Integrand size = 15, antiderivative size = 186 \[ \int \frac {1}{\sqrt {x} \left (a+b x^2\right )^3} \, dx=\frac {\sqrt {x}}{4 a \left (a+b x^2\right )^2}+\frac {7 \sqrt {x}}{16 a^2 \left (a+b x^2\right )}-\frac {21 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{11/4} \sqrt [4]{b}}+\frac {21 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{11/4} \sqrt [4]{b}}+\frac {21 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{32 \sqrt {2} a^{11/4} \sqrt [4]{b}} \] Output:
1/4*x^(1/2)/a/(b*x^2+a)^2+7/16*x^(1/2)/a^2/(b*x^2+a)-21/64*arctan(1-2^(1/2 )*b^(1/4)*x^(1/2)/a^(1/4))*2^(1/2)/a^(11/4)/b^(1/4)+21/64*arctan(1+2^(1/2) *b^(1/4)*x^(1/2)/a^(1/4))*2^(1/2)/a^(11/4)/b^(1/4)+21/64*arctanh(2^(1/2)*a ^(1/4)*b^(1/4)*x^(1/2)/(a^(1/2)+b^(1/2)*x))*2^(1/2)/a^(11/4)/b^(1/4)
Time = 0.20 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.74 \[ \int \frac {1}{\sqrt {x} \left (a+b x^2\right )^3} \, dx=\frac {\frac {4 a^{3/4} \sqrt {x} \left (11 a+7 b x^2\right )}{\left (a+b x^2\right )^2}-\frac {21 \sqrt {2} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{\sqrt [4]{b}}+\frac {21 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt [4]{b}}}{64 a^{11/4}} \] Input:
Integrate[1/(Sqrt[x]*(a + b*x^2)^3),x]
Output:
((4*a^(3/4)*Sqrt[x]*(11*a + 7*b*x^2))/(a + b*x^2)^2 - (21*Sqrt[2]*ArcTan[( Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/b^(1/4) + (21*Sqr t[2]*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/b^( 1/4))/(64*a^(11/4))
Time = 0.42 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.45, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.733, Rules used = {253, 253, 266, 755, 1476, 1082, 217, 1479, 25, 27, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{\sqrt {x} \left (a+b x^2\right )^3} \, dx\) |
\(\Big \downarrow \) 253 |
\(\displaystyle \frac {7 \int \frac {1}{\sqrt {x} \left (b x^2+a\right )^2}dx}{8 a}+\frac {\sqrt {x}}{4 a \left (a+b x^2\right )^2}\) |
\(\Big \downarrow \) 253 |
\(\displaystyle \frac {7 \left (\frac {3 \int \frac {1}{\sqrt {x} \left (b x^2+a\right )}dx}{4 a}+\frac {\sqrt {x}}{2 a \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {x}}{4 a \left (a+b x^2\right )^2}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {7 \left (\frac {3 \int \frac {1}{b x^2+a}d\sqrt {x}}{2 a}+\frac {\sqrt {x}}{2 a \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {x}}{4 a \left (a+b x^2\right )^2}\) |
\(\Big \downarrow \) 755 |
\(\displaystyle \frac {7 \left (\frac {3 \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\int \frac {\sqrt {b} x+\sqrt {a}}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}\right )}{2 a}+\frac {\sqrt {x}}{2 a \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {x}}{4 a \left (a+b x^2\right )^2}\) |
\(\Big \downarrow \) 1476 |
\(\displaystyle \frac {7 \left (\frac {3 \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {b}}}{2 \sqrt {a}}\right )}{2 a}+\frac {\sqrt {x}}{2 a \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {x}}{4 a \left (a+b x^2\right )^2}\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle \frac {7 \left (\frac {3 \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{2 a}+\frac {\sqrt {x}}{2 a \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {x}}{4 a \left (a+b x^2\right )^2}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {7 \left (\frac {3 \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{2 a}+\frac {\sqrt {x}}{2 a \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {x}}{4 a \left (a+b x^2\right )^2}\) |
\(\Big \downarrow \) 1479 |
\(\displaystyle \frac {7 \left (\frac {3 \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{b} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{2 a}+\frac {\sqrt {x}}{2 a \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {x}}{4 a \left (a+b x^2\right )^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {7 \left (\frac {3 \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{b} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{2 a}+\frac {\sqrt {x}}{2 a \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {x}}{4 a \left (a+b x^2\right )^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {7 \left (\frac {3 \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt [4]{a} \sqrt {b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{2 a}+\frac {\sqrt {x}}{2 a \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {x}}{4 a \left (a+b x^2\right )^2}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {7 \left (\frac {3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{2 a}+\frac {\sqrt {x}}{2 a \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {x}}{4 a \left (a+b x^2\right )^2}\) |
