Integrand size = 22, antiderivative size = 343 \[ \int \frac {A+B x^2}{x^{7/2} \left (a+b x^2\right )^3} \, dx=-\frac {9 (13 A b-5 a B)}{80 a^3 b x^{5/2}}+\frac {9 (13 A b-5 a B)}{16 a^4 \sqrt {x}}+\frac {A b-a B}{4 a b x^{5/2} \left (a+b x^2\right )^2}+\frac {13 A b-5 a B}{16 a^2 b x^{5/2} \left (a+b x^2\right )}-\frac {9 \sqrt [4]{b} (13 A b-5 a B) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{17/4}}+\frac {9 \sqrt [4]{b} (13 A b-5 a B) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{17/4}}+\frac {9 \sqrt [4]{b} (13 A b-5 a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{17/4}}-\frac {9 \sqrt [4]{b} (13 A b-5 a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{17/4}} \]
-9/80*(13*A*b-5*B*a)/a^3/b/x^(5/2)+1/4*(A*b-B*a)/a/b/x^(5/2)/(b*x^2+a)^2+1 /16*(13*A*b-5*B*a)/a^2/b/x^(5/2)/(b*x^2+a)-9/64*b^(1/4)*(13*A*b-5*B*a)*arc tan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(17/4)*2^(1/2)+9/64*b^(1/4)*(13*A *b-5*B*a)*arctan(1+b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(17/4)*2^(1/2)+9/128 *b^(1/4)*(13*A*b-5*B*a)*ln(a^(1/2)+x*b^(1/2)-a^(1/4)*b^(1/4)*2^(1/2)*x^(1/ 2))/a^(17/4)*2^(1/2)-9/128*b^(1/4)*(13*A*b-5*B*a)*ln(a^(1/2)+x*b^(1/2)+a^( 1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(17/4)*2^(1/2)+9/16*(13*A*b-5*B*a)/a^4/x^( 1/2)
Time = 0.74 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.60 \[ \int \frac {A+B x^2}{x^{7/2} \left (a+b x^2\right )^3} \, dx=\frac {-\frac {4 \sqrt [4]{a} \left (-585 A b^3 x^6+32 a^3 \left (A+5 B x^2\right )+9 a b^2 x^4 \left (-117 A+25 B x^2\right )+a^2 \left (-416 A b x^2+405 b B x^4\right )\right )}{x^{5/2} \left (a+b x^2\right )^2}+45 \sqrt {2} \sqrt [4]{b} (-13 A b+5 a B) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+45 \sqrt {2} \sqrt [4]{b} (-13 A b+5 a B) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{320 a^{17/4}} \]
((-4*a^(1/4)*(-585*A*b^3*x^6 + 32*a^3*(A + 5*B*x^2) + 9*a*b^2*x^4*(-117*A + 25*B*x^2) + a^2*(-416*A*b*x^2 + 405*b*B*x^4)))/(x^(5/2)*(a + b*x^2)^2) + 45*Sqrt[2]*b^(1/4)*(-13*A*b + 5*a*B)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2 ]*a^(1/4)*b^(1/4)*Sqrt[x])] + 45*Sqrt[2]*b^(1/4)*(-13*A*b + 5*a*B)*ArcTanh [(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(320*a^(17/4))
Time = 0.52 (sec) , antiderivative size = 329, normalized size of antiderivative = 0.96, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.591, Rules used = {362, 253, 264, 264, 266, 826, 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.
