Integrand size = 32, antiderivative size = 350 \[ \int \frac {A+B x+C x^2+D x^3}{(c x)^{3/2} \left (a+b x^2\right )^2} \, dx=-\frac {2 A}{a c \sqrt {c x} \left (a+b x^2\right )}+\frac {\sqrt {c x} (a (b B-a D)-b (5 A b-a C) x)}{2 a^2 b c^2 \left (a+b x^2\right )}+\frac {\left (\sqrt {b} (5 A b-a C)-\sqrt {a} (3 b B+a D)\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )}{4 \sqrt {2} a^{9/4} b^{5/4} c^{3/2}}-\frac {\left (\sqrt {b} (5 A b-a C)-\sqrt {a} (3 b B+a D)\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )}{4 \sqrt {2} a^{9/4} b^{5/4} c^{3/2}}+\frac {\left (\sqrt {b} (5 A b-a C)+\sqrt {a} (3 b B+a D)\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{4 \sqrt {2} a^{9/4} b^{5/4} c^{3/2}} \] Output:
-2*A/a/c/(c*x)^(1/2)/(b*x^2+a)+1/2*(c*x)^(1/2)*(a*(B*b-D*a)-b*(5*A*b-C*a)* x)/a^2/b/c^2/(b*x^2+a)+1/8*(b^(1/2)*(5*A*b-C*a)-a^(1/2)*(3*B*b+D*a))*arcta n(1-2^(1/2)*b^(1/4)*(c*x)^(1/2)/a^(1/4)/c^(1/2))*2^(1/2)/a^(9/4)/b^(5/4)/c ^(3/2)-1/8*(b^(1/2)*(5*A*b-C*a)-a^(1/2)*(3*B*b+D*a))*arctan(1+2^(1/2)*b^(1 /4)*(c*x)^(1/2)/a^(1/4)/c^(1/2))*2^(1/2)/a^(9/4)/b^(5/4)/c^(3/2)+1/8*(b^(1 /2)*(5*A*b-C*a)+a^(1/2)*(3*B*b+D*a))*arctanh(2^(1/2)*a^(1/4)*b^(1/4)*(c*x) ^(1/2)/c^(1/2)/(a^(1/2)+b^(1/2)*x))*2^(1/2)/a^(9/4)/b^(5/4)/c^(3/2)
Time = 1.42 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.68 \[ \int \frac {A+B x+C x^2+D x^3}{(c x)^{3/2} \left (a+b x^2\right )^2} \, dx=\frac {x \left (-\frac {4 \sqrt [4]{a} \sqrt [4]{b} \left (a^2 D x+5 A b^2 x^2+a b (4 A-x (B+C x))\right )}{a+b x^2}-\sqrt {2} \left (-5 A b^{3/2}+3 \sqrt {a} b B+a \sqrt {b} C+a^{3/2} D\right ) \sqrt {x} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+\sqrt {2} \left (5 A b^{3/2}+3 \sqrt {a} b B-a \sqrt {b} C+a^{3/2} D\right ) \sqrt {x} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{8 a^{9/4} b^{5/4} (c x)^{3/2}} \] Input:
Integrate[(A + B*x + C*x^2 + D*x^3)/((c*x)^(3/2)*(a + b*x^2)^2),x]
Output:
(x*((-4*a^(1/4)*b^(1/4)*(a^2*D*x + 5*A*b^2*x^2 + a*b*(4*A - x*(B + C*x)))) /(a + b*x^2) - Sqrt[2]*(-5*A*b^(3/2) + 3*Sqrt[a]*b*B + a*Sqrt[b]*C + a^(3/ 2)*D)*Sqrt[x]*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x ])] + Sqrt[2]*(5*A*b^(3/2) + 3*Sqrt[a]*b*B - a*Sqrt[b]*C + a^(3/2)*D)*Sqrt [x]*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]))/(8* a^(9/4)*b^(5/4)*(c*x)^(3/2))
Time = 0.83 (sec) , antiderivative size = 413, normalized size of antiderivative = 1.18, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {2337, 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.
