Integrand size = 14, antiderivative size = 170 \[ \int \frac {1}{\left (a+b x+c x^2\right )^{3/4}} \, dx=\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )^2}} \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{\sqrt [4]{c} (b+2 c x)} \]
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Time = 0.07 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {637, 226} \[ \int \frac {1}{\left (a+b x+c x^2\right )^{3/4}} \, dx=\frac {\sqrt {2} \sqrt [4]{b^2-4 a c} \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{\sqrt [4]{c} (b+2 c x)} \]
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Rule 226
Rule 637
Rubi steps \begin{align*} \text {integral}& = \frac {\left (4 \sqrt {(b+2 c x)^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b^2-4 a c+4 c x^4}} \, dx,x,\sqrt [4]{a+b x+c x^2}\right )}{b+2 c x} \\ & = \frac {\sqrt {2} \sqrt [4]{b^2-4 a c} \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )^2}} \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} (b+2 c x)} \\ \end{align*}
Time = 10.03 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.52 \[ \int \frac {1}{\left (a+b x+c x^2\right )^{3/4}} \, dx=\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (\frac {c (a+x (b+c x))}{-b^2+4 a c}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arcsin \left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ),2\right )}{c (a+x (b+c x))^{3/4}} \]
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\[\int \frac {1}{\left (c \,x^{2}+b x +a \right )^{\frac {3}{4}}}d x\]
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\[ \int \frac {1}{\left (a+b x+c x^2\right )^{3/4}} \, dx=\int { \frac {1}{{\left (c x^{2} + b x + a\right )}^{\frac {3}{4}}} \,d x } \]
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\[ \int \frac {1}{\left (a+b x+c x^2\right )^{3/4}} \, dx=\int \frac {1}{\left (a + b x + c x^{2}\right )^{\frac {3}{4}}}\, dx \]
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\[ \int \frac {1}{\left (a+b x+c x^2\right )^{3/4}} \, dx=\int { \frac {1}{{\left (c x^{2} + b x + a\right )}^{\frac {3}{4}}} \,d x } \]
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\[ \int \frac {1}{\left (a+b x+c x^2\right )^{3/4}} \, dx=\int { \frac {1}{{\left (c x^{2} + b x + a\right )}^{\frac {3}{4}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (a+b x+c x^2\right )^{3/4}} \, dx=\int \frac {1}{{\left (c\,x^2+b\,x+a\right )}^{3/4}} \,d x \]
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