Input:
Int[1/(Sqrt[x]*(a + b*x^2)^3),x]
Output:
Sqrt[x]/(4*a*(a + b*x^2)^2) + (7*(Sqrt[x]/(2*a*(a + b*x^2)) + (3*((-(ArcTa n[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)]/(Sqrt[2]*a^(1/4)*b^(1/4))) + ArcT an[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)]/(Sqrt[2]*a^(1/4)*b^(1/4)))/(2*Sq rt[a]) + (-1/2*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]/ (Sqrt[2]*a^(1/4)*b^(1/4)) + Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]/(2*Sqrt[2]*a^(1/4)*b^(1/4)))/(2*Sqrt[a])))/(2*a)))/(8*a)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c*x )^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 2*a*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m }, x] && LtQ[p, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[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]]))
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[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])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[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]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 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]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Time = 0.26 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.79
method | result | size |
derivativedivides | \(\frac {\sqrt {x}}{4 a \left (b \,x^{2}+a \right )^{2}}+\frac {\frac {7 \sqrt {x}}{16 a \left (b \,x^{2}+a \right )}+\frac {21 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{128 a^{2}}}{a}\) | \(147\) |
default | \(\frac {\sqrt {x}}{4 a \left (b \,x^{2}+a \right )^{2}}+\frac {\frac {7 \sqrt {x}}{16 a \left (b \,x^{2}+a \right )}+\frac {21 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{128 a^{2}}}{a}\) | \(147\) |
Input:
int(1/x^(1/2)/(b*x^2+a)^3,x,method=_RETURNVERBOSE)
Output:
1/4*x^(1/2)/a/(b*x^2+a)^2+7/4/a*(1/4*x^(1/2)/a/(b*x^2+a)+3/32/a^2*(a/b)^(1 /4)*2^(1/2)*(ln((x+(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2))/(x-(a/b)^(1/4) *x^(1/2)*2^(1/2)+(a/b)^(1/2)))+2*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+2*a rctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)))
Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.44 \[ \int \frac {1}{\sqrt {x} \left (a+b x^2\right )^3} \, dx=\frac {21 \, {\left (a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + a^{4}\right )} \left (-\frac {1}{a^{11} b}\right )^{\frac {1}{4}} \log \left (a^{3} \left (-\frac {1}{a^{11} b}\right )^{\frac {1}{4}} + \sqrt {x}\right ) - 21 \, {\left (-i \, a^{2} b^{2} x^{4} - 2 i \, a^{3} b x^{2} - i \, a^{4}\right )} \left (-\frac {1}{a^{11} b}\right )^{\frac {1}{4}} \log \left (i \, a^{3} \left (-\frac {1}{a^{11} b}\right )^{\frac {1}{4}} + \sqrt {x}\right ) - 21 \, {\left (i \, a^{2} b^{2} x^{4} + 2 i \, a^{3} b x^{2} + i \, a^{4}\right )} \left (-\frac {1}{a^{11} b}\right )^{\frac {1}{4}} \log \left (-i \, a^{3} \left (-\frac {1}{a^{11} b}\right )^{\frac {1}{4}} + \sqrt {x}\right ) - 21 \, {\left (a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + a^{4}\right )} \left (-\frac {1}{a^{11} b}\right )^{\frac {1}{4}} \log \left (-a^{3} \left (-\frac {1}{a^{11} b}\right )^{\frac {1}{4}} + \sqrt {x}\right ) + 4 \, {\left (7 \, b x^{2} + 11 \, a\right )} \sqrt {x}}{64 \, {\left (a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + a^{4}\right )}} \] Input:
integrate(1/x^(1/2)/(b*x^2+a)^3,x, algorithm="fricas")
Output:
1/64*(21*(a^2*b^2*x^4 + 2*a^3*b*x^2 + a^4)*(-1/(a^11*b))^(1/4)*log(a^3*(-1 /(a^11*b))^(1/4) + sqrt(x)) - 21*(-I*a^2*b^2*x^4 - 2*I*a^3*b*x^2 - I*a^4)* (-1/(a^11*b))^(1/4)*log(I*a^3*(-1/(a^11*b))^(1/4) + sqrt(x)) - 21*(I*a^2*b ^2*x^4 + 2*I*a^3*b*x^2 + I*a^4)*(-1/(a^11*b))^(1/4)*log(-I*a^3*(-1/(a^11*b ))^(1/4) + sqrt(x)) - 21*(a^2*b^2*x^4 + 2*a^3*b*x^2 + a^4)*(-1/(a^11*b))^( 1/4)*log(-a^3*(-1/(a^11*b))^(1/4) + sqrt(x)) + 4*(7*b*x^2 + 11*a)*sqrt(x)) /(a^2*b^2*x^4 + 2*a^3*b*x^2 + a^4)
Leaf count of result is larger than twice the leaf count of optimal. 627 vs. \(2 (172) = 344\).
