Integrand size = 38, antiderivative size = 133 \[ \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} (d+e x)^{7/2}} \, dx=\frac {6 d^2 \sqrt {a+b x}}{5 (b d-a e) (d+e x)^{5/2}}+\frac {8 d (8 b d-5 a e) \sqrt {a+b x}}{15 (b d-a e)^2 (d+e x)^{3/2}}+\frac {16 \left (23 b^2 d^2-35 a b d e+15 a^2 e^2\right ) \sqrt {a+b x}}{15 (b d-a e)^3 \sqrt {d+e x}} \]
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Time = 0.08 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used = {963, 79, 37} \[ \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} (d+e x)^{7/2}} \, dx=\frac {16 \sqrt {a+b x} \left (15 a^2 e^2-35 a b d e+23 b^2 d^2\right )}{15 \sqrt {d+e x} (b d-a e)^3}+\frac {6 d^2 \sqrt {a+b x}}{5 (d+e x)^{5/2} (b d-a e)}+\frac {8 d \sqrt {a+b x} (8 b d-5 a e)}{15 (d+e x)^{3/2} (b d-a e)^2} \]
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Rule 37
Rule 79
Rule 963
Rubi steps \begin{align*} \text {integral}& = \frac {6 d^2 \sqrt {a+b x}}{5 (b d-a e) (d+e x)^{5/2}}+\frac {2 \int \frac {6 d (6 b d-5 a e)+20 e (b d-a e) x}{\sqrt {a+b x} (d+e x)^{5/2}} \, dx}{5 (b d-a e)} \\ & = \frac {6 d^2 \sqrt {a+b x}}{5 (b d-a e) (d+e x)^{5/2}}+\frac {8 d (8 b d-5 a e) \sqrt {a+b x}}{15 (b d-a e)^2 (d+e x)^{3/2}}+\frac {\left (8 \left (23 b^2 d^2-35 a b d e+15 a^2 e^2\right )\right ) \int \frac {1}{\sqrt {a+b x} (d+e x)^{3/2}} \, dx}{15 (b d-a e)^2} \\ & = \frac {6 d^2 \sqrt {a+b x}}{5 (b d-a e) (d+e x)^{5/2}}+\frac {8 d (8 b d-5 a e) \sqrt {a+b x}}{15 (b d-a e)^2 (d+e x)^{3/2}}+\frac {16 \left (23 b^2 d^2-35 a b d e+15 a^2 e^2\right ) \sqrt {a+b x}}{15 (b d-a e)^3 \sqrt {d+e x}} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.86 \[ \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} (d+e x)^{7/2}} \, dx=\frac {2 \sqrt {a+b x} \left (225 b^2 d^2-300 a b d e+120 a^2 e^2+\frac {9 d^2 e^2 (a+b x)^2}{(d+e x)^2}-\frac {50 b d^2 e (a+b x)}{d+e x}+\frac {20 a d e^2 (a+b x)}{d+e x}\right )}{15 (b d-a e)^3 \sqrt {d+e x}} \]
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Time = 0.48 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.92
method | result | size |
default | \(-\frac {2 \sqrt {b x +a}\, \left (120 a^{2} e^{4} x^{2}-280 a b d \,e^{3} x^{2}+184 b^{2} d^{2} e^{2} x^{2}+260 a^{2} d \,e^{3} x -612 a b \,d^{2} e^{2} x +400 b^{2} d^{3} e x +149 a^{2} d^{2} e^{2}-350 a b \,d^{3} e +225 b^{2} d^{4}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} \left (a e -b d \right )^{3}}\) | \(122\) |
gosper | \(-\frac {2 \sqrt {b x +a}\, \left (120 a^{2} e^{4} x^{2}-280 a b d \,e^{3} x^{2}+184 b^{2} d^{2} e^{2} x^{2}+260 a^{2} d \,e^{3} x -612 a b \,d^{2} e^{2} x +400 b^{2} d^{3} e x +149 a^{2} d^{2} e^{2}-350 a b \,d^{3} e +225 b^{2} d^{4}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}\) | \(150\) |
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Leaf count of result is larger than twice the leaf count of optimal. 293 vs. \(2 (115) = 230\).
