Integrand size = 23, antiderivative size = 664 \[ \int \frac {A+B x}{x^{3/2} \left (a+b x+c x^2\right )^3} \, dx=\frac {3 \left (a b B \left (b^2-8 a c\right )-A \left (5 b^4-37 a b^2 c+60 a^2 c^2\right )\right )}{4 a^3 \left (b^2-4 a c\right )^2 \sqrt {x}}+\frac {A b^2-a b B-2 a A c+(A b-2 a B) c x}{2 a \left (b^2-4 a c\right ) \sqrt {x} \left (a+b x+c x^2\right )^2}-\frac {a b B \left (b^2-16 a c\right )-A \left (5 b^4-35 a b^2 c+36 a^2 c^2\right )+c \left (a B \left (b^2-28 a c\right )-A \left (5 b^3-32 a b c\right )\right ) x}{4 a^2 \left (b^2-4 a c\right )^2 \sqrt {x} \left (a+b x+c x^2\right )}+\frac {3 \sqrt {c} \left (a B \left (b^4-10 a b^2 c+56 a^2 c^2+b^3 \sqrt {b^2-4 a c}-8 a b c \sqrt {b^2-4 a c}\right )-A \left (5 b^5-47 a b^3 c+124 a^2 b c^2+5 b^4 \sqrt {b^2-4 a c}-37 a b^2 c \sqrt {b^2-4 a c}+60 a^2 c^2 \sqrt {b^2-4 a c}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{4 \sqrt {2} a^3 \left (b^2-4 a c\right )^{5/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {3 \sqrt {c} \left (a B \left (b^4-10 a b^2 c+56 a^2 c^2-b^3 \sqrt {b^2-4 a c}+8 a b c \sqrt {b^2-4 a c}\right )-A \left (5 b^5-47 a b^3 c+124 a^2 b c^2-5 b^4 \sqrt {b^2-4 a c}+37 a b^2 c \sqrt {b^2-4 a c}-60 a^2 c^2 \sqrt {b^2-4 a c}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{4 \sqrt {2} a^3 \left (b^2-4 a c\right )^{5/2} \sqrt {b+\sqrt {b^2-4 a c}}} \]
3/4*(a*b*B*(-8*a*c+b^2)-A*(60*a^2*c^2-37*a*b^2*c+5*b^4))/a^3/(-4*a*c+b^2)^ 2/x^(1/2)+1/2*(A*b^2-a*b*B-2*A*a*c+(A*b-2*B*a)*c*x)/a/(-4*a*c+b^2)/(c*x^2+ b*x+a)^2/x^(1/2)+1/4*(-a*b*B*(-16*a*c+b^2)+A*(36*a^2*c^2-35*a*b^2*c+5*b^4) -c*(a*B*(-28*a*c+b^2)-A*(-32*a*b*c+5*b^3))*x)/a^2/(-4*a*c+b^2)^2/(c*x^2+b* x+a)/x^(1/2)+3/8*arctan(2^(1/2)*c^(1/2)*x^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/ 2))*c^(1/2)*(a*B*(b^4-10*a*b^2*c+56*a^2*c^2+b^3*(-4*a*c+b^2)^(1/2)-8*a*b*c *(-4*a*c+b^2)^(1/2))-A*(5*b^5-47*a*b^3*c+124*a^2*b*c^2+5*b^4*(-4*a*c+b^2)^ (1/2)-37*a*b^2*c*(-4*a*c+b^2)^(1/2)+60*a^2*c^2*(-4*a*c+b^2)^(1/2)))/a^3/(- 4*a*c+b^2)^(5/2)*2^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2)-3/8*arctan(2^(1/2)*c ^(1/2)*x^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2))*c^(1/2)*(a*B*(b^4-10*a*b^2*c+ 56*a^2*c^2-b^3*(-4*a*c+b^2)^(1/2)+8*a*b*c*(-4*a*c+b^2)^(1/2))-A*(5*b^5-47* a*b^3*c+124*a^2*b*c^2-5*b^4*(-4*a*c+b^2)^(1/2)+37*a*b^2*c*(-4*a*c+b^2)^(1/ 2)-60*a^2*c^2*(-4*a*c+b^2)^(1/2)))/a^3/(-4*a*c+b^2)^(5/2)*2^(1/2)/(b+(-4*a *c+b^2)^(1/2))^(1/2)
