Integrand size = 15, antiderivative size = 177 \[ \int \frac {1}{x^{5/2} \left (a+b x^2\right )^2} \, dx=-\frac {7}{6 a^2 x^{3/2}}+\frac {1}{2 a x^{3/2} \left (a+b x^2\right )}+\frac {7 b^{3/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{11/4}}-\frac {7 b^{3/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{11/4}}-\frac {7 b^{3/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{4 \sqrt {2} a^{11/4}} \] Output:
-7/6/a^2/x^(3/2)+1/2/a/x^(3/2)/(b*x^2+a)+7/8*b^(3/4)*arctan(1-2^(1/2)*b^(1 /4)*x^(1/2)/a^(1/4))*2^(1/2)/a^(11/4)-7/8*b^(3/4)*arctan(1+2^(1/2)*b^(1/4) *x^(1/2)/a^(1/4))*2^(1/2)/a^(11/4)-7/8*b^(3/4)*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)
Time = 0.26 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^{5/2} \left (a+b x^2\right )^2} \, dx=\frac {-\frac {4 a^{3/4} \left (4 a+7 b x^2\right )}{x^{3/2} \left (a+b x^2\right )}+21 \sqrt {2} b^{3/4} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-21 \sqrt {2} b^{3/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{24 a^{11/4}} \] Input:
Integrate[1/(x^(5/2)*(a + b*x^2)^2),x]
Output:
((-4*a^(3/4)*(4*a + 7*b*x^2))/(x^(3/2)*(a + b*x^2)) + 21*Sqrt[2]*b^(3/4)*A rcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] - 21*Sqrt[2 ]*b^(3/4)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)] )/(24*a^(11/4))
Time = 0.41 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.47, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.733, Rules used = {253, 264, 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.
\(\displaystyle \int \frac {1}{x^{5/2} \left (a+b x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 253 |
\(\displaystyle \frac {7 \int \frac {1}{x^{5/2} \left (b x^2+a\right )}dx}{4 a}+\frac {1}{2 a x^{3/2} \left (a+b x^2\right )}\) |
\(\Big \downarrow \) 264 |
\(\displaystyle \frac {7 \left (-\frac {b \int \frac {1}{\sqrt {x} \left (b x^2+a\right )}dx}{a}-\frac {2}{3 a x^{3/2}}\right )}{4 a}+\frac {1}{2 a x^{3/2} \left (a+b x^2\right )}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {7 \left (-\frac {2 b \int \frac {1}{b x^2+a}d\sqrt {x}}{a}-\frac {2}{3 a x^{3/2}}\right )}{4 a}+\frac {1}{2 a x^{3/2} \left (a+b x^2\right )}\) |
\(\Big \downarrow \) 755 |
\(\displaystyle \frac {7 \left (-\frac {2 b \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\int \frac {\sqrt {b} x+\sqrt {a}}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{4 a}+\frac {1}{2 a x^{3/2} \left (a+b x^2\right )}\) |
\(\Big \downarrow \) 1476 |
\(\displaystyle \frac {7 \left (-\frac {2 b \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {b}}}{2 \sqrt {a}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{4 a}+\frac {1}{2 a x^{3/2} \left (a+b x^2\right )}\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle \frac {7 \left (-\frac {2 b \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{4 a}+\frac {1}{2 a x^{3/2} \left (a+b x^2\right )}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {7 \left (-\frac {2 b \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{4 a}+\frac {1}{2 a x^{3/2} \left (a+b x^2\right )}\) |
\(\Big \downarrow \) 1479 |
\(\displaystyle \frac {7 \left (-\frac {2 b \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{b} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{4 a}+\frac {1}{2 a x^{3/2} \left (a+b x^2\right )}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {7 \left (-\frac {2 b \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{b} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{4 a}+\frac {1}{2 a x^{3/2} \left (a+b x^2\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {7 \left (-\frac {2 b \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt [4]{a} \sqrt {b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{4 a}+\frac {1}{2 a x^{3/2} \left (a+b x^2\right )}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {7 \left (-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{4 a}+\frac {1}{2 a x^{3/2} \left (a+b x^2\right )}\) |