| \(\displaystyle \int \frac {A+B x+C x^2+D x^3}{(c x)^{3/2} \left (a+b x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 2337 |
| \(\displaystyle \frac {x \left (B-\frac {a D}{b}\right )-\frac {a C}{b}+A}{2 a c \sqrt {c x} \left (a+b x^2\right )}-\frac {\int -\frac {b \left (5 A-\frac {a C}{b}\right )+(3 b B+a D) x}{2 b (c x)^{3/2} \left (b x^2+a\right )}dx}{2 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {5 A b-a C+(3 b B+a D) x}{(c x)^{3/2} \left (b x^2+a\right )}dx}{4 a b}+\frac {x \left (B-\frac {a D}{b}\right )-\frac {a C}{b}+A}{2 a c \sqrt {c x} \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 553 |
| \(\displaystyle \frac {-\frac {2 \int -\frac {a (3 b B+a D)-b (5 A b-a C) x}{2 \sqrt {c x} \left (b x^2+a\right )}dx}{a c}-\frac {2 (5 A b-a C)}{a c \sqrt {c x}}}{4 a b}+\frac {x \left (B-\frac {a D}{b}\right )-\frac {a C}{b}+A}{2 a c \sqrt {c x} \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {a (3 b B+a D)-b (5 A b-a C) x}{\sqrt {c x} \left (b x^2+a\right )}dx}{a c}-\frac {2 (5 A b-a C)}{a c \sqrt {c x}}}{4 a b}+\frac {x \left (B-\frac {a D}{b}\right )-\frac {a C}{b}+A}{2 a c \sqrt {c x} \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 554 |
| \(\displaystyle \frac {\frac {2 \int \frac {a c (3 b B+a D)-b c (5 A b-a C) x}{b x^2 c^2+a c^2}d\sqrt {c x}}{a c}-\frac {2 (5 A b-a C)}{a c \sqrt {c x}}}{4 a b}+\frac {x \left (B-\frac {a D}{b}\right )-\frac {a C}{b}+A}{2 a c \sqrt {c x} \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {\frac {2 \left (\frac {1}{2} \left (\frac {\sqrt {a} (a D+3 b B)}{\sqrt {b}}-a C+5 A b\right ) \int \frac {\sqrt {b} \left (\sqrt {a} c-\sqrt {b} c x\right )}{b x^2 c^2+a c^2}d\sqrt {c x}-\frac {1}{2} \left (-\frac {\sqrt {a} (a D+3 b B)}{\sqrt {b}}-a C+5 A b\right ) \int \frac {\sqrt {b} \left (\sqrt {b} x c+\sqrt {a} c\right )}{b x^2 c^2+a c^2}d\sqrt {c x}\right )}{a c}-\frac {2 (5 A b-a C)}{a c \sqrt {c x}}}{4 a b}+\frac {x \left (B-\frac {a D}{b}\right )-\frac {a C}{b}+A}{2 a c \sqrt {c x} \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 \left (\frac {1}{2} \sqrt {b} \left (\frac {\sqrt {a} (a D+3 b B)}{\sqrt {b}}-a C+5 A b\right ) \int \frac {\sqrt {a} c-\sqrt {b} c x}{b x^2 c^2+a c^2}d\sqrt {c x}-\frac {1}{2} \sqrt {b} \left (-\frac {\sqrt {a} (a D+3 b B)}{\sqrt {b}}-a C+5 A b\right ) \int \frac {\sqrt {b} x c+\sqrt {a} c}{b x^2 c^2+a c^2}d\sqrt {c x}\right )}{a c}-\frac {2 (5 A b-a C)}{a c \sqrt {c x}}}{4 a b}+\frac {x \left (B-\frac {a D}{b}\right )-\frac {a C}{b}+A}{2 a c \sqrt {c x} \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\frac {2 \left (\frac {1}{2} \sqrt {b} \left (\frac {\sqrt {a} (a D+3 b B)}{\sqrt {b}}-a C+5 A b\right ) \int \frac {\sqrt {a} c-\sqrt {b} c x}{b x^2 c^2+a c^2}d\sqrt {c x}-\frac {1}{2} \sqrt {b} \left (-\frac {\sqrt {a} (a D+3 b B)}{\sqrt {b}}-a C+5 A b\right ) \left (\frac {\int \frac {1}{x c+\frac {\sqrt {a} c}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c x} \sqrt {c}}{\sqrt [4]{b}}}d\sqrt {c x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x c+\frac {\sqrt {a} c}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c x} \sqrt {c}}{\sqrt [4]{b}}}d\sqrt {c x}}{2 \sqrt {b}}\right )\right )}{a c}-\frac {2 (5 A b-a C)}{a c \sqrt {c x}}}{4 a b}+\frac {x \left (B-\frac {a D}{b}\right )-\frac {a C}{b}+A}{2 a c \sqrt {c x} \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {2 \left (\frac {1}{2} \sqrt {b} \left (\frac {\sqrt {a} (a