Integrand size = 13, antiderivative size = 384 \[ \int \frac {1}{\left (b x+c x^2\right )^{5/3}} \, dx=\frac {3 (b+2 c x) \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{5/3}}{2 c \left (-\frac {c x (b+c x)}{b^2}\right )^{2/3} \left (b x+c x^2\right )^{5/3}}+\frac {\sqrt [3]{2} 3^{3/4} \sqrt {2-\sqrt {3}} b^2 \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{5/3} \left (1-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right ) \sqrt {\frac {1+2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}+2 \sqrt [3]{2} \left (-\frac {c x (b+c x)}{b^2}\right )^{2/3}}{\left (1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}}{1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}}\right ),-7+4 \sqrt {3}\right )}{c (b+2 c x) \left (b x+c x^2\right )^{5/3} \sqrt {-\frac {1-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}}{\left (1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right )^2}}} \]
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Time = 0.33 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {636, 633, 205, 242, 225} \[ \int \frac {1}{\left (b x+c x^2\right )^{5/3}} \, dx=\frac {\sqrt [3]{2} 3^{3/4} \sqrt {2-\sqrt {3}} b^2 \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{5/3} \left (1-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right ) \sqrt {\frac {2 \sqrt [3]{2} \left (-\frac {c x (b+c x)}{b^2}\right )^{2/3}+2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}+1}{\left (-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}+\sqrt {3}+1}{-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{c (b+2 c x) \left (b x+c x^2\right )^{5/3} \sqrt {-\frac {1-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}}{\left (-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}-\sqrt {3}+1\right )^2}}}+\frac {3 (b+2 c x) \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{5/3}}{2 c \left (-\frac {c x (b+c x)}{b^2}\right )^{2/3} \left (b x+c x^2\right )^{5/3}} \]
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Rule 205
Rule 225
Rule 242
Rule 633
Rule 636
Rubi steps \begin{align*} \text {integral}& = \frac {\left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{5/3} \int \frac {1}{\left (-\frac {c x}{b}-\frac {c^2 x^2}{b^2}\right )^{5/3}} \, dx}{\left (b x+c x^2\right )^{5/3}} \\ & = -\frac {\left (4 \sqrt [3]{2} b^2 \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{5/3}\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {b^2 x^2}{c^2}\right )^{5/3}} \, dx,x,-\frac {c}{b}-\frac {2 c^2 x}{b^2}\right )}{c^2 \left (b x+c x^2\right )^{5/3}} \\ & = \frac {3 (b+2 c x) \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{5/3}}{2 c \left (-\frac {c x (b+c x)}{b^2}\right )^{2/3} \left (b x+c x^2\right )^{5/3}}-\frac {\left (\sqrt [3]{2} b^2 \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{5/3}\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {b^2 x^2}{c^2}\right )^{2/3}} \, dx,x,-\frac {c}{b}-\frac {2 c^2 x}{b^2}\right )}{c^2 \left (b x+c x^2\right )^{5/3}} \\ & = \frac {3 (b+2 c x) \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{5/3}}{2 c \left (-\frac {c x (b+c x)}{b^2}\right )^{2/3} \left (b x+c x^2\right )^{5/3}}+\frac {\left (3 \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{5/3} \sqrt {-1-\frac {4 c x}{b}-\frac {4 c^2 x^2}{b^2}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^3}} \, dx,x,2^{2/3} \sqrt [3]{-\frac {c x \left (1+\frac {c x}{b}\right )}{b}}\right )}{2^{2/3} \left (-\frac {c}{b}-\frac {2 c^2 x}{b^2}\right ) \left (b x+c x^2\right )^{5/3}} \\ & = \frac {3 (b+2 c x) \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{5/3}}{2 c \left (-\frac {c x (b+c x)}{b^2}\right )^{2/3} \left (b x+c x^2\right )^{5/3}}+\frac {\sqrt [3]{2} 3^{3/4} \sqrt {2-\sqrt {3}} b^2 \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{5/3} \sqrt {-1-\frac {4 c x}{b}-\frac {4 c^2 x^2}{b^2}} \left (1-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right ) \sqrt {\frac {1+2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}+2 \sqrt [3]{2} \left (-\frac {c x (b+c x)}{b^2}\right )^{2/3}}{\left (1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}}{1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}}\right )|-7+4 \sqrt {3}\right )}{c (b+2 c x) \left (b x+c x^2\right )^{5/3} \sqrt {-1-\frac {4 c x (b+c x)}{b^2}} \sqrt {-\frac {1-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}}{\left (1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right )^2}}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.02 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.12 \[ \int \frac {1}{\left (b x+c x^2\right )^{5/3}} \, dx=-\frac {3 \left (1+\frac {c x}{b}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {5}{3},\frac {1}{3},-\frac {c x}{b}\right )}{2 b (x (b+c x))^{2/3}} \]
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\[\int \frac {1}{\left (c \,x^{2}+b x \right )^{\frac {5}{3}}}d x\]
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\[ \int \frac {1}{\left (b x+c x^2\right )^{5/3}} \, dx=\int { \frac {1}{{\left (c x^{2} + b x\right )}^{\frac {5}{3}}} \,d x } \]
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\[ \int \frac {1}{\left (b x+c x^2\right )^{5/3}} \, dx=\int \frac {1}{\left (b x + c x^{2}\right )^{\frac {5}{3}}}\, dx \]
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\[ \int \frac {1}{\left (b x+c x^2\right )^{5/3}} \, dx=\int { \frac {1}{{\left (c x^{2} + b x\right )}^{\frac {5}{3}}} \,d x } \]
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\[ \int \frac {1}{\left (b x+c x^2\right )^{5/3}} \, dx=\int { \frac {1}{{\left (c x^{2} + b x\right )}^{\frac {5}{3}}} \,d x } \]
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Time = 9.07 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.09 \[ \int \frac {1}{\left (b x+c x^2\right )^{5/3}} \, dx=-\frac {3\,x\,{\left (\frac {c\,x}{b}+1\right )}^{5/3}\,{{}}_2{\mathrm {F}}_1\left (-\frac {2}{3},\frac {5}{3};\ \frac {1}{3};\ -\frac {c\,x}{b}\right )}{2\,{\left (c\,x^2+b\,x\right )}^{5/3}} \]
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