3.2.31 \(\int \frac {x^{3/2}}{(a+b x^2)^3} \, dx\) [131]

Optimal. Leaf size=139 \[ \frac {\sqrt {x} \left (-3 a+b x^2\right )}{16 a b \left (a+b x^2\right )^2}+\frac {3 \left (\arctan \left (\frac {\sqrt {2} \sqrt [4]{\frac {a}{b}} \sqrt {x}}{\sqrt {\frac {a}{b}}-x}\right )+\log \left (\frac {\sqrt {\frac {a}{b}}+\sqrt {2} \sqrt [4]{\frac {a}{b}} \sqrt {x}+x}{\sqrt {a+b x^2}}\right )\right )}{32 \sqrt {2} a \left (\frac {a}{b}\right )^{3/4} b^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.11, antiderivative size = 242, normalized size of antiderivative = 1.74, 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 = {294, 296, 335, 217, 1179, 642, 1176, 631, 210} \begin {gather*} -\frac {3 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{7/4} b^{5/4}}+\frac {3 \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt {2} a^{7/4} b^{5/4}}-\frac {3 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{64 \sqrt {2} a^{7/4} b^{5/4}}+\frac {3 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{64 \sqrt {2} a^{7/4} b^{5/4}}+\frac {\sqrt {x}}{16 a b \left (a+b x^2\right )}-\frac {\sqrt {x}}{4 b \left (a+b x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(3/2)/(a + b*x^2)^3,x]

[Out]

-1/4*Sqrt[x]/(b*(a + b*x^2)^2) + Sqrt[x]/(16*a*b*(a + b*x^2)) - (3*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4
)])/(32*Sqrt[2]*a^(7/4)*b^(5/4)) + (3*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(32*Sqrt[2]*a^(7/4)*b^(5/
4)) - (3*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(64*Sqrt[2]*a^(7/4)*b^(5/4)) + (3*Log[Sqr
t[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(64*Sqrt[2]*a^(7/4)*b^(5/4))

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 217

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

Rule 294

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

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 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 {\sqrt {x}}{4 b \left (a+b x^2\right )^2}+\frac {\int \frac {1}{\sqrt {x} \left (a+b x^2\right )^2} \, dx}{8 b}\\ &=-\frac {\sqrt {x}}{4 b \left (a+b x^2\right )^2}+\frac {\sqrt {x}}{16 a b \left (a+b x^2\right )}+\frac {3 \int \frac {1}{\sqrt {x} \left (a+b x^2\right )} \, dx}{32 a b}\\ &=-\frac {\sqrt {x}}{4 b \left (a+b x^2\right )^2}+\frac {\sqrt {x}}{16 a b \left (a+b x^2\right )}+\frac {3 \text {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{16 a b}\\ &=-\frac {\sqrt {x}}{4 b \left (a+b x^2\right )^2}+\frac {\sqrt {x}}{16 a b \left (a+b x^2\right )}+\frac {3 \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{32 a^{3/2} b}+\frac {3 \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{32 a^{3/2} b}\\ &=-\frac {\sqrt {x}}{4 b \left (a+b x^2\right )^2}+\frac {\sqrt {x}}{16 a b \left (a+b x^2\right )}+\frac {3 \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^{3/2} b^{3/2}}+\frac {3 \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^{3/2} b^{3/2}}-\frac {3 \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^{7/4} b^{5/4}}-\frac {3 \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^{7/4} b^{5/4}}\\ &=-\frac {\sqrt {x}}{4 b \left (a+b x^2\right )^2}+\frac {\sqrt {x}}{16 a b \left (a+b x^2\right )}-\frac {3 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{7/4} b^{5/4}}+\frac {3 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{7/4} b^{5/4}}+\frac {3 \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^{7/4} b^{5/4}}-\frac {3 \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^{7/4} b^{5/4}}\\ &=-\frac {\sqrt {x}}{4 b \left (a+b x^2\right )^2}+\frac {\sqrt {x}}{16 a b \left (a+b x^2\right )}-\frac {3 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{7/4} b^{5/4}}+\frac {3 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{7/4} b^{5/4}}-\frac {3 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{7/4} b^{5/4}}+\frac {3 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{7/4} b^{5/4}}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]
time = 0.29, size = 137, normalized size = 0.99 \begin {gather*} \frac {\frac {4 a^{3/4} \sqrt [4]{b} \sqrt {x} \left (-3 a+b x^2\right )}{\left (a+b x^2\right )^2}-3 \sqrt {2} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+3 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{64 a^{7/4} b^{5/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(3/2)/(a + b*x^2)^3,x]

[Out]

((4*a^(3/4)*b^(1/4)*Sqrt[x]*(-3*a + b*x^2))/(a + b*x^2)^2 - 3*Sqrt[2]*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^
(1/4)*b^(1/4)*Sqrt[x])] + 3*Sqrt[2]*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(64*a^(7
/4)*b^(5/4))

