Integrand size = 26, antiderivative size = 48 \[ \int \frac {(b d+2 c d x)^3}{\left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 d^3 (b+2 c x)^2}{\sqrt {a+b x+c x^2}}+16 c d^3 \sqrt {a+b x+c x^2} \]
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Time = 0.02 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {700, 643} \[ \int \frac {(b d+2 c d x)^3}{\left (a+b x+c x^2\right )^{3/2}} \, dx=16 c d^3 \sqrt {a+b x+c x^2}-\frac {2 d^3 (b+2 c x)^2}{\sqrt {a+b x+c x^2}} \]
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Rule 643
Rule 700
Rubi steps \begin{align*} \text {integral}& = -\frac {2 d^3 (b+2 c x)^2}{\sqrt {a+b x+c x^2}}+\left (8 c d^2\right ) \int \frac {b d+2 c d x}{\sqrt {a+b x+c x^2}} \, dx \\ & = -\frac {2 d^3 (b+2 c x)^2}{\sqrt {a+b x+c x^2}}+16 c d^3 \sqrt {a+b x+c x^2} \\ \end{align*}
Time = 0.43 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.83 \[ \int \frac {(b d+2 c d x)^3}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {d^3 \left (-2 b^2+8 b c x+8 c \left (2 a+c x^2\right )\right )}{\sqrt {a+x (b+c x)}} \]
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Time = 2.52 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.85
method | result | size |
gosper | \(\frac {2 d^{3} \left (4 c^{2} x^{2}+4 b c x +8 a c -b^{2}\right )}{\sqrt {c \,x^{2}+b x +a}}\) | \(41\) |
trager | \(\frac {2 d^{3} \left (4 c^{2} x^{2}+4 b c x +8 a c -b^{2}\right )}{\sqrt {c \,x^{2}+b x +a}}\) | \(41\) |
pseudoelliptic | \(\frac {2 d^{3} \left (4 c^{2} x^{2}+4 b c x +8 a c -b^{2}\right )}{\sqrt {c \,x^{2}+b x +a}}\) | \(41\) |
risch | \(8 c \,d^{3} \sqrt {c \,x^{2}+b x +a}+\frac {2 \left (4 a c -b^{2}\right ) d^{3}}{\sqrt {c \,x^{2}+b x +a}}\) | \(47\) |
default | \(d^{3} \left (\frac {2 b^{3} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+8 c^{3} \left (\frac {x^{2}}{c \sqrt {c \,x^{2}+b x +a}}-\frac {3 b \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )}{2 c}-\frac {2 a \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{c}\right )+6 b^{2} c \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )+12 b \,c^{2} \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )\right )\) | \(414\) |
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Time = 0.32 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.98 \[ \int \frac {(b d+2 c d x)^3}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {2 \, {\left (4 \, c^{2} d^{3} x^{2} + 4 \, b c d^{3} x - {\left (b^{2} - 8 \, a c\right )} d^{3}\right )}}{\sqrt {c x^{2} + b x + a}} \]
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Time = 0.13 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.92 \[ \int \frac {(b d+2 c d x)^3}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {16 a c d^{3}}{\sqrt {a + b x + c x^{2}}} - \frac {2 b^{2} d^{3}}{\sqrt {a + b x + c x^{2}}} + \frac {8 b c d^{3} x}{\sqrt {a + b x + c x^{2}}} + \frac {8 c^{2} d^{3} x^{2}}{\sqrt {a + b x + c x^{2}}} \]
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Exception generated. \[ \int \frac {(b d+2 c d x)^3}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.26 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00 \[ \int \frac {(b d+2 c d x)^3}{\left (a+b x+c x^2\right )^{3/2}} \, dx=8 \, \sqrt {c x^{2} + b x + a} c d^{3} - \frac {2 \, {\left (b^{2} d^{3} - 4 \, a c d^{3}\right )}}{\sqrt {c x^{2} + b x + a}} \]
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Time = 9.50 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.83 \[ \int \frac {(b d+2 c d x)^3}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {2\,d^3\,\left (-b^2+4\,b\,c\,x+4\,c^2\,x^2+8\,a\,c\right )}{\sqrt {c\,x^2+b\,x+a}} \]
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