D+3 b B)}{\sqrt {b}}-a C+5 A b\right ) \int \frac {\sqrt {a} c-\sqrt {b} c x}{b x^2 c^2+a c^2}d\sqrt {c x}-\frac {1}{2} \sqrt {b} \left (-\frac {\sqrt {a} (a D+3 b B)}{\sqrt {b}}-a C+5 A b\right ) \left (\frac {\int \frac {1}{-c x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}}-\frac {\int \frac {1}{-c x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}}\right )\right )}{a c}-\frac {2 (5 A b-a C)}{a c \sqrt {c x}}}{4 a b}+\frac {x \left (B-\frac {a D}{b}\right )-\frac {a C}{b}+A}{2 a c \sqrt {c x} \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {2 \left (\frac {1}{2} \sqrt {b} \left (\frac {\sqrt {a} (a D+3 b B)}{\sqrt {b}}-a C+5 A b\right ) \int \frac {\sqrt {a} c-\sqrt {b} c x}{b x^2 c^2+a c^2}d\sqrt {c x}-\frac {1}{2} \sqrt {b} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}}\right ) \left (-\frac {\sqrt {a} (a D+3 b B)}{\sqrt {b}}-a C+5 A b\right )\right )}{a c}-\frac {2 (5 A b-a C)}{a c \sqrt {c x}}}{4 a b}+\frac {x \left (B-\frac {a D}{b}\right )-\frac {a C}{b}+A}{2 a c \sqrt {c x} \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\frac {2 \left (\frac {1}{2} \sqrt {b} \left (\frac {\sqrt {a} (a D+3 b B)}{\sqrt {b}}-a C+5 A b\right ) \left (-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c}-2 \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{b} \left (x c+\frac {\sqrt {a} c}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c x} \sqrt {c}}{\sqrt [4]{b}}\right )}d\sqrt {c x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {c}+\sqrt {2} \sqrt [4]{b} \sqrt {c x}\right )}{\sqrt [4]{b} \left (x c+\frac {\sqrt {a} c}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c x} \sqrt {c}}{\sqrt [4]{b}}\right )}d\sqrt {c x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}}\right )-\frac {1}{2} \sqrt {b} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}}\right ) \left (-\frac {\sqrt {a} (a D+3 b B)}{\sqrt {b}}-a C+5 A b\right )\right )}{a c}-\frac {2 (5 A b-a C)}{a c \sqrt {c x}}}{4 a b}+\frac {x \left (B-\frac {a D}{b}\right )-\frac {a C}{b}+A}{2 a c \sqrt {c x} \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 \left (\frac {1}{2} \sqrt {b} \left (\frac {\sqrt {a} (a D+3 b B)}{\sqrt {b}}-a C+5 A b\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {c}-2 \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{b} \left (x c+\frac {\sqrt {a} c}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c x} \sqrt {c}}{\sqrt [4]{b}}\right )}d\sqrt {c x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}}+\frac {\int \frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {c}+\sqrt {2} \sqrt [4]{b} \sqrt {c x}\right )}{\sqrt [4]{b} \left (x c+\frac {\sqrt {a} c}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c x} \sqrt {c}}{\sqrt [4]{b}}\right )}d\sqrt {c x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}}\right )-\frac {1}{2} \sqrt {b} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}}\right ) \left (-\frac {\sqrt {a} (a D+3 b B)}{\sqrt {b}}-a C+5 A b\right )\right )}{a c}-\frac {2 (5 A b-a C)}{a c \sqrt {c x}}}{4 a b}+\frac {x \left (B-\frac {a D}{b}\right )-\frac {a C}{b}+A}{2 a c \sqrt {c x} \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 \left (\frac {1}{2} \sqrt {b} \left (\frac {\sqrt {a} (a D+3 b B)}{\sqrt {b}}-a C+5 A b\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {c}-2 \sqrt [4]{b} \sqrt {c x}}{x c+\frac {\sqrt {a} c}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c