Integrand size = 20, antiderivative size = 333 \[ \int \frac {A+B x}{x^{3/2} \left (a+c x^2\right )^3} \, dx=-\frac {45 A}{16 a^3 \sqrt {x}}+\frac {A+B x}{4 a \sqrt {x} \left (a+c x^2\right )^2}+\frac {9 A+7 B x}{16 a^2 \sqrt {x} \left (a+c x^2\right )}-\frac {3 \left (7 \sqrt {a} B-15 A \sqrt {c}\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{13/4} \sqrt [4]{c}}+\frac {3 \left (7 \sqrt {a} B-15 A \sqrt {c}\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{13/4} \sqrt [4]{c}}-\frac {3 \left (7 \sqrt {a} B+15 A \sqrt {c}\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} a^{13/4} \sqrt [4]{c}}+\frac {3 \left (7 \sqrt {a} B+15 A \sqrt {c}\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} a^{13/4} \sqrt [4]{c}} \]
-3/64*arctan(1-c^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))*(7*B*a^(1/2)-15*A*c^(1/2)) /a^(13/4)/c^(1/4)*2^(1/2)+3/64*arctan(1+c^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))*( 7*B*a^(1/2)-15*A*c^(1/2))/a^(13/4)/c^(1/4)*2^(1/2)-3/128*ln(a^(1/2)+x*c^(1 /2)-a^(1/4)*c^(1/4)*2^(1/2)*x^(1/2))*(7*B*a^(1/2)+15*A*c^(1/2))/a^(13/4)/c ^(1/4)*2^(1/2)+3/128*ln(a^(1/2)+x*c^(1/2)+a^(1/4)*c^(1/4)*2^(1/2)*x^(1/2)) *(7*B*a^(1/2)+15*A*c^(1/2))/a^(13/4)/c^(1/4)*2^(1/2)-45/16*A/a^3/x^(1/2)+1 /4*(B*x+A)/a/(c*x^2+a)^2/x^(1/2)+1/16*(7*B*x+9*A)/a^2/(c*x^2+a)/x^(1/2)
Time = 0.72 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.59 \[ \int \frac {A+B x}{x^{3/2} \left (a+c x^2\right )^3} \, dx=\frac {-\frac {4 \sqrt [4]{a} \left (45 A c^2 x^4+a^2 (32 A-11 B x)+a c x^2 (81 A-7 B x)\right )}{\sqrt {x} \left (a+c x^2\right )^2}+\frac {3 \sqrt {2} \left (-7 \sqrt {a} B+15 A \sqrt {c}\right ) \arctan \left (\frac {\sqrt {a}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}}\right )}{\sqrt [4]{c}}+\frac {3 \sqrt {2} \left (7 \sqrt {a} B+15 A \sqrt {c}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}}{\sqrt {a}+\sqrt {c} x}\right )}{\sqrt [4]{c}}}{64 a^{13/4}} \]
((-4*a^(1/4)*(45*A*c^2*x^4 + a^2*(32*A - 11*B*x) + a*c*x^2*(81*A - 7*B*x)) )/(Sqrt[x]*(a + c*x^2)^2) + (3*Sqrt[2]*(-7*Sqrt[a]*B + 15*A*Sqrt[c])*ArcTa n[(Sqrt[a] - Sqrt[c]*x)/(Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x])])/c^(1/4) + (3*S qrt[2]*(7*Sqrt[a]*B + 15*A*Sqrt[c])*ArcTanh[(Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[ x])/(Sqrt[a] + Sqrt[c]*x)])/c^(1/4))/(64*a^(13/4))
Time = 0.58 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {551, 27, 551, 27, 553, 27, 554, 1482, 27, 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.
