Integrand size = 38, antiderivative size = 122 \[ \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} \sqrt {d+e x}} \, dx=\frac {2 (7 b d-5 a e) \sqrt {a+b x} \sqrt {d+e x}}{b^2}+\frac {4 e (a+b x)^{3/2} \sqrt {d+e x}}{b^2}+\frac {2 \left (8 b^2 d^2-8 a b d e+3 a^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{b^{5/2} \sqrt {e}} \]
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Time = 0.08 (sec) , antiderivative size = 122, 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 = {965, 81, 65, 223, 212} \[ \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} \sqrt {d+e x}} \, dx=\frac {2 \left (3 a^2 e^2-8 a b d e+8 b^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{b^{5/2} \sqrt {e}}+\frac {4 e (a+b x)^{3/2} \sqrt {d+e x}}{b^2}+\frac {2 \sqrt {a+b x} \sqrt {d+e x} (7 b d-5 a e)}{b^2} \]
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Rule 65
Rule 81
Rule 212
Rule 223
Rule 965
Rubi steps \begin{align*} \text {integral}& = \frac {4 e (a+b x)^{3/2} \sqrt {d+e x}}{b^2}+\frac {\int \frac {2 e \left (15 b^2 d^2-6 a b d e-2 a^2 e^2\right )+4 b e^2 (7 b d-5 a e) x}{\sqrt {a+b x} \sqrt {d+e x}} \, dx}{2 b^2 e} \\ & = \frac {2 (7 b d-5 a e) \sqrt {a+b x} \sqrt {d+e x}}{b^2}+\frac {4 e (a+b x)^{3/2} \sqrt {d+e x}}{b^2}+\frac {\left (8 b^2 d^2-8 a b d e+3 a^2 e^2\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}} \, dx}{b^2} \\ & = \frac {2 (7 b d-5 a e) \sqrt {a+b x} \sqrt {d+e x}}{b^2}+\frac {4 e (a+b x)^{3/2} \sqrt {d+e x}}{b^2}+\frac {\left (2 \left (8 b^2 d^2-8 a b d e+3 a^2 e^2\right )\right ) \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^3} \\ & = \frac {2 (7 b d-5 a e) \sqrt {a+b x} \sqrt {d+e x}}{b^2}+\frac {4 e (a+b x)^{3/2} \sqrt {d+e x}}{b^2}+\frac {\left (2 \left (8 b^2 d^2-8 a b d e+3 a^2 e^2\right )\right ) \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^3} \\ & = \frac {2 (7 b d-5 a e) \sqrt {a+b x} \sqrt {d+e x}}{b^2}+\frac {4 e (a+b x)^{3/2} \sqrt {d+e x}}{b^2}+\frac {2 \left (8 b^2 d^2-8 a b d e+3 a^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{b^{5/2} \sqrt {e}} \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.84 \[ \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} \sqrt {d+e x}} \, dx=\frac {2 \sqrt {a+b x} \sqrt {d+e x} (7 b d-3 a e+2 b e x)}{b^2}+\frac {2 \left (8 b^2 d^2-8 a b d e+3 a^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {e} \sqrt {a+b x}}\right )}{b^{5/2} \sqrt {e}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(246\) vs. \(2(102)=204\).