x} \sqrt {c}}{\sqrt [4]{b}}}d\sqrt {c x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b} \sqrt {c}}+\frac {\int \frac {\sqrt [4]{a} \sqrt {c}+\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{x c+\frac {\sqrt {a} c}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c x} \sqrt {c}}{\sqrt [4]{b}}}d\sqrt {c x}}{2 \sqrt [4]{a} \sqrt {b} \sqrt {c}}\right )-\frac {1}{2} \sqrt {b} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}}\right ) \left (-\frac {\sqrt {a} (a D+3 b B)}{\sqrt {b}}-a C+5 A b\right )\right )}{a c}-\frac {2 (5 A b-a C)}{a c \sqrt {c x}}}{4 a b}+\frac {x \left (B-\frac {a D}{b}\right )-\frac {a C}{b}+A}{2 a c \sqrt {c x} \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {2 \left (\frac {1}{2} \sqrt {b} \left (\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {c} \sqrt {c x}+\sqrt {a} c+\sqrt {b} c x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {c} \sqrt {c x}+\sqrt {a} c+\sqrt {b} c x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}}\right ) \left (\frac {\sqrt {a} (a D+3 b B)}{\sqrt {b}}-a C+5 A b\right )-\frac {1}{2} \sqrt {b} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}}\right ) \left (-\frac {\sqrt {a} (a D+3 b B)}{\sqrt {b}}-a C+5 A b\right )\right )}{a c}-\frac {2 (5 A b-a C)}{a c \sqrt {c x}}}{4 a b}+\frac {x \left (B-\frac {a D}{b}\right )-\frac {a C}{b}+A}{2 a c \sqrt {c x} \left (a+b x^2\right )}\) |
Input:
Int[(A + B*x + C*x^2 + D*x^3)/((c*x)^(3/2)*(a + b*x^2)^2),x]
Output:
(A - (a*C)/b + (B - (a*D)/b)*x)/(2*a*c*Sqrt[c*x]*(a + b*x^2)) + ((-2*(5*A* b - a*C))/(a*c*Sqrt[c*x]) + (2*(-1/2*(Sqrt[b]*(5*A*b - a*C - (Sqrt[a]*(3*b *B + a*D))/Sqrt[b])*(-(ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[c*x])/(a^(1/4)*Sqr t[c])]/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[c])) + ArcTan[1 + (Sqrt[2]*b^(1/4)*Sq rt[c*x])/(a^(1/4)*Sqrt[c])]/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[c]))) + (Sqrt[b] *(5*A*b - a*C + (Sqrt[a]*(3*b*B + a*D))/Sqrt[b])*(-1/2*Log[Sqrt[a]*c + Sqr t[b]*c*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[c]*Sqrt[c*x]]/(Sqrt[2]*a^(1/4)*b^( 1/4)*Sqrt[c]) + Log[Sqrt[a]*c + Sqrt[b]*c*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt [c]*Sqrt[c*x]]/(2*Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[c])))/2))/(a*c))/(4*a*b)
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[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]
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ {Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq , a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(-(c*x)^(m + 1))*(f + g*x)*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))) , x] + Simp[1/(2*a*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1)*ExpandToSum[2 *a*(p + 1)*Q + f*(m + 2*p + 3) + g*(m + 2*p + 4)*x, x], x], x]] /; FreeQ[{a , b, c, m}, x] && PolyQ[Pq, x] && LtQ[p, -1] && !GtQ[m, 0]
Time = 0.74 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.11
| method | result | size |
