Integrand size = 38, antiderivative size = 136 \[ \int \frac {(g+h x) \sqrt {a+b x+c x^2}}{\left (a d+b d x+c d x^2\right )^{3/2}} \, dx=-\frac {(2 c g-b h) \sqrt {a+b x+c x^2} \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c \sqrt {b^2-4 a c} d \sqrt {a d+b d x+c d x^2}}+\frac {h \sqrt {a+b x+c x^2} \log \left (a+b x+c x^2\right )}{2 c d \sqrt {a d+b d x+c d x^2}} \]
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Time = 0.07 (sec) , antiderivative size = 136, 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 = {1013, 648, 632, 212, 642} \[ \int \frac {(g+h x) \sqrt {a+b x+c x^2}}{\left (a d+b d x+c d x^2\right )^{3/2}} \, dx=\frac {h \sqrt {a+b x+c x^2} \log \left (a+b x+c x^2\right )}{2 c d \sqrt {a d+b d x+c d x^2}}-\frac {\sqrt {a+b x+c x^2} (2 c g-b h) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c d \sqrt {b^2-4 a c} \sqrt {a d+b d x+c d x^2}} \]
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Rule 212
Rule 632
Rule 642
Rule 648
Rule 1013
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a+b x+c x^2} \int \frac {g+h x}{a d+b d x+c d x^2} \, dx}{\sqrt {a d+b d x+c d x^2}} \\ & = \frac {\left (h \sqrt {a+b x+c x^2}\right ) \int \frac {b d+2 c d x}{a d+b d x+c d x^2} \, dx}{2 c d \sqrt {a d+b d x+c d x^2}}+\frac {\left ((2 c d g-b d h) \sqrt {a+b x+c x^2}\right ) \int \frac {1}{a d+b d x+c d x^2} \, dx}{2 c d \sqrt {a d+b d x+c d x^2}} \\ & = \frac {h \sqrt {a+b x+c x^2} \log \left (a+b x+c x^2\right )}{2 c d \sqrt {a d+b d x+c d x^2}}-\frac {\left ((2 c d g-b d h) \sqrt {a+b x+c x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (b^2-4 a c\right ) d^2-x^2} \, dx,x,b d+2 c d x\right )}{c d \sqrt {a d+b d x+c d x^2}} \\ & = -\frac {(2 c g-b h) \sqrt {a+b x+c x^2} \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c \sqrt {b^2-4 a c} d \sqrt {a d+b d x+c d x^2}}+\frac {h \sqrt {a+b x+c x^2} \log \left (a+b x+c x^2\right )}{2 c d \sqrt {a d+b d x+c d x^2}} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.79 \[ \int \frac {(g+h x) \sqrt {a+b x+c x^2}}{\left (a d+b d x+c d x^2\right )^{3/2}} \, dx=\frac {(a+x (b+c x))^{3/2} \left ((4 c g-2 b h) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )+\sqrt {-b^2+4 a c} h \log (a+x (b+c x))\right )}{2 c \sqrt {-b^2+4 a c} (d (a+x (b+c x)))^{3/2}} \]
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Time = 0.90 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.89
method | result | size |
default | \(\frac {\sqrt {d \left (c \,x^{2}+b x +a \right )}\, \left (h \ln \left (c \,x^{2}+b x +a \right ) \sqrt {4 a c -b^{2}}-2 \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) b h +4 \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) c g \right )}{2 \sqrt {c \,x^{2}+b x +a}\, d^{2} c \sqrt {4 a c -b^{2}}}\) | \(121\) |
risch | \(\frac {\sqrt {c \,x^{2}+b x +a}\, \left (4 a c h -b^{2} h +\sqrt {-\left (b h -2 c g \right )^{2} \left (4 a c -b^{2}\right )}\right ) \ln \left (-4 a b c h +8 a \,c^{2} g +b^{3} h -2 b^{2} c g -2 \sqrt {-\left (b h -2 c g \right )^{2} \left (4 a c -b^{2}\right )}\, c x -\sqrt {-\left (b h -2 c g \right )^{2} \left (4 a c -b^{2}\right )}\, b \right )}{2 d \sqrt {d \left (c \,x^{2}+b x +a \right )}\, c \left (4 a c -b^{2}\right )}-\frac {\sqrt {c \,x^{2}+b x +a}\, \left (-4 a c h +b^{2} h +\sqrt {-\left (b h -2 c g \right )^{2} \left (4 a c -b^{2}\right )}\right ) \ln \left (-4 a b c h +8 a \,c^{2} g +b^{3} h -2 b^{2} c g +2 \sqrt {-\left (b h -2 c g \right )^{2} \left (4 a c -b^{2}\right )}\, c x +\sqrt {-\left (b h -2 c g \right )^{2} \left (4 a c -b^{2}\right )}\, b \right )}{2 d \sqrt {d \left (c \,x^{2}+b x +a \right )}\, c \left (4 a c -b^{2}\right )}\) | \(328\) |
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\[ \int \frac {(g+h x) \sqrt {a+b x+c x^2}}{\left (a d+b d x+c d x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {c x^{2} + b x + a} {\left (h x + g\right )}}{{\left (c d x^{2} + b d x + a d\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(g+h x) \sqrt {a+b x+c x^2}}{\left (a d+b d x+c d x^2\right )^{3/2}} \, dx=\int \frac {\left (g + h x\right ) \sqrt {a + b x + c x^{2}}}{\left (d \left (a + b x + c x^{2}\right )\right )^{\frac {3}{2}}}\, dx \]
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Exception generated. \[ \int \frac {(g+h x) \sqrt {a+b x+c x^2}}{\left (a d+b d x+c d x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {(g+h x) \sqrt {a+b x+c x^2}}{\left (a d+b d x+c d x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {c x^{2} + b x + a} {\left (h x + g\right )}}{{\left (c d x^{2} + b d x + a d\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(g+h x) \sqrt {a+b x+c x^2}}{\left (a d+b d x+c d x^2\right )^{3/2}} \, dx=\int \frac {\left (g+h\,x\right )\,\sqrt {c\,x^2+b\,x+a}}{{\left (c\,d\,x^2+b\,d\,x+a\,d\right )}^{3/2}} \,d x \]
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