Integrand size = 15, antiderivative size = 198 \[ \int \frac {1}{x^{5/2} \left (a+b x^2\right )^3} \, dx=-\frac {77}{48 a^3 x^{3/2}}+\frac {1}{4 a x^{3/2} \left (a+b x^2\right )^2}+\frac {11}{16 a^2 x^{3/2} \left (a+b x^2\right )}+\frac {77 b^{3/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{15/4}}-\frac {77 b^{3/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{15/4}}-\frac {77 b^{3/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{32 \sqrt {2} a^{15/4}} \] Output:
-77/48/a^3/x^(3/2)+1/4/a/x^(3/2)/(b*x^2+a)^2+11/16/a^2/x^(3/2)/(b*x^2+a)+7 7/64*b^(3/4)*arctan(1-2^(1/2)*b^(1/4)*x^(1/2)/a^(1/4))*2^(1/2)/a^(15/4)-77 /64*b^(3/4)*arctan(1+2^(1/2)*b^(1/4)*x^(1/2)/a^(1/4))*2^(1/2)/a^(15/4)-77/ 64*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^(15/4)
Time = 0.33 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x^{5/2} \left (a+b x^2\right )^3} \, dx=\frac {-\frac {4 a^{3/4} \left (32 a^2+121 a b x^2+77 b^2 x^4\right )}{x^{3/2} \left (a+b x^2\right )^2}+231 \sqrt {2} b^{3/4} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-231 \sqrt {2} b^{3/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{192 a^{15/4}} \] Input:
Integrate[1/(x^(5/2)*(a + b*x^2)^3),x]
Output:
((-4*a^(3/4)*(32*a^2 + 121*a*b*x^2 + 77*b^2*x^4))/(x^(3/2)*(a + b*x^2)^2) + 231*Sqrt[2]*b^(3/4)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4 )*Sqrt[x])] - 231*Sqrt[2]*b^(3/4)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] )/(Sqrt[a] + Sqrt[b]*x)])/(192*a^(15/4))
Time = 0.43 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.46, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {253, 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.
| \(\displaystyle \int \frac {1}{x^{5/2} \left (a+b x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {11 \int \frac {1}{x^{5/2} \left (b x^2+a\right )^2}dx}{8 a}+\frac {1}{4 a x^{3/2} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {11 \left (\frac {7 \int \frac {1}{x^{5/2} \left (b x^2+a\right )}dx}{4 a}+\frac {1}{2 a x^{3/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a x^{3/2} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {11 \left (\frac {7 \left (-\frac {b \int \frac {1}{\sqrt {x} \left (b x^2+a\right )}dx}{a}-\frac {2}{3 a x^{3/2}}\right )}{4 a}+\frac {1}{2 a x^{3/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a x^{3/2} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {11 \left (\frac {7 \left (-\frac {2 b \int \frac {1}{b x^2+a}d\sqrt {x}}{a}-\frac {2}{3 a x^{3/2}}\right )}{4 a}+\frac {1}{2 a x^{3/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a x^{3/2} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {11 \left (\frac {7 \left (-\frac {2 b \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\int \frac {\sqrt {b} x+\sqrt {a}}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{4 a}+\frac {1}{2 a x^{3/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a x^{3/2} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {11 \left (\frac {7 \left (-\frac {2 b \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {b}}}{2 \sqrt {a}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{4 a}+\frac {1}{2 a x^{3/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a x^{3/2} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {11 \left (\frac {7 \left (-\frac {2 b \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{4 a}+\frac {1}{2 a x^{3/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a x^{3/2} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {11 \left (\frac {7 \left (-\frac {2 b \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{4 a}+\frac {1}{2 a x^{3/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a x^{3/2} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {11 \left (\frac {7 \left (-\frac {2 b \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{b} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{4 a}+\frac {1}{2 a x^{3/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a x^{3/2} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {11 \left (\frac {7 \left (-\frac {2 b \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{b} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{4 