Integrand size = 32, antiderivative size = 384 \[ \int \frac {A+B x+C x^2+D x^3}{(c x)^{5/2} \left (a+b x^2\right )^2} \, dx=-\frac {5 b B-a D}{2 a^2 b c^2 \sqrt {c x}}-\frac {2 A}{3 a c (c x)^{3/2} \left (a+b x^2\right )}+\frac {3 a (b B-a D)-b (7 A b-3 a C) x}{6 a^2 b c^2 \sqrt {c x} \left (a+b x^2\right )}+\frac {\left (\sqrt {b} (7 A b-3 a C)+\sqrt {a} (5 b B-a D)\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )}{4 \sqrt {2} a^{11/4} b^{3/4} c^{5/2}}-\frac {\left (\sqrt {b} (7 A b-3 a C)+\sqrt {a} (5 b B-a D)\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )}{4 \sqrt {2} a^{11/4} b^{3/4} c^{5/2}}-\frac {\left (\sqrt {b} (7 A b-3 a C)-\sqrt {a} (5 b B-a D)\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{4 \sqrt {2} a^{11/4} b^{3/4} c^{5/2}} \] Output:
-1/2*(5*B*b-D*a)/a^2/b/c^2/(c*x)^(1/2)-2/3*A/a/c/(c*x)^(3/2)/(b*x^2+a)+1/6 *(3*a*(B*b-D*a)-b*(7*A*b-3*C*a)*x)/a^2/b/c^2/(c*x)^(1/2)/(b*x^2+a)+1/8*(b^ (1/2)*(7*A*b-3*C*a)+a^(1/2)*(5*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^(11/4)/b^(3/4)/c^(5/2)-1/8*(b^(1/2)*(7*A*b- 3*C*a)+a^(1/2)*(5*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^(11/4)/b^(3/4)/c^(5/2)-1/8*(b^(1/2)*(7*A*b-3*C*a)-a^(1/2 )*(5*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^(11/4)/b^(3/4)/c^(5/2)
Time = 1.36 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.63 \[ \int \frac {A+B x+C x^2+D x^3}{(c x)^{5/2} \left (a+b x^2\right )^2} \, dx=\frac {x \left (-\frac {4 a^{3/4} \left (4 a A+b x^2 (7 A+15 B x)-3 a x (-4 B+x (C+D x))\right )}{a+b x^2}+\frac {3 \sqrt {2} \left (7 A b^{3/2}+5 \sqrt {a} b B-3 a \sqrt {b} C-a^{3/2} D\right ) x^{3/2} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{b^{3/4}}-\frac {3 \sqrt {2} \left (7 A b^{3/2}-5 \sqrt {a} b B-3 a \sqrt {b} C+a^{3/2} D\right ) x^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{b^{3/4}}\right )}{24 a^{11/4} (c x)^{5/2}} \] Input:
Integrate[(A + B*x + C*x^2 + D*x^3)/((c*x)^(5/2)*(a + b*x^2)^2),x]
Output:
(x*((-4*a^(3/4)*(4*a*A + b*x^2*(7*A + 15*B*x) - 3*a*x*(-4*B + x*(C + D*x)) ))/(a + b*x^2) + (3*Sqrt[2]*(7*A*b^(3/2) + 5*Sqrt[a]*b*B - 3*a*Sqrt[b]*C - a^(3/2)*D)*x^(3/2)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)* Sqrt[x])])/b^(3/4) - (3*Sqrt[2]*(7*A*b^(3/2) - 5*Sqrt[a]*b*B - 3*a*Sqrt[b] *C + a^(3/2)*D)*x^(3/2)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/b^(3/4)))/(24*a^(11/4)*(c*x)^(5/2))
Time = 0.90 (sec) , antiderivative size = 446, normalized size of antiderivative = 1.16, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2337, 27, 553, 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.
