Optimal. Leaf size=322 \[ \frac {\sqrt [3]{2} 3^{3/4} \sqrt {2-\sqrt {3}} b^2 \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{2/3} \left (1-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right ) \sqrt {\frac {2 \sqrt [3]{2} \left (-\frac {c x (b+c x)}{b^2}\right )^{2/3}+2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}+1}{\left (-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}-\sqrt {3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac {-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}+\sqrt {3}+1}{-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{c (b+2 c x) \left (b x+c x^2\right )^{2/3} \sqrt {-\frac {1-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}}{\left (-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}-\sqrt {3}+1\right )^2}}} \]
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Rubi [A] time = 0.39, antiderivative size = 322, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {622, 619, 236, 219} \[ \frac {\sqrt [3]{2} 3^{3/4} \sqrt {2-\sqrt {3}} b^2 \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{2/3} \left (1-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}\right ) \sqrt {\frac {2 \sqrt [3]{2} \left (-\frac {c x (b+c x)}{b^2}\right )^{2/3}+2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}+1}{\left (-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}-\sqrt {3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac {-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}+\sqrt {3}+1}{-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{c (b+2 c x) \left (b x+c x^2\right )^{2/3} \sqrt {-\frac {1-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}}{\left (-2^{2/3} \sqrt [3]{-\frac {c x (b+c x)}{b^2}}-\sqrt {3}+1\right )^2}}} \]
Antiderivative was successfully verified.
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Rule 219
Rule 236
Rule 619
Rule 622
Rubi steps
\begin {align*} \int \frac {1}{\left (b x+c x^2\right )^{2/3}} \, dx &=\frac {\left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{2/3} \int \frac {1}{\left (-\frac {c x}{b}-\frac {c^2 x^2}{b^2}\right )^{2/3}} \, dx}{\left (b x+c x^2\right )^{2/3}}\\ &=-\frac {\left (\sqrt [3]{2} b^2 \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{2/3}\right ) \operatorname {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 )^{2/3}}\\ &=\frac {\left (3 \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{2/3} \sqrt {-1-\frac {4 c x}{b}-\frac {4 c^2 x^2}{b^2}}\right ) \operatorname {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 )^{2/3}}\\ &=\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 )^{2/3} \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 )^{2/3} \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*}
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Mathematica [C] time = 0.01, size = 43, normalized size = 0.13 \[ \frac {3 x \left (\frac {c x}{b}+1\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-\frac {c x}{b}\right )}{(x (b+c x))^{2/3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.00, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{{\left (c x^{2} + b x\right )}^{\frac {2}{3}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (c x^{2} + b x\right )}^{\frac {2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.77, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c \,x^{2}+b x \right )^{\frac {2}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (c x^{2} + b x\right )}^{\frac {2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.22, size = 36, normalized size = 0.11 \[ \frac {3\,x\,{\left (\frac {c\,x}{b}+1\right )}^{2/3}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{3},\frac {2}{3};\ \frac {4}{3};\ -\frac {c\,x}{b}\right )}{{\left (c\,x^2+b\,x\right )}^{2/3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b x + c x^{2}\right )^{\frac {2}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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