a}+\frac {1}{2 a x^{3/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a x^{3/2} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {11 \left (\frac {7 \left (-\frac {2 b \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt [4]{a} \sqrt {b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{4 a}+\frac {1}{2 a x^{3/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a x^{3/2} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {11 \left (\frac {7 \left (-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{4 a}+\frac {1}{2 a x^{3/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a x^{3/2} \left (a+b x^2\right )^2}\) |
Input:
Int[1/(x^(5/2)*(a + b*x^2)^3),x]
Output:
1/(4*a*x^(3/2)*(a + b*x^2)^2) + (11*(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)]/(Sq rt[2]*a^(1/4)*b^(1/4))) + ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)]/(S qrt[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] + Sq rt[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)))/(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] :> 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 = 0.28 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.73
| method | result | size |
| derivativedivides | \(-\frac {2}{3 a^{3} x^{\frac {3}{2}}}-\frac {2 b \left (\frac {\frac {15 b \,x^{\frac {5}{2}}}{32}+\frac {19 a \sqrt {x}}{32}}{\left (b \,x^{2}+a \right )^{2}}+\frac {77 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{256 a}\right )}{a^{3}}\) | \(145\) |
| default | \(-\frac {2}{3 a^{3} x^{\frac {3}{2}}}-\frac {2 b \left (\frac {\frac {15 b \,x^{\frac {5}{2}}}{32}+\frac {19 a \sqrt {x}}{32}}{\left (b \,x^{2}+a \right )^{2}}+\frac {77 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{256 a}\right )}{a^{3}}\) | \(145\) |
| risch | \(-\frac {2}{3 a^{3} x^{\frac {3}{2}}}-\frac {b \left (\frac {\frac {15 b \,x^{\frac {5}{2}}}{16}+\frac {19 a \sqrt {x}}{16}}{\left (b \,x^{2}+a \right )^{2}}+\frac {77 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{128 a}\right )}{a^{3}}\) | \(146\) |
Input:
int(1/x^(5/2)/(b*x^2+a)^3,x,method=_RETURNVERBOSE)
Output:
-2/3/a^3/x^(3/2)-2/a^3*b*((15/32*b*x^(5/2)+19/32*a*x^(1/2))/(b*x^2+a)^2+77 /256*(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)))
Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.57 \[ \int \frac {1}{x^{5/2} \left (a+b x^2\right )^3} \, dx=-\frac {231 \, {\left (a^{3} b^{2} x^{6} + 2 \, a^{4} b x^{4} + a^{5} x^{2}\right )} \left (-\frac {b^{3}}{a^{15}}\right )^{\frac {1}{4}} \log \left (77 \, a^{4} \left (-\frac {b^{3}}{a^{15}}\right )^{\frac {1}{4}} + 77 \, b \sqrt {x}\right ) + 231 \, {\left (i \, a^{3} b^{2} x^{6} + 2 i \, a^{4} b x^{4} + i \, a^{5} x^{2}\right )} \left (-\frac {b^{3}}{a^{15}}\right )^{\frac {1}{4}} \log \left (77 i \, a^{4} \left (-\frac {b^{3}}{a^{15}}\right )^{\frac {1}{4}} + 77 \, b \sqrt {x}\right ) + 231 \, {\left (-i \, a^{3} b^{2} x^{6} - 2 i \, a^{4} b x^{4} - i \, a^{5} x^{2}\right )} \left (-\frac {b^{3}}{a^{15}}\right )^{\frac {1}{4}} \log \left (-77 i \, a^{4} \left (-\frac {b^{3}}{a^{15}}\right )^{\frac {1}{4}} + 77 \, b \sqrt {x}\right ) - 231 \, {\left (a^{3} b^{2} x^{6} + 2 \, a^{4} b x^{4} + a^{5} x^{2}\right )} \left (-\frac {b^{3}}{a^{15}}\right )^{\frac {1}{4}} \log \left (-77 \, a^{4} \left (-\frac {b^{3}}{a^{15}}\right )^{\frac {1}{4}} + 77 \, b \sqrt {x}\right ) + 4 \, {\left (77 \, b^{2} x^{4} + 121 \, a b x^{2} + 32 \, a^{2}\right )} \sqrt {x}}{192 \, {\left (a^{3} b^{2} x^{6} + 2 \, a^{4} b x^{4} + a^{5} x^{2}\right )}} \] Input:
integrate(1/x^(5/2)/(b*x^2+a)^3,x, algorithm="fricas")
Output:
-1/192*(231*(a^3*b^2*x^6 + 2*a^4*b*x^4 + a^5*x^2)*(-b^3/a^15)^(1/4)*log(77 *a^4*(-b^3/a^15)^(1/4) + 77*b*sqrt(x)) + 231*(I*a^3*b^2*x^6 + 2*I*a^4*b*x^ 4 + I*a^5*x^2)*(-b^3/a^15)^(1/4)*log(77*I*a^4*(-b^3/a^15)^(1/4) + 77*b*sqr t(x)) + 231*(-I*a^3*b^2*x^6 - 2*I*a^4*b*x^4 - I*a^5*x^2)*(-b^3/a^15)^(1/4) *log(-77*I*a^4*(-b^3/a^15)^(1/4) + 77*b*sqrt(x)) - 231*(a^3*b^2*x^6 + 2*a^ 4*b*x^4 + a^5*x^2)*(-b^3/a^15)^(1/4)*log(-77*a^4*(-b^3/a^15)^(1/4) + 77*b* sqrt(x)) + 4*(77*b^2*x^4 + 121*a*b*x^2 + 32*a^2)*sqrt(x))/(a^3*b^2*x^6 + 2 *a^4*b*x^4 + a^5*x^2)