| \(\displaystyle \int \frac {A+B x+C x^2+D x^3}{(c x)^{5/2} \left (a+b x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 2337 |
| \(\displaystyle \frac {x \left (B-\frac {a D}{b}\right )-\frac {a C}{b}+A}{2 a c (c x)^{3/2} \left (a+b x^2\right )}-\frac {\int -\frac {7 A-\frac {3 a C}{b}+\left (5 B-\frac {a D}{b}\right ) x}{2 (c x)^{5/2} \left (b x^2+a\right )}dx}{2 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {7 A-\frac {3 a C}{b}+\left (5 B-\frac {a D}{b}\right ) x}{(c x)^{5/2} \left (b x^2+a\right )}dx}{4 a}+\frac {x \left (B-\frac {a D}{b}\right )-\frac {a C}{b}+A}{2 a c (c x)^{3/2} \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 553 |
| \(\displaystyle \frac {-\frac {2 \int -\frac {3 (a (5 b B-a D)-b (7 A b-3 a C) x)}{2 b (c x)^{3/2} \left (b x^2+a\right )}dx}{3 a c}-\frac {2 \left (7 A-\frac {3 a C}{b}\right )}{3 a c (c x)^{3/2}}}{4 a}+\frac {x \left (B-\frac {a D}{b}\right )-\frac {a C}{b}+A}{2 a c (c x)^{3/2} \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {a (5 b B-a D)-b (7 A b-3 a C) x}{(c x)^{3/2} \left (b x^2+a\right )}dx}{a b c}-\frac {2 \left (7 A-\frac {3 a C}{b}\right )}{3 a c (c x)^{3/2}}}{4 a}+\frac {x \left (B-\frac {a D}{b}\right )-\frac {a C}{b}+A}{2 a c (c x)^{3/2} \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 553 |
| \(\displaystyle \frac {\frac {-\frac {2 \int \frac {a b (7 A b-3 a C+(5 b B-a D) x)}{2 \sqrt {c x} \left (b x^2+a\right )}dx}{a c}-\frac {2 (5 b B-a D)}{c \sqrt {c x}}}{a b c}-\frac {2 \left (7 A-\frac {3 a C}{b}\right )}{3 a c (c x)^{3/2}}}{4 a}+\frac {x \left (B-\frac {a D}{b}\right )-\frac {a C}{b}+A}{2 a c (c x)^{3/2} \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {b \int \frac {7 A b-3 a C+(5 b B-a D) x}{\sqrt {c x} \left (b x^2+a\right )}dx}{c}-\frac {2 (5 b B-a D)}{c \sqrt {c x}}}{a b c}-\frac {2 \left (7 A-\frac {3 a C}{b}\right )}{3 a c (c x)^{3/2}}}{4 a}+\frac {x \left (B-\frac {a D}{b}\right )-\frac {a C}{b}+A}{2 a c (c x)^{3/2} \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 554 |
| \(\displaystyle \frac {\frac {-\frac {2 b \int \frac {c (7 A b-3 a C)+c (5 b B-a D) x}{b x^2 c^2+a c^2}d\sqrt {c x}}{c}-\frac {2 (5 b B-a D)}{c \sqrt {c x}}}{a b c}-\frac {2 \left (7 A-\frac {3 a C}{b}\right )}{3 a c (c x)^{3/2}}}{4 a}+\frac {x \left (B-\frac {a D}{b}\right )-\frac {a C}{b}+A}{2 a c (c x)^{3/2} \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {\frac {-\frac {2 b \left (\frac {\left (\frac {\sqrt {b} (7 A b-3 a C)}{\sqrt {a}}-a D+5 b B\right ) \int \frac {\sqrt {b} \left (\sqrt {b} x c+\sqrt {a} c\right )}{b x^2 c^2+a c^2}d\sqrt {c x}}{2 b}-\frac {\left (-\frac {\sqrt {b} (7 A b-3 a C)}{\sqrt {a}}-a D+5 b B\right ) \int \frac {\sqrt {b} \left (\sqrt {a} c-\sqrt {b} c x\right )}{b x^2 c^2+a c^2}d\sqrt {c x}}{2 b}\right )}{c}-\frac {2 (5 b B-a D)}{c \sqrt {c x}}}{a b c}-\frac {2 \left (7 A-\frac {3 a C}{b}\right )}{3 a c (c x)^{3/2}}}{4 a}+\frac {x \left (B-\frac {a D}{b}\right )-\frac {a C}{b}+A}{2 a c (c x)^{3/2} \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {2 b \left (\frac {\left (\frac {\sqrt {b} (7 A b-3 a C)}{\sqrt {a}}-a D+5 b B\right ) \int \frac {\sqrt {b} x c+\sqrt {a} c}{b x^2 c^2+a c^2}d\sqrt {c x}}{2 \sqrt {b}}-\frac {\left (-\frac {\sqrt {b} (7 A b-3 a C)}{\sqrt {a}}-a D+5 b B\right ) \int \frac {\sqrt {a} c-\sqrt {b} c x}{b x^2 c^2+a c^2}d\sqrt {c x}}{2 \sqrt {b}}\right )}{c}-\frac {2 (5 b B-a D)}{c \sqrt {c x}}}{a b c}-\frac {2 \left (7 A-\frac {3 a C}{b}\right )}{3 a c (c x)^{3/2}}}{4 a}+\frac {x \left (B-\frac {a D}{b}\right )-\frac {a C}{b}+A}{2 a c (c x)^{3/2} \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\frac {-\frac {2 b \left (\frac {\left (\frac {\sqrt {b} (7 A b-3 a C)}{\sqrt {a}}-a D+5 b B\right ) \left (\frac {\int \frac {1}{x c+\frac {\sqrt {a} c}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c x} \sqrt {c}}{\sqrt [4]{b}}}d\sqrt {c x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x c+\frac {\sqrt {a} c}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c x} \sqrt {c}}{\sqrt [4]{b}}}d\sqrt {c x}}{2 \sqrt {b}}\right )}{2 \sqrt {b}}-\frac {\left (-\frac {\sqrt {b} (7 A b-3 a C)}{\sqrt {a}}-a D+5 b B\right ) \int \frac {\sqrt {a} c-\sqrt {b} c x}{b x^2 c^2+a c^2}d\sqrt {c x}}{2 \sqrt {b}}\right )}{c}-\frac {2 (5 b B-a D)}{c \sqrt {c x}}}{a b c}-\frac {2 \left (7 A-\frac {3 a C}{b}\right )}{3 a c (c x)^{3/2}}}{4 a}+\frac {x \left (B-\frac {a D}{b}\right )-\frac {a C}{b}+A}{2 a c (c x)^{3/2} \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {-\frac {2 b \left (\frac {\left (\frac {\sqrt {b} (7 A b-3 a C)}{\sqrt {a}}-a D+5 b B\right ) \left (\frac {\int \frac {1}{-c x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}}-\frac {\int \frac {1}{-c x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}}\right )}{2 \sqrt {b}}-\frac {\left (-\frac {\sqrt {b} (7 A b-3 a C)}{\sqrt {a}}-a D+5 b B\right ) \int \frac {\sqrt {a} c-\sqrt {b} c x}{b x^2 c^2+a c^2}d\sqrt {c x}}{2 \sqrt {b}}\right )}{c}-\frac {2 (5 b B-a D)}{c \sqrt {c x}}}{a b c}-\frac {2 \left (7 A-\frac {3 a C}{b}\right )}{3 a c (c x)^{3/2}}}{4 a}+\frac {x \left (B-\frac {a D}{b}\right )-\frac {a C}{b}+A}{2 a c (c x)^{3/2} \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {-\frac {2 b \left (\frac {\left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}}\right ) \left (\frac {\sqrt {b} (7 A b-3 a C)}{\sqrt {a}}-a D+5 b B\right )}{2 \sqrt {b}}-\frac {\left (-\frac {\sqrt {b} (7 A b-3 a C)}{\sqrt {a}}-a D+5 b B\right ) \int \frac {\sqrt {a} c-\sqrt {b} c x}{b x^2 c^2+a c^2}d\sqrt {c x}}{2 \sqrt {b}}\right )}{c}-\frac {2 (5 b B-a D)}{c \sqrt {c x}}}{a b c}-\frac {2 \left (7 A-\frac {3 a C}{b}\right )}{3 a c (c x)^{3/2}}}{4 a}+\frac {x \left (B-\frac {a D}{b}\right )-\frac {a C}{b}+A}{2 a c (c x)^{3/2} \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\frac {-\frac {2 b \left (\frac {\left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}}\right ) \left (\frac {\sqrt {b} (7 A b-3 a C)}{\sqrt {a}}-a D+5 b B\right )}{2 \sqrt {b}}-\frac {\left (-\frac {\sqrt {b} (7 A b-3 a C)}{\sqrt {a}}-a D+5 b B\right ) \left (-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c}-2 \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{b} \left (x c+\frac {\sqrt {a} c}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c x} \sqrt {c}}{\sqrt [4]{b}}\right )}d\sqrt {c x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {c}+\sqrt {2} \sqrt [4]{b} \sqrt {c x}\right )}{\sqrt [4]{b} \left (x c+\frac {\sqrt {a} c}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c x} \sqrt {c}}{\sqrt [4]{b}}\right )}d\sqrt {c x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}}\right )}{2 \sqrt {b}}\right )}{c}-\frac {2 (5 b B-a D)}{c \sqrt {c x}}}{a b c}-\frac {2 \left (7 A-\frac {3 a C}{b}\right )}{3 a c (c x)^{3/2}}}{4 a}+\frac {x \left (B-\frac {a D}{b}\right )-\frac {a C}{b}+A}{2 a c (c x)^{3/2} \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {-\frac {2 b \left (\frac {\left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}}\right ) \left (\frac {\sqrt {b} (7 A b-3 a C)}{\sqrt {a}}-a D+5 b B\right )}{2 \sqrt {b}}-\frac {\left (-\frac {\sqrt {b} (7 A b-3 a C)}{\sqrt {a}}-a D+5 b B\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {c}-2 \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{b} \left (x c+\frac {\sqrt {a} c}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c x} \sqrt {c}}{\sqrt [4]{b}}\right )}d\sqrt {c x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}}+\frac {\int \frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {c}+\sqrt {2} \sqrt [4]{b} \sqrt {c x}\right )}{\sqrt [4]{b} \left (x c+\frac {\sqrt {a} c}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c x} \sqrt {c}}{\sqrt [4]{b}}\right )}d\sqrt {c x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}}\right )}{2 \sqrt {b}}\right )}{c}-\frac {2 (5 b B-a D)}{c \sqrt {c x}}}{a b c}-\frac {2 \left (7 A-\frac {3 a C}{b}\right )}{3 a c (c x)^{3/2}}}{4 a}+\frac {x \left (B-\frac {a D}{b}\right )-\frac {a C}{b}+A}{2 a c (c x)^{3/2} \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {2 b \left (\frac {\left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}}\right ) \left (\frac {\sqrt {b} (7 A b-3 a C)}{\sqrt {a}}-a D+5 b B\right )}{2 \sqrt {b}}-\frac {\left (-\frac {\sqrt {b} (7 A b-3 a C)}{\sqrt {a}}-a D+5 b B\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {c}-2 \sqrt [4]{b} \sqrt {c x}}{x c+\frac {\sqrt {a} c}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c x} \sqrt {c}}{\sqrt [4]{b}}}d\sqrt {c x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b} \sqrt {c}}+\frac {\int \frac {\sqrt [4]{a} \sqrt {c}+\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{x c+\frac {\sqrt {a} c}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c x} \sqrt {c}}{\sqrt [4]{b}}}d\sqrt {c x}}{2 \sqrt [4]{a} \sqrt {b} \sqrt {c}}\right )}{2 \sqrt {b}}\right )}{c}-\frac {2 (5 b B-a D)}{c \sqrt {c x}}}{a b c}-\frac {2 \left (7 A-\frac {3 a C}{b}\right )}{3 a c (c x)^{3/2}}}{4 a}+\frac {x \left (B-\frac {a D}{b}\right )-\frac {a C}{b}+A}{2 a c (c x)^{3/2} \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {-\frac {2 b \left (\frac {\left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}}\right ) \left (\frac {\sqrt {b} (7 A b-3 a C)}{\sqrt {a}}-a D+5 b B\right )}{2 \sqrt {b}}-\frac {\left (\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {c} \sqrt {c x}+\sqrt {a} c+\sqrt {b} c x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {c} \sqrt {c x}+\sqrt {a} c+\sqrt {b} c x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}}\right ) \left (-\frac {\sqrt {b} (7 A b-3 a C)}{\sqrt {a}}-a D+5 b B\right )}{2 \sqrt {b}}\right )}{c}-\frac {2 (5 b B-a D)}{c \sqrt {c x}}}{a b c}-\frac {2 \left (7 A-\frac {3 a C}{b}\right )}{3 a c (c x)^{3/2}}}{4 a}+\frac {x \left (B-\frac {a D}{b}\right )-\frac {a C}{b}+A}{2 a c (c x)^{3/2} \left (a+b x^2\right )}\) |
Input:
Int[(A + B*x + C*x^2 + D*x^3)/((c*x)^(5/2)*(a + b*x^2)^2),x]
Output:
(A - (a*C)/b + (B - (a*D)/b)*x)/(2*a*c*(c*x)^(3/2)*(a + b*x^2)) + ((-2*(7* A - (3*a*C)/b))/(3*a*c*(c*x)^(3/2)) + ((-2*(5*b*B - a*D))/(c*Sqrt[c*x]) - (2*b*(((5*b*B + (Sqrt[b]*(7*A*b - 3*a*C))/Sqrt[a] - a*D)*(-(ArcTan[1 - (Sq rt[2]*b^(1/4)*Sqrt[c*x])/(a^(1/4)*Sqrt[c])]/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[ c])) + ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[c*x])/(a^(1/4)*Sqrt[c])]/(Sqrt[2]* a^(1/4)*b^(1/4)*Sqrt[c])))/(2*Sqrt[b]) - ((5*b*B - (Sqrt[b]*(7*A*b - 3*a*C ))/Sqrt[a] - a*D)*(-1/2*Log[Sqrt[a]*c + Sqrt[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*Sqrt[b])))/c)/(a*b*c))/(4*a)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((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 = 0.86 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.01
| method | result | size |
| derivativedivides | \(-\frac {2 A}{3 c \,a^{2} \left (c x \right )^{\frac {3}{2}}}-\frac {2 B}{c^{2} a^{2} \sqrt {c x}}-\frac {2 \left (\frac {\left (\frac {B b}{4}-\frac {D a}{4}\right ) \left (c x \right )^{\frac {3}{2}}+\left (\frac {1}{4} A b c -\frac {1}{4} C a c \right ) \sqrt {c x}}{b \,c^{2} x^{2}+a \,c^{2}}+\frac {\left (7 A b c -3 C a c \right ) \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {c x +\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}{c x -\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c x}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c x}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a \,c^{2}}+\frac {\left (5 B b -D a \right ) \sqrt {2}\, \left (\ln \left (\frac {c x -\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}{c x +\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c x}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c x}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 b \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{2} c^{2}}\) | \(388\) |