| \(\displaystyle \int \frac {A+B x}{x^{3/2} \left (a+c x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 551 |
| \(\displaystyle \frac {A+B x}{4 a \sqrt {x} \left (a+c x^2\right )^2}-\frac {\int -\frac {9 A+7 B x}{2 x^{3/2} \left (c x^2+a\right )^2}dx}{4 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {9 A+7 B x}{x^{3/2} \left (c x^2+a\right )^2}dx}{8 a}+\frac {A+B x}{4 a \sqrt {x} \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 551 |
| \(\displaystyle \frac {\frac {9 A+7 B x}{2 a \sqrt {x} \left (a+c x^2\right )}-\frac {\int -\frac {3 (15 A+7 B x)}{2 x^{3/2} \left (c x^2+a\right )}dx}{2 a}}{8 a}+\frac {A+B x}{4 a \sqrt {x} \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3 \int \frac {15 A+7 B x}{x^{3/2} \left (c x^2+a\right )}dx}{4 a}+\frac {9 A+7 B x}{2 a \sqrt {x} \left (a+c x^2\right )}}{8 a}+\frac {A+B x}{4 a \sqrt {x} \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 553 |
| \(\displaystyle \frac {\frac {3 \left (-\frac {2 \int -\frac {7 a B-15 A c x}{2 \sqrt {x} \left (c x^2+a\right )}dx}{a}-\frac {30 A}{a \sqrt {x}}\right )}{4 a}+\frac {9 A+7 B x}{2 a \sqrt {x} \left (a+c x^2\right )}}{8 a}+\frac {A+B x}{4 a \sqrt {x} \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3 \left (\frac {\int \frac {7 a B-15 A c x}{\sqrt {x} \left (c x^2+a\right )}dx}{a}-\frac {30 A}{a \sqrt {x}}\right )}{4 a}+\frac {9 A+7 B x}{2 a \sqrt {x} \left (a+c x^2\right )}}{8 a}+\frac {A+B x}{4 a \sqrt {x} \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 554 |
| \(\displaystyle \frac {\frac {3 \left (\frac {2 \int \frac {7 a B-15 A c x}{c x^2+a}d\sqrt {x}}{a}-\frac {30 A}{a \sqrt {x}}\right )}{4 a}+\frac {9 A+7 B x}{2 a \sqrt {x} \left (a+c x^2\right )}}{8 a}+\frac {A+B x}{4 a \sqrt {x} \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {\frac {3 \left (\frac {2 \left (\frac {1}{2} \left (\frac {7 \sqrt {a} B}{\sqrt {c}}+15 A\right ) \int \frac {\sqrt {c} \left (\sqrt {a}-\sqrt {c} x\right )}{c x^2+a}d\sqrt {x}-\frac {1}{2} \left (15 A-\frac {7 \sqrt {a} B}{\sqrt {c}}\right ) \int \frac {\sqrt {c} \left (\sqrt {c} x+\sqrt {a}\right )}{c x^2+a}d\sqrt {x}\right )}{a}-\frac {30 A}{a \sqrt {x}}\right )}{4 a}+\frac {9 A+7 B x}{2 a \sqrt {x} \left (a+c x^2\right )}}{8 a}+\frac {A+B x}{4 a \sqrt {x} \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3 \left (\frac {2 \left (\frac {1}{2} \sqrt {c} \left (\frac {7 \sqrt {a} B}{\sqrt {c}}+15 A\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{c x^2+a}d\sqrt {x}-\frac {1}{2} \sqrt {c} \left (15 A-\frac {7 \sqrt {a} B}{\sqrt {c}}\right ) \int \frac {\sqrt {c} x+\sqrt {a}}{c x^2+a}d\sqrt {x}\right )}{a}-\frac {30 A}{a \sqrt {x}}\right )}{4 a}+\frac {9 A+7 B x}{2 a \sqrt {x} \left (a+c x^2\right )}}{8 a}+\frac {A+B x}{4 a \sqrt {x} \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\frac {3 \left (\frac {2 \left (\frac {1}{2} \sqrt {c} \left (\frac {7 \sqrt {a} B}{\sqrt {c}}+15 A\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{c x^2+a}d\sqrt {x}-\frac {1}{2} \sqrt {c} \left (15 A-\frac {7 \sqrt {a} B}{\sqrt {c}}\right ) \left (\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {c}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {c}}\right )\right )}{a}-\frac {30 A}{a \sqrt {x}}\right )}{4 a}+\frac {9 A+7 B x}{2 a \sqrt {x} \left (a+c x^2\right )}}{8 a}+\frac {A+B x}{4 a \sqrt {x} \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {3 \left (\frac {2 \left (\frac {1}{2} \sqrt {c} \left (\frac {7 \sqrt {a} B}{\sqrt {c}}+15 