| \(\displaystyle \int \frac {A+B x^2}{x^{7/2} \left (a+b x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 362 |
| \(\displaystyle \frac {(13 A b-5 a B) \int \frac {1}{x^{7/2} \left (b x^2+a\right )^2}dx}{8 a b}+\frac {A b-a B}{4 a b x^{5/2} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {(13 A b-5 a B) \left (\frac {9 \int \frac {1}{x^{7/2} \left (b x^2+a\right )}dx}{4 a}+\frac {1}{2 a x^{5/2} \left (a+b x^2\right )}\right )}{8 a b}+\frac {A b-a B}{4 a b x^{5/2} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {(13 A b-5 a B) \left (\frac {9 \left (-\frac {b \int \frac {1}{x^{3/2} \left (b x^2+a\right )}dx}{a}-\frac {2}{5 a x^{5/2}}\right )}{4 a}+\frac {1}{2 a x^{5/2} \left (a+b x^2\right )}\right )}{8 a b}+\frac {A b-a B}{4 a b x^{5/2} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {(13 A b-5 a B) \left (\frac {9 \left (-\frac {b \left (-\frac {b \int \frac {\sqrt {x}}{b x^2+a}dx}{a}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2}{5 a x^{5/2}}\right )}{4 a}+\frac {1}{2 a x^{5/2} \left (a+b x^2\right )}\right )}{8 a b}+\frac {A b-a B}{4 a b x^{5/2} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {(13 A b-5 a B) \left (\frac {9 \left (-\frac {b \left (-\frac {2 b \int \frac {x}{b x^2+a}d\sqrt {x}}{a}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2}{5 a x^{5/2}}\right )}{4 a}+\frac {1}{2 a x^{5/2} \left (a+b x^2\right )}\right )}{8 a b}+\frac {A b-a B}{4 a b x^{5/2} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {(13 A b-5 a B) \left (\frac {9 \left (-\frac {b \left (-\frac {2 b \left (\frac {\int \frac {\sqrt {b} x+\sqrt {a}}{b x^2+a}d\sqrt {x}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {b}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2}{5 a x^{5/2}}\right )}{4 a}+\frac {1}{2 a x^{5/2} \left (a+b x^2\right )}\right )}{8 a b}+\frac {A b-a B}{4 a b x^{5/2} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {(13 A b-5 a B) \left (\frac {9 \left (-\frac {b \left (-\frac {2 b \left (\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {b}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {b}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2}{5 a x^{5/2}}\right )}{4 a}+\frac {1}{2 a x^{5/2} \left (a+b x^2\right )}\right )}{8 a b}+\frac {A b-a B}{4 a b x^{5/2} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {(13 A b-5 a B) \left (\frac {9 \left (-\frac {b \left (-\frac {2 b \left (\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {b}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2}{5 a x^{5/2}}\right )}{4 a}+\frac {1}{2 a x^{5/2} \left (a+b x^2\right )}\right )}{8 a b}+\frac {A b-a B}{4 a b x^{5/2} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {(13 A b-5 a B) \left (\frac {9 \left (-\frac {b \left (-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {b}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2}{5 a x^{5/2}}\right )}{4 a}+\frac {1}{2 a x^{5/2} \left (a+b x^2\right )}\right )}{8 a b}+\frac {A b-a B}{4 a b x^{5/2} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {(13 A b-5 a B) \left (\frac {9 \left (-\frac {b \left (-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{b} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2}{5 a x^{5/2}}\right )}{4 a}+\frac {1}{2 a x^{5/2} \left (a+b x^2\right )}\right )}{8 a b}+\frac {A b-a B}{4 a b x^{5/2} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(13 A b-5 a B) \left (\frac {9 \left (-\frac {b \left (-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{b} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2}{5 a x^{5/2}}\right )}{4 a}+\frac {1}{2 a x^{5/2} \left (a+b x^2\right )}\right )}{8 a b}+\frac {A b-a B}{4 a b x^{5/2} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(13 A b-5 a B) \left (\frac {9 \left (-\frac {b \left (-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt [4]{a} \sqrt {b}}}{2 \sqrt {b}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2}{5 a x^{5/2}}\right )}{4 a}+\frac {1}{2 a x^{5/2} \left (a+b x^2\right )}\right )}{8 a b}+\frac {A b-a B}{4 a b x^{5/2} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {(13 A b-5 a B) \left (\frac {9 \left (-\frac {b \left (-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2}{5 a x^{5/2}}\right )}{4 a}+\frac {1}{2 a x^{5/2} \left (a+b x^2\right )}\right )}{8 a b}+\frac {A b-a B}{4 a b x^{5/2} \left (a+b x^2\right )^2}\) |
(A*b - a*B)/(4*a*b*x^(5/2)*(a + b*x^2)^2) + ((13*A*b - 5*a*B)*(1/(2*a*x^(5 /2)*(a + b*x^2)) + (9*(-2/(5*a*x^(5/2)) - (b*(-2/(a*Sqrt[x]) - (2*b*((-(Ar cTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)]/(Sqrt[2]*a^(1/4)*b^(1/4))) + A rcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)]/(Sqrt[2]*a^(1/4)*b^(1/4)))/(2 *Sqrt[b]) - (-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[b])))/a))/a))/(4*a))) /(8*a*b)
3.4.90.3.1 Defintions of rubi rules used
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[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[(-(b*c - a*d))*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*b*e *(p + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(2*a*b*(p + 1)) I nt[(e*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && N eQ[b*c - a*d, 0] && LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) || !RationalQ[m] || (ILtQ[p + 1/2, 0] && LeQ[-1, m, -2*(p + 1)]))
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s) Int[(r + s*x^2)/(a + b*x^ 4), x], x] - Simp[1/(2*s) 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 = 2.67 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.55
| method | result | size |
| derivativedivides | \(-\frac {2 A}{5 a^{3} x^{\frac {5}{2}}}-\frac {2 \left (-3 A b +B a \right )}{a^{4} \sqrt {x}}+\frac {2 b \left (\frac {\left (\frac {21}{32} b^{2} A -\frac {13}{32} a b B \right ) x^{\frac {7}{2}}+\frac {a \left (25 A b -17 B a \right ) x^{\frac {3}{2}}}{32}}{\left (b \,x^{2}+a \right )^{2}}+\frac {\left (\frac {117 A b}{32}-\frac {45 B a}{32}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{a^{4}}\) | \(190\) |
| default | \(-\frac {2 A}{5 a^{3} x^{\frac {5}{2}}}-\frac {2 \left (-3 A b +B a \right )}{a^{4} \sqrt {x}}+\frac {2 b \left (\frac {\left (\frac {21}{32} b^{2} A -\frac {13}{32} a b B \right ) x^{\frac {7}{2}}+\frac {a \left (25 A b -17 B a \right ) x^{\frac {3}{2}}}{32}}{\left (b \,x^{2}+a \right )^{2}}+\frac {\left (\frac {117 A b}{32}-\frac {45 B a}{32}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{a^{4}}\) | \(190\) |