Time = 10.43 (sec) , antiderivative size = 676, normalized size of antiderivative = 1.02 \[ \int \frac {A+B x}{x^{3/2} \left (a+b x+c x^2\right )^3} \, dx=-\frac {\frac {2 \left (4 a^4 c^2 (32 A-11 B x)+15 A b^4 x^2 (b+c x)^2-a b^2 x (b+c x) \left (3 b B x (b+c x)+A \left (-25 b^2+116 b c x+111 c^2 x^2\right )\right )+a^3 c \left (B x \left (37 b^2+4 b c x-28 c^2 x^2\right )+A \left (-64 b^2+364 b c x+324 c^2 x^2\right )\right )+a^2 \left (b B x \left (-5 b^3+20 b^2 c x+49 b c^2 x^2+24 c^3 x^3\right )+A \left (8 b^4-194 b^3 c x+25 b^2 c^2 x^2+392 b c^3 x^3+180 c^4 x^4\right )\right )\right )}{\sqrt {x} (a+x (b+c x))^2}+\frac {3 \sqrt {2} \sqrt {c} \left (-a B \left (b^4-10 a b^2 c+56 a^2 c^2+b^3 \sqrt {b^2-4 a c}-8 a b c \sqrt {b^2-4 a c}\right )+A \left (5 b^5-47 a b^3 c+124 a^2 b c^2+5 b^4 \sqrt {b^2-4 a c}-37 a b^2 c \sqrt {b^2-4 a c}+60 a^2 c^2 \sqrt {b^2-4 a c}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {3 \sqrt {2} \sqrt {c} \left (a B \left (b^4-10 a b^2 c+56 a^2 c^2-b^3 \sqrt {b^2-4 a c}+8 a b c \sqrt {b^2-4 a c}\right )+A \left (-5 b^5+47 a b^3 c-124 a^2 b c^2+5 b^4 \sqrt {b^2-4 a c}-37 a b^2 c \sqrt {b^2-4 a c}+60 a^2 c^2 \sqrt {b^2-4 a c}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}}}}{8 a^3 \left (b^2-4 a c\right )^2} \]
-1/8*((2*(4*a^4*c^2*(32*A - 11*B*x) + 15*A*b^4*x^2*(b + c*x)^2 - a*b^2*x*( b + c*x)*(3*b*B*x*(b + c*x) + A*(-25*b^2 + 116*b*c*x + 111*c^2*x^2)) + a^3 *c*(B*x*(37*b^2 + 4*b*c*x - 28*c^2*x^2) + A*(-64*b^2 + 364*b*c*x + 324*c^2 *x^2)) + a^2*(b*B*x*(-5*b^3 + 20*b^2*c*x + 49*b*c^2*x^2 + 24*c^3*x^3) + A* (8*b^4 - 194*b^3*c*x + 25*b^2*c^2*x^2 + 392*b*c^3*x^3 + 180*c^4*x^4))))/(S qrt[x]*(a + x*(b + c*x))^2) + (3*Sqrt[2]*Sqrt[c]*(-(a*B*(b^4 - 10*a*b^2*c + 56*a^2*c^2 + b^3*Sqrt[b^2 - 4*a*c] - 8*a*b*c*Sqrt[b^2 - 4*a*c])) + A*(5* b^5 - 47*a*b^3*c + 124*a^2*b*c^2 + 5*b^4*Sqrt[b^2 - 4*a*c] - 37*a*b^2*c*Sq rt[b^2 - 4*a*c] + 60*a^2*c^2*Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[2]*Sqrt[c]*S qrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (3*Sqrt[2]*Sqrt[c]*(a*B*(b^4 - 10*a*b^2*c + 56*a^2*c^2 - b^3 *Sqrt[b^2 - 4*a*c] + 8*a*b*c*Sqrt[b^2 - 4*a*c]) + A*(-5*b^5 + 47*a*b^3*c - 124*a^2*b*c^2 + 5*b^4*Sqrt[b^2 - 4*a*c] - 37*a*b^2*c*Sqrt[b^2 - 4*a*c] + 60*a^2*c^2*Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b + S qrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(a^3* (b^2 - 4*a*c)^2)
Time = 1.22 (sec) , antiderivative size = 578, normalized size of antiderivative = 0.87, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {1235, 27, 1235, 27, 1198, 25, 1197, 1480, 218}
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.
