Integrand size = 38, antiderivative size = 116 \[ \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} (d+e x)^{5/2}} \, dx=\frac {2 d^2 \sqrt {a+b x}}{(b d-a e) (d+e x)^{3/2}}+\frac {4 d (3 b d-2 a e) \sqrt {a+b x}}{(b d-a e)^2 \sqrt {d+e x}}+\frac {16 \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{\sqrt {b} \sqrt {e}} \]
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Time = 0.07 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {963, 79, 65, 223, 212} \[ \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} (d+e x)^{5/2}} \, dx=\frac {16 \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{\sqrt {b} \sqrt {e}}+\frac {2 d^2 \sqrt {a+b x}}{(d+e x)^{3/2} (b d-a e)}+\frac {4 d \sqrt {a+b x} (3 b d-2 a e)}{\sqrt {d+e x} (b d-a e)^2} \]
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Rule 65
Rule 79
Rule 212
Rule 223
Rule 963
Rubi steps \begin{align*} \text {integral}& = \frac {2 d^2 \sqrt {a+b x}}{(b d-a e) (d+e x)^{3/2}}+\frac {2 \int \frac {3 d (7 b d-6 a e)+12 e (b d-a e) x}{\sqrt {a+b x} (d+e x)^{3/2}} \, dx}{3 (b d-a e)} \\ & = \frac {2 d^2 \sqrt {a+b x}}{(b d-a e) (d+e x)^{3/2}}+\frac {4 d (3 b d-2 a e) \sqrt {a+b x}}{(b d-a e)^2 \sqrt {d+e x}}+8 \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}} \, dx \\ & = \frac {2 d^2 \sqrt {a+b x}}{(b d-a e) (d+e x)^{3/2}}+\frac {4 d (3 b d-2 a e) \sqrt {a+b x}}{(b d-a e)^2 \sqrt {d+e x}}+\frac {16 \text {Subst}\left (\int \frac {1}{\sqrt {d-\frac {a e}{b}+\frac {e x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{b} \\ & = \frac {2 d^2 \sqrt {a+b x}}{(b d-a e) (d+e x)^{3/2}}+\frac {4 d (3 b d-2 a e) \sqrt {a+b x}}{(b d-a e)^2 \sqrt {d+e x}}+\frac {16 \text {Subst}\left (\int \frac {1}{1-\frac {e x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {d+e x}}\right )}{b} \\ & = \frac {2 d^2 \sqrt {a+b x}}{(b d-a e) (d+e x)^{3/2}}+\frac {4 d (3 b d-2 a e) \sqrt {a+b x}}{(b d-a e)^2 \sqrt {d+e x}}+\frac {16 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{\sqrt {b} \sqrt {e}} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.85 \[ \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} (d+e x)^{5/2}} \, dx=\frac {2 d \sqrt {a+b x} \left (7 b d-4 a e-\frac {d e (a+b x)}{d+e x}\right )}{(b d-a e)^2 \sqrt {d+e x}}+\frac {16 \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{\sqrt {b} \sqrt {e}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(600\) vs. \(2(96)=192\).
Time = 0.48 (sec) , antiderivative size = 601, normalized size of antiderivative = 5.18
method | result | size |
default | \(\frac {2 \sqrt {b x +a}\, \left (4 \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{2} e^{4} x^{2}-8 \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a b d \,e^{3} x^{2}+4 \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{2} d^{2} e^{2} x^{2}+8 \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{2} d \,e^{3} x -16 \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a b \,d^{2} e^{2} x +8 \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{2} d^{3} e x +4 \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{2} d^{2} e^{2}-8 \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a b \,d^{3} e +4 \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{2} d^{4}-4 a d \,e^{2} x \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+6 b \,d^{2} e x \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-5 a \,d^{2} e \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+7 b \,d^{3} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\right )}{\sqrt {b e}\, \left (a e -b d \right )^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \left (e x +d \right )^{\frac {3}{2}}}\) | \(601\) |
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Leaf count of result is larger than twice the leaf count of optimal. 326 vs. \(2 (96) = 192\).
