Optimal. Leaf size=146 \[ -\frac {448 c^2 (b+2 c x)}{15 b^6 \sqrt [4]{b x+c x^2}}+\frac {112 c (b+2 c x)}{45 b^4 \left (b x+c x^2\right )^{5/4}}-\frac {4 (b+2 c x)}{9 b^2 \left (b x+c x^2\right )^{9/4}}+\frac {448 \sqrt {2} c^2 \sqrt [4]{-\frac {c \left (b x+c x^2\right )}{b^2}} E\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {2 c x}{b}+1\right )\right |2\right )}{15 b^5 \sqrt [4]{b x+c x^2}} \]
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Rubi [A] time = 0.06, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {614, 622, 619, 228} \[ -\frac {448 c^2 (b+2 c x)}{15 b^6 \sqrt [4]{b x+c x^2}}+\frac {448 \sqrt {2} c^2 \sqrt [4]{-\frac {c \left (b x+c x^2\right )}{b^2}} E\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {2 c x}{b}+1\right )\right |2\right )}{15 b^5 \sqrt [4]{b x+c x^2}}+\frac {112 c (b+2 c x)}{45 b^4 \left (b x+c x^2\right )^{5/4}}-\frac {4 (b+2 c x)}{9 b^2 \left (b x+c x^2\right )^{9/4}} \]
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 )^{13/4}} \, dx &=-\frac {4 (b+2 c x)}{9 b^2 \left (b x+c x^2\right )^{9/4}}-\frac {(28 c) \int \frac {1}{\left (b x+c x^2\right )^{9/4}} \, dx}{9 b^2}\\ &=-\frac {4 (b+2 c x)}{9 b^2 \left (b x+c x^2\right )^{9/4}}+\frac {112 c (b+2 c x)}{45 b^4 \left (b x+c x^2\right )^{5/4}}+\frac {\left (112 c^2\right ) \int \frac {1}{\left (b x+c x^2\right )^{5/4}} \, dx}{15 b^4}\\ &=-\frac {4 (b+2 c x)}{9 b^2 \left (b x+c x^2\right )^{9/4}}+\frac {112 c (b+2 c x)}{45 b^4 \left (b x+c x^2\right )^{5/4}}-\frac {448 c^2 (b+2 c x)}{15 b^6 \sqrt [4]{b x+c x^2}}+\frac {\left (448 c^3\right ) \int \frac {1}{\sqrt [4]{b x+c x^2}} \, dx}{15 b^6}\\ &=-\frac {4 (b+2 c x)}{9 b^2 \left (b x+c x^2\right )^{9/4}}+\frac {112 c (b+2 c x)}{45 b^4 \left (b x+c x^2\right )^{5/4}}-\frac {448 c^2 (b+2 c x)}{15 b^6 \sqrt [4]{b x+c x^2}}+\frac {\left (448 c^3 \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}{15 b^6 \sqrt [4]{b x+c x^2}}\\ &=-\frac {4 (b+2 c x)}{9 b^2 \left (b x+c x^2\right )^{9/4}}+\frac {112 c (b+2 c x)}{45 b^4 \left (b x+c x^2\right )^{5/4}}-\frac {448 c^2 (b+2 c x)}{15 b^6 \sqrt [4]{b x+c x^2}}-\frac {\left (224 \sqrt {2} c \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 )}{15 b^4 \sqrt [4]{b x+c x^2}}\\ &=-\frac {4 (b+2 c x)}{9 b^2 \left (b x+c x^2\right )^{9/4}}+\frac {112 c (b+2 c x)}{45 b^4 \left (b x+c x^2\right )^{5/4}}-\frac {448 c^2 (b+2 c x)}{15 b^6 \sqrt [4]{b x+c x^2}}+\frac {448 \sqrt {2} c^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 )}{15 b^5 \sqrt [4]{b x+c x^2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 50, normalized size = 0.34 \[ -\frac {4 \sqrt [4]{\frac {c x}{b}+1} \, _2F_1\left (-\frac {9}{4},\frac {13}{4};-\frac {5}{4};-\frac {c x}{b}\right )}{9 b^3 x^2 \sqrt [4]{x (b+c x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.11, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c x^{2} + b x\right )}^{\frac {3}{4}}}{c^{4} x^{8} + 4 \, b c^{3} x^{7} + 6 \, b^{2} c^{2} x^{6} + 4 \, b^{3} c x^{5} + b^{4} x^{4}}, x\right ) \]
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 \[ \int \frac {1}{{\left (c x^{2} + b x\right )}^{\frac {13}{4}}}\,{d x} \]
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 \[ \int \frac {1}{\left (c \,x^{2}+b x \right )^{\frac {13}{4}}}\, dx \]
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 \[ \int \frac {1}{{\left (c x^{2} + b x\right )}^{\frac {13}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.29, size = 36, normalized size = 0.25 \[ -\frac {4\,x\,{\left (\frac {c\,x}{b}+1\right )}^{13/4}\,{{}}_2{\mathrm {F}}_1\left (-\frac {9}{4},\frac {13}{4};\ -\frac {5}{4};\ -\frac {c\,x}{b}\right )}{9\,{\left (c\,x^2+b\,x\right )}^{13/4}} \]
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 \[ \int \frac {1}{\left (b x + c x^{2}\right )^{\frac {13}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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