A\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{c x^2+a}d\sqrt {x}-\frac {1}{2} \sqrt {c} \left (15 A-\frac {7 \sqrt {a} B}{\sqrt {c}}\right ) \left (\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )\right )}{a}-\frac {30 A}{a \sqrt {x}}\right )}{4 a}+\frac {9 A+7 B x}{2 a \sqrt {x} \left (a+c x^2\right )}}{8 a}+\frac {A+B x}{4 a \sqrt {x} \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {3 \left (\frac {2 \left (\frac {1}{2} \sqrt {c} \left (\frac {7 \sqrt {a} B}{\sqrt {c}}+15 A\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{c x^2+a}d\sqrt {x}-\frac {1}{2} \sqrt {c} \left (15 A-\frac {7 \sqrt {a} B}{\sqrt {c}}\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )\right )}{a}-\frac {30 A}{a \sqrt {x}}\right )}{4 a}+\frac {9 A+7 B x}{2 a \sqrt {x} \left (a+c x^2\right )}}{8 a}+\frac {A+B x}{4 a \sqrt {x} \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\frac {3 \left (\frac {2 \left (\frac {1}{2} \sqrt {c} \left (\frac {7 \sqrt {a} B}{\sqrt {c}}+15 A\right ) \left (-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{c} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{c} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )-\frac {1}{2} \sqrt {c} \left (15 A-\frac {7 \sqrt {a} B}{\sqrt {c}}\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )\right )}{a}-\frac {30 A}{a \sqrt {x}}\right )}{4 a}+\frac {9 A+7 B x}{2 a \sqrt {x} \left (a+c x^2\right )}}{8 a}+\frac {A+B x}{4 a \sqrt {x} \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {3 \left (\frac {2 \left (\frac {1}{2} \sqrt {c} \left (\frac {7 \sqrt {a} B}{\sqrt {c}}+15 A\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{c} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{c} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )-\frac {1}{2} \sqrt {c} \left (15 A-\frac {7 \sqrt {a} B}{\sqrt {c}}\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )\right )}{a}-\frac {30 A}{a \sqrt {x}}\right )}{4 a}+\frac {9 A+7 B x}{2 a \sqrt {x} \left (a+c x^2\right )}}{8 a}+\frac {A+B x}{4 a \sqrt {x} \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3 \left (\frac {2 \left (\frac {1}{2} \sqrt {c} \left (\frac {7 \sqrt {a} B}{\sqrt {c}}+15 A\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {c}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{a}}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt [4]{a} \sqrt {c}}\right )-\frac {1}{2} \sqrt {c} \left (15 A-\frac {7 \sqrt {a} B}{\sqrt {c}}\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )\right )}{a}-\frac {30 A}{a \sqrt {x}}\right )}{4 a}+\frac {9 A+7 B x}{2 a \sqrt {x} \left (a+c x^2\right )}}{8 a}+\frac {A+B x}{4 a \sqrt {x} \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {3 \left (\frac {2 \left (\frac {1}{2} \sqrt {c} \left (\frac {7 \sqrt {a} B}{\sqrt {c}}+15 A\right ) \left (\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )-\frac {1}{2} \sqrt {c} \left (15 A-\frac {7 \sqrt {a} B}{\sqrt {c}}\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )\right )}{a}-\frac {30 A}{a \sqrt {x}}\right )}{4 a}+\frac {9 A+7 B x}{2 a \sqrt {x} \left (a+c x^2\right )}}{8 a}+\frac {A+B x}{4 a \sqrt {x} \left (a+c x^2\right )^2}\) |
(A + B*x)/(4*a*Sqrt[x]*(a + c*x^2)^2) + ((9*A + 7*B*x)/(2*a*Sqrt[x]*(a + c *x^2)) + (3*((-30*A)/(a*Sqrt[x]) + (2*(-1/2*((15*A - (7*Sqrt[a]*B)/Sqrt[c] )*Sqrt[c]*(-(ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)]/(Sqrt[2]*a^(1/4 )*c^(1/4))) + ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)]/(Sqrt[2]*a^(1/ 4)*c^(1/4)))) + ((15*A + (7*Sqrt[a]*B)/Sqrt[c])*Sqrt[c]*(-1/2*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x]/(Sqrt[2]*a^(1/4)*c^(1/4)) + Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x]/(2*Sqrt[2]*a^( 1/4)*c^(1/4))))/2))/a))/(4*a))/(8*a)
