Integrand size = 29, antiderivative size = 164 \[ \int \frac {A+B x+C x^2}{\sqrt {c+d x} \sqrt {e+f x}} \, dx=-\frac {(3 C d e+5 c C f-4 B d f) \sqrt {c+d x} \sqrt {e+f x}}{4 d^2 f^2}+\frac {C (c+d x)^{3/2} \sqrt {e+f x}}{2 d^2 f}+\frac {\left (C \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )+4 d f (2 A d f-B (d e+c f))\right ) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right )}{4 d^{5/2} f^{5/2}} \]
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Time = 0.11 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {965, 81, 65, 223, 212} \[ \int \frac {A+B x+C x^2}{\sqrt {c+d x} \sqrt {e+f x}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right ) \left (4 d f (2 A d f-B (c f+d e))+C \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right )}{4 d^{5/2} f^{5/2}}-\frac {\sqrt {c+d x} \sqrt {e+f x} (-4 B d f+5 c C f+3 C d e)}{4 d^2 f^2}+\frac {C (c+d x)^{3/2} \sqrt {e+f x}}{2 d^2 f} \]
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Rule 65
Rule 81
Rule 212
Rule 223
Rule 965
Rubi steps \begin{align*} \text {integral}& = \frac {C (c+d x)^{3/2} \sqrt {e+f x}}{2 d^2 f}+\frac {\int \frac {\frac {1}{2} \left (-3 c C d e-c^2 C f+4 A d^2 f\right )-\frac {1}{2} d (3 C d e+5 c C f-4 B d f) x}{\sqrt {c+d x} \sqrt {e+f x}} \, dx}{2 d^2 f} \\ & = -\frac {(3 C d e+5 c C f-4 B d f) \sqrt {c+d x} \sqrt {e+f x}}{4 d^2 f^2}+\frac {C (c+d x)^{3/2} \sqrt {e+f x}}{2 d^2 f}+\frac {\left (C \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )+4 d f (2 A d f-B (d e+c f))\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x}} \, dx}{8 d^2 f^2} \\ & = -\frac {(3 C d e+5 c C f-4 B d f) \sqrt {c+d x} \sqrt {e+f x}}{4 d^2 f^2}+\frac {C (c+d x)^{3/2} \sqrt {e+f x}}{2 d^2 f}+\frac {\left (C \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )+4 d f (2 A d f-B (d e+c f))\right ) \text {Subst}\left (\int \frac {1}{\sqrt {e-\frac {c f}{d}+\frac {f x^2}{d}}} \, dx,x,\sqrt {c+d x}\right )}{4 d^3 f^2} \\ & = -\frac {(3 C d e+5 c C f-4 B d f) \sqrt {c+d x} \sqrt {e+f x}}{4 d^2 f^2}+\frac {C (c+d x)^{3/2} \sqrt {e+f x}}{2 d^2 f}+\frac {\left (C \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )+4 d f (2 A d f-B (d e+c f))\right ) \text {Subst}\left (\int \frac {1}{1-\frac {f x^2}{d}} \, dx,x,\frac {\sqrt {c+d x}}{\sqrt {e+f x}}\right )}{4 d^3 f^2} \\ & = -\frac {(3 C d e+5 c C f-4 B d f) \sqrt {c+d x} \sqrt {e+f x}}{4 d^2 f^2}+\frac {C (c+d x)^{3/2} \sqrt {e+f x}}{2 d^2 f}+\frac {\left (C \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )+4 d f (2 A d f-B (d e+c f))\right ) \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right )}{4 d^{5/2} f^{5/2}} \\ \end{align*}
Time = 0.33 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.86 \[ \int \frac {A+B x+C x^2}{\sqrt {c+d x} \sqrt {e+f x}} \, dx=\frac {\sqrt {c+d x} \sqrt {e+f x} (4 B d f+C (-3 d e-3 c f+2 d f x))}{4 d^2 f^2}+\frac {\left (C \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )+4 d f (2 A d f-B (d e+c f))\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {f} \sqrt {c+d x}}\right )}{4 d^{5/2} f^{5/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(424\) vs. \(2(138)=276\).