| default | \(-\frac {2 A}{3 c \,a^{2} \left (c x \right )^{\frac {3}{2}}}-\frac {2 B}{c^{2} a^{2} \sqrt {c x}}-\frac {2 \left (\frac {\left (\frac {B b}{4}-\frac {D a}{4}\right ) \left (c x \right )^{\frac {3}{2}}+\left (\frac {1}{4} A b c -\frac {1}{4} C a c \right ) \sqrt {c x}}{b \,c^{2} x^{2}+a \,c^{2}}+\frac {\left (7 A b c -3 C a c \right ) \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {c x +\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}{c x -\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c x}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c x}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a \,c^{2}}+\frac {\left (5 B b -D a \right ) \sqrt {2}\, \left (\ln \left (\frac {c x -\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}{c x +\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c x}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c x}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 b \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{2} c^{2}}\) | \(388\) |
| pseudoelliptic | \(-\frac {2 \left (\frac {21 \left (b \,x^{2}+a \right ) \left (\ln \left (\frac {c x +\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}{c x -\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c x}-\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c x}+\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}\right )\right ) \left (c x \right )^{\frac {3}{2}} \sqrt {2}\, \left (A b -\frac {3 C a}{7}\right ) b \sqrt {\frac {a \,c^{2}}{b}}}{32}+\left (\frac {15 \left (b \,x^{2}+a \right ) \left (\ln \left (\frac {c x -\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}{c x +\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} \sqrt {c x}\, \sqrt {2}+\sqrt {\frac {a \,c^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c x}-\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c x}+\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}}}\right )\right ) \sqrt {2}\, \left (B b -\frac {D a}{5}\right ) \left (c x \right )^{\frac {3}{2}}}{32}+\left (\left (-\frac {3}{4} D x^{3}-\frac {3}{4} C \,x^{2}+3 B x +A \right ) a +\frac {7 \left (\frac {15 B x}{7}+A \right ) x^{2} b}{4}\right ) \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} c b \right ) c a \right )}{3 \left (c x \right )^{\frac {3}{2}} \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{4}} c^{3} a^{3} \left (b \,x^{2}+a \right ) b}\) | \(432\) |
Input:
int((D*x^3+C*x^2+B*x+A)/(c*x)^(5/2)/(b*x^2+a)^2,x,method=_RETURNVERBOSE)
Output:
-2/3*A/c/a^2/(c*x)^(3/2)-2*B/c^2/a^2/(c*x)^(1/2)-2/a^2/c^2*(((1/4*B*b-1/4* D*a)*(c*x)^(3/2)+(1/4*A*b*c-1/4*C*a*c)*(c*x)^(1/2))/(b*c^2*x^2+a*c^2)+1/32 *(7*A*b*c-3*C*a*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*ar ctan(2^(1/2)/(a*c^2/b)^(1/4)*(c*x)^(1/2)-1))+1/32*(5*B*b-D*a)/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*arctan(2^(1/2)/(a*c^2/b)^(1/4)*(c*x)^ (1/2)-1)))
Leaf count of result is larger than twice the leaf count of optimal. 3722 vs. \(2 (291) = 582\).
Time = 0.42 (sec) , antiderivative size = 3722, normalized size of antiderivative = 9.69 \[ \int \frac {A+B x+C x^2+D x^3}{(c x)^{5/2} \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:
integrate((D*x^3+C*x^2+B*x+A)/(c*x)^(5/2)/(b*x^2+a)^2,x, algorithm="fricas ")
Output:
Too large to include
Result contains complex when optimal does not.