________________________________________________________________________________________

Maple [A]
time = 0.05, size = 138, normalized size = 0.99

method	result	size
derivativedivides	\(\frac {\frac {x^{\frac {5}{2}}}{16 a}-\frac {3 \sqrt {x}}{16 b}}{\left (x^{2} b +a \right )^{2}}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{128 b \,a^{2}}\)	\(138\)
default	\(\frac {\frac {x^{\frac {5}{2}}}{16 a}-\frac {3 \sqrt {x}}{16 b}}{\left (x^{2} b +a \right )^{2}}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{128 b \,a^{2}}\)	\(138\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(1/32/a*x^(5/2)-3/32*x^(1/2)/b)/(b*x^2+a)^2+3/128/b/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)))+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))

________________________________________________________________________________________

Maxima [A]
time = 0.47, size = 221, normalized size = 1.59 \begin {gather*} \frac {b x^{\frac {5}{2}} - 3 \, a \sqrt {x}}{16 \, {\left (a b^{3} x^{4} + 2 \, a^{2} b^{2} x^{2} + a^{3} b\right )}} + \frac {3 \, {\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 {a} \sqrt {\sqrt {a} \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 {a} \sqrt {\sqrt {a} \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 {3}{4}} b^{\frac {1}{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 {3}{4}} b^{\frac {1}{4}}}\right )}}{128 \, a b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(b*x^(5/2) - 3*a*sqrt(x))/(a*b^3*x^4 + 2*a^2*b^2*x^2 + a^3*b) + 3/128*(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(a)*sqrt(sqrt(a)*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(a)*sqrt(sqrt(a)*s
qrt(b))) + sqrt(2)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)) - sqrt(2)*log(
-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)))/(a*b)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 257 vs. \(2 (112) = 224\).
time = 0.62, size = 257, normalized size = 1.85 \begin {gather*} \frac {12 \, {\left (a b^{3} x^{4} + 2 \, a^{2} b^{2} x^{2} + a^{3} b\right )} \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {1}{4}} \arctan \left (\sqrt {a^{4} b^{2} \sqrt {-\frac {1}{a^{7} b^{5}}} + x} a^{5} b^{4} \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {3}{4}} - a^{5} b^{4} \sqrt {x} \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {3}{4}}\right ) + 3 \, {\left (a b^{3} x^{4} + 2 \, a^{2} b^{2} x^{2} + a^{3} b\right )} \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {1}{4}} \log \left (a^{2} b \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {1}{4}} + \sqrt {x}\right ) - 3 \, {\left (a b^{3} x^{4} + 2 \, a^{2} b^{2} x^{2} + a^{3} b\right )} \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {1}{4}} \log \left (-a^{2} b \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {1}{4}} + \sqrt {x}\right ) + 4 \, {\left (b x^{2} - 3 \, a\right )} \sqrt {x}}{64 \, {\left (a b^{3} x^{4} + 2 \, a^{2} b^{2} x^{2} + a^{3} b\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*(12*(a*b^3*x^4 + 2*a^2*b^2*x^2 + a^3*b)*(-1/(a^7*b^5))^(1/4)*arctan(sqrt(a^4*b^2*sqrt(-1/(a^7*b^5)) + x)*
a^5*b^4*(-1/(a^7*b^5))^(3/4) - a^5*b^4*sqrt(x)*(-1/(a^7*b^5))^(3/4)) + 3*(a*b^3*x^4 + 2*a^2*b^2*x^2 + a^3*b)*(
-1/(a^7*b^5))^(1/4)*log(a^2*b*(-1/(a^7*b^5))^(1/4) + sqrt(x)) - 3*(a*b^3*x^4 + 2*a^2*b^2*x^2 + a^3*b)*(-1/(a^7
*b^5))^(1/4)*log(-a^2*b*(-1/(a^7*b^5))^(1/4) + sqrt(x)) + 4*(b*x^2 - 3*a)*sqrt(x))/(a*b^3*x^4 + 2*a^2*b^2*x^2
+ a^3*b)

