Integrand size = 22, antiderivative size = 118 \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 (d+e x)^2 (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {16 \left (c d^2-b d e+a e^2\right ) (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}} \]
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Time = 0.03 (sec) , antiderivative size = 118, 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 = {736, 650} \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {16 \left (a e^2-b d e+c d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {2 (d+e x)^2 (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]
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Rule 650
Rule 736
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (d+e x)^2 (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {\left (8 \left (c d^2-b d e+a e^2\right )\right ) \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 (d+e x)^2 (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {16 \left (c d^2-b d e+a e^2\right ) (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 = 1.17 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.61 \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 \left (-6 b^2 (a e-c d x) \left (d^2-6 d e x+e^2 x^2\right )+b^3 \left (-d^3-9 d^2 e x+9 d e^2 x^2+e^3 x^3\right )+12 b (d-e x) \left (2 a^2 e^2+2 c^2 d^2 x^2+a c (d-e x)^2\right )-8 \left (2 a^3 e^3-2 c^3 d^3 x^3+3 a^2 c e \left (d^2+e^2 x^2\right )-3 a c^2 d x \left (d^2+e^2 x^2\right )\right )\right )}{3 \left (b^2-4 a c\right )^2 (a+x (b+c x))^{3/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(286\) vs. \(2(110)=220\).
Time = 0.42 (sec) , antiderivative size = 287, normalized size of antiderivative = 2.43
method | result | size |
trager | \(-\frac {2 \left (12 a b c \,e^{3} x^{3}-24 a \,c^{2} d \,e^{2} x^{3}-b^{3} e^{3} x^{3}-6 b^{2} c d \,e^{2} x^{3}+24 b \,c^{2} d^{2} e \,x^{3}-16 c^{3} d^{3} x^{3}+24 a^{2} c \,e^{3} x^{2}+6 a \,b^{2} e^{3} x^{2}-36 a b c d \,e^{2} x^{2}-9 b^{3} d \,e^{2} x^{2}+36 b^{2} c \,d^{2} e \,x^{2}-24 b \,c^{2} d^{3} x^{2}+24 a^{2} b \,e^{3} x -36 a \,b^{2} d \,e^{2} x +36 a b c \,d^{2} e x -24 a \,c^{2} d^{3} x +9 b^{3} d^{2} e x -6 b^{2} c \,d^{3} x +16 a^{3} e^{3}-24 a^{2} b d \,e^{2}+24 a^{2} c \,d^{2} e +6 a \,b^{2} d^{2} e -12 a b c \,d^{3}+b^{3} d^{3}\right )}{3 \left (4 a c -b^{2}\right )^{2} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}\) | \(287\) |
gosper | \(-\frac {2 \left (12 a b c \,e^{3} x^{3}-24 a \,c^{2} d \,e^{2} x^{3}-b^{3} e^{3} x^{3}-6 b^{2} c d \,e^{2} x^{3}+24 b \,c^{2} d^{2} e \,x^{3}-16 c^{3} d^{3} x^{3}+24 a^{2} c \,e^{3} x^{2}+6 a \,b^{2} e^{3} x^{2}-36 a b c d \,e^{2} x^{2}-9 b^{3} d \,e^{2} x^{2}+36 b^{2} c \,d^{2} e \,x^{2}-24 b \,c^{2} d^{3} x^{2}+24 a^{2} b \,e^{3} x -36 a \,b^{2} d \,e^{2} x +36 a b c \,d^{2} e x -24 a \,c^{2} d^{3} x +9 b^{3} d^{2} e x -6 b^{2} c \,d^{3} x +16 a^{3} e^{3}-24 a^{2} b d \,e^{2}+24 a^{2} c \,d^{2} e +6 a \,b^{2} d^{2} e -12 a b c \,d^{3}+b^{3} d^{3}\right )}{3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}\) | \(296\) |
default | \(d^{3} \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^{3} \left (-\frac {x^{2}}{c \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {b \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 c}+\frac {2 a \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 )}{c}\right )+3 d \,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 )+3 d^{2} 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 )\) | \(675\) |
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Leaf count of result is larger than twice the leaf count of optimal. 366 vs. \(2 (110) = 220\).