Time = 147.06 (sec) , antiderivative size = 627, normalized size of antiderivative = 3.37 \[ \int \frac {1}{\sqrt {x} \left (a+b x^2\right )^3} \, dx=\begin {cases} \frac {\tilde {\infty }}{x^{\frac {11}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 \sqrt {x}}{a^{3}} & \text {for}\: b = 0 \\- \frac {2}{11 b^{3} x^{\frac {11}{2}}} & \text {for}\: a = 0 \\\frac {44 a^{2} \sqrt {x}}{64 a^{5} + 128 a^{4} b x^{2} + 64 a^{3} b^{2} x^{4}} - \frac {21 a^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{5} + 128 a^{4} b x^{2} + 64 a^{3} b^{2} x^{4}} + \frac {21 a^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{5} + 128 a^{4} b x^{2} + 64 a^{3} b^{2} x^{4}} + \frac {42 a^{2} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{64 a^{5} + 128 a^{4} b x^{2} + 64 a^{3} b^{2} x^{4}} + \frac {28 a b x^{\frac {5}{2}}}{64 a^{5} + 128 a^{4} b x^{2} + 64 a^{3} b^{2} x^{4}} - \frac {42 a b x^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{5} + 128 a^{4} b x^{2} + 64 a^{3} b^{2} x^{4}} + \frac {42 a b x^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{5} + 128 a^{4} b x^{2} + 64 a^{3} b^{2} x^{4}} + \frac {84 a b x^{2} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{64 a^{5} + 128 a^{4} b x^{2} + 64 a^{3} b^{2} x^{4}} - \frac {21 b^{2} x^{4} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{5} + 128 a^{4} b x^{2} + 64 a^{3} b^{2} x^{4}} + \frac {21 b^{2} x^{4} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{5} + 128 a^{4} b x^{2} + 64 a^{3} b^{2} x^{4}} + \frac {42 b^{2} x^{4} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{64 a^{5} + 128 a^{4} b x^{2} + 64 a^{3} b^{2} x^{4}} & \text {otherwise} \end {cases} \] Input:
integrate(1/x**(1/2)/(b*x**2+a)**3,x)
Output:
Piecewise((zoo/x**(11/2), Eq(a, 0) & Eq(b, 0)), (2*sqrt(x)/a**3, Eq(b, 0)) , (-2/(11*b**3*x**(11/2)), Eq(a, 0)), (44*a**2*sqrt(x)/(64*a**5 + 128*a**4 *b*x**2 + 64*a**3*b**2*x**4) - 21*a**2*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)* *(1/4))/(64*a**5 + 128*a**4*b*x**2 + 64*a**3*b**2*x**4) + 21*a**2*(-a/b)** (1/4)*log(sqrt(x) + (-a/b)**(1/4))/(64*a**5 + 128*a**4*b*x**2 + 64*a**3*b* *2*x**4) + 42*a**2*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/(64*a**5 + 12 8*a**4*b*x**2 + 64*a**3*b**2*x**4) + 28*a*b*x**(5/2)/(64*a**5 + 128*a**4*b *x**2 + 64*a**3*b**2*x**4) - 42*a*b*x**2*(-a/b)**(1/4)*log(sqrt(x) - (-a/b )**(1/4))/(64*a**5 + 128*a**4*b*x**2 + 64*a**3*b**2*x**4) + 42*a*b*x**2*(- a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/4))/(64*a**5 + 128*a**4*b*x**2 + 64*a **3*b**2*x**4) + 84*a*b*x**2*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/(64 *a**5 + 128*a**4*b*x**2 + 64*a**3*b**2*x**4) - 21*b**2*x**4*(-a/b)**(1/4)* log(sqrt(x) - (-a/b)**(1/4))/(64*a**5 + 128*a**4*b*x**2 + 64*a**3*b**2*x** 4) + 21*b**2*x**4*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/4))/(64*a**5 + 12 8*a**4*b*x**2 + 64*a**3*b**2*x**4) + 42*b**2*x**4*(-a/b)**(1/4)*atan(sqrt( x)/(-a/b)**(1/4))/(64*a**5 + 128*a**4*b*x**2 + 64*a**3*b**2*x**4), True))
Time = 0.11 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.17 \[ \int \frac {1}{\sqrt {x} \left (a+b x^2\right )^3} \, dx=\frac {7 \, b x^{\frac {5}{2}} + 11 \, a \sqrt {x}}{16 \, {\left (a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + a^{4}\right )}} + \frac {21 \, {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}\right )}}{128 \, a^{2}} \] Input:
integrate(1/x^(1/2)/(b*x^2+a)^3,x, algorithm="maxima")
Output:
1/16*(7*b*x^(5/2) + 11*a*sqrt(x))/(a^2*b^2*x^4 + 2*a^3*b*x^2 + a^4) + 21/1 28*(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)*sqrt(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(-s qrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)))/a ^2