| \(\displaystyle \int \frac {A+B x}{x^{3/2} \left (a+b x+c x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle \frac {c x (A b-2 a B)-2 a A c-a b B+A b^2}{2 a \sqrt {x} \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {\int -\frac {5 A b^2-a B b-18 a A c+7 (A b-2 a B) c x}{2 x^{3/2} \left (c x^2+b x+a\right )^2}dx}{2 a \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {5 A b^2-a B b-18 a A c+7 (A b-2 a B) c x}{x^{3/2} \left (c x^2+b x+a\right )^2}dx}{4 a \left (b^2-4 a c\right )}+\frac {c x (A b-2 a B)-2 a A c-a b B+A b^2}{2 a \sqrt {x} \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle \frac {-\frac {\int \frac {3 \left (a b B \left (b^2-8 a c\right )-A \left (5 b^4-37 a c b^2+60 a^2 c^2\right )+c \left (a B \left (b^2-28 a c\right )-A \left (5 b^3-32 a b c\right )\right ) x\right )}{2 x^{3/2} \left (c x^2+b x+a\right )}dx}{a \left (b^2-4 a c\right )}-\frac {-A \left (36 a^2 c^2-35 a b^2 c+5 b^4\right )+c x \left (a B \left (b^2-28 a c\right )-A \left (5 b^3-32 a b c\right )\right )+a b B \left (b^2-16 a c\right )}{a \sqrt {x} \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{4 a \left (b^2-4 a c\right )}+\frac {c x (A b-2 a B)-2 a A c-a b B+A b^2}{2 a \sqrt {x} \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {3 \int \frac {a b B \left (b^2-8 a c\right )-A \left (5 b^4-37 a c b^2+60 a^2 c^2\right )+c \left (a B \left (b^2-28 a c\right )-A \left (5 b^3-32 a b c\right )\right ) x}{x^{3/2} \left (c x^2+b x+a\right )}dx}{2 a \left (b^2-4 a c\right )}-\frac {-A \left (36 a^2 c^2-35 a b^2 c+5 b^4\right )+c x \left (a B \left (b^2-28 a c\right )-A \left (5 b^3-32 a b c\right )\right )+a b B \left (b^2-16 a c\right )}{a \sqrt {x} \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{4 a \left (b^2-4 a c\right )}+\frac {c x (A b-2 a B)-2 a A c-a b B+A b^2}{2 a \sqrt {x} \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1198 |
| \(\displaystyle \frac {-\frac {3 \left (\frac {\int -\frac {a B \left (b^4-9 a c b^2+28 a^2 c^2\right )-A \left (5 b^5-42 a c b^3+92 a^2 c^2 b\right )+c \left (a b B \left (b^2-8 a c\right )-A \left (5 b^4-37 a c b^2+60 a^2 c^2\right )\right ) x}{\sqrt {x} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 \left (a b B \left (b^2-8 a c\right )-A \left (60 a^2 c^2-37 a b^2 c+5 b^4\right )\right )}{a \sqrt {x}}\right )}{2 a \left (b^2-4 a c\right )}-\frac {-A \left (36 a^2 c^2-35 a b^2 c+5 b^4\right )+c x \left (a B \left (b^2-28 a c\right )-A \left (5 b^3-32 a b c\right )\right )+a b B \left (b^2-16 a c\right )}{a \sqrt {x} \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{4 a \left (b^2-4 a c\right )}+\frac {c x (A b-2 a B)-2 a A c-a b B+A b^2}{2 a \sqrt {x} \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {3 \left (-\frac {\int \frac {a B \left (b^4-9 a c b^2+28 a^2 c^2\right )-A \left (5 b^5-42 a c b^3+92 a^2 c^2 b\right )+c \left (a b B \left (b^2-8 a c\right )-A \left (5 b^4-37 a c b^2+60 a^2 c^2\right )\right ) x}{\sqrt {x} \left (c x^2+b x+a\right )}dx}{a}-\frac {2 \left (a b B \left (b^2-8 a c\right )-A \left (60 a^2 c^2-37 