| derivativedivides | \(-\frac {2 \left (\frac {\left (\frac {A b}{4}-\frac {C a}{4}\right ) \left (c x \right )^{\frac {3}{2}}-\frac {c a \left (B b -D a \right ) \sqrt {c x}}{4 b}}{b \,c^{2} x^{2}+a \,c^{2}}+\frac {\frac {\left (-3 a b B c -D a^{2} c \right ) \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {c x +\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}{c x -\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c x}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c x}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a \,c^{2}}+\frac {\left (5 b^{2} A -C a b \right ) \sqrt {2}\, \left (\ln \left (\frac {c x -\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}{c x +\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c x}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c x}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}}{4 b}\right )}{c \,a^{2}}-\frac {2 A}{c \,a^{2} \sqrt {c x}}\) | \(389\) |
| default | \(-\frac {2 \left (\frac {\left (\frac {A b}{4}-\frac {C a}{4}\right ) \left (c x \right )^{\frac {3}{2}}-\frac {c a \left (B b -D a \right ) \sqrt {c x}}{4 b}}{b \,c^{2} x^{2}+a \,c^{2}}+\frac {\frac {\left (-3 a b B c -D a^{2} c \right ) \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {c x +\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}{c x -\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c x}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c x}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a \,c^{2}}+\frac {\left (5 b^{2} A -C a b \right ) \sqrt {2}\, \left (\ln \left (\frac {c x -\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}{c x +\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c x}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c x}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}}{4 b}\right )}{c \,a^{2}}-\frac {2 A}{c \,a^{2} \sqrt {c x}}\) | \(389\) |
| pseudoelliptic | \(-\frac {5 \left (-\frac {3 \left (b \,x^{2}+a \right ) \sqrt {2}\, \left (B b +\frac {D a}{3}\right ) \left (\arctan \left (\frac {\sqrt {2}\, \sqrt {c x}-\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}\right )+\arctan \left (\frac {\sqrt {2}\, \sqrt {c x}+\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}\right )+\frac {\ln \left (\frac {c x +\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}{c x -\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}\right )}{2}\right ) \sqrt {c x}\, \sqrt {\frac {a \,c^{2}}{b}}}{5}+\left (\left (b \,x^{2}+a \right ) \sqrt {2}\, \left (\arctan \left (\frac {\sqrt {2}\, \sqrt {c x}-\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}\right )+\arctan \left (\frac {\sqrt {2}\, \sqrt {c x}+\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}\right )+\frac {\ln \left (\frac {c x -\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}{c x +\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}\right )}{2}\right ) \left (A b -\frac {C a}{5}\right ) \sqrt {c x}+\frac {16 \left (\frac {D a^{2} x}{4}+b \left (-\frac {1}{4} C \,x^{2}-\frac {1}{4} B x +A \right ) a +\frac {5 A \,b^{2} x^{2}}{4}\right ) \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}{5}\right ) c \right )}{8 \sqrt {c x}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} c^{2} \left (b \,x^{2}+a \right ) a^{2} b}\) | \(423\) |
Input:
int((D*x^3+C*x^2+B*x+A)/(c*x)^(3/2)/(b*x^2+a)^2,x,method=_RETURNVERBOSE)
Output:
-2/c/a^2*(((1/4*A*b-1/4*C*a)*(c*x)^(3/2)-1/4*c*a*(B*b-D*a)/b*(c*x)^(1/2))/ (b*c^2*x^2+a*c^2)+1/4/b*(1/8*(-3*B*a*b*c-D*a^2*c)*(a*c^2/b)^(1/4)/a/c^2*2^ (1/2)*(ln((c*x+(a*c^2/b)^(1/4)*(c*x)^(1/2)*2^(1/2)+(a*c^2/b)^(1/2))/(c*x-( a*c^2/b)^(1/4)*(c*x)^(1/2)*2^(1/2)+(a*c^2/b)^(1/2)))+2*arctan(2^(1/2)/(a*c ^2/b)^(1/4)*(c*x)^(1/2)+1)+2*arctan(2^(1/2)/(a*c^2/b)^(1/4)*(c*x)^(1/2)-1) )+1/8*(5*A*b^2-C*a*b)/b/(a*c^2/b)^(1/4)*2^(1/2)*(ln((c*x-(a*c^2/b)^(1/4)*( c*x)^(1/2)*2^(1/2)+(a*c^2/b)^(1/2))/(c*x+(a*c^2/b)^(1/4)*(c*x)^(1/2)*2^(1/ 2)+(a*c^2/b)^(1/2)))+2*arctan(2^(1/2)/(a*c^2/b)^(1/4)*(c*x)^(1/2)+1)+2*arc tan(2^(1/2)/(a*c^2/b)^(1/4)*(c*x)^(1/2)-1))))-2*A/c/a^2/(c*x)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 3700 vs. \(2 (263) = 526\).
