3.301 \(\int \frac {1}{x^{3/2} (a+b x^2)^2} \, dx\)

Optimal. Leaf size=230 \[ -\frac {5 \sqrt [4]{b} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4}}+\frac {5 \sqrt [4]{b} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4}}+\frac {5 \sqrt [4]{b} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{9/4}}-\frac {5 \sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} a^{9/4}}-\frac {5}{2 a^2 \sqrt {x}}+\frac {1}{2 a \sqrt {x} \left (a+b x^2\right )} \]

[Out]

5/8*b^(1/4)*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(9/4)*2^(1/2)-5/8*b^(1/4)*arctan(1+b^(1/4)*2^(1/2)*x^(
1/2)/a^(1/4))/a^(9/4)*2^(1/2)-5/16*b^(1/4)*ln(a^(1/2)+x*b^(1/2)-a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(9/4)*2^(1/
2)+5/16*b^(1/4)*ln(a^(1/2)+x*b^(1/2)+a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(9/4)*2^(1/2)-5/2/a^2/x^(1/2)+1/2/a/(b
*x^2+a)/x^(1/2)

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Rubi [A]  time = 0.19, antiderivative size = 230, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {290, 325, 329, 297, 1162, 617, 204, 1165, 628} \[ -\frac {5 \sqrt [4]{b} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4}}+\frac {5 \sqrt [4]{b} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4}}+\frac {5 \sqrt [4]{b} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{9/4}}-\frac {5 \sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} a^{9/4}}-\frac {5}{2 a^2 \sqrt {x}}+\frac {1}{2 a \sqrt {x} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^(3/2)*(a + b*x^2)^2),x]

[Out]

-5/(2*a^2*Sqrt[x]) + 1/(2*a*Sqrt[x]*(a + b*x^2)) + (5*b^(1/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(
4*Sqrt[2]*a^(9/4)) - (5*b^(1/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(9/4)) - (5*b^(1/4
)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(9/4)) + (5*b^(1/4)*Log[Sqrt[a] + S
qrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(9/4))

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 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 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 325

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

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 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]

Rubi steps

\begin {align*} \int \frac {1}{x^{3/2} \left (a+b x^2\right )^2} \, dx &=\frac {1}{2 a \sqrt {x} \left (a+b x^2\right )}+\frac {5 \int \frac {1}{x^{3/2} \left (a+b x^2\right )} \, dx}{4 a}\\ &=-\frac {5}{2 a^2 \sqrt {x}}+\frac {1}{2 a \sqrt {x} \left (a+b x^2\right )}-\frac {(5 b) \int \frac {\sqrt {x}}{a+b x^2} \, dx}{4 a^2}\\ &=-\frac {5}{2 a^2 \sqrt {x}}+\frac {1}{2 a \sqrt {x} \left (a+b x^2\right )}-\frac {(5 b) \operatorname {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 a^2}\\ &=-\frac {5}{2 a^2 \sqrt {x}}+\frac {1}{2 a \sqrt {x} \left (a+b x^2\right )}+\frac {\left (5 \sqrt {b}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 a^2}-\frac {\left (5 \sqrt {b}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 a^2}\\ &=-\frac {5}{2 a^2 \sqrt {x}}+\frac {1}{2 a \sqrt {x} \left (a+b x^2\right )}-\frac {5 \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^2}-\frac {5 \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^2}-\frac {\left (5 \sqrt [4]{b}\right ) \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^{9/4}}-\frac {\left (5 \sqrt [4]{b}\right ) \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^{9/4}}\\ &=-\frac {5}{2 a^2 \sqrt {x}}+\frac {1}{2 a \sqrt {x} \left (a+b x^2\right )}-\frac {5 \sqrt [4]{b} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4}}+\frac {5 \sqrt [4]{b} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4}}-\frac {\left (5 \sqrt [4]{b}\right ) \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^{9/4}}+\frac {\left (5 \sqrt [4]{b}\right ) \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^{9/4}}\\ &=-\frac {5}{2 a^2 \sqrt {x}}+\frac {1}{2 a \sqrt {x} \left (a+b x^2\right )}+\frac {5 \sqrt [4]{b} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{9/4}}-\frac {5 \sqrt [4]{b} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{9/4}}-\frac {5 \sqrt [4]{b} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4}}+\frac {5 \sqrt [4]{b} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4}}\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 27, normalized size = 0.12 \[ -\frac {2 \, _2F_1\left (-\frac {1}{4},2;\frac {3}{4};-\frac {b x^2}{a}\right )}{a^2 \sqrt {x}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(3/2)*(a + b*x^2)^2),x]