Input:
Int[1/(x^(5/2)*(a + b*x^2)^2),x]
Output:
1/(2*a*x^(3/2)*(a + b*x^2)) + (7*(-2/(3*a*x^(3/2)) - (2*b*((-(ArcTan[1 - ( Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)]/(Sqrt[2]*a^(1/4)*b^(1/4))) + ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)]/(Sqrt[2]*a^(1/4)*b^(1/4)))/(2*Sqrt[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])))/a))/(4*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] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -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 = 0.37 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.77
method | result | size |
derivativedivides | \(-\frac {2 b \left (\frac {\sqrt {x}}{4 b \,x^{2}+4 a}+\frac {7 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a}\right )}{a^{2}}-\frac {2}{3 a^{2} x^{\frac {3}{2}}}\) | \(136\) |
default | \(-\frac {2 b \left (\frac {\sqrt {x}}{4 b \,x^{2}+4 a}+\frac {7 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a}\right )}{a^{2}}-\frac {2}{3 a^{2} x^{\frac {3}{2}}}\) | \(136\) |
risch | \(-\frac {2}{3 a^{2} x^{\frac {3}{2}}}-\frac {b \left (\frac {\sqrt {x}}{2 b \,x^{2}+2 a}+\frac {7 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{16 a}\right )}{a^{2}}\) | \(136\) |
Input:
int(1/x^(5/2)/(b*x^2+a)^2,x,method=_RETURNVERBOSE)
Output:
-2*b/a^2*(1/4*x^(1/2)/(b*x^2+a)+7/32*(a/b)^(1/4)/a*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*arctan(2^(1/2)/(a/b)^(1/4)*x ^(1/2)-1)))-2/3/a^2/x^(3/2)
Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.38 \[ \int \frac {1}{x^{5/2} \left (a+b x^2\right )^2} \, dx=-\frac {21 \, {\left (a^{2} b x^{4} + a^{3} x^{2}\right )} \left (-\frac {b^{3}}{a^{11}}\right )^{\frac {1}{4}} \log \left (7 \, a^{3} \left (-\frac {b^{3}}{a^{11}}\right )^{\frac {1}{4}} + 7 \, b \sqrt {x}\right ) + 21 \, {\left (i \, a^{2} b x^{4} + i \, a^{3} x^{2}\right )} \left (-\frac {b^{3}}{a^{11}}\right )^{\frac {1}{4}} \log \left (7 i \, a^{3} \left (-\frac {b^{3}}{a^{11}}\right )^{\frac {1}{4}} + 7 \, b \sqrt {x}\right ) + 21 \, {\left (-i \, a^{2} b x^{4} - i \, a^{3} x^{2}\right )} \left (-\frac {b^{3}}{a^{11}}\right )^{\frac {1}{4}} \log \left (-7 i \, a^{3} \left (-\frac {b^{3}}{a^{11}}\right )^{\frac {1}{4}} + 7 \, b \sqrt {x}\right ) - 21 \, {\left (a^{2} b x^{4} + a^{3} x^{2}\right )} \left (-\frac {b^{3}}{a^{11}}\right )^{\frac {1}{4}} \log \left (-7 \, a^{3} \left (-\frac {b^{3}}{a^{11}}\right )^{\frac {1}{4}} + 7 \, b \sqrt {x}\right ) + 4 \, {\left (7 \, b x^{2} + 4 \, a\right )} \sqrt {x}}{24 \, {\left (a^{2} b x^{4} + a^{3} x^{2}\right )}} \] Input:
integrate(1/x^(5/2)/(b*x^2+a)^2,x, algorithm="fricas")
Output:
-1/24*(21*(a^2*b*x^4 + a^3*x^2)*(-b^3/a^11)^(1/4)*log(7*a^3*(-b^3/a^11)^(1 /4) + 7*b*sqrt(x)) + 21*(I*a^2*b*x^4 + I*a^3*x^2)*(-b^3/a^11)^(1/4)*log(7* I*a^3*(-b^3/a^11)^(1/4) + 7*b*sqrt(x)) + 21*(-I*a^2*b*x^4 - I*a^3*x^2)*(-b ^3/a^11)^(1/4)*log(-7*I*a^3*(-b^3/a^11)^(1/4) + 7*b*sqrt(x)) - 21*(a^2*b*x ^4 + a^3*x^2)*(-b^3/a^11)^(1/4)*log(-7*a^3*(-b^3/a^11)^(1/4) + 7*b*sqrt(x) ) + 4*(7*b*x^2 + 4*a)*sqrt(x))/(a^2*b*x^4 + a^3*x^2)
Leaf count of result is larger than twice the leaf count of optimal. 425 vs. \(2 (165) = 330\).