Time = 1.23 (sec) , antiderivative size = 366, normalized size of antiderivative = 3.10 \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (24 \, a^{2} b d e^{2} - 16 \, a^{3} e^{3} - {\left (b^{3} - 12 \, a b c\right )} d^{3} - 6 \, {\left (a b^{2} + 4 \, a^{2} c\right )} d^{2} e + {\left (16 \, c^{3} d^{3} - 24 \, b c^{2} d^{2} e + 6 \, {\left (b^{2} c + 4 \, a c^{2}\right )} d e^{2} + {\left (b^{3} - 12 \, a b c\right )} e^{3}\right )} x^{3} + 3 \, {\left (8 \, b c^{2} d^{3} - 12 \, b^{2} c d^{2} e + 3 \, {\left (b^{3} + 4 \, a b c\right )} d e^{2} - 2 \, {\left (a b^{2} + 4 \, a^{2} c\right )} e^{3}\right )} x^{2} + 3 \, {\left (12 \, a b^{2} d e^{2} - 8 \, a^{2} b e^{3} + 2 \, {\left (b^{2} c + 4 \, a c^{2}\right )} d^{3} - 3 \, {\left (b^{3} + 4 \, a b c\right )} d^{2} e\right )} x\right )} \sqrt {c x^{2} + b x + a}}{3 \, {\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2} + {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{4} + 2 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{3} + {\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} x^{2} + 2 \, {\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} x\right )}} \]
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Timed out. \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 333 vs. \(2 (110) = 220\).
Time = 0.29 (sec) , antiderivative size = 333, normalized size of antiderivative = 2.82 \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 \, {\left ({\left ({\left (\frac {{\left (16 \, c^{3} d^{3} - 24 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} + 24 \, a c^{2} d e^{2} + b^{3} e^{3} - 12 \, a b c e^{3}\right )} x}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}} + \frac {3 \, {\left (8 \, b c^{2} d^{3} - 12 \, b^{2} c d^{2} e + 3 \, b^{3} d e^{2} + 12 \, a b c d e^{2} - 2 \, a b^{2} e^{3} - 8 \, a^{2} c e^{3}\right )}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x + \frac {3 \, {\left (2 \, b^{2} c d^{3} + 8 \, a c^{2} d^{3} - 3 \, b^{3} d^{2} e - 12 \, a b c d^{2} e + 12 \, a b^{2} d e^{2} - 8 \, a^{2} b e^{3}\right )}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x - \frac {b^{3} d^{3} - 12 \, a b c d^{3} + 6 \, a b^{2} d^{2} e + 24 \, a^{2} c d^{2} e - 24 \, a^{2} b d e^{2} + 16 \, a^{3} e^{3}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )}}{3 \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \]
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Time = 10.53 (sec) , antiderivative size = 528, normalized size of antiderivative = 4.47 \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2\,a\,b^4\,e^3+2\,b^5\,e^3\,x-2\,b^4\,e^3\,\left (c\,x^2+b\,x+a\right )+16\,a^3\,c^2\,e^3-2\,b^3\,c^2\,d^3-12\,a^2\,b^2\,c\,e^3-48\,a^2\,c^3\,d^2\,e-4\,b^2\,c^3\,d^3\,x-48\,a^2\,c^2\,e^3\,\left (c\,x^2+b\,x+a\right )+8\,a\,b\,c^3\,d^3+16\,a\,c^4\,d^3\,x+16\,b\,c^3\,d^3\,\left (c\,x^2+b\,x+a\right )+32\,c^4\,d^3\,x\,\left (c\,x^2+b\,x+a\right )-6\,a\,b^3\,c\,d\,e^2-14\,a\,b^3\,c\,e^3\,x-6\,b^4\,c\,d\,e^2\,x+12\,a\,b^2\,c\,e^3\,\left (c\,x^2+b\,x+a\right )+6\,b^3\,c\,d\,e^2\,\left (c\,x^2+b\,x+a\right )+2\,b^3\,c\,e^3\,x\,\left (c\,x^2+b\,x+a\right )+12\,a\,b^2\,c^2\,d^2\,e+24\,a^2\,b\,c^2\,d\,e^2+24\,a^2\,b\,c^2\,e^3\,x-48\,a^2\,c^3\,d\,e^2\,x+6\,b^3\,c^2\,d^2\,e\,x-24\,b^2\,c^2\,d^2\,e\,\left (c\,x^2+b\,x+a\right )+12\,b^2\,c^2\,d\,e^2\,x\,\left (c\,x^2+b\,x+a\right )-24\,a\,b\,c^3\,d^2\,e\,x+24\,a\,b\,c^2\,d\,e^2\,\left (c\,x^2+b\,x+a\right )-24\,a\,b\,c^2\,e^3\,x\,\left (c\,x^2+b\,x+a\right )+48\,a\,c^3\,d\,e^2\,x\,\left (c\,x^2+b\,x+a\right )-48\,b\,c^3\,d^2\,e\,x\,\left (c\,x^2+b\,x+a\right )+36\,a\,b^2\,c^2\,d\,e^2\,x}{\left (48\,a^2\,c^4-24\,a\,b^2\,c^3+3\,b^4\,c^2\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \]
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