a b^2 c+5 b^4\right )\right )}{a \sqrt {x}}\right )}{2 a \left (b^2-4 a c\right )}-\frac {-A \left (36 a^2 c^2-35 a b^2 c+5 b^4\right )+c x \left (a B \left (b^2-28 a c\right )-A \left (5 b^3-32 a b c\right )\right )+a b B \left (b^2-16 a c\right )}{a \sqrt {x} \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{4 a \left (b^2-4 a c\right )}+\frac {c x (A b-2 a B)-2 a A c-a b B+A b^2}{2 a \sqrt {x} \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1197 |
| \(\displaystyle \frac {-\frac {3 \left (-\frac {2 \int \frac {a B \left (b^4-9 a c b^2+28 a^2 c^2\right )-A \left (5 b^5-42 a c b^3+92 a^2 c^2 b\right )+c \left (a b B \left (b^2-8 a c\right )-A \left (5 b^4-37 a c b^2+60 a^2 c^2\right )\right ) x}{c x^2+b x+a}d\sqrt {x}}{a}-\frac {2 \left (a b B \left (b^2-8 a c\right )-A \left (60 a^2 c^2-37 a b^2 c+5 b^4\right )\right )}{a \sqrt {x}}\right )}{2 a \left (b^2-4 a c\right )}-\frac {-A \left (36 a^2 c^2-35 a b^2 c+5 b^4\right )+c x \left (a B \left (b^2-28 a c\right )-A \left (5 b^3-32 a b c\right )\right )+a b B \left (b^2-16 a c\right )}{a \sqrt {x} \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{4 a \left (b^2-4 a c\right )}+\frac {c x (A b-2 a B)-2 a A c-a b B+A b^2}{2 a \sqrt {x} \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {-\frac {3 \left (-\frac {2 \left (\frac {1}{2} c \left (-A \left (60 a^2 c^2-37 a b^2 c+5 b^4\right )+\frac {a B \left (56 a^2 c^2-10 a b^2 c+b^4\right )-A \left (124 a^2 b c^2-47 a b^3 c+5 b^5\right )}{\sqrt {b^2-4 a c}}+a b B \left (b^2-8 a c\right )\right ) \int \frac {1}{\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )+c x}d\sqrt {x}+\frac {1}{2} c \left (-A \left (60 a^2 c^2-37 a b^2 c+5 b^4\right )-\frac {a B \left (56 a^2 c^2-10 a b^2 c+b^4\right )-A \left (124 a^2 b c^2-47 a b^3 c+5 b^5\right )}{\sqrt {b^2-4 a c}}+a b B \left (b^2-8 a c\right )\right ) \int \frac {1}{\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )+c x}d\sqrt {x}\right )}{a}-\frac {2 \left (a b B \left (b^2-8 a c\right )-A \left (60 a^2 c^2-37 a b^2 c+5 b^4\right )\right )}{a \sqrt {x}}\right )}{2 a \left (b^2-4 a c\right )}-\frac {-A \left (36 a^2 c^2-35 a b^2 c+5 b^4\right )+c x \left (a B \left (b^2-28 a c\right )-A \left (5 b^3-32 a b c\right )\right )+a b B \left (b^2-16 a c\right )}{a \sqrt {x} \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{4 a \left (b^2-4 a c\right )}+\frac {c x (A b-2 a B)-2 a A c-a b B+A b^2}{2 a \sqrt {x} \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {-\frac {3 \left (-\frac {2 \left (\frac {\sqrt {c} \left (-A \left (60 a^2 c^2-37 a b^2 c+5 b^4\right )+\frac {a B \left (56 a^2 c^2-10 a b^2 c+b^4\right )-A \left (124 a^2 b c^2-47 a b^3 c+5 b^5\right )}{\sqrt {b^2-4 a c}}+a b B \left (b^2-8 a c\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {c} \left (-A \left (60 a^2 c^2-37 a b^2 c+5 b^4\right )-\frac {a B \left (56 a^2 c^2-10 a b^2 c+b^4\right )-A \left (124 a^2 b c^2-47 a b^3 c+5 b^5\right )}{\sqrt {b^2-4 a c}}+a b B \left (b^2-8 a c\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {2} \sqrt {\sqrt {b^2-4 