Time = 5.70 (sec) , antiderivative size = 425, normalized size of antiderivative = 2.59
method | result | size |
default | \(\frac {\left (8 A \ln \left (\frac {2 d f x +2 \sqrt {\left (d x +c \right ) \left (f x +e \right )}\, \sqrt {d f}+c f +d e}{2 \sqrt {d f}}\right ) d^{2} f^{2}-4 B \ln \left (\frac {2 d f x +2 \sqrt {\left (d x +c \right ) \left (f x +e \right )}\, \sqrt {d f}+c f +d e}{2 \sqrt {d f}}\right ) c d \,f^{2}-4 B \ln \left (\frac {2 d f x +2 \sqrt {\left (d x +c \right ) \left (f x +e \right )}\, \sqrt {d f}+c f +d e}{2 \sqrt {d f}}\right ) d^{2} e f +4 C \sqrt {\left (d x +c \right ) \left (f x +e \right )}\, \sqrt {d f}\, d f x +3 C \ln \left (\frac {2 d f x +2 \sqrt {\left (d x +c \right ) \left (f x +e \right )}\, \sqrt {d f}+c f +d e}{2 \sqrt {d f}}\right ) c^{2} f^{2}+2 C \ln \left (\frac {2 d f x +2 \sqrt {\left (d x +c \right ) \left (f x +e \right )}\, \sqrt {d f}+c f +d e}{2 \sqrt {d f}}\right ) c d e f +3 C \ln \left (\frac {2 d f x +2 \sqrt {\left (d x +c \right ) \left (f x +e \right )}\, \sqrt {d f}+c f +d e}{2 \sqrt {d f}}\right ) d^{2} e^{2}+8 B \sqrt {d f}\, \sqrt {\left (d x +c \right ) \left (f x +e \right )}\, d f -6 C \sqrt {d f}\, \sqrt {\left (d x +c \right ) \left (f x +e \right )}\, c f -6 C \sqrt {d f}\, \sqrt {\left (d x +c \right ) \left (f x +e \right )}\, d e \right ) \sqrt {d x +c}\, \sqrt {f x +e}}{8 \sqrt {d f}\, f^{2} d^{2} \sqrt {\left (d x +c \right ) \left (f x +e \right )}}\) | \(425\) |
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Time = 0.32 (sec) , antiderivative size = 380, normalized size of antiderivative = 2.32 \[ \int \frac {A+B x+C x^2}{\sqrt {c+d x} \sqrt {e+f x}} \, dx=\left [\frac {{\left (3 \, C d^{2} e^{2} + 2 \, {\left (C c d - 2 \, B d^{2}\right )} e f + {\left (3 \, C c^{2} - 4 \, B c d + 8 \, A d^{2}\right )} f^{2}\right )} \sqrt {d f} \log \left (8 \, d^{2} f^{2} x^{2} + d^{2} e^{2} + 6 \, c d e f + c^{2} f^{2} + 4 \, {\left (2 \, d f x + d e + c f\right )} \sqrt {d f} \sqrt {d x + c} \sqrt {f x + e} + 8 \, {\left (d^{2} e f + c d f^{2}\right )} x\right ) + 4 \, {\left (2 \, C d^{2} f^{2} x - 3 \, C d^{2} e f - {\left (3 \, C c d - 4 \, B d^{2}\right )} f^{2}\right )} \sqrt {d x + c} \sqrt {f x + e}}{16 \, d^{3} f^{3}}, -\frac {{\left (3 \, C d^{2} e^{2} + 2 \, {\left (C c d - 2 \, B d^{2}\right )} e f + {\left (3 \, C c^{2} - 4 \, B c d + 8 \, A d^{2}\right )} f^{2}\right )} \sqrt {-d f} \arctan \left (\frac {{\left (2 \, d f x + d e + c f\right )} \sqrt {-d f} \sqrt {d x + c} \sqrt {f x + e}}{2 \, {\left (d^{2} f^{2} x^{2} + c d e f + {\left (d^{2} e f + c d f^{2}\right )} x\right )}}\right ) - 2 \, {\left (2 \, C d^{2} f^{2} x - 3 \, C d^{2} e f - {\left (3 \, C c d - 4 \, B d^{2}\right )} f^{2}\right )} \sqrt {d x + c} \sqrt {f x + e}}{8 \, d^{3} f^{3}}\right ] \]
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\[ \int \frac {A+B x+C x^2}{\sqrt {c+d x} \sqrt {e+f x}} \, dx=\int \frac {A + B x + C x^{2}}{\sqrt {c + d x} \sqrt {e + f x}}\, dx \]
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Exception generated. \[ \int \frac {A+B x+C x^2}{\sqrt {c+d x} \sqrt {e+f x}} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.29 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.16 \[ \int \frac {A+B x+C x^2}{\sqrt {c+d x} \sqrt {e+f x}} \, dx=\frac {{\left (\sqrt {d^{2} e + {\left (d x + c\right )} d f - c d f} \sqrt {d x + c} {\left (\frac {2 \, {\left (d x + c\right )} C}{d^{3} f} - \frac {3 \, C d^{6} e f + 5 \, C c d^{5} f^{2} - 4 \, B d^{6} f^{2}}{d^{8} f^{3}}\right )} - \frac {{\left (3 \, C d^{2} e^{2} + 2 \, C c d e f - 4 \, B d^{2} e f + 3 \, C c^{2} f^{2} - 4 \, B c d f^{2} + 8 \, A d^{2} f^{2}\right )} \log \left ({\left | -\sqrt {d f} \sqrt {d x + c} + \sqrt {d^{2} e + {\left (d x + c\right )} d f - c d f} \right |}\right )}{\sqrt {d f} d^{2} f^{2}}\right )} d}{4 \, {\left | d \right |}} \]