3.5.31.3.1 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[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym bol] :> Simp[(-(e*x)^(m + 1))*(c + d*x)*((a + b*x^2)^(p + 1)/(2*a*e*(p + 1) )), x] + Simp[1/(2*a*(p + 1)) Int[(e*x)^m*(c*(m + 2*p + 3) + d*(m + 2*p + 4)*x)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && LtQ[p , -1] && LtQ[m, 0]
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym bol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + Simp [1/(a*e*(m + 1)) Int[(e*x)^(m + 1)*(a + b*x^2)^p*(a*d*(m + 1) - b*c*(m + 2*p + 3)*x), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && LtQ[m, -1]
Int[((c_) + (d_.)*(x_))/(Sqrt[(e_.)*(x_)]*((a_) + (b_.)*(x_)^2)), x_Symbol] :> Simp[2 Subst[Int[(e*c + d*x^2)/(a*e^2 + b*x^4), x], x, Sqrt[e*x]], x] /; FreeQ[{a, b, c, d, e}, x]
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]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ a*c, 2]}, Simp[(d*q + a*e)/(2*a*c) Int[(q + c*x^2)/(a + c*x^4), x], x] + Simp[(d*q - a*e)/(2*a*c) Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ[{a , c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(- a)*c]
Time = 0.14 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.80
| method | result | size |
| derivativedivides | \(-\frac {2 \left (\frac {\frac {13 A \,c^{2} x^{\frac {7}{2}}}{32}-\frac {7 a B c \,x^{\frac {5}{2}}}{32}+\frac {17 a A c \,x^{\frac {3}{2}}}{32}-\frac {11 a^{2} B \sqrt {x}}{32}}{\left (c \,x^{2}+a \right )^{2}}-\frac {21 B \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{256}+\frac {45 A \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{256 \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{a^{3}}-\frac {2 A}{a^{3} \sqrt {x}}\) | \(268\) |
| default | \(-\frac {2 \left (\frac {\frac {13 A \,c^{2} x^{\frac {7}{2}}}{32}-\frac {7 a B c \,x^{\frac {5}{2}}}{32}+\frac {17 a A c \,x^{\frac {3}{2}}}{32}-\frac {11 a^{2} B \sqrt {x}}{32}}{\left (c \,x^{2}+a \right )^{2}}-\frac {21 B \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{256}+\frac {45 A \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{256 \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{a^{3}}-\frac {2 A}{a^{3} \sqrt {x}}\) | \(268\) |
| risch | \(-\frac {2 A}{a^{3} \sqrt {x}}-\frac {\frac {\frac {13 A \,c^{2} x^{\frac {7}{2}}}{16}-\frac {7 a B c \,x^{\frac {5}{2}}}{16}+\frac {17 a A c \,x^{\frac {3}{2}}}{16}-\frac {11 a^{2} B \sqrt {x}}{16}}{\left (c \,x^{2}+a \right )^{2}}-\frac {21 B \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{128}+\frac {45 A \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{128 \left (\frac {a}{c}\right )^{\frac {1}{4}}}}{a^{3}}\) | \(269\) |
-2/a^3*((13/32*A*c^2*x^(7/2)-7/32*a*B*c*x^(5/2)+17/32*a*A*c*x^(3/2)-11/32* a^2*B*x^(1/2))/(c*x^2+a)^2-21/256*B*(a/c)^(1/4)*2^(1/2)*(ln((x+(a/c)^(1/4) *x^(1/2)*2^(1/2)+(a/c)^(1/2))/(x-(a/c)^(1/4)*x^(1/2)*2^(1/2)+(a/c)^(1/2))) +2*arctan(2^(1/2)/(a/c)^(1/4)*x^(1/2)+1)+2*arctan(2^(1/2)/(a/c)^(1/4)*x^(1 /2)-1))+45/256*A/(a/c)^(1/4)*2^(1/2)*(ln((x-(a/c)^(1/4)*x^(1/2)*2^(1/2)+(a /c)^(1/2))/(x+(a/c)^(1/4)*x^(1/2)*2^(1/2)+(a/c)^(1/2)))+2*arctan(2^(1/2)/( a/c)^(1/4)*x^(1/2)+1)+2*arctan(2^(1/2)/(a/c)^(1/4)*x^(1/2)-1)))-2*A/a^3/x^ (1/2)
Leaf count of result is larger than twice the leaf count of optimal. 958 vs. \(2 (233) = 466\).