| risch | \(-\frac {2 \left (-15 A b \,x^{2}+5 B a \,x^{2}+A a \right )}{5 a^{4} x^{\frac {5}{2}}}+\frac {b \left (\frac {2 \left (\frac {21}{32} b^{2} A -\frac {13}{32} a b B \right ) x^{\frac {7}{2}}+\frac {a \left (25 A b -17 B a \right ) x^{\frac {3}{2}}}{16}}{\left (b \,x^{2}+a \right )^{2}}+\frac {\left (\frac {117 A b}{32}-\frac {45 B a}{32}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{a^{4}}\) | \(191\) |
-2/5*A/a^3/x^(5/2)-2*(-3*A*b+B*a)/a^4/x^(1/2)+2/a^4*b*(((21/32*b^2*A-13/32 *a*b*B)*x^(7/2)+1/32*a*(25*A*b-17*B*a)*x^(3/2))/(b*x^2+a)^2+1/8*(117/32*A* b-45/32*B*a)/b/(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*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)))
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 918, normalized size of antiderivative = 2.68 \[ \int \frac {A+B x^2}{x^{7/2} \left (a+b x^2\right )^3} \, dx=\frac {45 \, {\left (a^{4} b^{2} x^{7} + 2 \, a^{5} b x^{5} + a^{6} x^{3}\right )} \left (-\frac {625 \, B^{4} a^{4} b - 6500 \, A B^{3} a^{3} b^{2} + 25350 \, A^{2} B^{2} a^{2} b^{3} - 43940 \, A^{3} B a b^{4} + 28561 \, A^{4} b^{5}}{a^{17}}\right )^{\frac {1}{4}} \log \left (729 \, a^{13} \left (-\frac {625 \, B^{4} a^{4} b - 6500 \, A B^{3} a^{3} b^{2} + 25350 \, A^{2} B^{2} a^{2} b^{3} - 43940 \, A^{3} B a b^{4} + 28561 \, A^{4} b^{5}}{a^{17}}\right )^{\frac {3}{4}} - 729 \, {\left (125 \, B^{3} a^{3} b - 975 \, A B^{2} a^{2} b^{2} + 2535 \, A^{2} B a b^{3} - 2197 \, A^{3} b^{4}\right )} \sqrt {x}\right ) - 45 \, {\left (i \, a^{4} b^{2} x^{7} + 2 i \, a^{5} b x^{5} + i \, a^{6} x^{3}\right )} \left (-\frac {625 \, B^{4} a^{4} b - 6500 \, A B^{3} a^{3} b^{2} + 25350 \, A^{2} B^{2} a^{2} b^{3} - 43940 \, A^{3} B a b^{4} + 28561 \, A^{4} b^{5}}{a^{17}}\right )^{\frac {1}{4}} \log \left (729 i \, a^{13} \left (-\frac {625 \, B^{4} a^{4} b - 6500 \, A B^{3} a^{3} b^{2} + 25350 \, A^{2} B^{2} a^{2} b^{3} - 43940 \, A^{3} B a b^{4} + 28561 \, A^{4} b^{5}}{a^{17}}\right )^{\frac {3}{4}} - 729 \, {\left (125 \, B^{3} a^{3} b - 975 \, A B^{2} a^{2} b^{2} + 2535 \, A^{2} B a b^{3} - 2197 \, A^{3} b^{4}\right )} \sqrt {x}\right ) - 45 \, {\left (-i \, a^{4} b^{2} x^{7} - 2 i \, a^{5} b x^{5} - i \, a^{6} x^{3}\right )} \left (-\frac {625 \, B^{4} a^{4} b - 6500 \, A B^{3} a^{3} b^{2} + 25350 \, A^{2} B^{2} a^{2} b^{3} - 43940 \, A^{3} B a b^{4} + 28561 \, A^{4} b^{5}}{a^{17}}\right )^{\frac {1}{4}} \log \left (-729 i \, a^{13} \left (-\frac {625 \, B^{4} a^{4} b - 6500 \, A B^{3} a^{3} b^{2} + 25350 \, A^{2} B^{2} a^{2} b^{3} - 43940 \, A^{3} B a b^{4} + 28561 \, A^{4} b^{5}}{a^{17}}\right )^{\frac {3}{4}} - 729 \, {\left (125 \, B^{3} a^{3} b - 975 \, A B^{2} a^{2} b^{2} + 2535 \, A^{2} B a b^{3} - 2197 \, A^{3} b^{4}\right )} \sqrt {x}\right ) - 45 \, {\left (a^{4} b^{2} x^{7} + 2 \, a^{5} b x^{5} + a^{6} x^{3}\right )} \left (-\frac {625 \, B^{4} a^{4} b - 6500 \, A B^{3} a^{3} b^{2} + 