Optimal. Leaf size=83 \[ \frac {4 \sqrt {2} \sqrt [4]{-\frac {c \left (b x+c x^2\right )}{b^2}} E\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {2 c x}{b}+1\right )\right |2\right )}{b \sqrt [4]{b x+c x^2}}-\frac {4 (b+2 c x)}{b^2 \sqrt [4]{b x+c x^2}} \]
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Rubi [A] time = 0.03, antiderivative size = 83, 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 = {614, 622, 619, 228} \[ \frac {4 \sqrt {2} \sqrt [4]{-\frac {c \left (b x+c x^2\right )}{b^2}} E\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {2 c x}{b}+1\right )\right |2\right )}{b \sqrt [4]{b x+c x^2}}-\frac {4 (b+2 c x)}{b^2 \sqrt [4]{b x+c x^2}} \]
Antiderivative was successfully verified.
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Rule 228
Rule 614
Rule 619
Rule 622
Rubi steps
\begin {align*} \int \frac {1}{\left (b x+c x^2\right )^{5/4}} \, dx &=-\frac {4 (b+2 c x)}{b^2 \sqrt [4]{b x+c x^2}}+\frac {(4 c) \int \frac {1}{\sqrt [4]{b x+c x^2}} \, dx}{b^2}\\ &=-\frac {4 (b+2 c x)}{b^2 \sqrt [4]{b x+c x^2}}+\frac {\left (4 c \sqrt [4]{-\frac {c \left (b x+c x^2\right )}{b^2}}\right ) \int \frac {1}{\sqrt [4]{-\frac {c x}{b}-\frac {c^2 x^2}{b^2}}} \, dx}{b^2 \sqrt [4]{b x+c x^2}}\\ &=-\frac {4 (b+2 c x)}{b^2 \sqrt [4]{b x+c x^2}}-\frac {\left (2 \sqrt {2} \sqrt [4]{-\frac {c \left (b x+c x^2\right )}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1-\frac {b^2 x^2}{c^2}}} \, dx,x,-\frac {c}{b}-\frac {2 c^2 x}{b^2}\right )}{c \sqrt [4]{b x+c x^2}}\\ &=-\frac {4 (b+2 c x)}{b^2 \sqrt [4]{b x+c x^2}}+\frac {4 \sqrt {2} \sqrt [4]{-\frac {c \left (b x+c x^2\right )}{b^2}} E\left (\left .\frac {1}{2} \sin ^{-1}\left (1+\frac {2 c x}{b}\right )\right |2\right )}{b \sqrt [4]{b x+c x^2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 45, normalized size = 0.54 \[ -\frac {4 \sqrt [4]{\frac {c x}{b}+1} \, _2F_1\left (-\frac {1}{4},\frac {5}{4};\frac {3}{4};-\frac {c x}{b}\right )}{b \sqrt [4]{x (b+c x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.97, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c x^{2} + b x\right )}^{\frac {3}{4}}}{c^{2} x^{4} + 2 \, b c x^{3} + b^{2} x^{2}}, x\right ) \]
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 \[ \int \frac {1}{{\left (c x^{2} + b x\right )}^{\frac {5}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.92, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c \,x^{2}+b x \right )^{\frac {5}{4}}}\, dx \]
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 \[ \int \frac {1}{{\left (c x^{2} + b x\right )}^{\frac {5}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.22, size = 36, normalized size = 0.43 \[ -\frac {4\,x\,{\left (\frac {c\,x}{b}+1\right )}^{5/4}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {5}{4};\ \frac {3}{4};\ -\frac {c\,x}{b}\right )}{{\left (c\,x^2+b\,x\right )}^{5/4}} \]
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 \[ \int \frac {1}{\left (b x + c x^{2}\right )^{\frac {5}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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