Time = 0.51 (sec) , antiderivative size = 3700, normalized size of antiderivative = 10.57 \[ \int \frac {A+B x+C x^2+D x^3}{(c x)^{3/2} \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:
integrate((D*x^3+C*x^2+B*x+A)/(c*x)^(3/2)/(b*x^2+a)^2,x, algorithm="fricas ")
Output:
Too large to include
Result contains complex when optimal does not.
Time = 82.54 (sec) , antiderivative size = 4014, normalized size of antiderivative = 11.47 \[ \int \frac {A+B x+C x^2+D x^3}{(c x)^{3/2} \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:
integrate((D*x**3+C*x**2+B*x+A)/(c*x)**(3/2)/(b*x**2+a)**2,x)
Output:
A*(16*a**(5/4)*exp(3*I*pi/4)*gamma(-1/4)/(32*a**(13/4)*c**(3/2)*sqrt(x)*ex p(3*I*pi/4)*gamma(3/4) + 32*a**(9/4)*b*c**(3/2)*x**(5/2)*exp(3*I*pi/4)*gam ma(3/4)) + 20*a**(1/4)*b*x**2*exp(3*I*pi/4)*gamma(-1/4)/(32*a**(13/4)*c**( 3/2)*sqrt(x)*exp(3*I*pi/4)*gamma(3/4) + 32*a**(9/4)*b*c**(3/2)*x**(5/2)*ex p(3*I*pi/4)*gamma(3/4)) - 5*a*b**(1/4)*sqrt(x)*log(1 - b**(1/4)*sqrt(x)*ex p_polar(I*pi/4)/a**(1/4))*gamma(-1/4)/(32*a**(13/4)*c**(3/2)*sqrt(x)*exp(3 *I*pi/4)*gamma(3/4) + 32*a**(9/4)*b*c**(3/2)*x**(5/2)*exp(3*I*pi/4)*gamma( 3/4)) - 5*I*a*b**(1/4)*sqrt(x)*log(1 - b**(1/4)*sqrt(x)*exp_polar(3*I*pi/4 )/a**(1/4))*gamma(-1/4)/(32*a**(13/4)*c**(3/2)*sqrt(x)*exp(3*I*pi/4)*gamma (3/4) + 32*a**(9/4)*b*c**(3/2)*x**(5/2)*exp(3*I*pi/4)*gamma(3/4)) + 5*a*b* *(1/4)*sqrt(x)*log(1 - b**(1/4)*sqrt(x)*exp_polar(5*I*pi/4)/a**(1/4))*gamm a(-1/4)/(32*a**(13/4)*c**(3/2)*sqrt(x)*exp(3*I*pi/4)*gamma(3/4) + 32*a**(9 /4)*b*c**(3/2)*x**(5/2)*exp(3*I*pi/4)*gamma(3/4)) + 5*I*a*b**(1/4)*sqrt(x) *log(1 - b**(1/4)*sqrt(x)*exp_polar(7*I*pi/4)/a**(1/4))*gamma(-1/4)/(32*a* *(13/4)*c**(3/2)*sqrt(x)*exp(3*I*pi/4)*gamma(3/4) + 32*a**(9/4)*b*c**(3/2) *x**(5/2)*exp(3*I*pi/4)*gamma(3/4)) - 5*b**(5/4)*x**(5/2)*log(1 - b**(1/4) *sqrt(x)*exp_polar(I*pi/4)/a**(1/4))*gamma(-1/4)/(32*a**(13/4)*c**(3/2)*sq rt(x)*exp(3*I*pi/4)*gamma(3/4) + 32*a**(9/4)*b*c**(3/2)*x**(5/2)*exp(3*I*p i/4)*gamma(3/4)) - 5*I*b**(5/4)*x**(5/2)*log(1 - b**(1/4)*sqrt(x)*exp_pola r(3*I*pi/4)/a**(1/4))*gamma(-1/4)/(32*a**(13/4)*c**(3/2)*sqrt(x)*exp(3*...
Time = 0.13 (sec) , antiderivative size = 454, normalized size of antiderivative = 1.30 \[ \int \frac {A+B x+C x^2+D x^3}{(c x)^{3/2} \left (a+b x^2\right )^2} \, dx=-\frac {\frac {8 \, {\left (4 \, A a b c^{2} - {\left (C a b - 5 \, A b^{2}\right )} c^{2} x^{2} + {\left (D a^{2} - B a b\right )} c^{2} x\right )}}{\left (c x\right )^{\frac {5}{2}} a^{2} b^{2} + \sqrt {c x} a^{3} b c^{2}} + \frac {\frac {\sqrt {2} {\left ({\left (C a b - 5 \, A b^{2}\right )} \sqrt {a} c - {\left (D a^{2} \sqrt {b} + 3 \, B a b^{\frac {3}{2}}\right )} c\right )} \log \left (\sqrt {b} c x + \sqrt {2} \left (a c^{2}\right )^{\frac {1}{4}} \sqrt {c x} b^{\frac {1}{4}} + \sqrt {a} c\right )}{\left (a c^{2}\right )^{\frac {3}{4}} b^{\frac {3}{4}}} - \frac {\sqrt {2} {\left ({\left (C a b - 5 \, A b^{2}\right )} \sqrt {a} c - {\left (D a^{2} \sqrt {b} + 3 \, B a b^{\frac {3}{2}}\right )} c\right )} \log \left (\sqrt {b} c x - \sqrt {2} \left (a c^{2}\right )^{\frac {1}{4}} \sqrt {c x} b^{\frac {1}{4}} + \sqrt {a} c\right )}{\left (a c^{2}\right )^{\frac {3}{4}} b^{\frac {3}{4}}} - \frac {2 \, \sqrt {2} {\left ({\left (C a b - 5 \, A b^{2}\right )} \sqrt {a} c + {\left (D a^{2} \sqrt {b} + 3 \, B a b^{\frac {3}{2}}\right )} c\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a c^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {c x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} c}}\right )}{\sqrt {\sqrt {a} \sqrt {b} c} \sqrt {a} \sqrt {b} c} - \frac {2 \, \sqrt {2} {\left ({\left (C a b - 5 \, A b^{2}\right )} \sqrt {a} c + {\left (D a^{2} \sqrt {b} + 3 \, B a b^{\frac {3}{2}}\right )} c\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a c^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {c x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} c}}\right )}{\sqrt {\sqrt {a} \sqrt {b} c} \sqrt {a} \sqrt {b} c}}{a^{2} b}}{16 \, c} \] Input:
integrate((D*x^3+C*x^2+B*x+A)/(c*x)^(3/2)/(b*x^2+a)^2,x, algorithm="maxima ")
Output:
-1/16*(8*(4*A*a*b*c^2 - (C*a*b - 5*A*b^2)*c^2*x^2 + (D*a^2 - B*a*b)*c^2*x) /((c*x)^(5/2)*a^2*b^2 + sqrt(c*x)*a^3*b*c^2) + (sqrt(2)*((C*a*b - 5*A*b^2) *sqrt(a)*c - (D*a^2*sqrt(b) + 3*B*a*b^(3/2))*c)*log(sqrt(b)*c*x + sqrt(2)* (a*c^2)^(1/4)*sqrt(c*x)*b^(1/4) + sqrt(a)*c)/((a*c^2)^(3/4)*b^(3/4)) - sqr t(2)*((C*a*b - 5*A*b^2)*sqrt(a)*c - (D*a^2*sqrt(b) + 3*B*a*b^(3/2))*c)*log (sqrt(b)*c*x - sqrt(2)*(a*c^2)^(1/4)*sqrt(c*x)*b^(1/4) + sqrt(a)*c)/((a*c^ 2)^(3/4)*b^(3/4)) - 2*sqrt(2)*((C*a*b - 5*A*b^2)*sqrt(a)*c + (D*a^2*sqrt(b ) + 3*B*a*b^(3/2))*c)*arctan(1/2*sqrt(2)*(sqrt(2)*(a*c^2)^(1/4)*b^(1/4) + 2*sqrt(c*x)*sqrt(b))/sqrt(sqrt(a)*sqrt(b)*c))/(sqrt(sqrt(a)*sqrt(b)*c)*sqr t(a)*sqrt(b)*c) - 2*sqrt(2)*((C*a*b - 5*A*b^2)*sqrt(a)*c + (D*a^2*sqrt(b) + 3*B*a*b^(3/2))*c)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*c^2)^(1/4)*b^(1/4) - 2 *sqrt(c*x)*sqrt(b))/sqrt(sqrt(a)*sqrt(b)*c))/(sqrt(sqrt(a)*sqrt(b)*c)*sqrt (a)*sqrt(b)*c))/(a^2*b))/c
Leaf count of result is larger than twice the leaf count of optimal. 543 vs. \(2 (263) = 526\).
Time = 0.14 (sec) , antiderivative size = 543, normalized size of antiderivative = 1.55 \[ \int \frac {A+B x+C x^2+D x^3}{(c x)^{3/2} \left (a+b x^2\right )^2} \, dx=\frac {\frac {8 \, {\left (C a b c^{2} x^{2} - 5 \, A b^{2} c^{2} x^{2} - D a^{2} c^{2} x + B a b c^{2} x - 4 \, A a b c^{2}\right )}}{{\left (\sqrt {c x} b c^{2} x^{2} + \sqrt {c x} a c^{2}\right )} a^{2} b} + \frac {2 \, \sqrt {2} {\left (\left (a b^{3} c^{2}\right )^{\frac {1}{4}} D a^{2} b c + 3 \, \left (a b^{3} c^{2}\right )^{\frac {1}{4}} B a b^{2} c + \left (a b^{3} c^{2}\right )^{\frac {3}{4}} C a - 5 \, \left (a b^{3} c^{2}\right )^{\frac {3}{4}} A b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a c^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {c x}\right )}}{2 \, \left (\frac {a c^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{3} b^{3} c^{2}} + \frac {2 \, \sqrt {2} {\left (\left (a b^{3} c^{2}\right )^{\frac {1}{4}} D a^{2} b c + 3 \, \left (a b^{3} c^{2}\right )^{\frac {1}{4}} B a b^{2} c + \left (a b^{3} c^{2}\right )^{\frac {3}{4}} C a - 5 \, \left (a b^{3} c^{2}\right )^{\frac {3}{4}} A b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a c^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {c x}\right )}}{2 \, \left (\frac {a c^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{3} b^{3} c^{2}} + \frac {\sqrt {2} {\left (\left (a b^{3} c^{2}\right )^{\frac {1}{4}} D a^{2} b c + 3 \, \left (a b^{3} c^{2}\right )^{\frac {1}{4}} B a b^{2} c - \left (a b^{3} c^{2}\right )^{\frac {3}{4}} C a + 5 \, \left (a b^{3} c^{2}\right )^{\frac {3}{4}} A b\right )} \log \left (c x + \sqrt {2} \left (\frac {a c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x} + \sqrt {\frac {a c^{2}}{b}}\right )}{a^{3} b^{3} c^{2}} - \frac {\sqrt {2} {\left (\left (a b^{3} c^{2}\right )^{\frac {1}{4}} D a^{2} b c + 3 \, \left (a b^{3} c^{2}\right )^{\frac {1}{4}} B a b^{2} c - \left (a b^{3} c^{2}\right )^{\frac {3}{4}} C a + 5 \, \left (a b^{3} c^{2}\right )^{\frac {3}{4}} A b\right )} \log \left (c x - \sqrt {2} \left (\frac {a c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x} + \sqrt {\frac {a c^{2}}{b}}\right )}{a^{3} b^{3} c^{2}}}{16 \, c} \] Input:
integrate((D*x^3+C*x^2+B*x+A)/(c*x)^(3/2)/(b*x^2+a)^2,x, algorithm="giac")
Output:
1/16*(8*(C*a*b*c^2*x^2 - 5*A*b^2*c^2*x^2 - D*a^2*c^2*x + B*a*b*c^2*x - 4*A *a*b*c^2)/((sqrt(c*x)*b*c^2*x^2 + sqrt(c*x)*a*c^2)*a^2*b) + 2*sqrt(2)*((a* b^3*c^2)^(1/4)*D*a^2*b*c + 3*(a*b^3*c^2)^(1/4)*B*a*b^2*c + (a*b^3*c^2)^(3/ 4)*C*a - 5*(a*b^3*c^2)^(3/4)*A*b)*arctan(1/2*sqrt(2)*(sqrt(2)*(a*c^2/b)^(1 /4) + 2*sqrt(c*x))/(a*c^2/b)^(1/4))/(a^3*b^3*c^2) + 2*sqrt(2)*((a*b^3*c^2) ^(1/4)*D*a^2*b*c + 3*(a*b^3*c^2)^(1/4)*B*a*b^2*c + (a*b^3*c^2)^(3/4)*C*a - 5*(a*b^3*c^2)^(3/4)*A*b)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*c^2/b)^(1/4) - 2 *sqrt(c*x))/(a*c^2/b)^(1/4))/(a^3*b^3*c^2) + sqrt(2)*((a*b^3*c^2)^(1/4)*D* a^2*b*c + 3*(a*b^3*c^2)^(1/4)*B*a*b^2*c - (a*b^3*c^2)^(3/4)*C*a + 5*(a*b^3 *c^2)^(3/4)*A*b)*log(c*x + sqrt(2)*(a*c^2/b)^(1/4)*sqrt(c*x) + sqrt(a*c^2/ b))/(a^3*b^3*c^2) - sqrt(2)*((a*b^3*c^2)^(1/4)*D*a^2*b*c + 3*(a*b^3*c^2)^( 1/4)*B*a*b^2*c - (a*b^3*c^2)^(3/4)*C*a + 5*(a*b^3*c^2)^(3/4)*A*b)*log(c*x - sqrt(2)*(a*c^2/b)^(1/4)*sqrt(c*x) + sqrt(a*c^2/b))/(a^3*b^3*c^2))/c
Timed out. \[ \int \frac {A+B x+C x^2+D x^3}{(c x)^{3/2} \left (a+b x^2\right )^2} \, dx=\int \frac {A+B\,x+C\,x^2+x^3\,D}{{\left (c\,x\right )}^{3/2}\,{\left (b\,x^2+a\right )}^2} \,d x \] Input:
int((A + B*x + C*x^2 + x^3*D)/((c*x)^(3/2)*(a + b*x^2)^2),x)
Output:
int((A + B*x + C*x^2 + x^3*D)/((c*x)^(3/2)*(a + b*x^2)^2), x)
Time = 0.16 (sec) , antiderivative size = 1273, normalized size of antiderivative = 3.64 \[ \int \frac {A+B x+C x^2+D x^3}{(c x)^{3/2} \left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:
int((D*x^3+C*x^2+B*x+A)/(c*x)^(3/2)/(b*x^2+a)^2,x)
Output:
(sqrt(c)*(10*sqrt(x)*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqr t(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a*b**2 - 2*sqrt(x)* b**(1/4)*a**(3/4)*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*c + 10*sqrt(x)*b**(1/4)*a**(3/4)*sqr t(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/ 4)*sqrt(2)))*b**3*x**2 - 2*sqrt(x)*b**(1/4)*a**(3/4)*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* c*x**2 - 2*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*d - 6*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**2 - 2*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*d*x**2 - 6*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**3*x **2 - 10*sqrt(x)*b**(1/4)*a**(3/4)*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**2 + 2*sqrt(x)*b**( 1/4)*a**(3/4)*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*c - 10*sqrt(x)*b**(1/4)*a**(3/4)*sqrt(2) *atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*s qrt(2)))*b**3*x**2 + 2*sqrt(x)*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)...