x**4*(-a/b)**(1/4)) + 10*b**2*x**4*atan(sqrt(x)/(-a/b)**(1/4))/(64*a**4*b*(-a/b)**(1/4) + 128*a**3*b**2*
x**2*(-a/b)**(1/4) + 64*a**2*b**3*x**4*(-a/b)**(1/4)), True))

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Giac [A]
time = 0.46, size = 209, normalized size = 1.42 \begin {gather*} \frac {5 \, b x^{\frac {7}{2}} + 9 \, a x^{\frac {3}{2}}}{16 \, {\left (b x^{2} + a\right )}^{2} a^{2}} + \frac {5 \, \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 )}{64 \, a^{3} b^{3}} + \frac {5 \, \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 )}{64 \, a^{3} b^{3}} - \frac {5 \, \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 )}{128 \, a^{3} b^{3}} + \frac {5 \, \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 )}{128 \, a^{3} b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(5*b*x^(7/2) + 9*a*x^(3/2))/((b*x^2 + a)^2*a^2) + 5/64*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^3*b^3) + 5/64*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^3*b^3) - 5/128*sqrt(2)*(a*b^3)^(3/4)*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x
+ sqrt(a/b))/(a^3*b^3) + 5/128*sqrt(2)*(a*b^3)^(3/4)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^3*b^
3)

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Mupad [B]
time = 0.12, size = 86, normalized size = 0.59 \begin {gather*} \frac {\frac {9\,x^{3/2}}{16\,a}+\frac {5\,b\,x^{7/2}}{16\,a^2}}{a^2+2\,a\,b\,x^2+b^2\,x^4}+\frac {5\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {x}}{{\left (-a\right )}^{1/4}}\right )}{32\,{\left (-a\right )}^{9/4}\,b^{3/4}}-\frac {5\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {x}}{{\left (-a\right )}^{1/4}}\right )}{32\,{\left (-a\right )}^{9/4}\,b^{3/4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)/(a + b*x^2)^3,x)

[Out]

((9*x^(3/2))/(16*a) + (5*b*x^(7/2))/(16*a^2))/(a^2 + b^2*x^4 + 2*a*b*x^2) + (5*atan((b^(1/4)*x^(1/2))/(-a)^(1/
4)))/(32*(-a)^(9/4)*b^(3/4)) - (5*atanh((b^(1/4)*x^(1/2))/(-a)^(1/4)))/(32*(-a)^(9/4)*b^(3/4))

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