Integrand size = 22, antiderivative size = 98 \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 (b+2 c x) (d+e x)^2}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {8 (2 c d-b e) (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {742, 650} \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {8 (2 c d-b e) (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {2 (b+2 c x) (d+e x)^2}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]
[In]
[Out]
Rule 650
Rule 742
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (b+2 c x) (d+e x)^2}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {(4 (2 c d-b e)) \int \frac {d+e x}{\left (a+b x+c x^2\right )^{3/2}} \, dx}{3 \left (b^2-4 a c\right )} \\ & = -\frac {2 (b+2 c x) (d+e x)^2}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {8 (2 c d-b e) (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}} \\ \end{align*}
Time = 1.01 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.70 \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 \left (-b^3 \left (d^2+6 d e x-3 e^2 x^2\right )+4 b \left (2 a^2 e^2+2 c^2 d x^2 (3 d-2 e x)+3 a c (d-e x)^2\right )+8 c \left (-2 a^2 d e+2 c^2 d^2 x^3+a c x \left (3 d^2+e^2 x^2\right )\right )+b^2 \left (-4 a e (d-3 e x)+2 c x \left (3 d^2-12 d e x+e^2 x^2\right )\right )\right )}{3 \left (b^2-4 a c\right )^2 (a+x (b+c x))^{3/2}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(205\) vs. \(2(90)=180\).
Time = 0.32 (sec) , antiderivative size = 206, normalized size of antiderivative = 2.10
method | result | size |
trager | \(\frac {\frac {16}{3} a \,c^{2} e^{2} x^{3}+\frac {4}{3} b^{2} c \,e^{2} x^{3}-\frac {32}{3} b \,c^{2} d e \,x^{3}+\frac {32}{3} c^{3} d^{2} x^{3}+8 a b c \,e^{2} x^{2}+2 b^{3} e^{2} x^{2}-16 b^{2} c d e \,x^{2}+16 b \,c^{2} d^{2} x^{2}+8 a \,b^{2} e^{2} x -16 a b c d e x +16 a \,c^{2} d^{2} x -4 b^{3} d e x +4 b^{2} c \,d^{2} x +\frac {16}{3} a^{2} b \,e^{2}-\frac {32}{3} a^{2} c d e -\frac {8}{3} a \,b^{2} d e +8 a b c \,d^{2}-\frac {2}{3} b^{3} d^{2}}{\left (4 a c -b^{2}\right )^{2} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}\) | \(206\) |
gosper | \(\frac {\frac {16}{3} a \,c^{2} e^{2} x^{3}+\frac {4}{3} b^{2} c \,e^{2} x^{3}-\frac {32}{3} b \,c^{2} d e \,x^{3}+\frac {32}{3} c^{3} d^{2} x^{3}+8 a b c \,e^{2} x^{2}+2 b^{3} e^{2} x^{2}-16 b^{2} c d e \,x^{2}+16 b \,c^{2} d^{2} x^{2}+8 a \,b^{2} e^{2} x -16 a b c d e x +16 a \,c^{2} d^{2} x -4 b^{3} d e x +4 b^{2} c \,d^{2} x +\frac {16}{3} a^{2} b \,e^{2}-\frac {32}{3} a^{2} c d e -\frac {8}{3} a \,b^{2} d e +8 a b c \,d^{2}-\frac {2}{3} b^{3} d^{2}}{\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}\) | \(215\) |
default | \(d^{2} \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )+e^{2} \left (-\frac {x}{2 c \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {1}{3 c \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {b \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}\right )}{4 c}+\frac {a \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}\right )+2 d e \left (-\frac {1}{3 c \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {b \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}\right )\) | \(357\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 305 vs. \(2 (90) = 180\).
Time = 0.79 (sec) , antiderivative size = 305, normalized size of antiderivative = 3.11 \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (8 \, a^{2} b e^{2} + 2 \, {\left (8 \, c^{3} d^{2} - 8 \, b c^{2} d e + {\left (b^{2} c + 4 \, a c^{2}\right )} e^{2}\right )} x^{3} - {\left (b^{3} - 12 \, a b c\right )} d^{2} - 4 \, {\left (a b^{2} + 4 \, a^{2} c\right )} d e + 3 \, {\left (8 \, b c^{2} d^{2} - 8 \, b^{2} c d e + {\left (b^{3} + 4 \, a b c\right )} e^{2}\right )} x^{2} + 6 \, {\left (2 \, a b^{2} e^{2} + {\left (b^{2} c + 4 \, a c^{2}\right )} d^{2} - {\left (b^{3} + 4 \, a b c\right )} d e\right )} x\right )} \sqrt {c x^{2} + b x + a}}{3 \, {\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2} + {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{4} + 2 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{3} + {\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} x^{2} + 2 \, {\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} x\right )}} \]
[In]
[Out]
Timed out. \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Timed out} \]
[In]
[Out]
Exception generated. \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 263 vs. \(2 (90) = 180\).
Time = 0.31 (sec) , antiderivative size = 263, normalized size of antiderivative = 2.68 \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 \, {\left ({\left ({\left (\frac {2 \, {\left (8 \, c^{3} d^{2} - 8 \, b c^{2} d e + b^{2} c e^{2} + 4 \, a c^{2} e^{2}\right )} x}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}} + \frac {3 \, {\left (8 \, b c^{2} d^{2} - 8 \, b^{2} c d e + b^{3} e^{2} + 4 \, a b c e^{2}\right )}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x + \frac {6 \, {\left (b^{2} c d^{2} + 4 \, a c^{2} d^{2} - b^{3} d e - 4 \, a b c d e + 2 \, a b^{2} e^{2}\right )}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x - \frac {b^{3} d^{2} - 12 \, a b c d^{2} + 4 \, a b^{2} d e + 16 \, a^{2} c d e - 8 \, a^{2} b e^{2}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )}}{3 \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \]
[In]
[Out]
Time = 10.16 (sec) , antiderivative size = 321, normalized size of antiderivative = 3.28 \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2\,b^3\,e^2\,\left (c\,x^2+b\,x+a\right )-2\,b^3\,c\,d^2-2\,b^4\,e^2\,x-2\,a\,b^3\,e^2-16\,a^2\,c^2\,e^2\,x-4\,b^2\,c^2\,d^2\,x+8\,a\,b\,c^2\,d^2+8\,a^2\,b\,c\,e^2-32\,a^2\,c^2\,d\,e+16\,a\,c^3\,d^2\,x+16\,b\,c^2\,d^2\,\left (c\,x^2+b\,x+a\right )+32\,c^3\,d^2\,x\,\left (c\,x^2+b\,x+a\right )+12\,a\,b^2\,c\,e^2\,x+16\,a\,c^2\,e^2\,x\,\left (c\,x^2+b\,x+a\right )+4\,b^2\,c\,e^2\,x\,\left (c\,x^2+b\,x+a\right )+8\,a\,b^2\,c\,d\,e+4\,b^3\,c\,d\,e\,x+8\,a\,b\,c\,e^2\,\left (c\,x^2+b\,x+a\right )-16\,b^2\,c\,d\,e\,\left (c\,x^2+b\,x+a\right )-16\,a\,b\,c^2\,d\,e\,x-32\,b\,c^2\,d\,e\,x\,\left (c\,x^2+b\,x+a\right )}{\left (48\,a^2\,c^3-24\,a\,b^2\,c^2+3\,b^4\,c\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \]
[In]
[Out]