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 666 vs. \(2 (112) = 224\).
time = 145.38, size = 666, normalized size = 4.79 \begin {gather*} \begin {cases} \frac {\tilde {\infty }}{x^{\frac {7}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 x^{\frac {5}{2}}}{5 a^{3}} & \text {for}\: b = 0 \\- \frac {2}{7 b^{3} x^{\frac {7}{2}}} & \text {for}\: a = 0 \\- \frac {12 a^{2} \sqrt {x}}{64 a^{4} b + 128 a^{3} b^{2} x^{2} + 64 a^{2} b^{3} x^{4}} - \frac {3 a^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{4} b + 128 a^{3} b^{2} x^{2} + 64 a^{2} b^{3} x^{4}} + \frac {3 a^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{4} b + 128 a^{3} b^{2} x^{2} + 64 a^{2} b^{3} x^{4}} + \frac {6 a^{2} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{64 a^{4} b + 128 a^{3} b^{2} x^{2} + 64 a^{2} b^{3} x^{4}} + \frac {4 a b x^{\frac {5}{2}}}{64 a^{4} b + 128 a^{3} b^{2} x^{2} + 64 a^{2} b^{3} x^{4}} - \frac {6 a b x^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{4} b + 128 a^{3} b^{2} x^{2} + 64 a^{2} b^{3} x^{4}} + \frac {6 a b x^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{4} b + 128 a^{3} b^{2} x^{2} + 64 a^{2} b^{3} x^{4}} + \frac {12 a b x^{2} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{64 a^{4} b + 128 a^{3} b^{2} x^{2} + 64 a^{2} b^{3} x^{4}} - \frac {3 b^{2} x^{4} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{4} b + 128 a^{3} b^{2} x^{2} + 64 a^{2} b^{3} x^{4}} + \frac {3 b^{2} x^{4} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{4} b + 128 a^{3} b^{2} x^{2} + 64 a^{2} b^{3} x^{4}} + \frac {6 b^{2} x^{4} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{64 a^{4} b + 128 a^{3} b^{2} x^{2} + 64 a^{2} b^{3} x^{4}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo/x**(7/2), Eq(a, 0) & Eq(b, 0)), (2*x**(5/2)/(5*a**3), Eq(b, 0)), (-2/(7*b**3*x**(7/2)), Eq(a, 0
)), (-12*a**2*sqrt(x)/(64*a**4*b + 128*a**3*b**2*x**2 + 64*a**2*b**3*x**4) - 3*a**2*(-a/b)**(1/4)*log(sqrt(x)
- (-a/b)**(1/4))/(64*a**4*b + 128*a**3*b**2*x**2 + 64*a**2*b**3*x**4) + 3*a**2*(-a/b)**(1/4)*log(sqrt(x) + (-a
/b)**(1/4))/(64*a**4*b + 128*a**3*b**2*x**2 + 64*a**2*b**3*x**4) + 6*a**2*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(
1/4))/(64*a**4*b + 128*a**3*b**2*x**2 + 64*a**2*b**3*x**4) + 4*a*b*x**(5/2)/(64*a**4*b + 128*a**3*b**2*x**2 +
64*a**2*b**3*x**4) - 6*a*b*x**2*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(64*a**4*b + 128*a**3*b**2*x**2 + 6
4*a**2*b**3*x**4) + 6*a*b*x**2*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/4))/(64*a**4*b + 128*a**3*b**2*x**2 + 64
*a**2*b**3*x**4) + 12*a*b*x**2*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/(64*a**4*b + 128*a**3*b**2*x**2 + 64*
a**2*b**3*x**4) - 3*b**2*x**4*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(64*a**4*b + 128*a**3*b**2*x**2 + 64*
a**2*b**3*x**4) + 3*b**2*x**4*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/4))/(64*a**4*b + 128*a**3*b**2*x**2 + 64*
a**2*b**3*x**4) + 6*b**2*x**4*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/(64*a**4*b + 128*a**3*b**2*x**2 + 64*a
**2*b**3*x**4), True))

________________________________________________________________________________________

Giac [A]
time = 0.44, size = 211, normalized size = 1.52 \begin {gather*} \frac {3 \, \sqrt {2} \left (a b^{3}\right )^{\frac {1}{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^{2} b^{2}} + \frac {3 \, \sqrt {2} \left (a b^{3}\right )^{\frac {1}{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^{2} b^{2}} + \frac {3 \, \sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{128 \, a^{2} b^{2}} - \frac {3 \, \sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{128 \, a^{2} b^{2}} + \frac {b x^{\frac {5}{2}} - 3 \, a \sqrt {x}}{16 \, {\left (b x^{2} + a\right )}^{2} a b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/64*sqrt(2)*(a*b^3)^(1/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(a^2*b^2) + 3/64*
sqrt(2)*(a*b^3)^(1/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(a^2*b^2) + 3/128*sqr
t(2)*(a*b^3)^(1/4)*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^2*b^2) - 3/128*sqrt(2)*(a*b^3)^(1/4)*lo
g(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^2*b^2) + 1/16*(b*x^(5/2) - 3*a*sqrt(x))/((b*x^2 + a)^2*a*b)

________________________________________________________________________________________

Mupad [B]
time = 0.09, size = 85, normalized size = 0.61 \begin {gather*} \frac {\frac {x^{5/2}}{16\,a}-\frac {3\,\sqrt {x}}{16\,b}}{a^2+2\,a\,b\,x^2+b^2\,x^4}+\frac {3\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {x}}{{\left (-a\right )}^{1/4}}\right )}{32\,{\left (-a\right )}^{7/4}\,b^{5/4}}+\frac {3\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {x}}{{\left (-a\right )}^{1/4}}\right )}{32\,{\left (-a\right )}^{7/4}\,b^{5/4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(3/2)/(a + b*x^2)^3,x)

[Out]

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

________________________________________________________________________________________