Time = 0.64 (sec) , antiderivative size = 665, normalized size of antiderivative = 5.73 \[ \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} (d+e x)^{5/2}} \, dx=\left [\frac {2 \, {\left (2 \, {\left (b^{2} d^{4} - 2 \, a b d^{3} e + a^{2} d^{2} e^{2} + {\left (b^{2} d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} x^{2} + 2 \, {\left (b^{2} d^{3} e - 2 \, a b d^{2} e^{2} + a^{2} d e^{3}\right )} x\right )} \sqrt {b e} \log \left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} + 4 \, {\left (2 \, b e x + b d + a e\right )} \sqrt {b e} \sqrt {b x + a} \sqrt {e x + d} + 8 \, {\left (b^{2} d e + a b e^{2}\right )} x\right ) + {\left (7 \, b^{2} d^{3} e - 5 \, a b d^{2} e^{2} + 2 \, {\left (3 \, b^{2} d^{2} e^{2} - 2 \, a b d e^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}\right )}}{b^{3} d^{4} e - 2 \, a b^{2} d^{3} e^{2} + a^{2} b d^{2} e^{3} + {\left (b^{3} d^{2} e^{3} - 2 \, a b^{2} d e^{4} + a^{2} b e^{5}\right )} x^{2} + 2 \, {\left (b^{3} d^{3} e^{2} - 2 \, a b^{2} d^{2} e^{3} + a^{2} b d e^{4}\right )} x}, -\frac {2 \, {\left (4 \, {\left (b^{2} d^{4} - 2 \, a b d^{3} e + a^{2} d^{2} e^{2} + {\left (b^{2} d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} x^{2} + 2 \, {\left (b^{2} d^{3} e - 2 \, a b d^{2} e^{2} + a^{2} d e^{3}\right )} x\right )} \sqrt {-b e} \arctan \left (\frac {{\left (2 \, b e x + b d + a e\right )} \sqrt {-b e} \sqrt {b x + a} \sqrt {e x + d}}{2 \, {\left (b^{2} e^{2} x^{2} + a b d e + {\left (b^{2} d e + a b e^{2}\right )} x\right )}}\right ) - {\left (7 \, b^{2} d^{3} e - 5 \, a b d^{2} e^{2} + 2 \, {\left (3 \, b^{2} d^{2} e^{2} - 2 \, a b d e^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}\right )}}{b^{3} d^{4} e - 2 \, a b^{2} d^{3} e^{2} + a^{2} b d^{2} e^{3} + {\left (b^{3} d^{2} e^{3} - 2 \, a b^{2} d e^{4} + a^{2} b e^{5}\right )} x^{2} + 2 \, {\left (b^{3} d^{3} e^{2} - 2 \, a b^{2} d^{2} e^{3} + a^{2} b d e^{4}\right )} x}\right ] \]
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\[ \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} (d+e x)^{5/2}} \, dx=\int \frac {15 d^{2} + 20 d e x + 8 e^{2} x^{2}}{\sqrt {a + b x} \left (d + e x\right )^{\frac {5}{2}}}\, dx \]
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Exception generated. \[ \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} (d+e x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 220 vs. \(2 (96) = 192\).
Time = 0.35 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.90 \[ \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} (d+e x)^{5/2}} \, dx=-\frac {16 \, b \log \left ({\left | -\sqrt {b e} \sqrt {b x + a} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \right |}\right )}{\sqrt {b e} {\left | b \right |}} + \frac {2 \, \sqrt {b x + a} {\left (\frac {2 \, {\left (3 \, b^{6} d^{2} e^{2} - 2 \, a b^{5} d e^{3}\right )} {\left (b x + a\right )}}{b^{4} d^{2} e {\left | b \right |} - 2 \, a b^{3} d e^{2} {\left | b \right |} + a^{2} b^{2} e^{3} {\left | b \right |}} + \frac {7 \, b^{7} d^{3} e - 11 \, a b^{6} d^{2} e^{2} + 4 \, a^{2} b^{5} d e^{3}}{b^{4} d^{2} e {\left | b \right |} - 2 \, a b^{3} d e^{2} {\left | b \right |} + a^{2} b^{2} e^{3} {\left | b \right |}}\right )}}{{\left (b^{2} d + {\left (b x + a\right )} b e - a b e\right )}^{\frac {3}{2}}} \]
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Timed out. \[ \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} (d+e x)^{5/2}} \, dx=\int \frac {15\,d^2+20\,d\,e\,x+8\,e^2\,x^2}{\sqrt {a+b\,x}\,{\left (d+e\,x\right )}^{5/2}} \,d x \]
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