Integrand size = 22, antiderivative size = 114 \[ \int \frac {A+C x^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \left (b c \left (A+\frac {a C}{c}\right )+\left (2 A c^2+\left (b^2-2 a c\right ) C\right ) x\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {2 \left (8 A c+4 a C+\frac {b^2 C}{c}\right ) (b+2 c x)}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1674, 12, 627} \[ \int \frac {A+C x^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 (b+2 c x) \left (4 a C+8 A c+\frac {b^2 C}{c}\right )}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {2 \left (x \left (C \left (b^2-2 a c\right )+2 A c^2\right )+b c \left (\frac {a C}{c}+A\right )\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]
[In]
[Out]
Rule 12
Rule 627
Rule 1674
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \left (b c \left (A+\frac {a C}{c}\right )+\left (2 A c^2+\left (b^2-2 a c\right ) C\right ) x\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \int \frac {8 A c+4 a C+\frac {b^2 C}{c}}{2 \left (a+b x+c x^2\right )^{3/2}} \, dx}{3 \left (b^2-4 a c\right )} \\ & = -\frac {2 \left (b c \left (A+\frac {a C}{c}\right )+\left (2 A c^2+\left (b^2-2 a c\right ) C\right ) x\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {\left (8 A c+4 a C+\frac {b^2 C}{c}\right ) \int \frac {1}{\left (a+b x+c x^2\right )^{3/2}} \, dx}{3 \left (b^2-4 a c\right )} \\ & = -\frac {2 \left (b c \left (A+\frac {a C}{c}\right )+\left (2 A c^2+\left (b^2-2 a c\right ) C\right ) x\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {2 \left (8 A c+4 a C+\frac {b^2 C}{c}\right ) (b+2 c x)}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}} \\ \end{align*}
Time = 0.81 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.94 \[ \int \frac {A+C x^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {-2 A (b+2 c x) \left (b^2-8 b c x-4 c \left (3 a+2 c x^2\right )\right )+2 C \left (8 a^2 b+b^2 x^2 (3 b+2 c x)+4 a x \left (3 b^2+3 b c x+2 c^2 x^2\right )\right )}{3 \left (b^2-4 a c\right )^2 (a+x (b+c x))^{3/2}} \]
[In]
[Out]
Time = 0.68 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.12
method | result | size |
trager | \(\frac {\frac {32}{3} A \,c^{3} x^{3}+\frac {16}{3} C a \,c^{2} x^{3}+\frac {4}{3} C \,b^{2} c \,x^{3}+16 A b \,c^{2} x^{2}+8 C a b c \,x^{2}+2 C \,b^{3} x^{2}+16 a A \,c^{2} x +4 A \,b^{2} c x +8 C a \,b^{2} x +8 A a b c -\frac {2}{3} A \,b^{3}+\frac {16}{3} b \,a^{2} C}{\left (4 a c -b^{2}\right )^{2} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}\) | \(128\) |
gosper | \(\frac {\frac {32}{3} A \,c^{3} x^{3}+\frac {16}{3} C a \,c^{2} x^{3}+\frac {4}{3} C \,b^{2} c \,x^{3}+16 A b \,c^{2} x^{2}+8 C a b c \,x^{2}+2 C \,b^{3} x^{2}+16 a A \,c^{2} x +4 A \,b^{2} c x +8 C a \,b^{2} x +8 A a b c -\frac {2}{3} A \,b^{3}+\frac {16}{3} b \,a^{2} C}{\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}\) | \(137\) |
default | \(A \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )+C \left (-\frac {x}{2 c \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {1}{3 c \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {b \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}\right )}{4 c}+\frac {a \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}\right )\) | \(259\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 242 vs. \(2 (106) = 212\).
Time = 0.67 (sec) , antiderivative size = 242, normalized size of antiderivative = 2.12 \[ \int \frac {A+C x^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (8 \, C a^{2} b - A b^{3} + 12 \, A a b c + 2 \, {\left (C b^{2} c + 4 \, C a c^{2} + 8 \, A c^{3}\right )} x^{3} + 3 \, {\left (C b^{3} + 4 \, C a b c + 8 \, A b c^{2}\right )} x^{2} + 6 \, {\left (2 \, C a b^{2} + A b^{2} c + 4 \, A a c^{2}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{3 \, {\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2} + {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{4} + 2 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{3} + {\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} x^{2} + 2 \, {\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} x\right )}} \]
[In]
[Out]
Timed out. \[ \int \frac {A+C x^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Timed out} \]
[In]
[Out]
Exception generated. \[ \int \frac {A+C x^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.69 \[ \int \frac {A+C x^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 \, {\left ({\left ({\left (\frac {2 \, {\left (C b^{2} c + 4 \, C a c^{2} + 8 \, A c^{3}\right )} x}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}} + \frac {3 \, {\left (C b^{3} + 4 \, C a b c + 8 \, A b c^{2}\right )}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x + \frac {6 \, {\left (2 \, C a b^{2} + A b^{2} c + 4 \, A a c^{2}\right )}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x + \frac {8 \, C a^{2} b - A b^{3} + 12 \, A a b c}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )}}{3 \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \]
[In]
[Out]
Time = 13.39 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.11 \[ \int \frac {A+C x^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2\,\left (8\,C\,a^2\,b+12\,C\,a\,b^2\,x+12\,C\,a\,b\,c\,x^2+12\,A\,a\,b\,c+8\,C\,a\,c^2\,x^3+24\,A\,a\,c^2\,x+3\,C\,b^3\,x^2-A\,b^3+2\,C\,b^2\,c\,x^3+6\,A\,b^2\,c\,x+24\,A\,b\,c^2\,x^2+16\,A\,c^3\,x^3\right )}{3\,{\left (4\,a\,c-b^2\right )}^2\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \]
[In]
[Out]