Time = 0.30 (sec) , antiderivative size = 958, normalized size of antiderivative = 2.88 \[ \int \frac {A+B x}{x^{3/2} \left (a+c x^2\right )^3} \, dx=-\frac {3 \, {\left (a^{3} c^{2} x^{5} + 2 \, a^{4} c x^{3} + a^{5} x\right )} \sqrt {\frac {a^{6} \sqrt {-\frac {2401 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 50625 \, A^{4} c^{2}}{a^{13} c}} + 210 \, A B}{a^{6}}} \log \left (-9 \, {\left (2401 \, B^{4} a^{2} - 50625 \, A^{4} c^{2}\right )} \sqrt {x} + 9 \, {\left (15 \, A a^{10} c \sqrt {-\frac {2401 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 50625 \, A^{4} c^{2}}{a^{13} c}} + 343 \, B^{3} a^{5} - 1575 \, A^{2} B a^{4} c\right )} \sqrt {\frac {a^{6} \sqrt {-\frac {2401 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 50625 \, A^{4} c^{2}}{a^{13} c}} + 210 \, A B}{a^{6}}}\right ) - 3 \, {\left (a^{3} c^{2} x^{5} + 2 \, a^{4} c x^{3} + a^{5} x\right )} \sqrt {\frac {a^{6} \sqrt {-\frac {2401 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 50625 \, A^{4} c^{2}}{a^{13} c}} + 210 \, A B}{a^{6}}} \log \left (-9 \, {\left (2401 \, B^{4} a^{2} - 50625 \, A^{4} c^{2}\right )} \sqrt {x} - 9 \, {\left (15 \, A a^{10} c \sqrt {-\frac {2401 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 50625 \, A^{4} c^{2}}{a^{13} c}} + 343 \, B^{3} a^{5} - 1575 \, A^{2} B a^{4} c\right )} \sqrt {\frac {a^{6} \sqrt {-\frac {2401 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 50625 \, A^{4} c^{2}}{a^{13} c}} + 210 \, A B}{a^{6}}}\right ) - 3 \, {\left (a^{3} c^{2} x^{5} + 2 \, a^{4} c x^{3} + a^{5} x\right )} \sqrt {-\frac {a^{6} \sqrt {-\frac {2401 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 50625 \, A^{4} c^{2}}{a^{13} c}} - 210 \, A B}{a^{6}}} \log \left (-9 \, {\left (2401 \, B^{4} a^{2} - 50625 \, A^{4} c^{2}\right )} \sqrt {x} + 9 \, {\left (15 \, A a^{10} c \sqrt {-\frac {2401 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 50625 \, A^{4} c^{2}}{a^{13} c}} - 343 \, B^{3} a^{5} + 1575 \, A^{2} B a^{4} c\right )} \sqrt {-\frac {a^{6} \sqrt {-\frac {2401 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 50625 \, A^{4} c^{2}}{a^{13} c}} - 210 \, A B}{a^{6}}}\right ) + 3 \, {\left (a^{3} c^{2} x^{5} + 2 \, a^{4} c x^{3} + a^{5} x\right )} \sqrt {-\frac {a^{6} \sqrt {-\frac {2401 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 50625 \, A^{4} c^{2}}{a^{13} c}} - 210 \, A B}{a^{6}}} \log \left (-9 \, {\left (2401 \, B^{4} a^{2} - 50625 \, A^{4} c^{2}\right )} \sqrt {x} - 9 \, {\left (15 \, A a^{10} c \sqrt {-\frac {2401 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 50625 \, A^{4} c^{2}}{a^{13} c}} - 343 \, B^{3} a^{5} + 1575 \, A^{2} B a^{4} c\right )} \sqrt {-\frac {a^{6} \sqrt {-\frac {2401 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 50625 \, A^{4} c^{2}}{a^{13} c}} - 210 \, A B}{a^{6}}}\right ) + 4 \, {\left (45 \, A c^{2} x^{4} - 7 \, B a c x^{3} + 81 \, A a c x^{2} - 11 \, B a^{2} x + 32 \, A a^{2}\right )} \sqrt {x}}{64 \, {\left (a^{3} c^{2} x^{5} + 2 \, a^{4} c x^{3} + a^{5} x\right )}} \]
-1/64*(3*(a^3*c^2*x^5 + 2*a^4*c*x^3 + a^5*x)*sqrt((a^6*sqrt(-(2401*B^4*a^2 - 22050*A^2*B^2*a*c + 50625*A^4*c^2)/(a^13*c)) + 210*A*B)/a^6)*log(-9*(24 01*B^4*a^2 - 50625*A^4*c^2)*sqrt(x) + 9*(15*A*a^10*c*sqrt(-(2401*B^4*a^2 - 22050*A^2*B^2*a*c + 50625*A^4*c^2)/(a^13*c)) + 343*B^3*a^5 - 1575*A^2*B*a ^4*c)*sqrt((a^6*sqrt(-(2401*B^4*a^2 - 22050*A^2*B^2*a*c + 50625*A^4*c^2)/( a^13*c)) + 210*A*B)/a^6)) - 3*(a^3*c^2*x^5 + 2*a^4*c*x^3 + a^5*x)*sqrt((a^ 6*sqrt(-(2401*B^4*a^2 - 22050*A^2*B^2*a*c + 50625*A^4*c^2)/(a^13*c)) + 210 *A*B)/a^6)*log(-9*(2401*B^4*a^2 - 50625*A^4*c^2)*sqrt(x) - 9*(15*A*a^10*c* sqrt(-(2401*B^4*a^2 - 22050*A^2*B^2*a*c + 50625*A^4*c^2)/(a^13*c)) + 343*B ^3*a^5 - 1575*A^2*B*a^4*c)*sqrt((a^6*sqrt(-(2401*B^4*a^2 - 22050*A^2*B^2*a *c + 50625*A^4*c^2)/(a^13*c)) + 210*A*B)/a^6)) - 3*(a^3*c^2*x^5 + 2*a^4*c* x^3 + a^5*x)*sqrt(-(a^6*sqrt(-(2401*B^4*a^2 - 22050*A^2*B^2*a*c + 50625*A^ 4*c^2)/(a^13*c)) - 210*A*B)/a^6)*log(-9*(2401*B^4*a^2 - 50625*A^4*c^2)*sqr t(x) + 9*(15*A*a^10*c*sqrt(-(2401*B^4*a^2 - 22050*A^2*B^2*a*c + 50625*A^4* c^2)/(a^13*c)) - 343*B^3*a^5 + 1575*A^2*B*a^4*c)*sqrt(-(a^6*sqrt(-(2401*B^ 4*a^2 - 22050*A^2*B^2*a*c + 50625*A^4*c^2)/(a^13*c)) - 210*A*B)/a^6)) + 3* (a^3*c^2*x^5 + 2*a^4*c*x^3 + a^5*x)*sqrt(-(a^6*sqrt(-(2401*B^4*a^2 - 22050 *A^2*B^2*a*c + 50625*A^4*c^2)/(a^13*c)) - 210*A*B)/a^6)*log(-9*(2401*B^4*a ^2 - 50625*A^4*c^2)*sqrt(x) - 9*(15*A*a^10*c*sqrt(-(2401*B^4*a^2 - 22050*A ^2*B^2*a*c + 50625*A^4*c^2)/(a^13*c)) - 343*B^3*a^5 + 1575*A^2*B*a^4*c)...
