Integrand size = 22, antiderivative size = 127 \[ \int \frac {x^2}{\sqrt {a+b x} \sqrt {c+d x}} \, dx=-\frac {3 (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b^2 d^2}+\frac {x \sqrt {a+b x} \sqrt {c+d x}}{2 b d}-\frac {\left (4 a b c d-3 (b c+a d)^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{5/2} d^{5/2}} \]
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Time = 0.07 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {92, 81, 65, 223, 212} \[ \int \frac {x^2}{\sqrt {a+b x} \sqrt {c+d x}} \, dx=-\frac {\left (4 a b c d-3 (a d+b c)^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{5/2} d^{5/2}}-\frac {3 \sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{4 b^2 d^2}+\frac {x \sqrt {a+b x} \sqrt {c+d x}}{2 b d} \]
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Rule 65
Rule 81
Rule 92
Rule 212
Rule 223
Rubi steps \begin{align*} \text {integral}& = \frac {x \sqrt {a+b x} \sqrt {c+d x}}{2 b d}+\frac {\int \frac {-a c-\frac {3}{2} (b c+a d) x}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 b d} \\ & = -\frac {3 (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b^2 d^2}+\frac {x \sqrt {a+b x} \sqrt {c+d x}}{2 b d}-\frac {\left (4 a b c d-3 (b c+a d)^2\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 b^2 d^2} \\ & = -\frac {3 (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b^2 d^2}+\frac {x \sqrt {a+b x} \sqrt {c+d x}}{2 b d}-\frac {\left (4 a b c d-3 (b c+a d)^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{4 b^3 d^2} \\ & = -\frac {3 (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b^2 d^2}+\frac {x \sqrt {a+b x} \sqrt {c+d x}}{2 b d}-\frac {\left (4 a b c d-3 (b c+a d)^2\right ) \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{4 b^3 d^2} \\ & = -\frac {3 (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b^2 d^2}+\frac {x \sqrt {a+b x} \sqrt {c+d x}}{2 b d}-\frac {\left (4 a b c d-3 (b c+a d)^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{5/2} d^{5/2}} \\ \end{align*}
Time = 0.35 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.87 \[ \int \frac {x^2}{\sqrt {a+b x} \sqrt {c+d x}} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} (-3 b c-3 a d+2 b d x)}{4 b^2 d^2}+\frac {\left (3 b^2 c^2+2 a b c d+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{4 b^{5/2} d^{5/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(250\) vs. \(2(101)=202\).
Time = 0.58 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.98
method | result | size |
default | \(\frac {\left (3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} d^{2}+2 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a b c d +3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{2} c^{2}+4 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b d x -6 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a d -6 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b c \right ) \sqrt {b x +a}\, \sqrt {d x +c}}{8 \sqrt {b d}\, d^{2} b^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}}\) | \(251\) |
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Time = 0.25 (sec) , antiderivative size = 308, normalized size of antiderivative = 2.43 \[ \int \frac {x^2}{\sqrt {a+b x} \sqrt {c+d x}} \, dx=\left [\frac {{\left (3 \, b^{2} c^{2} + 2 \, a b c d + 3 \, a^{2} d^{2}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \, {\left (2 \, b^{2} d^{2} x - 3 \, b^{2} c d - 3 \, a b d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, b^{3} d^{3}}, -\frac {{\left (3 \, b^{2} c^{2} + 2 \, a b c d + 3 \, a^{2} d^{2}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) - 2 \, {\left (2 \, b^{2} d^{2} x - 3 \, b^{2} c d - 3 \, a b d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, b^{3} d^{3}}\right ] \]
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\[ \int \frac {x^2}{\sqrt {a+b x} \sqrt {c+d x}} \, dx=\int \frac {x^{2}}{\sqrt {a + b x} \sqrt {c + d x}}\, dx \]
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Exception generated. \[ \int \frac {x^2}{\sqrt {a+b x} \sqrt {c+d x}} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.30 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.18 \[ \int \frac {x^2}{\sqrt {a+b x} \sqrt {c+d x}} \, dx=\frac {{\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a} {\left (\frac {2 \, {\left (b x + a\right )}}{b^{3} d} - \frac {3 \, b^{6} c d + 5 \, a b^{5} d^{2}}{b^{8} d^{3}}\right )} - \frac {{\left (3 \, b^{2} c^{2} + 2 \, a b c d + 3 \, a^{2} d^{2}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} b^{2} d^{2}}\right )} b}{4 \, {\left | b \right |}} \]
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Time = 16.52 (sec) , antiderivative size = 508, normalized size of antiderivative = 4.00 \[ \int \frac {x^2}{\sqrt {a+b x} \sqrt {c+d x}} \, dx=\frac {\mathrm {atanh}\left (\frac {\sqrt {d}\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{\sqrt {b}\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )}\right )\,\left (3\,a^2\,d^2+2\,a\,b\,c\,d+3\,b^2\,c^2\right )}{2\,b^{5/2}\,d^{5/2}}-\frac {\frac {\left (\sqrt {a+b\,x}-\sqrt {a}\right )\,\left (\frac {3\,a^2\,b\,d^2}{2}+a\,b^2\,c\,d+\frac {3\,b^3\,c^2}{2}\right )}{d^6\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )}-\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3\,\left (\frac {11\,a^2\,d^2}{2}+25\,a\,b\,c\,d+\frac {11\,b^2\,c^2}{2}\right )}{d^5\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^3}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^7\,\left (\frac {3\,a^2\,d^2}{2}+a\,b\,c\,d+\frac {3\,b^2\,c^2}{2}\right )}{b^2\,d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^7}-\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^5\,\left (\frac {11\,a^2\,d^2}{2}+25\,a\,b\,c\,d+\frac {11\,b^2\,c^2}{2}\right )}{b\,d^4\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^5}+\frac {\sqrt {a}\,\sqrt {c}\,\left (32\,a\,d+32\,b\,c\right )\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}{d^4\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^4}}{\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^8}{{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^8}+\frac {b^4}{d^4}-\frac {4\,b^3\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}+\frac {6\,b^2\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}{d^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^4}-\frac {4\,b\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^6}{d\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^6}} \]
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