a c}+b}}\right )}{a}-\frac {2 \left (a b B \left (b^2-8 a c\right )-A \left (60 a^2 c^2-37 a b^2 c+5 b^4\right )\right )}{a \sqrt {x}}\right )}{2 a \left (b^2-4 a c\right )}-\frac {-A \left (36 a^2 c^2-35 a b^2 c+5 b^4\right )+c x \left (a B \left (b^2-28 a c\right )-A \left (5 b^3-32 a b c\right )\right )+a b B \left (b^2-16 a c\right )}{a \sqrt {x} \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{4 a \left (b^2-4 a c\right )}+\frac {c x (A b-2 a B)-2 a A c-a b B+A b^2}{2 a \sqrt {x} \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
(A*b^2 - a*b*B - 2*a*A*c + (A*b - 2*a*B)*c*x)/(2*a*(b^2 - 4*a*c)*Sqrt[x]*( a + b*x + c*x^2)^2) + (-((a*b*B*(b^2 - 16*a*c) - A*(5*b^4 - 35*a*b^2*c + 3 6*a^2*c^2) + c*(a*B*(b^2 - 28*a*c) - A*(5*b^3 - 32*a*b*c))*x)/(a*(b^2 - 4* a*c)*Sqrt[x]*(a + b*x + c*x^2))) - (3*((-2*(a*b*B*(b^2 - 8*a*c) - A*(5*b^4 - 37*a*b^2*c + 60*a^2*c^2)))/(a*Sqrt[x]) - (2*((Sqrt[c]*(a*b*B*(b^2 - 8*a *c) - A*(5*b^4 - 37*a*b^2*c + 60*a^2*c^2) + (a*B*(b^4 - 10*a*b^2*c + 56*a^ 2*c^2) - A*(5*b^5 - 47*a*b^3*c + 124*a^2*b*c^2))/Sqrt[b^2 - 4*a*c])*ArcTan [(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[c]*(a*b*B*(b^2 - 8*a*c) - A*(5*b^4 - 37*a*b^2 *c + 60*a^2*c^2) - (a*B*(b^4 - 10*a*b^2*c + 56*a^2*c^2) - A*(5*b^5 - 47*a* b^3*c + 124*a^2*b*c^2))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x] )/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[b + Sqrt[b^2 - 4*a*c]])))/a) )/(2*a*(b^2 - 4*a*c)))/(4*a*(b^2 - 4*a*c))
3.11.28.3.1 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/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)), x_Symbol] :> Simp[2 Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; Fr eeQ[{a, b, c, d, e, f, g}, x]
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*f - d*g)*((d + e*x)^(m + 1)/((m + 1)*(c *d^2 - b*d*e + a*e^2))), x] + Simp[1/(c*d^2 - b*d*e + a*e^2) Int[(d + e*x )^(m + 1)*(Simp[c*d*f - f*b*e + a*e*g - c*(e*f - d*g)*x, x]/(a + b*x + c*x^ 2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && FractionQ[m] && LtQ[m, -1 ]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2 *a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^m *(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p] )
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/( b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
Time = 0.47 (sec) , antiderivative size = 762, normalized size of antiderivative = 1.15
| method | result | size |
| derivativedivides | \(-\frac {2 \left (\frac {\frac {c^{2} \left (52 A \,a^{2} c^{2}-47 A a \,b^{2} c +7 A \,b^{4}+24 a^{2} b B c -3 B \,b^{3} a \right ) x^{\frac {7}{2}}}{128 a^{2} c^{2}-64 a \,b^{2} c +8 b^{4}}+\frac {c \left (136 A \,a^{2} b \,c^{2}-99 A a \,b^{3} c +14 A \,b^{5}-28 B \,a^{3} c^{2}+49 B \,a^{2} b^{2} c -6 B a \,b^{4}\right ) x^{\frac {5}{2}}}{128 a^{2} c^{2}-64 a \,b^{2} c +8 b^{4}}+\frac {\left (68 A \,a^{3} c^{3}+25 A \,a^{2} b^{2} c^{2}-43 A a \,b^{4} c +7 A \,b^{6}+4 B \,a^{3} b \,c^{2}+20 B \,a^{2} b^{3} c -3 B a \,b^{5}\right ) x^{\frac {3}{2}}}{128 a^{2} c^{2}-64 a \,b^{2} c +8 b^{4}}+\frac {a \left (108 A \,a^{2} b \,c^{2}-66 A a \,b^{3} c +9 A \,b^{5}-44 B \,a^{3} c^{2}+37 B \,a^{2} b^{2} c -5 B a \,b^{4}\right ) \sqrt {x}}{128 a^{2} c^{2}-64 a \,b^{2} c +8 b^{4}}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {3 c \left (-\frac {\left (60 A \,a^{2} c^{2} \sqrt {-4 a c +b^{2}}-37 A a \,b^{2} c \sqrt {-4 a c +b^{2}}+5 A \,b^{4} \sqrt {-4 a c +b^{2}}+124 A \,a^{2} b \,c^{2}-47 A a \,b^{3} c +5 A \,b^{5}+8 a^{2} b B c \sqrt {-4 a c +b^{2}}-B \,b^{3} a \sqrt {-4 a c +b^{2}}-56 B \,a^{3} c^{2}+10 B \,a^{2} b^{2} c -B a \,b^{4}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (60 A \,a^{2} c^{2} \sqrt {-4 a c +b^{2}}-37 A a \,b^{2} c \sqrt {-4 a c +b^{2}}+5 A \,b^{4} \sqrt {-4 a c +b^{2}}-124 A \,a^{2} b \,c^{2}+47 A a \,b^{3} c -5 A \,b^{5}+8 a^{2} b B c \sqrt {-4 a c +b^{2}}-B \,b^{3} a \sqrt {-4 a c +b^{2}}+56 B \,a^{3} c^{2}-10 B \,a^{2} b^{2} c +B a \,b^{4}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}\right )}{a^{3}}-\frac {2 A}{a^{3} \sqrt {x}}\) | \(762\) |
| default | \(-\frac {2 \left (\frac {\frac {c^{2} \left (52 A \,a^{2} c^{2}-47 A a \,b^{2} c +7 A \,b^{4}+24 a^{2} b B c -3 B \,b^{3} a \right ) x^{\frac {7}{2}}}{128 a^{2} c^{2}-64 a \,b^{2} c +8 b^{4}}+\frac {c \left (136 A \,a^{2} b \,c^{2}-99 A a \,b^{3} c +14 A \,b^{5}-28 B \,a^{3} c^{2}+49 B \,a^{2} b^{2} c -6 B a \,b^{4}\right ) x^{\frac {5}{2}}}{128 a^{2} c^{2}-64 a \,b^{2} c +8 b^{4}}+\frac {\left (68 A \,a^{3} c^{3}+25 A \,a^{2} b^{2} c^{2}-43 A a \,b^{4} c +7 A \,b^{6}+4 B \,a^{3} b \,c^{2}+20 B \,a^{2} b^{3} c -3 B a \,b^{5}\right ) x^{\frac {3}{2}}}{128 a^{2} c^{2}-64 a \,b^{2} c +8 b^{4}}+\frac {a \left (108 A \,a^{2} b \,c^{2}-66 A a \,b^{3} c +9 A \,b^{5}-44 B \,a^{3} c^{2}+37 B \,a^{2} b^{2} c -5 B a \,b^{4}\right ) \sqrt {x}}{128 a^{2} c^{2}-64 a \,b^{2} c +8 b^{4}}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {3 c \left (-\frac {\left (60 A \,a^{2} c^{2} \sqrt {-4 a c +b^{2}}-37 A a \,b^{2} c \sqrt {-4 a c +b^{2}}+5 A \,b^{4} \sqrt {-4 a c +b^{2}}+124 A \,a^{2} b \,c^{2}-47 A a \,b^{3} c +5 A \,b^{5}+8 a^{2} b B c \sqrt {-4 a c +b^{2}}-B \,b^{3} a \sqrt {-4 a c +b^{2}}-56 B \,a^{3} c^{2}+10 B \,a^{2} b^{2} c -B a \,b^{4}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (60 A \,a^{2} c^{2} \sqrt {-4 a c +b^{2}}-37 A a \,b^{2} c \sqrt {-4 a c +b^{2}}+5 A \,b^{4} \sqrt {-4 a c +b^{2}}-124 A \,a^{2} b \,c^{2}+47 A a \,b^{3} c -5 A \,b^{5}+8 a^{2} b B c \sqrt {-4 a c +b^{2}}-B \,b^{3} a \sqrt {-4 a c +b^{2}}+56 B \,a^{3} c^{2}-10 B \,a^{2} b^{2} c +B a \,b^{4}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}\right )}{a^{3}}-\frac {2 A}{a^{3} \sqrt {x}}\) | \(762\) |