[Out]

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

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fricas [A]  time = 0.95, size = 208, normalized size = 0.90 \[ \frac {20 \, {\left (a^{2} b x^{3} + a^{3} x\right )} \left (-\frac {b}{a^{9}}\right )^{\frac {1}{4}} \arctan \left (-\frac {125 \, a^{2} b \sqrt {x} \left (-\frac {b}{a^{9}}\right )^{\frac {1}{4}} - \sqrt {-15625 \, a^{5} b \sqrt {-\frac {b}{a^{9}}} + 15625 \, b^{2} x} a^{2} \left (-\frac {b}{a^{9}}\right )^{\frac {1}{4}}}{125 \, b}\right ) - 5 \, {\left (a^{2} b x^{3} + a^{3} x\right )} \left (-\frac {b}{a^{9}}\right )^{\frac {1}{4}} \log \left (125 \, a^{7} \left (-\frac {b}{a^{9}}\right )^{\frac {3}{4}} + 125 \, b \sqrt {x}\right ) + 5 \, {\left (a^{2} b x^{3} + a^{3} x\right )} \left (-\frac {b}{a^{9}}\right )^{\frac {1}{4}} \log \left (-125 \, a^{7} \left (-\frac {b}{a^{9}}\right )^{\frac {3}{4}} + 125 \, b \sqrt {x}\right ) - 4 \, {\left (5 \, b x^{2} + 4 \, a\right )} \sqrt {x}}{8 \, {\left (a^{2} b x^{3} + a^{3} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(20*(a^2*b*x^3 + a^3*x)*(-b/a^9)^(1/4)*arctan(-1/125*(125*a^2*b*sqrt(x)*(-b/a^9)^(1/4) - sqrt(-15625*a^5*b
*sqrt(-b/a^9) + 15625*b^2*x)*a^2*(-b/a^9)^(1/4))/b) - 5*(a^2*b*x^3 + a^3*x)*(-b/a^9)^(1/4)*log(125*a^7*(-b/a^9
)^(3/4) + 125*b*sqrt(x)) + 5*(a^2*b*x^3 + a^3*x)*(-b/a^9)^(1/4)*log(-125*a^7*(-b/a^9)^(3/4) + 125*b*sqrt(x)) -
4*(5*b*x^2 + 4*a)*sqrt(x))/(a^2*b*x^3 + a^3*x)

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giac [A]  time = 0.62, size = 210, normalized size = 0.91 \[ -\frac {5 \, b x^{2} + 4 \, a}{2 \, {\left (b x^{\frac {5}{2}} + a \sqrt {x}\right )} a^{2}} - \frac {5 \, \sqrt {2} \left (a b^{3}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{3} b^{2}} - \frac {5 \, \sqrt {2} \left (a b^{3}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{3} b^{2}} + \frac {5 \, \sqrt {2} \left (a b^{3}\right )^{\frac {3}{4}} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, a^{3} b^{2}} - \frac {5 \, \sqrt {2} \left (a b^{3}\right )^{\frac {3}{4}} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, a^{3} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(5*b*x^2 + 4*a)/((b*x^(5/2) + a*sqrt(x))*a^2) - 5/8*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^3*b^2) - 5/8*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^3*b^2) + 5/16*sqrt(2)*(a*b^3)^(3/4)*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqr
t(a/b))/(a^3*b^2) - 5/16*sqrt(2)*(a*b^3)^(3/4)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^3*b^2)