Timed out. \[ \int \frac {1}{x^{5/2} \left (a+b x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate(1/x**(5/2)/(b*x**2+a)**3,x)
Output:
Timed out
Time = 0.11 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.17 \[ \int \frac {1}{x^{5/2} \left (a+b x^2\right )^3} \, dx=-\frac {77 \, b^{2} x^{4} + 121 \, a b x^{2} + 32 \, a^{2}}{48 \, {\left (a^{3} b^{2} x^{\frac {11}{2}} + 2 \, a^{4} b x^{\frac {7}{2}} + a^{5} x^{\frac {3}{2}}\right )}} - \frac {77 \, {\left (\frac {2 \, \sqrt {2} b \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} b \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} b^{\frac {3}{4}} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}}} - \frac {\sqrt {2} b^{\frac {3}{4}} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}}}\right )}}{128 \, a^{3}} \] Input:
integrate(1/x^(5/2)/(b*x^2+a)^3,x, algorithm="maxima")
Output:
-1/48*(77*b^2*x^4 + 121*a*b*x^2 + 32*a^2)/(a^3*b^2*x^(11/2) + 2*a^4*b*x^(7 /2) + a^5*x^(3/2)) - 77/128*(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)))/(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)))/(sqrt(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^3
Time = 0.13 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.05 \[ \int \frac {1}{x^{5/2} \left (a+b x^2\right )^3} \, dx=-\frac {77 \, \sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{64 \, a^{4}} - \frac {77 \, \sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{64 \, a^{4}} - \frac {77 \, \sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{128 \, a^{4}} + \frac {77 \, \sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{128 \, a^{4}} - \frac {15 \, b^{2} x^{\frac {5}{2}} + 19 \, a b \sqrt {x}}{16 \, {\left (b x^{2} + a\right )}^{2} a^{3}} - \frac {2}{3 \, a^{3} x^{\frac {3}{2}}} \] Input:
integrate(1/x^(5/2)/(b*x^2+a)^3,x, algorithm="giac")
Output:
-77/64*sqrt(2)*(a*b^3)^(1/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*s qrt(x))/(a/b)^(1/4))/a^4 - 77/64*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^4 - 77/128*sqrt(2)*(a*b^ 3)^(1/4)*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/a^4 + 77/128*sqr t(2)*(a*b^3)^(1/4)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/a^4 - 1/16*(15*b^2*x^(5/2) + 19*a*b*sqrt(x))/((b*x^2 + a)^2*a^3) - 2/3/(a^3*x^( 3/2))
Time = 0.13 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.50 \[ \int \frac {1}{x^{5/2} \left (a+b x^2\right )^3} \, dx=\frac {77\,{\left (-b\right )}^{3/4}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {x}}{a^{1/4}}\right )}{32\,a^{15/4}}-\frac {\frac {2}{3\,a}+\frac {121\,b\,x^2}{48\,a^2}+\frac {77\,b^2\,x^4}{48\,a^3}}{a^2\,x^{3/2}+b^2\,x^{11/2}+2\,a\,b\,x^{7/2}}+\frac {77\,{\left (-b\right )}^{3/4}\,\mathrm {atanh}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {x}}{a^{1/4}}\right )}{32\,a^{15/4}} \] Input:
int(1/(x^(5/2)*(a + b*x^2)^3),x)
Output:
(77*(-b)^(3/4)*atan(((-b)^(1/4)*x^(1/2))/a^(1/4)))/(32*a^(15/4)) - (2/(3*a ) + (121*b*x^2)/(48*a^2) + (77*b^2*x^4)/(48*a^3))/(a^2*x^(3/2) + b^2*x^(11 /2) + 2*a*b*x^(7/2)) + (77*(-b)^(3/4)*atanh(((-b)^(1/4)*x^(1/2))/a^(1/4))) /(32*a^(15/4))
Time = 0.24 (sec) , antiderivative size = 513, normalized size of antiderivative = 2.59 \[ \int \frac {1}{x^{5/2} \left (a+b x^2\right )^3} \, dx =\text {Too large to display} \] Input:
int(1/x^(5/2)/(b*x^2+a)^3,x)
Output:
(462*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**2*x + 924*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*b*x**3 + 462*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**2*x**5 - 462*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**2* x - 924*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*b*x**3 - 462*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**2*x**5 + 231*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 **2*x + 462*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*b*x**3 + 231*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**2*x**5 - 231*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**2*x - 462*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* b*x**3 - 231*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**2*x**5 - 256*a**3 - 968*a**2*b*x**...