25350 \, A^{2} B^{2} a^{2} b^{3} - 43940 \, A^{3} B a b^{4} + 28561 \, A^{4} b^{5}}{a^{17}}\right )^{\frac {1}{4}} \log \left (-729 \, a^{13} \left (-\frac {625 \, B^{4} a^{4} b - 6500 \, A B^{3} a^{3} b^{2} + 25350 \, A^{2} B^{2} a^{2} b^{3} - 43940 \, A^{3} B a b^{4} + 28561 \, A^{4} b^{5}}{a^{17}}\right )^{\frac {3}{4}} - 729 \, {\left (125 \, B^{3} a^{3} b - 975 \, A B^{2} a^{2} b^{2} + 2535 \, A^{2} B a b^{3} - 2197 \, A^{3} b^{4}\right )} \sqrt {x}\right ) - 4 \, {\left (45 \, {\left (5 \, B a b^{2} - 13 \, A b^{3}\right )} x^{6} + 81 \, {\left (5 \, B a^{2} b - 13 \, A a b^{2}\right )} x^{4} + 32 \, A a^{3} + 32 \, {\left (5 \, B a^{3} - 13 \, A a^{2} b\right )} x^{2}\right )} \sqrt {x}}{320 \, {\left (a^{4} b^{2} x^{7} + 2 \, a^{5} b x^{5} + a^{6} x^{3}\right )}} \]
1/320*(45*(a^4*b^2*x^7 + 2*a^5*b*x^5 + a^6*x^3)*(-(625*B^4*a^4*b - 6500*A* B^3*a^3*b^2 + 25350*A^2*B^2*a^2*b^3 - 43940*A^3*B*a*b^4 + 28561*A^4*b^5)/a ^17)^(1/4)*log(729*a^13*(-(625*B^4*a^4*b - 6500*A*B^3*a^3*b^2 + 25350*A^2* B^2*a^2*b^3 - 43940*A^3*B*a*b^4 + 28561*A^4*b^5)/a^17)^(3/4) - 729*(125*B^ 3*a^3*b - 975*A*B^2*a^2*b^2 + 2535*A^2*B*a*b^3 - 2197*A^3*b^4)*sqrt(x)) - 45*(I*a^4*b^2*x^7 + 2*I*a^5*b*x^5 + I*a^6*x^3)*(-(625*B^4*a^4*b - 6500*A*B ^3*a^3*b^2 + 25350*A^2*B^2*a^2*b^3 - 43940*A^3*B*a*b^4 + 28561*A^4*b^5)/a^ 17)^(1/4)*log(729*I*a^13*(-(625*B^4*a^4*b - 6500*A*B^3*a^3*b^2 + 25350*A^2 *B^2*a^2*b^3 - 43940*A^3*B*a*b^4 + 28561*A^4*b^5)/a^17)^(3/4) - 729*(125*B ^3*a^3*b - 975*A*B^2*a^2*b^2 + 2535*A^2*B*a*b^3 - 2197*A^3*b^4)*sqrt(x)) - 45*(-I*a^4*b^2*x^7 - 2*I*a^5*b*x^5 - I*a^6*x^3)*(-(625*B^4*a^4*b - 6500*A *B^3*a^3*b^2 + 25350*A^2*B^2*a^2*b^3 - 43940*A^3*B*a*b^4 + 28561*A^4*b^5)/ a^17)^(1/4)*log(-729*I*a^13*(-(625*B^4*a^4*b - 6500*A*B^3*a^3*b^2 + 25350* A^2*B^2*a^2*b^3 - 43940*A^3*B*a*b^4 + 28561*A^4*b^5)/a^17)^(3/4) - 729*(12 5*B^3*a^3*b - 975*A*B^2*a^2*b^2 + 2535*A^2*B*a*b^3 - 2197*A^3*b^4)*sqrt(x) ) - 45*(a^4*b^2*x^7 + 2*a^5*b*x^5 + a^6*x^3)*(-(625*B^4*a^4*b - 6500*A*B^3 *a^3*b^2 + 25350*A^2*B^2*a^2*b^3 - 43940*A^3*B*a*b^4 + 28561*A^4*b^5)/a^17 )^(1/4)*log(-729*a^13*(-(625*B^4*a^4*b - 6500*A*B^3*a^3*b^2 + 25350*A^2*B^ 2*a^2*b^3 - 43940*A^3*B*a*b^4 + 28561*A^4*b^5)/a^17)^(3/4) - 729*(125*B^3* a^3*b - 975*A*B^2*a^2*b^2 + 2535*A^2*B*a*b^3 - 2197*A^3*b^4)*sqrt(x)) -...
Timed out. \[ \int \frac {A+B x^2}{x^{7/2} \left (a+b x^2\right )^3} \, dx=\text {Timed out} \]
Time = 0.30 (sec) , antiderivative size = 285, normalized size of antiderivative = 0.83 \[ \int \frac {A+B x^2}{x^{7/2} \left (a+b x^2\right )^3} \, dx=-\frac {45 \, {\left (5 \, B a b^{2} - 13 \, A b^{3}\right )} x^{6} + 81 \, {\left (5 \, B a^{2} b - 13 \, A a b^{2}\right )} x^{4} + 32 \, A a^{3} + 32 \, {\left (5 \, B a^{3} - 13 \, A a^{2} b\right )} x^{2}}{80 \, {\left (a^{4} b^{2} x^{\frac {13}{2}} + 2 \, a^{5} b x^{\frac {9}{2}} + a^{6} x^{\frac {5}{2}}\right )}} - \frac {9 \, {\left (5 \, B a b - 13 \, A b^{2}\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{128 \, a^{4}} \]