| risch | \(-\frac {2 A}{a^{3} \sqrt {x}}-\frac {\frac {\frac {2 c^{2} \left (52 A \,a^{2} c^{2}-47 A a \,b^{2} c +7 A \,b^{4}+24 a^{2} b B c -3 B \,b^{3} a \right ) x^{\frac {7}{2}}}{128 a^{2} c^{2}-64 a \,b^{2} c +8 b^{4}}+\frac {2 c \left (136 A \,a^{2} b \,c^{2}-99 A a \,b^{3} c +14 A \,b^{5}-28 B \,a^{3} c^{2}+49 B \,a^{2} b^{2} c -6 B a \,b^{4}\right ) x^{\frac {5}{2}}}{128 a^{2} c^{2}-64 a \,b^{2} c +8 b^{4}}+\frac {2 \left (68 A \,a^{3} c^{3}+25 A \,a^{2} b^{2} c^{2}-43 A a \,b^{4} c +7 A \,b^{6}+4 B \,a^{3} b \,c^{2}+20 B \,a^{2} b^{3} c -3 B a \,b^{5}\right ) x^{\frac {3}{2}}}{128 a^{2} c^{2}-64 a \,b^{2} c +8 b^{4}}+\frac {2 a \left (108 A \,a^{2} b \,c^{2}-66 A a \,b^{3} c +9 A \,b^{5}-44 B \,a^{3} c^{2}+37 B \,a^{2} b^{2} c -5 B a \,b^{4}\right ) \sqrt {x}}{128 a^{2} c^{2}-64 a \,b^{2} c +8 b^{4}}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {3 c \left (-\frac {\left (60 A \,a^{2} c^{2} \sqrt {-4 a c +b^{2}}-37 A a \,b^{2} c \sqrt {-4 a c +b^{2}}+5 A \,b^{4} \sqrt {-4 a c +b^{2}}+124 A \,a^{2} b \,c^{2}-47 A a \,b^{3} c +5 A \,b^{5}+8 a^{2} b B c \sqrt {-4 a c +b^{2}}-B \,b^{3} a \sqrt {-4 a c +b^{2}}-56 B \,a^{3} c^{2}+10 B \,a^{2} b^{2} c -B a \,b^{4}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (60 A \,a^{2} c^{2} \sqrt {-4 a c +b^{2}}-37 A a \,b^{2} c \sqrt {-4 a c +b^{2}}+5 A \,b^{4} \sqrt {-4 a c +b^{2}}-124 A \,a^{2} b \,c^{2}+47 A a \,b^{3} c -5 A \,b^{5}+8 a^{2} b B c \sqrt {-4 a c +b^{2}}-B \,b^{3} a \sqrt {-4 a c +b^{2}}+56 B \,a^{3} c^{2}-10 B \,a^{2} b^{2} c +B a \,b^{4}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}}{a^{3}}\) | \(763\) |
-2/a^3*((1/8*c^2*(52*A*a^2*c^2-47*A*a*b^2*c+7*A*b^4+24*B*a^2*b*c-3*B*a*b^3 )/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(7/2)+1/8*c*(136*A*a^2*b*c^2-99*A*a*b^3*c+1 4*A*b^5-28*B*a^3*c^2+49*B*a^2*b^2*c-6*B*a*b^4)/(16*a^2*c^2-8*a*b^2*c+b^4)* x^(5/2)+1/8*(68*A*a^3*c^3+25*A*a^2*b^2*c^2-43*A*a*b^4*c+7*A*b^6+4*B*a^3*b* c^2+20*B*a^2*b^3*c-3*B*a*b^5)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(3/2)+1/8*a*(10 8*A*a^2*b*c^2-66*A*a*b^3*c+9*A*b^5-44*B*a^3*c^2+37*B*a^2*b^2*c-5*B*a*b^4)/ (16*a^2*c^2-8*a*b^2*c+b^4)*x^(1/2))/(c*x^2+b*x+a)^2+3/2/(16*a^2*c^2-8*a*b^ 2*c+b^4)*c*(-1/8*(60*A*a^2*c^2*(-4*a*c+b^2)^(1/2)-37*A*a*b^2*c*(-4*a*c+b^2 )^(1/2)+5*A*b^4*(-4*a*c+b^2)^(1/2)+124*A*a^2*b*c^2-47*A*a*b^3*c+5*A*b^5+8* a^2*b*B*c*(-4*a*c+b^2)^(1/2)-B*b^3*a*(-4*a*c+b^2)^(1/2)-56*B*a^3*c^2+10*B* a^2*b^2*c-B*a*b^4)/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^ (1/2)*arctanh(c*x^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))+1/8*(60 *A*a^2*c^2*(-4*a*c+b^2)^(1/2)-37*A*a*b^2*c*(-4*a*c+b^2)^(1/2)+5*A*b^4*(-4* a*c+b^2)^(1/2)-124*A*a^2*b*c^2+47*A*a*b^3*c-5*A*b^5+8*a^2*b*B*c*(-4*a*c+b^ 2)^(1/2)-B*b^3*a*(-4*a*c+b^2)^(1/2)+56*B*a^3*c^2-10*B*a^2*b^2*c+B*a*b^4)/( -4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x^(1/2 )*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))))-2*A/a^3/x^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 12534 vs. \(2 (589) = 1178\).
Time = 147.66 (sec) , antiderivative size = 12534, normalized size of antiderivative = 18.88 \[ \int \frac {A+B x}{x^{3/2} \left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {A+B x}{x^{3/2} \left (a+b x+c x^2\right )^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {A+B x}{x^{3/2} \left (a+b x+c x^2\right )^3} \, dx=\text {Exception raised: RuntimeError} \]
Leaf count of result is larger than twice the leaf count of optimal. 9534 vs. \(2 (589) = 1178\).