Timed out. \[ \int \frac {A+B x}{x^{3/2} \left (a+c x^2\right )^3} \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 313, normalized size of antiderivative = 0.94 \[ \int \frac {A+B x}{x^{3/2} \left (a+c x^2\right )^3} \, dx=-\frac {45 \, A c^{2} x^{4} - 7 \, B a c x^{3} + 81 \, A a c x^{2} - 11 \, B a^{2} x + 32 \, A a^{2}}{16 \, {\left (a^{3} c^{2} x^{\frac {9}{2}} + 2 \, a^{4} c x^{\frac {5}{2}} + a^{5} \sqrt {x}\right )}} + \frac {3 \, {\left (\frac {2 \, \sqrt {2} {\left (7 \, B a \sqrt {c} - 15 \, A \sqrt {a} c\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} + 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} {\left (7 \, B a \sqrt {c} - 15 \, A \sqrt {a} c\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} - 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {\sqrt {2} {\left (7 \, B a \sqrt {c} + 15 \, A \sqrt {a} c\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (7 \, B a \sqrt {c} + 15 \, A \sqrt {a} c\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}}\right )}}{128 \, a^{3}} \]
-1/16*(45*A*c^2*x^4 - 7*B*a*c*x^3 + 81*A*a*c*x^2 - 11*B*a^2*x + 32*A*a^2)/ (a^3*c^2*x^(9/2) + 2*a^4*c*x^(5/2) + a^5*sqrt(x)) + 3/128*(2*sqrt(2)*(7*B* a*sqrt(c) - 15*A*sqrt(a)*c)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*c^(1/4) + 2*sqrt(c)*sqrt(x))/sqrt(sqrt(a)*sqrt(c)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(c))*s qrt(c)) + 2*sqrt(2)*(7*B*a*sqrt(c) - 15*A*sqrt(a)*c)*arctan(-1/2*sqrt(2)*( sqrt(2)*a^(1/4)*c^(1/4) - 2*sqrt(c)*sqrt(x))/sqrt(sqrt(a)*sqrt(c)))/(sqrt( a)*sqrt(sqrt(a)*sqrt(c))*sqrt(c)) + sqrt(2)*(7*B*a*sqrt(c) + 15*A*sqrt(a)* c)*log(sqrt(2)*a^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x + sqrt(a))/(a^(3/4)*c^( 3/4)) - sqrt(2)*(7*B*a*sqrt(c) + 15*A*sqrt(a)*c)*log(-sqrt(2)*a^(1/4)*c^(1 /4)*sqrt(x) + sqrt(c)*x + sqrt(a))/(a^(3/4)*c^(3/4)))/a^3
Time = 0.28 (sec) , antiderivative size = 304, normalized size of antiderivative = 0.91 \[ \int \frac {A+B x}{x^{3/2} \left (a+c x^2\right )^3} \, dx=-\frac {2 \, A}{a^{3} \sqrt {x}} + \frac {3 \, \sqrt {2} {\left (7 \, \left (a c^{3}\right )^{\frac {1}{4}} B a c - 15 \, \left (a c^{3}\right )^{\frac {3}{4}} A\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{64 \, a^{4} c^{2}} + \frac {3 \, \sqrt {2} {\left (7 \, \left (a c^{3}\right )^{\frac {1}{4}} B a c - 15 \, \left (a c^{3}\right )^{\frac {3}{4}} A\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{64 \, a^{4} c^{2}} + \frac {3 \, \sqrt {2} {\left (7 \, \left (a c^{3}\right )^{\frac {1}{4}} B a c + 15 \, \left (a c^{3}\right )^{\frac {3}{4}} A\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{c}}\right )}{128 \, a^{4} c^{2}} - \frac {3 \, \sqrt {2} {\left (7 \, \left (a c^{3}\right )^{\frac {1}{4}} B a c + 15 \, \left (a c^{3}\right )^{\frac {3}{4}} A\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{c}}\right )}{128 \, a^{4} c^{2}} - \frac {13 \, A c^{2} x^{\frac {7}{2}} - 7 \, B a c x^{\frac {5}{2}} + 17 \, A a c x^{\frac {3}{2}} - 11 \, B a^{2} \sqrt {x}}{16 \, {\left (c x^{2} + a\right )}^{2} a^{3}} \]
-2*A/(a^3*sqrt(x)) + 3/64*sqrt(2)*(7*(a*c^3)^(1/4)*B*a*c - 15*(a*c^3)^(3/4 )*A)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/c)^(1/4) + 2*sqrt(x))/(a/c)^(1/4))/(a^ 4*c^2) + 3/64*sqrt(2)*(7*(a*c^3)^(1/4)*B*a*c - 15*(a*c^3)^(3/4)*A)*arctan( -1/2*sqrt(2)*(sqrt(2)*(a/c)^(1/4) - 2*sqrt(x))/(a/c)^(1/4))/(a^4*c^2) + 3/ 128*sqrt(2)*(7*(a*c^3)^(1/4)*B*a*c + 15*(a*c^3)^(3/4)*A)*log(sqrt(2)*sqrt( x)*(a/c)^(1/4) + x + sqrt(a/c))/(a^4*c^2) - 3/128*sqrt(2)*(7*(a*c^3)^(1/4) *B*a*c + 15*(a*c^3)^(3/4)*A)*log(-sqrt(2)*sqrt(x)*(a/c)^(1/4) + x + sqrt(a /c))/(a^4*c^2) - 1/16*(13*A*c^2*x^(7/2) - 7*B*a*c*x^(5/2) + 17*A*a*c*x^(3/ 2) - 11*B*a^2*sqrt(x))/((c*x^2 + a)^2*a^3)
Time = 0.29 (sec) , antiderivative size = 673, normalized size of antiderivative = 2.02 \[ \int \frac {A+B x}{x^{3/2} \left (a+c x^2\right )^3} \, dx=2\,\mathrm {atanh}\left (\frac {66355200\,A^2\,a^{10}\,c^4\,\sqrt {x}\,\sqrt {\frac {945\,A\,B}{2048\,a^6}-\frac {2025\,A^2\,\sqrt {-a^{13}\,c}}{4096\,a^{13}}+\frac {441\,B^2\,\sqrt {-a^{13}\,c}}{4096\,a^{12}\,c}}}{46656000\,A^3\,a^7\,c^4-4741632\,B^3\,a^2\,c^2\,\sqrt {-a^{13}\,c}-10160640\,A\,B^2\,a^8\,c^3+21772800\,A^2\,B\,a\,c^3\,\sqrt {-a^{13}\,c}}-\frac {14450688\,B^2\,a^{11}\,c^3\,\sqrt {x}\,\sqrt {\frac {945\,A\,B}{2048\,a^6}-\frac {2025\,A^2\,\sqrt {-a^{13}\,c}}{4096\,a^{13}}+\frac {441\,B^2\,\sqrt {-a^{13}\,c}}{4096\,a^{12}\,c}}}{46656000\,A^3\,a^7\,c^4-4741632\,B^3\,a^2\,c^2\,\sqrt {-a^{13}\,c}-10160640\,A\,B^2\,a^8\,c^3+21772800\,A^2\,B\,a\,c^3\,\sqrt {-a^{13}\,c}}\right )\,\sqrt {\frac {9\,\left (49\,B^2\,a\,\sqrt {-a^{13}\,c}-225\,A^2\,c\,\sqrt {-a^{13}\,c}+210\,A\,B\,a^7\,c\right )}{4096\,a^{13}\,c}}+2\,\mathrm {atanh}\left (\frac {66355200\,A^2\,a^{10}\,c^4\,\sqrt {x}\,\sqrt {\frac {2025\,A^2\,\sqrt {-a^{13}\,c}}{4096\,a^{13}}+\frac {945\,A\,B}{2048\,a^6}-\frac {441\,B^2\,\sqrt {-a^{13}\,c}}{4096\,a^{12}\,c}}}{46656000\,A^3\,a^7\,c^4+4741632\,B^3\,a^2\,c^2\,\sqrt {-a^{13}\,c}-10160640\,A\,B^2\,a^8\,c^3-21772800\,A^2\,B\,a\,c^3\,\sqrt {-a^{13}\,c}}-\frac {14450688\,B^2\,a^{11}\,c^3\,\sqrt {x}\,\sqrt {\frac {2025\,A^2\,\sqrt {-a^{13}\,c}}{4096\,a^{13}}+\frac {945\,A\,B}{2048\,a^6}-\frac {441\,B^2\,\sqrt {-a^{13}\,c}}{4096\,a^{12}\,c}}}{46656000\,A^3\,a^7\,c^4+4741632\,B^3\,a^2\,c^2\,\sqrt {-a^{13}\,c}-10160640\,A\,B^2\,a^8\,c^3-21772800\,A^2\,B\,a\,c^3\,\sqrt {-a^{13}\,c}}\right )\,\sqrt {\frac {9\,\left (225\,A^2\,c\,\sqrt {-a^{13}\,c}-49\,B^2\,a\,\sqrt {-a^{13}\,c}+210\,A\,B\,a^7\,c\right )}{4096\,a^{13}\,c}}-\frac {\frac {2\,A}{a}-\frac {11\,B\,x}{16\,a}+\frac {81\,A\,c\,x^2}{16\,a^2}-\frac {7\,B\,c\,x^3}{16\,a^2}+\frac {45\,A\,c^2\,x^4}{16\,a^3}}{a^2\,\sqrt {x}+c^2\,x^{9/2}+2\,a\,c\,x^{5/2}} \]
2*atanh((66355200*A^2*a^10*c^4*x^(1/2)*((945*A*B)/(2048*a^6) - (2025*A^2*( -a^13*c)^(1/2))/(4096*a^13) + (441*B^2*(-a^13*c)^(1/2))/(4096*a^12*c))^(1/ 2))/(46656000*A^3*a^7*c^4 - 4741632*B^3*a^2*c^2*(-a^13*c)^(1/2) - 10160640 *A*B^2*a^8*c^3 + 21772800*A^2*B*a*c^3*(-a^13*c)^(1/2)) - (14450688*B^2*a^1 1*c^3*x^(1/2)*((945*A*B)/(2048*a^6) - (2025*A^2*(-a^13*c)^(1/2))/(4096*a^1 3) + (441*B^2*(-a^13*c)^(1/2))/(4096*a^12*c))^(1/2))/(46656000*A^3*a^7*c^4 - 4741632*B^3*a^2*c^2*(-a^13*c)^(1/2) - 10160640*A*B^2*a^8*c^3 + 21772800 *A^2*B*a*c^3*(-a^13*c)^(1/2)))*((9*(49*B^2*a*(-a^13*c)^(1/2) - 225*A^2*c*( -a^13*c)^(1/2) + 210*A*B*a^7*c))/(4096*a^13*c))^(1/2) + 2*atanh((66355200* A^2*a^10*c^4*x^(1/2)*((2025*A^2*(-a^13*c)^(1/2))/(4096*a^13) + (945*A*B)/( 2048*a^6) - (441*B^2*(-a^13*c)^(1/2))/(4096*a^12*c))^(1/2))/(46656000*A^3* a^7*c^4 + 4741632*B^3*a^2*c^2*(-a^13*c)^(1/2) - 10160640*A*B^2*a^8*c^3 - 2 1772800*A^2*B*a*c^3*(-a^13*c)^(1/2)) - (14450688*B^2*a^11*c^3*x^(1/2)*((20 25*A^2*(-a^13*c)^(1/2))/(4096*a^13) + (945*A*B)/(2048*a^6) - (441*B^2*(-a^ 13*c)^(1/2))/(4096*a^12*c))^(1/2))/(46656000*A^3*a^7*c^4 + 4741632*B^3*a^2 *c^2*(-a^13*c)^(1/2) - 10160640*A*B^2*a^8*c^3 - 21772800*A^2*B*a*c^3*(-a^1 3*c)^(1/2)))*((9*(225*A^2*c*(-a^13*c)^(1/2) - 49*B^2*a*(-a^13*c)^(1/2) + 2 10*A*B*a^7*c))/(4096*a^13*c))^(1/2) - ((2*A)/a - (11*B*x)/(16*a) + (81*A*c *x^2)/(16*a^2) - (7*B*c*x^3)/(16*a^2) + (45*A*c^2*x^4)/(16*a^3))/(a^2*x^(1 /2) + c^2*x^(9/2) + 2*a*c*x^(5/2))