Time = 106.93 (sec) , antiderivative size = 4048, normalized size of antiderivative = 10.54 \[ \int \frac {A+B x+C x^2+D x^3}{(c x)^{5/2} \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:
integrate((D*x**3+C*x**2+B*x+A)/(c*x)**(5/2)/(b*x**2+a)**2,x)
Output:
A*(16*a**(7/4)*exp(I*pi/4)*gamma(-3/4)/(32*a**(15/4)*c**(5/2)*x**(3/2)*exp (I*pi/4)*gamma(1/4) + 32*a**(11/4)*b*c**(5/2)*x**(7/2)*exp(I*pi/4)*gamma(1 /4)) + 28*a**(3/4)*b*x**2*exp(I*pi/4)*gamma(-3/4)/(32*a**(15/4)*c**(5/2)*x **(3/2)*exp(I*pi/4)*gamma(1/4) + 32*a**(11/4)*b*c**(5/2)*x**(7/2)*exp(I*pi /4)*gamma(1/4)) - 21*a*b**(3/4)*x**(3/2)*log(1 - b**(1/4)*sqrt(x)*exp_pola r(I*pi/4)/a**(1/4))*gamma(-3/4)/(32*a**(15/4)*c**(5/2)*x**(3/2)*exp(I*pi/4 )*gamma(1/4) + 32*a**(11/4)*b*c**(5/2)*x**(7/2)*exp(I*pi/4)*gamma(1/4)) + 21*I*a*b**(3/4)*x**(3/2)*log(1 - b**(1/4)*sqrt(x)*exp_polar(3*I*pi/4)/a**( 1/4))*gamma(-3/4)/(32*a**(15/4)*c**(5/2)*x**(3/2)*exp(I*pi/4)*gamma(1/4) + 32*a**(11/4)*b*c**(5/2)*x**(7/2)*exp(I*pi/4)*gamma(1/4)) + 21*a*b**(3/4)* x**(3/2)*log(1 - b**(1/4)*sqrt(x)*exp_polar(5*I*pi/4)/a**(1/4))*gamma(-3/4 )/(32*a**(15/4)*c**(5/2)*x**(3/2)*exp(I*pi/4)*gamma(1/4) + 32*a**(11/4)*b* c**(5/2)*x**(7/2)*exp(I*pi/4)*gamma(1/4)) - 21*I*a*b**(3/4)*x**(3/2)*log(1 - b**(1/4)*sqrt(x)*exp_polar(7*I*pi/4)/a**(1/4))*gamma(-3/4)/(32*a**(15/4 )*c**(5/2)*x**(3/2)*exp(I*pi/4)*gamma(1/4) + 32*a**(11/4)*b*c**(5/2)*x**(7 /2)*exp(I*pi/4)*gamma(1/4)) - 21*b**(7/4)*x**(7/2)*log(1 - b**(1/4)*sqrt(x )*exp_polar(I*pi/4)/a**(1/4))*gamma(-3/4)/(32*a**(15/4)*c**(5/2)*x**(3/2)* exp(I*pi/4)*gamma(1/4) + 32*a**(11/4)*b*c**(5/2)*x**(7/2)*exp(I*pi/4)*gamm a(1/4)) + 21*I*b**(7/4)*x**(7/2)*log(1 - b**(1/4)*sqrt(x)*exp_polar(3*I*pi /4)/a**(1/4))*gamma(-3/4)/(32*a**(15/4)*c**(5/2)*x**(3/2)*exp(I*pi/4)*g...
