Integrand size = 44, antiderivative size = 210 \[ \int \frac {f+g x}{(d+e x)^3 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=-\frac {2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{5 e^2 (2 c d-b e) (d+e x)^3}-\frac {2 (4 c e f+6 c d g-5 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{15 e^2 (2 c d-b e)^2 (d+e x)^2}-\frac {4 c (4 c e f+6 c d g-5 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{15 e^2 (2 c d-b e)^3 (d+e x)} \]
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Time = 0.20 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.068, Rules used = {806, 672, 664} \[ \int \frac {f+g x}{(d+e x)^3 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=-\frac {2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{5 e^2 (d+e x)^3 (2 c d-b e)}-\frac {4 c \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-5 b e g+6 c d g+4 c e f)}{15 e^2 (d+e x) (2 c d-b e)^3}-\frac {2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-5 b e g+6 c d g+4 c e f)}{15 e^2 (d+e x)^2 (2 c d-b e)^2} \]
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Rule 664
Rule 672
Rule 806
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{5 e^2 (2 c d-b e) (d+e x)^3}+\frac {(4 c e f+6 c d g-5 b e g) \int \frac {1}{(d+e x)^2 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{5 e (2 c d-b e)} \\ & = -\frac {2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{5 e^2 (2 c d-b e) (d+e x)^3}-\frac {2 (4 c e f+6 c d g-5 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{15 e^2 (2 c d-b e)^2 (d+e x)^2}+\frac {(2 c (4 c e f+6 c d g-5 b e g)) \int \frac {1}{(d+e x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{15 e (2 c d-b e)^2} \\ & = -\frac {2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{5 e^2 (2 c d-b e) (d+e x)^3}-\frac {2 (4 c e f+6 c d g-5 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{15 e^2 (2 c d-b e)^2 (d+e x)^2}-\frac {4 c (4 c e f+6 c d g-5 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{15 e^2 (2 c d-b e)^3 (d+e x)} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.79 \[ \int \frac {f+g x}{(d+e x)^3 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=-\frac {2 (-c d+b e+c e x) \left (b^2 e^2 (3 e f+2 d g+5 e g x)-2 b c e \left (7 d^2 g+e^2 x (2 f+5 g x)+2 d e (4 f+9 g x)\right )+4 c^2 \left (3 d^3 g+2 e^3 f x^2+3 d e^2 x (2 f+g x)+d^2 e (7 f+9 g x)\right )\right )}{15 e^2 (-2 c d+b e)^3 (d+e x)^2 \sqrt {(d+e x) (-b e+c (d-e x))}} \]
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Time = 1.07 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.07
method | result | size |
trager | \(\frac {2 \left (-10 b c \,e^{3} g \,x^{2}+12 c^{2} d \,e^{2} g \,x^{2}+8 c^{2} e^{3} f \,x^{2}+5 b^{2} e^{3} g x -36 b c d \,e^{2} g x -4 b c \,e^{3} f x +36 c^{2} d^{2} e g x +24 c^{2} d \,e^{2} f x +2 b^{2} d \,e^{2} g +3 b^{2} e^{3} f -14 b c \,d^{2} e g -16 b c d \,e^{2} f +12 c^{2} d^{3} g +28 c^{2} d^{2} e f \right ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}{15 \left (b^{3} e^{3}-6 b^{2} c d \,e^{2}+12 b \,c^{2} d^{2} e -8 c^{3} d^{3}\right ) e^{2} \left (e x +d \right )^{3}}\) | \(224\) |
gosper | \(-\frac {2 \left (x c e +b e -c d \right ) \left (-10 b c \,e^{3} g \,x^{2}+12 c^{2} d \,e^{2} g \,x^{2}+8 c^{2} e^{3} f \,x^{2}+5 b^{2} e^{3} g x -36 b c d \,e^{2} g x -4 b c \,e^{3} f x +36 c^{2} d^{2} e g x +24 c^{2} d \,e^{2} f x +2 b^{2} d \,e^{2} g +3 b^{2} e^{3} f -14 b c \,d^{2} e g -16 b c d \,e^{2} f +12 c^{2} d^{3} g +28 c^{2} d^{2} e f \right )}{15 \left (e x +d \right )^{2} \left (b^{3} e^{3}-6 b^{2} c d \,e^{2}+12 b \,c^{2} d^{2} e -8 c^{3} d^{3}\right ) e^{2} \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}\) | \(236\) |