Time = 1.87 (sec) , antiderivative size = 9534, normalized size of antiderivative = 14.36 \[ \int \frac {A+B x}{x^{3/2} \left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]
-3/32*((10*b^6*c^2 - 114*a*b^4*c^3 + 416*a^2*b^2*c^4 - 480*a^3*c^5 - 5*sqr t(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^6 + 57*sqrt(2)*sq rt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^4*c + 10*sqrt(2)*sqrt( b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^5*c - 208*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^2 - 74*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c^2 - 5*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4*c^2 + 240*sqrt(2)*sqrt(b^2 - 4 *a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*c^3 + 120*sqrt(2)*sqrt(b^2 - 4*a *c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^3 + 37*sqrt(2)*sqrt(b^2 - 4*a* c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^3 - 60*sqrt(2)*sqrt(b^2 - 4*a*c )*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*c^4 - 10*(b^2 - 4*a*c)*b^4*c^2 + 74* (b^2 - 4*a*c)*a*b^2*c^3 - 120*(b^2 - 4*a*c)*a^2*c^4)*(a^3*b^4 - 8*a^4*b^2* c + 16*a^5*c^2)^2*A - (2*a*b^5*c^2 - 24*a^2*b^3*c^3 + 64*a^3*b*c^4 - sqrt( 2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^5 + 12*sqrt(2)*sq rt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^3*c + 2*sqrt(2)*sqrt (b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^4*c - 32*sqrt(2)*sqrt(b^ 2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b*c^2 - 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^2 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c^2 + 8*sqrt(2)*sqrt(b^2 - 4 *a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^3 - 2*(b^2 - 4*a*c)*a*b^3...
Time = 18.98 (sec) , antiderivative size = 29137, normalized size of antiderivative = 43.88 \[ \int \frac {A+B x}{x^{3/2} \left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]
- atan(((x^(1/2)*(33973862400*A^2*a^20*c^14 - 7398752256*B^2*a^21*c^13 - 2 8800*A^2*a^9*b^22*c^3 + 1232640*A^2*a^10*b^20*c^4 - 23879808*A^2*a^11*b^18 *c^5 + 275975424*A^2*a^12*b^16*c^6 - 2109763584*A^2*a^13*b^14*c^7 + 111718 56384*A^2*a^14*b^12*c^8 - 41653370880*A^2*a^15*b^10*c^9 + 108726976512*A^2 *a^16*b^8*c^10 - 192980975616*A^2*a^17*b^6*c^11 + 218414186496*A^2*a^18*b^ 4*c^12 - 137631891456*A^2*a^19*b^2*c^13 - 1152*B^2*a^11*b^20*c^3 + 50688*B ^2*a^12*b^18*c^4 - 1025280*B^2*a^13*b^16*c^5 + 12496896*B^2*a^14*b^14*c^6 - 101744640*B^2*a^15*b^12*c^7 + 579796992*B^2*a^16*b^10*c^8 - 2346319872*B ^2*a^17*b^8*c^9 + 6653214720*B^2*a^18*b^6*c^10 - 12608077824*B^2*a^19*b^4* c^11 + 14344519680*B^2*a^20*b^2*c^12 + 11520*A*B*a^10*b^21*c^3 - 499968*A* B*a^11*b^19*c^4 + 9900288*A*B*a^12*b^17*c^5 - 117559296*A*B*a^13*b^15*c^6 + 925433856*A*B*a^14*b^13*c^7 - 5038866432*A*B*a^15*b^11*c^8 + 19191693312 *A*B*a^16*b^9*c^9 - 50422874112*A*B*a^17*b^7*c^10 + 87350575104*A*B*a^18*b ^5*c^11 - 89992986624*A*B*a^19*b^3*c^12 + 41825599488*A*B*a^20*b*c^13) + ( -(9*(25*A^2*b^21 + B^2*a^2*b^19 + 25*A^2*b^6*(-(4*a*c - b^2)^15)^(1/2) - 1 0*A*B*a*b^20 + 17794*A^2*a^2*b^17*c^2 - 188095*A^2*a^3*b^15*c^3 + 1299860* A^2*a^4*b^13*c^4 - 6126640*A^2*a^5*b^11*c^5 + 19905600*A^2*a^6*b^9*c^6 - 4 3904256*A^2*a^7*b^7*c^7 + 62684160*A^2*a^8*b^5*c^8 - 52039680*A^2*a^9*b^3* c^9 - 225*A^2*a^3*c^3*(-(4*a*c - b^2)^15)^(1/2) + B^2*a^2*b^4*(-(4*a*c - b ^2)^15)^(1/2) + 769*B^2*a^4*b^15*c^2 - 8620*B^2*a^5*b^13*c^3 + 63440*B^...