Time = 0.12 (sec) , antiderivative size = 437, normalized size of antiderivative = 1.14 \[ \int \frac {A+B x+C x^2+D x^3}{(c x)^{5/2} \left (a+b x^2\right )^2} \, dx=\frac {\frac {8 \, {\left (3 \, {\left (D a - 5 \, B b\right )} c^{3} x^{3} - 12 \, B a c^{3} x + {\left (3 \, C a - 7 \, A b\right )} c^{3} x^{2} - 4 \, A a c^{3}\right )}}{\left (c x\right )^{\frac {7}{2}} a^{2} b c + \left (c x\right )^{\frac {3}{2}} a^{3} c^{3}} - \frac {3 \, {\left (\frac {\sqrt {2} {\left ({\left (D a - 5 \, B b\right )} \sqrt {a} c - {\left (3 \, C a \sqrt {b} - 7 \, A b^{\frac {3}{2}}\right )} c\right )} \log \left (\sqrt {b} c x + \sqrt {2} \left (a c^{2}\right )^{\frac {1}{4}} \sqrt {c x} b^{\frac {1}{4}} + \sqrt {a} c\right )}{\left (a c^{2}\right )^{\frac {3}{4}} b^{\frac {3}{4}}} - \frac {\sqrt {2} {\left ({\left (D a - 5 \, B b\right )} \sqrt {a} c - {\left (3 \, C a \sqrt {b} - 7 \, A b^{\frac {3}{2}}\right )} c\right )} \log \left (\sqrt {b} c x - \sqrt {2} \left (a c^{2}\right )^{\frac {1}{4}} \sqrt {c x} b^{\frac {1}{4}} + \sqrt {a} c\right )}{\left (a c^{2}\right )^{\frac {3}{4}} b^{\frac {3}{4}}} - \frac {2 \, \sqrt {2} {\left ({\left (D a - 5 \, B b\right )} \sqrt {a} c + {\left (3 \, C a \sqrt {b} - 7 \, A b^{\frac {3}{2}}\right )} c\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a c^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {c x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} c}}\right )}{\sqrt {\sqrt {a} \sqrt {b} c} \sqrt {a} \sqrt {b} c} - \frac {2 \, \sqrt {2} {\left ({\left (D a - 5 \, B b\right )} \sqrt {a} c + {\left (3 \, C a \sqrt {b} - 7 \, A b^{\frac {3}{2}}\right )} c\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a c^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {c x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} c}}\right )}{\sqrt {\sqrt {a} \sqrt {b} c} \sqrt {a} \sqrt {b} c}\right )}}{a^{2} c}}{48 \, c} \] Input:
integrate((D*x^3+C*x^2+B*x+A)/(c*x)^(5/2)/(b*x^2+a)^2,x, algorithm="maxima ")
Output:
1/48*(8*(3*(D*a - 5*B*b)*c^3*x^3 - 12*B*a*c^3*x + (3*C*a - 7*A*b)*c^3*x^2 - 4*A*a*c^3)/((c*x)^(7/2)*a^2*b*c + (c*x)^(3/2)*a^3*c^3) - 3*(sqrt(2)*((D* a - 5*B*b)*sqrt(a)*c - (3*C*a*sqrt(b) - 7*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 )) - sqrt(2)*((D*a - 5*B*b)*sqrt(a)*c - (3*C*a*sqrt(b) - 7*A*b^(3/2))*c)*l og(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)*((D*a - 5*B*b)*sqrt(a)*c + (3*C*a*sqrt(b) - 7*A*b^(3/2))*c)*arctan(1/2*sqrt(2)*(sqrt(2)*(a*c^2)^(1/4)*b^(1/4) + 2*sq rt(c*x)*sqrt(b))/sqrt(sqrt(a)*sqrt(b)*c))/(sqrt(sqrt(a)*sqrt(b)*c)*sqrt(a) *sqrt(b)*c) - 2*sqrt(2)*((D*a - 5*B*b)*sqrt(a)*c + (3*C*a*sqrt(b) - 7*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*c))/c
Time = 0.14 (sec) , antiderivative size = 546, normalized size of antiderivative = 1.42 \[ \int \frac {A+B x+C x^2+D x^3}{(c x)^{5/2} \left (a+b x^2\right )^2} \, dx=\frac {\sqrt {c x} D a c x - \sqrt {c x} B b c x + \sqrt {c x} C a c - \sqrt {c x} A b c}{2 \, {\left (b c^{2} x^{2} + a c^{2}\right )} a^{2} c^{2}} - \frac {2 \, {\left (3 \, B c x + A c\right )}}{3 \, \sqrt {c x} a^{2} c^{3} x} + \frac {\sqrt {2} {\left (3 \, \left (a b^{3} c^{2}\right )^{\frac {1}{4}} C a b^{2} c - 7 \, \left (a b^{3} c^{2}\right )^{\frac {1}{4}} A b^{3} c + \left (a b^{3} c^{2}\right )^{\frac {3}{4}} D a - 5 \, \left (a b^{3} c^{2}\right )^{\frac {3}{4}} B 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 )}{8 \, a^{3} b^{3} c^{4}} + \frac {\sqrt {2} {\left (3 \, \left (a b^{3} c^{2}\right )^{\frac {1}{4}} C a b^{2} c - 7 \, \left (a b^{3} c^{2}\right )^{\frac {1}{4}} A b^{3} c + \left (a b^{3} c^{2}\right )^{\frac {3}{4}} D a - 5 \, \left (a b^{3} c^{2}\right )^{\frac {3}{4}} B 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 )}{8 \, a^{3} b^{3} c^{4}} + \frac {\sqrt {2} {\left (3 \, \left (a b^{3} c^{2}\right )^{\frac {1}{4}} C a b^{2} c - 7 \, \left (a b^{3} c^{2}\right )^{\frac {1}{4}} A b^{3} c - \left (a b^{3} c^{2}\right )^{\frac {3}{4}} D a + 5 \, \left (a b^{3} c^{2}\right )^{\frac {3}{4}} B 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 )}{16 \, a^{3} b^{3} c^{4}} - \frac {\sqrt {2} {\left (3 \, \left (a b^{3} c^{2}\right )^{\frac {1}{4}} C a b^{2} c - 7 \, \left (a b^{3} c^{2}\right )^{\frac {1}{4}} A b^{3} c - \left (a b^{3} c^{2}\right )^{\frac {3}{4}} D a + 5 \, \left (a b^{3} c^{2}\right )^{\frac {3}{4}} B 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 )}{16 \, a^{3} b^{3} c^{4}} \] Input:
integrate((D*x^3+C*x^2+B*x+A)/(c*x)^(5/2)/(b*x^2+a)^2,x, algorithm="giac")
Output:
1/2*(sqrt(c*x)*D*a*c*x - sqrt(c*x)*B*b*c*x + sqrt(c*x)*C*a*c - sqrt(c*x)*A *b*c)/((b*c^2*x^2 + a*c^2)*a^2*c^2) - 2/3*(3*B*c*x + A*c)/(sqrt(c*x)*a^2*c ^3*x) + 1/8*sqrt(2)*(3*(a*b^3*c^2)^(1/4)*C*a*b^2*c - 7*(a*b^3*c^2)^(1/4)*A *b^3*c + (a*b^3*c^2)^(3/4)*D*a - 5*(a*b^3*c^2)^(3/4)*B*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^4) + 1/8*sqrt(2)*(3*(a*b^3*c^2)^(1/4)*C*a*b^2*c - 7*(a*b^3*c^2)^(1/4)*A*b^3*c + (a*b^3*c^2)^(3/4)*D*a - 5*(a*b^3*c^2)^(3/4)*B*b)*arctan(-1/2*sqrt(2)*(s qrt(2)*(a*c^2/b)^(1/4) - 2*sqrt(c*x))/(a*c^2/b)^(1/4))/(a^3*b^3*c^4) + 1/1 6*sqrt(2)*(3*(a*b^3*c^2)^(1/4)*C*a*b^2*c - 7*(a*b^3*c^2)^(1/4)*A*b^3*c - ( a*b^3*c^2)^(3/4)*D*a + 5*(a*b^3*c^2)^(3/4)*B*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^4) - 1/16*sqrt(2)*(3*(a*b^3* c^2)^(1/4)*C*a*b^2*c - 7*(a*b^3*c^2)^(1/4)*A*b^3*c - (a*b^3*c^2)^(3/4)*D*a + 5*(a*b^3*c^2)^(3/4)*B*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^4)
Timed out. \[ \int \frac {A+B x+C x^2+D x^3}{(c x)^{5/2} \left (a+b x^2\right )^2} \, dx=\int \frac {A+B\,x+C\,x^2+x^3\,D}{{\left (c\,x\right )}^{5/2}\,{\left (b\,x^2+a\right )}^2} \,d x \] Input:
int((A + B*x + C*x^2 + x^3*D)/((c*x)^(5/2)*(a + b*x^2)^2),x)
Output:
int((A + B*x + C*x^2 + x^3*D)/((c*x)^(5/2)*(a + b*x^2)^2), x)
Time = 0.16 (sec) , antiderivative size = 1309, normalized size of antiderivative = 3.41 \[ \int \frac {A+B x+C x^2+D x^3}{(c x)^{5/2} \left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:
int((D*x^3+C*x^2+B*x+A)/(c*x)^(5/2)/(b*x^2+a)^2,x)
Output:
(sqrt(c)*( - 6*sqrt(x)*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*s qrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**2*d*x + 30*sqr t(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*x - 6*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*d*x**3 + 30*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**3*x**3 + 42*sqrt(x)*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/ 4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**2*b*x - 18 *sqrt(x)*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqr t(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**2*c*x + 42*sqrt(x)*b**(3/4)* a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b** (1/4)*a**(1/4)*sqrt(2)))*a*b**2*x**3 - 18*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*c*x**3 + 6*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**2*d* x - 30*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*x + 6*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*d*x**3 - 30*sqrt(x)*b**(1/4)*a**(3/4)...