Time = 113.71 (sec) , antiderivative size = 425, normalized size of antiderivative = 2.40 \[ \int \frac {1}{x^{5/2} \left (a+b x^2\right )^2} \, dx=\begin {cases} \frac {\tilde {\infty }}{x^{\frac {11}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2}{3 a^{2} x^{\frac {3}{2}}} & \text {for}\: b = 0 \\- \frac {2}{11 b^{2} x^{\frac {11}{2}}} & \text {for}\: a = 0 \\- \frac {16 a^{2}}{24 a^{4} x^{\frac {3}{2}} + 24 a^{3} b x^{\frac {7}{2}}} + \frac {21 a b x^{\frac {3}{2}} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{24 a^{4} x^{\frac {3}{2}} + 24 a^{3} b x^{\frac {7}{2}}} - \frac {21 a b x^{\frac {3}{2}} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{24 a^{4} x^{\frac {3}{2}} + 24 a^{3} b x^{\frac {7}{2}}} - \frac {42 a b x^{\frac {3}{2}} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{24 a^{4} x^{\frac {3}{2}} + 24 a^{3} b x^{\frac {7}{2}}} - \frac {28 a b x^{2}}{24 a^{4} x^{\frac {3}{2}} + 24 a^{3} b x^{\frac {7}{2}}} + \frac {21 b^{2} x^{\frac {7}{2}} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{24 a^{4} x^{\frac {3}{2}} + 24 a^{3} b x^{\frac {7}{2}}} - \frac {21 b^{2} x^{\frac {7}{2}} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{24 a^{4} x^{\frac {3}{2}} + 24 a^{3} b x^{\frac {7}{2}}} - \frac {42 b^{2} x^{\frac {7}{2}} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{24 a^{4} x^{\frac {3}{2}} + 24 a^{3} b x^{\frac {7}{2}}} & \text {otherwise} \end {cases} \] Input:
integrate(1/x**(5/2)/(b*x**2+a)**2,x)
Output:
Piecewise((zoo/x**(11/2), Eq(a, 0) & Eq(b, 0)), (-2/(3*a**2*x**(3/2)), Eq( b, 0)), (-2/(11*b**2*x**(11/2)), Eq(a, 0)), (-16*a**2/(24*a**4*x**(3/2) + 24*a**3*b*x**(7/2)) + 21*a*b*x**(3/2)*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)** (1/4))/(24*a**4*x**(3/2) + 24*a**3*b*x**(7/2)) - 21*a*b*x**(3/2)*(-a/b)**( 1/4)*log(sqrt(x) + (-a/b)**(1/4))/(24*a**4*x**(3/2) + 24*a**3*b*x**(7/2)) - 42*a*b*x**(3/2)*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/(24*a**4*x**(3 /2) + 24*a**3*b*x**(7/2)) - 28*a*b*x**2/(24*a**4*x**(3/2) + 24*a**3*b*x**( 7/2)) + 21*b**2*x**(7/2)*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(24*a* *4*x**(3/2) + 24*a**3*b*x**(7/2)) - 21*b**2*x**(7/2)*(-a/b)**(1/4)*log(sqr t(x) + (-a/b)**(1/4))/(24*a**4*x**(3/2) + 24*a**3*b*x**(7/2)) - 42*b**2*x* *(7/2)*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/(24*a**4*x**(3/2) + 24*a* *3*b*x**(7/2)), True))
Time = 0.11 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.18 \[ \int \frac {1}{x^{5/2} \left (a+b x^2\right )^2} \, dx=-\frac {7 \, b x^{2} + 4 \, a}{6 \, {\left (a^{2} b x^{\frac {7}{2}} + a^{3} x^{\frac {3}{2}}\right )}} - \frac {7 \, {\left (\frac {2 \, \sqrt {2} b \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} b \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} b^{\frac {3}{4}} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}}} - \frac {\sqrt {2} b^{\frac {3}{4}} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}}}\right )}}{16 \, a^{2}} \] Input:
integrate(1/x^(5/2)/(b*x^2+a)^2,x, algorithm="maxima")
Output:
-1/6*(7*b*x^2 + 4*a)/(a^2*b*x^(7/2) + a^3*x^(3/2)) - 7/16*(2*sqrt(2)*b*arc tan(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)*b*arctan(-1/2*sqrt( 2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(s qrt(a)*sqrt(sqrt(a)*sqrt(b))) + sqrt(2)*b^(3/4)*log(sqrt(2)*a^(1/4)*b^(1/4 )*sqrt(x) + sqrt(b)*x + sqrt(a))/a^(3/4) - sqrt(2)*b^(3/4)*log(-sqrt(2)*a^ (1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/a^(3/4))/a^2