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maple [A]  time = 0.02, size = 158, normalized size = 0.69 \[ -\frac {b \,x^{\frac {3}{2}}}{2 \left (b \,x^{2}+a \right ) a^{2}}-\frac {5 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{8 \left (\frac {a}{b}\right )^{\frac {1}{4}} a^{2}}-\frac {5 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{8 \left (\frac {a}{b}\right )^{\frac {1}{4}} a^{2}}-\frac {5 \sqrt {2}\, \ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{16 \left (\frac {a}{b}\right )^{\frac {1}{4}} a^{2}}-\frac {2}{a^{2} \sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/a^2/x^(1/2)-1/2/a^2*b*x^(3/2)/(b*x^2+a)-5/16/a^2/(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)))-5/8/a^2/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x
^(1/2)+1)-5/8/a^2/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)

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maxima [A]  time = 2.98, size = 208, normalized size = 0.90 \[ -\frac {5 \, b x^{2} + 4 \, a}{2 \, {\left (a^{2} b x^{\frac {5}{2}} + a^{3} \sqrt {x}\right )}} - \frac {5 \, b {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{16 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(5*b*x^2 + 4*a)/(a^2*b*x^(5/2) + a^3*sqrt(x)) - 5/16*b*(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)*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)) - 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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mupad [B]  time = 0.08, size = 77, normalized size = 0.33 \[ \frac {5\,{\left (-b\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {x}}{a^{1/4}}\right )}{4\,a^{9/4}}-\frac {5\,{\left (-b\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {x}}{a^{1/4}}\right )}{4\,a^{9/4}}-\frac {\frac {2}{a}+\frac {5\,b\,x^2}{2\,a^2}}{a\,\sqrt {x}+b\,x^{5/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(3/2)*(a + b*x^2)^2),x)

[Out]