Time = 0.90 (sec) , antiderivative size = 293, normalized size of antiderivative = 2.20 \[ \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} (d+e x)^{7/2}} \, dx=\frac {2 \, {\left (225 \, b^{2} d^{4} - 350 \, a b d^{3} e + 149 \, a^{2} d^{2} e^{2} + 8 \, {\left (23 \, b^{2} d^{2} e^{2} - 35 \, a b d e^{3} + 15 \, a^{2} e^{4}\right )} x^{2} + 4 \, {\left (100 \, b^{2} d^{3} e - 153 \, a b d^{2} e^{2} + 65 \, a^{2} d e^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{15 \, {\left (b^{3} d^{6} - 3 \, a b^{2} d^{5} e + 3 \, a^{2} b d^{4} e^{2} - a^{3} d^{3} e^{3} + {\left (b^{3} d^{3} e^{3} - 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} b d e^{5} - a^{3} e^{6}\right )} x^{3} + 3 \, {\left (b^{3} d^{4} e^{2} - 3 \, a b^{2} d^{3} e^{3} + 3 \, a^{2} b d^{2} e^{4} - a^{3} d e^{5}\right )} x^{2} + 3 \, {\left (b^{3} d^{5} e - 3 \, a b^{2} d^{4} e^{2} + 3 \, a^{2} b d^{3} e^{3} - a^{3} d^{2} e^{4}\right )} x\right )}} \]
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\[ \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} (d+e x)^{7/2}} \, dx=\int \frac {15 d^{2} + 20 d e x + 8 e^{2} x^{2}}{\sqrt {a + b x} \left (d + e x\right )^{\frac {7}{2}}}\, dx \]
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Exception generated. \[ \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} (d+e x)^{7/2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 359 vs. \(2 (115) = 230\).
Time = 0.38 (sec) , antiderivative size = 359, normalized size of antiderivative = 2.70 \[ \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} (d+e x)^{7/2}} \, dx=\frac {2 \, {\left (4 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (23 \, b^{8} d^{2} e^{4} - 35 \, a b^{7} d e^{5} + 15 \, a^{2} b^{6} e^{6}\right )} {\left (b x + a\right )}}{b^{5} d^{3} e^{2} {\left | b \right |} - 3 \, a b^{4} d^{2} e^{3} {\left | b \right |} + 3 \, a^{2} b^{3} d e^{4} {\left | b \right |} - a^{3} b^{2} e^{5} {\left | b \right |}} + \frac {5 \, {\left (20 \, b^{9} d^{3} e^{3} - 49 \, a b^{8} d^{2} e^{4} + 41 \, a^{2} b^{7} d e^{5} - 12 \, a^{3} b^{6} e^{6}\right )}}{b^{5} d^{3} e^{2} {\left | b \right |} - 3 \, a b^{4} d^{2} e^{3} {\left | b \right |} + 3 \, a^{2} b^{3} d e^{4} {\left | b \right |} - a^{3} b^{2} e^{5} {\left | b \right |}}\right )} + \frac {15 \, {\left (15 \, b^{10} d^{4} e^{2} - 50 \, a b^{9} d^{3} e^{3} + 63 \, a^{2} b^{8} d^{2} e^{4} - 36 \, a^{3} b^{7} d e^{5} + 8 \, a^{4} b^{6} e^{6}\right )}}{b^{5} d^{3} e^{2} {\left | b \right |} - 3 \, a b^{4} d^{2} e^{3} {\left | b \right |} + 3 \, a^{2} b^{3} d e^{4} {\left | b \right |} - a^{3} b^{2} e^{5} {\left | b \right |}}\right )} \sqrt {b x + a}}{15 \, {\left (b^{2} d + {\left (b x + a\right )} b e - a b e\right )}^{\frac {5}{2}}} \]
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Time = 13.36 (sec) , antiderivative size = 268, normalized size of antiderivative = 2.02 \[ \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} (d+e x)^{7/2}} \, dx=-\frac {\sqrt {d+e\,x}\,\left (\frac {x^2\,\left (240\,a^3\,e^4-40\,a^2\,b\,d\,e^3-856\,a\,b^2\,d^2\,e^2+800\,b^3\,d^3\,e\right )}{15\,e^3\,{\left (a\,e-b\,d\right )}^3}+\frac {x\,\left (520\,a^3\,d\,e^3-926\,a^2\,b\,d^2\,e^2+100\,a\,b^2\,d^3\,e+450\,b^3\,d^4\right )}{15\,e^3\,{\left (a\,e-b\,d\right )}^3}+\frac {2\,a\,d^2\,\left (149\,a^2\,e^2-350\,a\,b\,d\,e+225\,b^2\,d^2\right )}{15\,e^3\,{\left (a\,e-b\,d\right )}^3}+\frac {16\,b\,x^3\,\left (15\,a^2\,e^2-35\,a\,b\,d\,e+23\,b^2\,d^2\right )}{15\,e\,{\left (a\,e-b\,d\right )}^3}\right )}{x^3\,\sqrt {a+b\,x}+\frac {d^3\,\sqrt {a+b\,x}}{e^3}+\frac {3\,d\,x^2\,\sqrt {a+b\,x}}{e}+\frac {3\,d^2\,x\,\sqrt {a+b\,x}}{e^2}} \]
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