Time = 0.13 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x^{5/2} \left (a+b x^2\right )^2} \, dx=-\frac {7 \, \sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{3}} - \frac {7 \, \sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{3}} - \frac {7 \, \sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, a^{3}} + \frac {7 \, \sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, a^{3}} - \frac {b \sqrt {x}}{2 \, {\left (b x^{2} + a\right )} a^{2}} - \frac {2}{3 \, a^{2} x^{\frac {3}{2}}} \] Input:
integrate(1/x^(5/2)/(b*x^2+a)^2,x, algorithm="giac")
Output:
-7/8*sqrt(2)*(a*b^3)^(1/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqr t(x))/(a/b)^(1/4))/a^3 - 7/8*sqrt(2)*(a*b^3)^(1/4)*arctan(-1/2*sqrt(2)*(sq rt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/a^3 - 7/16*sqrt(2)*(a*b^3)^(1/ 4)*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/a^3 + 7/16*sqrt(2)*(a* b^3)^(1/4)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/a^3 - 1/2*b*s qrt(x)/((b*x^2 + a)*a^2) - 2/3/(a^2*x^(3/2))
Time = 0.22 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.44 \[ \int \frac {1}{x^{5/2} \left (a+b x^2\right )^2} \, dx=\frac {7\,{\left (-b\right )}^{3/4}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {x}}{a^{1/4}}\right )}{4\,a^{11/4}}-\frac {\frac {2}{3\,a}+\frac {7\,b\,x^2}{6\,a^2}}{a\,x^{3/2}+b\,x^{7/2}}+\frac {7\,{\left (-b\right )}^{3/4}\,\mathrm {atanh}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {x}}{a^{1/4}}\right )}{4\,a^{11/4}} \] Input:
int(1/(x^(5/2)*(a + b*x^2)^2),x)
Output:
(7*(-b)^(3/4)*atan(((-b)^(1/4)*x^(1/2))/a^(1/4)))/(4*a^(11/4)) - (2/(3*a) + (7*b*x^2)/(6*a^2))/(a*x^(3/2) + b*x^(7/2)) + (7*(-b)^(3/4)*atanh(((-b)^( 1/4)*x^(1/2))/a^(1/4)))/(4*a^(11/4))
Time = 0.22 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.90 \[ \int \frac {1}{x^{5/2} \left (a+b x^2\right )^2} \, dx=\frac {42 \sqrt {x}\, b^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) x +42 \sqrt {x}\, b^{\frac {7}{4}} a^{\frac {1}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) x^{3}-42 \sqrt {x}\, b^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) x -42 \sqrt {x}\, b^{\frac {7}{4}} a^{\frac {1}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) x^{3}+21 \sqrt {x}\, b^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) x +21 \sqrt {x}\, b^{\frac {7}{4}} a^{\frac {1}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) x^{3}-21 \sqrt {x}\, b^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) x -21 \sqrt {x}\, b^{\frac {7}{4}} a^{\frac {1}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) x^{3}-32 a^{2}-56 a b \,x^{2}}{48 \sqrt {x}\, a^{3} x \left (b \,x^{2}+a \right )} \] Input:
int(1/x^(5/2)/(b*x^2+a)^2,x)
Output:
(42*sqrt(x)*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*x + 42*sqrt(x)*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*x**3 - 42*sqrt(x)*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*x - 42*sqrt(x)*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*x**3 + 21*sqrt(x)* 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*x + 21*sqrt(x)*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*x**3 - 21*sqrt(x)*b**(3 /4)*a**(1/4)*sqrt(2)*log(sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqr t(b)*x)*a*x - 21*sqrt(x)*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*x**3 - 32*a**2 - 56*a*b*x**2)/(48* sqrt(x)*a**3*x*(a + b*x**2))