Time = 0.13 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.12 \[ \int \frac {1}{\sqrt {x} \left (a+b x^2\right )^3} \, dx=\frac {21 \, \sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{64 \, a^{3} b} + \frac {21 \, \sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{64 \, a^{3} b} + \frac {21 \, \sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{128 \, a^{3} b} - \frac {21 \, \sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{128 \, a^{3} b} + \frac {7 \, b x^{\frac {5}{2}} + 11 \, a \sqrt {x}}{16 \, {\left (b x^{2} + a\right )}^{2} a^{2}} \] Input:
integrate(1/x^(1/2)/(b*x^2+a)^3,x, algorithm="giac")
Output:
21/64*sqrt(2)*(a*b^3)^(1/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sq rt(x))/(a/b)^(1/4))/(a^3*b) + 21/64*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^3*b) + 21/128*sqrt(2 )*(a*b^3)^(1/4)*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^3*b) - 21/128*sqrt(2)*(a*b^3)^(1/4)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt( a/b))/(a^3*b) + 1/16*(7*b*x^(5/2) + 11*a*sqrt(x))/((b*x^2 + a)^2*a^2)
Time = 0.20 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.46 \[ \int \frac {1}{\sqrt {x} \left (a+b x^2\right )^3} \, dx=\frac {\frac {11\,\sqrt {x}}{16\,a}+\frac {7\,b\,x^{5/2}}{16\,a^2}}{a^2+2\,a\,b\,x^2+b^2\,x^4}-\frac {21\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {x}}{{\left (-a\right )}^{1/4}}\right )}{32\,{\left (-a\right )}^{11/4}\,b^{1/4}}-\frac {21\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {x}}{{\left (-a\right )}^{1/4}}\right )}{32\,{\left (-a\right )}^{11/4}\,b^{1/4}} \] Input:
int(1/(x^(1/2)*(a + b*x^2)^3),x)
Output:
((11*x^(1/2))/(16*a) + (7*b*x^(5/2))/(16*a^2))/(a^2 + b^2*x^4 + 2*a*b*x^2) - (21*atan((b^(1/4)*x^(1/2))/(-a)^(1/4)))/(32*(-a)^(11/4)*b^(1/4)) - (21* atanh((b^(1/4)*x^(1/2))/(-a)^(1/4)))/(32*(-a)^(11/4)*b^(1/4))
Time = 0.20 (sec) , antiderivative size = 477, normalized size of antiderivative = 2.56 \[ \int \frac {1}{\sqrt {x} \left (a+b x^2\right )^3} \, dx =\text {Too large to display} \] Input:
int(1/x^(1/2)/(b*x^2+a)^3,x)
Output:
( - 42*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt( x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**2 - 84*b**(3/4)*a**(1/4)*sqrt( 2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4) *sqrt(2)))*a*b*x**2 - 42*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4) *sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*b**2*x**4 + 42* b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt (b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**2 + 84*b**(3/4)*a**(1/4)*sqrt(2)*atan ((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2 )))*a*b*x**2 + 42*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2 ) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*b**2*x**4 - 21*b**(3/4 )*a**(1/4)*sqrt(2)*log( - sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sq rt(b)*x)*a**2 - 42*b**(3/4)*a**(1/4)*sqrt(2)*log( - sqrt(x)*b**(1/4)*a**(1 /4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*a*b*x**2 - 21*b**(3/4)*a**(1/4)*sqrt(2) *log( - sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*b**2*x**4 + 21*b**(3/4)*a**(1/4)*sqrt(2)*log(sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sq rt(a) + sqrt(b)*x)*a**2 + 42*b**(3/4)*a**(1/4)*sqrt(2)*log(sqrt(x)*b**(1/4 )*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*a*b*x**2 + 21*b**(3/4)*a**(1/4)* sqrt(2)*log(sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*b**2* x**4 + 88*sqrt(x)*a**2*b + 56*sqrt(x)*a*b**2*x**2)/(128*a**3*b*(a**2 + 2*a *b*x**2 + b**2*x**4))