-1/80*(45*(5*B*a*b^2 - 13*A*b^3)*x^6 + 81*(5*B*a^2*b - 13*A*a*b^2)*x^4 + 3 2*A*a^3 + 32*(5*B*a^3 - 13*A*a^2*b)*x^2)/(a^4*b^2*x^(13/2) + 2*a^5*b*x^(9/ 2) + a^6*x^(5/2)) - 9/128*(5*B*a*b - 13*A*b^2)*(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)))/(s qrt(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)*sqr t(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^4
Time = 0.30 (sec) , antiderivative size = 326, normalized size of antiderivative = 0.95 \[ \int \frac {A+B x^2}{x^{7/2} \left (a+b x^2\right )^3} \, dx=-\frac {9 \, \sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {3}{4}} B a - 13 \, \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{64 \, a^{5} b^{2}} - \frac {9 \, \sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {3}{4}} B a - 13 \, \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{64 \, a^{5} b^{2}} + \frac {9 \, \sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {3}{4}} B a - 13 \, \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{128 \, a^{5} b^{2}} - \frac {9 \, \sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {3}{4}} B a - 13 \, \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{128 \, a^{5} b^{2}} - \frac {13 \, B a b^{2} x^{\frac {7}{2}} - 21 \, A b^{3} x^{\frac {7}{2}} + 17 \, B a^{2} b x^{\frac {3}{2}} - 25 \, A a b^{2} x^{\frac {3}{2}}}{16 \, {\left (b x^{2} + a\right )}^{2} a^{4}} - \frac {2 \, {\left (5 \, B a x^{2} - 15 \, A b x^{2} + A a\right )}}{5 \, a^{4} x^{\frac {5}{2}}} \]
-9/64*sqrt(2)*(5*(a*b^3)^(3/4)*B*a - 13*(a*b^3)^(3/4)*A*b)*arctan(1/2*sqrt (2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(a^5*b^2) - 9/64*sqrt(2 )*(5*(a*b^3)^(3/4)*B*a - 13*(a*b^3)^(3/4)*A*b)*arctan(-1/2*sqrt(2)*(sqrt(2 )*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(a^5*b^2) + 9/128*sqrt(2)*(5*(a*b^ 3)^(3/4)*B*a - 13*(a*b^3)^(3/4)*A*b)*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^5*b^2) - 9/128*sqrt(2)*(5*(a*b^3)^(3/4)*B*a - 13*(a*b^3)^(3 /4)*A*b)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^5*b^2) - 1/1 6*(13*B*a*b^2*x^(7/2) - 21*A*b^3*x^(7/2) + 17*B*a^2*b*x^(3/2) - 25*A*a*b^2 *x^(3/2))/((b*x^2 + a)^2*a^4) - 2/5*(5*B*a*x^2 - 15*A*b*x^2 + A*a)/(a^4*x^ (5/2))
Time = 5.52 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.44 \[ \int \frac {A+B x^2}{x^{7/2} \left (a+b x^2\right )^3} \, dx=\frac {\frac {2\,x^2\,\left (13\,A\,b-5\,B\,a\right )}{5\,a^2}-\frac {2\,A}{5\,a}+\frac {9\,b^2\,x^6\,\left (13\,A\,b-5\,B\,a\right )}{16\,a^4}+\frac {81\,b\,x^4\,\left (13\,A\,b-5\,B\,a\right )}{80\,a^3}}{a^2\,x^{5/2}+b^2\,x^{13/2}+2\,a\,b\,x^{9/2}}+\frac {9\,{\left (-b\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {x}}{a^{1/4}}\right )\,\left (13\,A\,b-5\,B\,a\right )}{32\,a^{17/4}}-\frac {9\,{\left (-b\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {x}}{a^{1/4}}\right )\,\left (13\,A\,b-5\,B\,a\right )}{32\,a^{17/4}} \]
((2*x^2*(13*A*b - 5*B*a))/(5*a^2) - (2*A)/(5*a) + (9*b^2*x^6*(13*A*b - 5*B *a))/(16*a^4) + (81*b*x^4*(13*A*b - 5*B*a))/(80*a^3))/(a^2*x^(5/2) + b^2*x ^(13/2) + 2*a*b*x^(9/2)) + (9*(-b)^(1/4)*atan(((-b)^(1/4)*x^(1/2))/a^(1/4) )*(13*A*b - 5*B*a))/(32*a^(17/4)) - (9*(-b)^(1/4)*atanh(((-b)^(1/4)*x^(1/2 ))/a^(1/4))*(13*A*b - 5*B*a))/(32*a^(17/4))