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Time = 50.86 (sec) , antiderivative size = 833, normalized size of antiderivative = 5.08 \[ \int \frac {A+B x+C x^2}{\sqrt {c+d x} \sqrt {e+f x}} \, dx=\frac {\frac {\left (2\,B\,c\,f+2\,B\,d\,e\right )\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )}{f^3\,\left (\sqrt {e+f\,x}-\sqrt {e}\right )}+\frac {\left (2\,B\,c\,f+2\,B\,d\,e\right )\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^3}{d\,f^2\,{\left (\sqrt {e+f\,x}-\sqrt {e}\right )}^3}-\frac {8\,B\,\sqrt {c}\,\sqrt {e}\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}{f^2\,{\left (\sqrt {e+f\,x}-\sqrt {e}\right )}^2}}{\frac {{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^4}{{\left (\sqrt {e+f\,x}-\sqrt {e}\right )}^4}+\frac {d^2}{f^2}-\frac {2\,d\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}{f\,{\left (\sqrt {e+f\,x}-\sqrt {e}\right )}^2}}-\frac {\frac {\left (\sqrt {c+d\,x}-\sqrt {c}\right )\,\left (\frac {3\,C\,c^2\,d\,f^2}{2}+C\,c\,d^2\,e\,f+\frac {3\,C\,d^3\,e^2}{2}\right )}{f^6\,\left (\sqrt {e+f\,x}-\sqrt {e}\right )}-\frac {{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^3\,\left (\frac {11\,C\,c^2\,f^2}{2}+25\,C\,c\,d\,e\,f+\frac {11\,C\,d^2\,e^2}{2}\right )}{f^5\,{\left (\sqrt {e+f\,x}-\sqrt {e}\right )}^3}+\frac {{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^7\,\left (\frac {3\,C\,c^2\,f^2}{2}+C\,c\,d\,e\,f+\frac {3\,C\,d^2\,e^2}{2}\right )}{d^2\,f^3\,{\left (\sqrt {e+f\,x}-\sqrt {e}\right )}^7}-\frac {{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^5\,\left (\frac {11\,C\,c^2\,f^2}{2}+25\,C\,c\,d\,e\,f+\frac {11\,C\,d^2\,e^2}{2}\right )}{d\,f^4\,{\left (\sqrt {e+f\,x}-\sqrt {e}\right )}^5}+\frac {\sqrt {c}\,\sqrt {e}\,\left (32\,C\,c\,f+32\,C\,d\,e\right )\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^4}{f^4\,{\left (\sqrt {e+f\,x}-\sqrt {e}\right )}^4}}{\frac {{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^8}{{\left (\sqrt {e+f\,x}-\sqrt {e}\right )}^8}+\frac {d^4}{f^4}-\frac {4\,d\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^6}{f\,{\left (\sqrt {e+f\,x}-\sqrt {e}\right )}^6}-\frac {4\,d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}{f^3\,{\left (\sqrt {e+f\,x}-\sqrt {e}\right )}^2}+\frac {6\,d^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^4}{f^2\,{\left (\sqrt {e+f\,x}-\sqrt {e}\right )}^4}}-\frac {4\,A\,\mathrm {atan}\left (\frac {d\,\left (\sqrt {e+f\,x}-\sqrt {e}\right )}{\sqrt {-d\,f}\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )}\right )}{\sqrt {-d\,f}}-\frac {2\,B\,\mathrm {atanh}\left (\frac {\sqrt {f}\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )}{\sqrt {d}\,\left (\sqrt {e+f\,x}-\sqrt {e}\right )}\right )\,\left (c\,f+d\,e\right )}{d^{3/2}\,f^{3/2}}+\frac {C\,\mathrm {atanh}\left (\frac {\sqrt {f}\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )}{\sqrt {d}\,\left (\sqrt {e+f\,x}-\sqrt {e}\right )}\right )\,\left (3\,c^2\,f^2+2\,c\,d\,e\,f+3\,d^2\,e^2\right )}{2\,d^{5/2}\,f^{5/2}} \]
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