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

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sympy [A]  time = 150.18, size = 700, normalized size = 3.04 \[ \begin {cases} \frac {\tilde {\infty }}{x^{\frac {9}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2}{9 b^{2} x^{\frac {9}{2}}} & \text {for}\: a = 0 \\- \frac {2}{a^{2} \sqrt {x}} & \text {for}\: b = 0 \\- \frac {16 \sqrt [4]{-1} a^{\frac {5}{4}} \sqrt [4]{\frac {1}{b}}}{8 \sqrt [4]{-1} a^{\frac {13}{4}} \sqrt {x} \sqrt [4]{\frac {1}{b}} + 8 \sqrt [4]{-1} a^{\frac {9}{4}} b x^{\frac {5}{2}} \sqrt [4]{\frac {1}{b}}} - \frac {20 \sqrt [4]{-1} \sqrt [4]{a} b x^{2} \sqrt [4]{\frac {1}{b}}}{8 \sqrt [4]{-1} a^{\frac {13}{4}} \sqrt {x} \sqrt [4]{\frac {1}{b}} + 8 \sqrt [4]{-1} a^{\frac {9}{4}} b x^{\frac {5}{2}} \sqrt [4]{\frac {1}{b}}} - \frac {5 a \sqrt {x} \log {\left (- \sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac {1}{b}} + \sqrt {x} \right )}}{8 \sqrt [4]{-1} a^{\frac {13}{4}} \sqrt {x} \sqrt [4]{\frac {1}{b}} + 8 \sqrt [4]{-1} a^{\frac {9}{4}} b x^{\frac {5}{2}} \sqrt [4]{\frac {1}{b}}} + \frac {5 a \sqrt {x} \log {\left (\sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac {1}{b}} + \sqrt {x} \right )}}{8 \sqrt [4]{-1} a^{\frac {13}{4}} \sqrt {x} \sqrt [4]{\frac {1}{b}} + 8 \sqrt [4]{-1} a^{\frac {9}{4}} b x^{\frac {5}{2}} \sqrt [4]{\frac {1}{b}}} + \frac {10 a \sqrt {x} \operatorname {atan}{\left (\frac {\left (-1\right )^{\frac {3}{4}} \sqrt {x}}{\sqrt [4]{a} \sqrt [4]{\frac {1}{b}}} \right )}}{8 \sqrt [4]{-1} a^{\frac {13}{4}} \sqrt {x} \sqrt [4]{\frac {1}{b}} + 8 \sqrt [4]{-1} a^{\frac {9}{4}} b x^{\frac {5}{2}} \sqrt [4]{\frac {1}{b}}} - \frac {5 b x^{\frac {5}{2}} \log {\left (- \sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac {1}{b}} + \sqrt {x} \right )}}{8 \sqrt [4]{-1} a^{\frac {13}{4}} \sqrt {x} \sqrt [4]{\frac {1}{b}} + 8 \sqrt [4]{-1} a^{\frac {9}{4}} b x^{\frac {5}{2}} \sqrt [4]{\frac {1}{b}}} + \frac {5 b x^{\frac {5}{2}} \log {\left (\sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac {1}{b}} + \sqrt {x} \right )}}{8 \sqrt [4]{-1} a^{\frac {13}{4}} \sqrt {x} \sqrt [4]{\frac {1}{b}} + 8 \sqrt [4]{-1} a^{\frac {9}{4}} b x^{\frac {5}{2}} \sqrt [4]{\frac {1}{b}}} + \frac {10 b x^{\frac {5}{2}} \operatorname {atan}{\left (\frac {\left (-1\right )^{\frac {3}{4}} \sqrt {x}}{\sqrt [4]{a} \sqrt [4]{\frac {1}{b}}} \right )}}{8 \sqrt [4]{-1} a^{\frac {13}{4}} \sqrt {x} \sqrt [4]{\frac {1}{b}} + 8 \sqrt [4]{-1} a^{\frac {9}{4}} b x^{\frac {5}{2}} \sqrt [4]{\frac {1}{b}}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo/x**(9/2), Eq(a, 0) & Eq(b, 0)), (-2/(9*b**2*x**(9/2)), Eq(a, 0)), (-2/(a**2*sqrt(x)), Eq(b, 0))
, (-16*(-1)**(1/4)*a**(5/4)*(1/b)**(1/4)/(8*(-1)**(1/4)*a**(13/4)*sqrt(x)*(1/b)**(1/4) + 8*(-1)**(1/4)*a**(9/4
)*b*x**(5/2)*(1/b)**(1/4)) - 20*(-1)**(1/4)*a**(1/4)*b*x**2*(1/b)**(1/4)/(8*(-1)**(1/4)*a**(13/4)*sqrt(x)*(1/b
)**(1/4) + 8*(-1)**(1/4)*a**(9/4)*b*x**(5/2)*(1/b)**(1/4)) - 5*a*sqrt(x)*log(-(-1)**(1/4)*a**(1/4)*(1/b)**(1/4
) + sqrt(x))/(8*(-1)**(1/4)*a**(13/4)*sqrt(x)*(1/b)**(1/4) + 8*(-1)**(1/4)*a**(9/4)*b*x**(5/2)*(1/b)**(1/4)) +
5*a*sqrt(x)*log((-1)**(1/4)*a**(1/4)*(1/b)**(1/4) + sqrt(x))/(8*(-1)**(1/4)*a**(13/4)*sqrt(x)*(1/b)**(1/4) +
8*(-1)**(1/4)*a**(9/4)*b*x**(5/2)*(1/b)**(1/4)) + 10*a*sqrt(x)*atan((-1)**(3/4)*sqrt(x)/(a**(1/4)*(1/b)**(1/4)
))/(8*(-1)**(1/4)*a**(13/4)*sqrt(x)*(1/b)**(1/4) + 8*(-1)**(1/4)*a**(9/4)*b*x**(5/2)*(1/b)**(1/4)) - 5*b*x**(5
/2)*log(-(-1)**(1/4)*a**(1/4)*(1/b)**(1/4) + sqrt(x))/(8*(-1)**(1/4)*a**(13/4)*sqrt(x)*(1/b)**(1/4) + 8*(-1)**
(1/4)*a**(9/4)*b*x**(5/2)*(1/b)**(1/4)) + 5*b*x**(5/2)*log((-1)**(1/4)*a**(1/4)*(1/b)**(1/4) + sqrt(x))/(8*(-1
)**(1/4)*a**(13/4)*sqrt(x)*(1/b)**(1/4) + 8*(-1)**(1/4)*a**(9/4)*b*x**(5/2)*(1/b)**(1/4)) + 10*b*x**(5/2)*atan
((-1)**(3/4)*sqrt(x)/(a**(1/4)*(1/b)**(1/4)))/(8*(-1)**(1/4)*a**(13/4)*sqrt(x)*(1/b)**(1/4) + 8*(-1)**(1/4)*a*
*(9/4)*b*x**(5/2)*(1/b)**(1/4)), True))

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