Optimal. Leaf size=59 \[ \frac {2 \sqrt {2} b \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{3/4} F\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {2 c x}{b}+1\right )\right |2\right )}{c \left (b x+c x^2\right )^{3/4}} \]
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Rubi [A] time = 0.02, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {622, 619, 232} \[ \frac {2 \sqrt {2} b \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{3/4} F\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {2 c x}{b}+1\right )\right |2\right )}{c \left (b x+c x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
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Rule 232
Rule 619
Rule 622
Rubi steps
\begin {align*} \int \frac {1}{\left (b x+c x^2\right )^{3/4}} \, dx &=\frac {\left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{3/4} \int \frac {1}{\left (-\frac {c x}{b}-\frac {c^2 x^2}{b^2}\right )^{3/4}} \, dx}{\left (b x+c x^2\right )^{3/4}}\\ &=-\frac {\left (\sqrt {2} b^2 \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {b^2 x^2}{c^2}\right )^{3/4}} \, dx,x,-\frac {c}{b}-\frac {2 c^2 x}{b^2}\right )}{c^2 \left (b x+c x^2\right )^{3/4}}\\ &=\frac {2 \sqrt {2} b \left (-\frac {c \left (b x+c x^2\right )}{b^2}\right )^{3/4} F\left (\left .\frac {1}{2} \sin ^{-1}\left (1+\frac {2 c x}{b}\right )\right |2\right )}{c \left (b x+c x^2\right )^{3/4}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 43, normalized size = 0.73 \[ \frac {4 x \left (\frac {c x}{b}+1\right )^{3/4} \, _2F_1\left (\frac {1}{4},\frac {3}{4};\frac {5}{4};-\frac {c x}{b}\right )}{(x (b+c x))^{3/4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{{\left (c x^{2} + b x\right )}^{\frac {3}{4}}}, x\right ) \]
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 \[ \int \frac {1}{{\left (c x^{2} + b x\right )}^{\frac {3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.72, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c \,x^{2}+b x \right )^{\frac {3}{4}}}\, dx \]
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 \[ \int \frac {1}{{\left (c x^{2} + b x\right )}^{\frac {3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 36, normalized size = 0.61 \[ \frac {4\,x\,{\left (\frac {c\,x}{b}+1\right )}^{3/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {3}{4};\ \frac {5}{4};\ -\frac {c\,x}{b}\right )}{{\left (c\,x^2+b\,x\right )}^{3/4}} \]
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 \[ \int \frac {1}{\left (b x + c x^{2}\right )^{\frac {3}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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