Time = 0.48 (sec) , antiderivative size = 247, normalized size of antiderivative = 2.02
method | result | size |
default | \(\frac {\left (4 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, b e x +3 \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{2} e^{2}-8 \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a b d e +8 \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{2} d^{2}-6 \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, a e +14 \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, b d \right ) \sqrt {e x +d}\, \sqrt {b x +a}}{\sqrt {b e}\, b^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}}\) | \(247\) |
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Time = 0.41 (sec) , antiderivative size = 308, normalized size of antiderivative = 2.52 \[ \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} \sqrt {d+e x}} \, dx=\left [\frac {{\left (8 \, b^{2} d^{2} - 8 \, a b d e + 3 \, a^{2} e^{2}\right )} \sqrt {b e} \log \left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} + 4 \, {\left (2 \, b e x + b d + a e\right )} \sqrt {b e} \sqrt {b x + a} \sqrt {e x + d} + 8 \, {\left (b^{2} d e + a b e^{2}\right )} x\right ) + 4 \, {\left (2 \, b^{2} e^{2} x + 7 \, b^{2} d e - 3 \, a b e^{2}\right )} \sqrt {b x + a} \sqrt {e x + d}}{2 \, b^{3} e}, -\frac {{\left (8 \, b^{2} d^{2} - 8 \, a b d e + 3 \, a^{2} e^{2}\right )} \sqrt {-b e} \arctan \left (\frac {{\left (2 \, b e x + b d + a e\right )} \sqrt {-b e} \sqrt {b x + a} \sqrt {e x + d}}{2 \, {\left (b^{2} e^{2} x^{2} + a b d e + {\left (b^{2} d e + a b e^{2}\right )} x\right )}}\right ) - 2 \, {\left (2 \, b^{2} e^{2} x + 7 \, b^{2} d e - 3 \, a b e^{2}\right )} \sqrt {b x + a} \sqrt {e x + d}}{b^{3} e}\right ] \]
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\[ \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} \sqrt {d+e x}} \, dx=\int \frac {15 d^{2} + 20 d e x + 8 e^{2} x^{2}}{\sqrt {a + b x} \sqrt {d + e x}}\, dx \]
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Exception generated. \[ \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} \sqrt {d+e x}} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.29 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.20 \[ \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} \sqrt {d+e x}} \, dx=\frac {2 \, {\left (\sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \sqrt {b x + a} {\left (\frac {2 \, {\left (b x + a\right )} e}{b^{3}} + \frac {7 \, b^{6} d e^{2} - 5 \, a b^{5} e^{3}}{b^{8} e^{2}}\right )} - \frac {{\left (8 \, b^{2} d^{2} - 8 \, a b d e + 3 \, a^{2} e^{2}\right )} \log \left ({\left | -\sqrt {b e} \sqrt {b x + a} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \right |}\right )}{\sqrt {b e} b^{2}}\right )} b}{{\left | b \right |}} \]
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Time = 32.86 (sec) , antiderivative size = 893, normalized size of antiderivative = 7.32 \[ \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} \sqrt {d+e x}} \, dx=\frac {\frac {\left (40\,b\,d^2+40\,a\,e\,d\right )\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{e^2\,\left (\sqrt {d+e\,x}-\sqrt {d}\right )}-\frac {160\,\sqrt {a}\,d^{3/2}\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{e\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}+\frac {\left (40\,b\,d^2+40\,a\,e\,d\right )\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3}{b\,e\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^3}}{\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}{{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^4}+\frac {b^2}{e^2}-\frac {2\,b\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{e\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}}-\frac {\frac {\left (\sqrt {a+b\,x}-\sqrt {a}\right )\,\left (12\,a^2\,b\,e^2+8\,a\,b^2\,d\,e+12\,b^3\,d^2\right )}{e^4\,\left (\sqrt {d+e\,x}-\sqrt {d}\right )}-\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3\,\left (44\,a^2\,e^2+200\,a\,b\,d\,e+44\,b^2\,d^2\right )}{e^3\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^3}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^7\,\left (12\,a^2\,e^2+8\,a\,b\,d\,e+12\,b^2\,d^2\right )}{b^2\,e\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^7}-\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^5\,\left (44\,a^2\,e^2+200\,a\,b\,d\,e+44\,b^2\,d^2\right )}{b\,e^2\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^5}+\frac {\sqrt {a}\,\sqrt {d}\,\left (256\,a\,e+256\,b\,d\right )\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}{e^2\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^4}}{\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^8}{{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^8}+\frac {b^4}{e^4}-\frac {4\,b^3\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{e^3\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}+\frac {6\,b^2\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}{e^2\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^4}-\frac {4\,b\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^6}{e\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^6}}-\frac {60\,d^2\,\mathrm {atan}\left (\frac {b\,\left (\sqrt {d+e\,x}-\sqrt {d}\right )}{\sqrt {-b\,e}\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}\right )}{\sqrt {-b\,e}}-\frac {2\,\ln \left (\frac {\sqrt {e}\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{\sqrt {d+e\,x}-\sqrt {d}}-\sqrt {b}\right )\,\left (3\,a^2\,e^2+2\,a\,b\,d\,e+3\,b^2\,d^2\right )}{b^{5/2}\,\sqrt {e}}+\frac {\ln \left (\sqrt {b}+\frac {\sqrt {e}\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{\sqrt {d+e\,x}-\sqrt {d}}\right )\,\left (6\,a^2\,e^2+4\,a\,b\,d\,e+6\,b^2\,d^2\right )}{b^{5/2}\,\sqrt {e}}-\frac {40\,d\,\mathrm {atanh}\left (\frac {\sqrt {e}\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{\sqrt {b}\,\left (\sqrt {d+e\,x}-\sqrt {d}\right )}\right )\,\left (a\,e+b\,d\right )}{b^{3/2}\,\sqrt {e}} \]
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