default | \(\frac {g \left (-\frac {2 \sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )}}{3 \left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )^{2}}-\frac {4 c \,e^{2} \sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )}}{3 \left (-b \,e^{2}+2 c d e \right )^{2} \left (x +\frac {d}{e}\right )}\right )}{e^{3}}+\frac {\left (-d g +e f \right ) \left (-\frac {2 \sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )}}{5 \left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )^{3}}+\frac {4 c \,e^{2} \left (-\frac {2 \sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )}}{3 \left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )^{2}}-\frac {4 c \,e^{2} \sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )}}{3 \left (-b \,e^{2}+2 c d e \right )^{2} \left (x +\frac {d}{e}\right )}\right )}{5 \left (-b \,e^{2}+2 c d e \right )}\right )}{e^{4}}\) | \(365\) |
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Time = 7.70 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.75 \[ \int \frac {f+g x}{(d+e x)^3 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=-\frac {2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, {\left (4 \, c^{2} e^{3} f + {\left (6 \, c^{2} d e^{2} - 5 \, b c e^{3}\right )} g\right )} x^{2} + {\left (28 \, c^{2} d^{2} e - 16 \, b c d e^{2} + 3 \, b^{2} e^{3}\right )} f + 2 \, {\left (6 \, c^{2} d^{3} - 7 \, b c d^{2} e + b^{2} d e^{2}\right )} g + {\left (4 \, {\left (6 \, c^{2} d e^{2} - b c e^{3}\right )} f + {\left (36 \, c^{2} d^{2} e - 36 \, b c d e^{2} + 5 \, b^{2} e^{3}\right )} g\right )} x\right )}}{15 \, {\left (8 \, c^{3} d^{6} e^{2} - 12 \, b c^{2} d^{5} e^{3} + 6 \, b^{2} c d^{4} e^{4} - b^{3} d^{3} e^{5} + {\left (8 \, c^{3} d^{3} e^{5} - 12 \, b c^{2} d^{2} e^{6} + 6 \, b^{2} c d e^{7} - b^{3} e^{8}\right )} x^{3} + 3 \, {\left (8 \, c^{3} d^{4} e^{4} - 12 \, b c^{2} d^{3} e^{5} + 6 \, b^{2} c d^{2} e^{6} - b^{3} d e^{7}\right )} x^{2} + 3 \, {\left (8 \, c^{3} d^{5} e^{3} - 12 \, b c^{2} d^{4} e^{4} + 6 \, b^{2} c d^{3} e^{5} - b^{3} d^{2} e^{6}\right )} x\right )}} \]
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\[ \int \frac {f+g x}{(d+e x)^3 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\int \frac {f + g x}{\sqrt {- \left (d + e x\right ) \left (b e - c d + c e x\right )} \left (d + e x\right )^{3}}\, dx \]
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Exception generated. \[ \int \frac {f+g x}{(d+e x)^3 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\text {Exception raised: ValueError} \]
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Exception generated. \[ \int \frac {f+g x}{(d+e x)^3 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\text {Exception raised: TypeError} \]
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Time = 12.21 (sec) , antiderivative size = 471, normalized size of antiderivative = 2.24 \[ \int \frac {f+g x}{(d+e x)^3 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\frac {\left (\frac {8\,c^2\,d\,g+16\,c^2\,e\,f-8\,b\,c\,e\,g}{15\,e^2\,{\left (b\,e-2\,c\,d\right )}^3}-\frac {8\,c^2\,d\,g}{15\,e^2\,{\left (b\,e-2\,c\,d\right )}^3}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{d+e\,x}-\frac {\left (\frac {2\,b\,g}{5\,\left (3\,b\,e^2-6\,c\,d\,e\right )\,\left (b\,e-2\,c\,d\right )}-\frac {4\,c\,d\,g}{5\,e\,\left (3\,b\,e^2-6\,c\,d\,e\right )\,\left (b\,e-2\,c\,d\right )}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{{\left (d+e\,x\right )}^2}-\frac {\left (\frac {12\,c\,d\,g-12\,b\,e\,g+8\,c\,e\,f}{5\,e\,\left (3\,b\,e^2-6\,c\,d\,e\right )\,\left (b\,e-2\,c\,d\right )}+\frac {4\,c\,d\,g}{5\,e\,\left (3\,b\,e^2-6\,c\,d\,e\right )\,\left (b\,e-2\,c\,d\right )}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{{\left (d+e\,x\right )}^2}+\frac {\left (\frac {2\,f}{5\,b\,e^2-10\,c\,d\,e}-\frac {2\,d\,g}{e\,\left (5\,b\,e^2-10\,c\,d\,e\right )}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{{\left (d+e\,x\right )}^3}-\frac {\left (\frac {4\,c\,g\,\left (3\,b\,e-4\,c\,d\right )}{15\,e^2\,{\left (b\,e-2\,c\,d\right )}^3}-\frac {8\,c^2\,d\,g}{15\,e^2\,{\left (b\,e-2\,c\,d\right )}^3}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{d+e\,x} \]
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