Optimal. Leaf size=155 \[ \frac {77 a^{5/2} c^6 \sqrt {c x} \sqrt [4]{\frac {a}{b x^2}+1} E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{20 b^{7/2} \sqrt [4]{a+b x^2}}+\frac {77 a^2 c^5 (c x)^{3/2}}{60 b^3 \sqrt [4]{a+b x^2}}-\frac {11 a c^3 (c x)^{7/2}}{30 b^2 \sqrt [4]{a+b x^2}}+\frac {c (c x)^{11/2}}{5 b \sqrt [4]{a+b x^2}} \]
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Rubi [A] time = 0.07, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {285, 284, 335, 196} \[ \frac {77 a^2 c^5 (c x)^{3/2}}{60 b^3 \sqrt [4]{a+b x^2}}+\frac {77 a^{5/2} c^6 \sqrt {c x} \sqrt [4]{\frac {a}{b x^2}+1} E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{20 b^{7/2} \sqrt [4]{a+b x^2}}-\frac {11 a c^3 (c x)^{7/2}}{30 b^2 \sqrt [4]{a+b x^2}}+\frac {c (c x)^{11/2}}{5 b \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
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Rule 196
Rule 284
Rule 285
Rule 335
Rubi steps
\begin {align*} \int \frac {(c x)^{13/2}}{\left (a+b x^2\right )^{5/4}} \, dx &=\frac {c (c x)^{11/2}}{5 b \sqrt [4]{a+b x^2}}-\frac {\left (11 a c^2\right ) \int \frac {(c x)^{9/2}}{\left (a+b x^2\right )^{5/4}} \, dx}{10 b}\\ &=-\frac {11 a c^3 (c x)^{7/2}}{30 b^2 \sqrt [4]{a+b x^2}}+\frac {c (c x)^{11/2}}{5 b \sqrt [4]{a+b x^2}}+\frac {\left (77 a^2 c^4\right ) \int \frac {(c x)^{5/2}}{\left (a+b x^2\right )^{5/4}} \, dx}{60 b^2}\\ &=\frac {77 a^2 c^5 (c x)^{3/2}}{60 b^3 \sqrt [4]{a+b x^2}}-\frac {11 a c^3 (c x)^{7/2}}{30 b^2 \sqrt [4]{a+b x^2}}+\frac {c (c x)^{11/2}}{5 b \sqrt [4]{a+b x^2}}-\frac {\left (77 a^3 c^6\right ) \int \frac {\sqrt {c x}}{\left (a+b x^2\right )^{5/4}} \, dx}{40 b^3}\\ &=\frac {77 a^2 c^5 (c x)^{3/2}}{60 b^3 \sqrt [4]{a+b x^2}}-\frac {11 a c^3 (c x)^{7/2}}{30 b^2 \sqrt [4]{a+b x^2}}+\frac {c (c x)^{11/2}}{5 b \sqrt [4]{a+b x^2}}-\frac {\left (77 a^3 c^6 \sqrt [4]{1+\frac {a}{b x^2}} \sqrt {c x}\right ) \int \frac {1}{\left (1+\frac {a}{b x^2}\right )^{5/4} x^2} \, dx}{40 b^4 \sqrt [4]{a+b x^2}}\\ &=\frac {77 a^2 c^5 (c x)^{3/2}}{60 b^3 \sqrt [4]{a+b x^2}}-\frac {11 a c^3 (c x)^{7/2}}{30 b^2 \sqrt [4]{a+b x^2}}+\frac {c (c x)^{11/2}}{5 b \sqrt [4]{a+b x^2}}+\frac {\left (77 a^3 c^6 \sqrt [4]{1+\frac {a}{b x^2}} \sqrt {c x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {a x^2}{b}\right )^{5/4}} \, dx,x,\frac {1}{x}\right )}{40 b^4 \sqrt [4]{a+b x^2}}\\ &=\frac {77 a^2 c^5 (c x)^{3/2}}{60 b^3 \sqrt [4]{a+b x^2}}-\frac {11 a c^3 (c x)^{7/2}}{30 b^2 \sqrt [4]{a+b x^2}}+\frac {c (c x)^{11/2}}{5 b \sqrt [4]{a+b x^2}}+\frac {77 a^{5/2} c^6 \sqrt [4]{1+\frac {a}{b x^2}} \sqrt {c x} E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{20 b^{7/2} \sqrt [4]{a+b x^2}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 87, normalized size = 0.56 \[ \frac {c^5 (c x)^{3/2} \left (-77 a^2 \sqrt [4]{\frac {b x^2}{a}+1} \, _2F_1\left (\frac {3}{4},\frac {5}{4};\frac {7}{4};-\frac {b x^2}{a}\right )+77 a^2-22 a b x^2+12 b^2 x^4\right )}{60 b^3 \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.80, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {c x} c^{6} x^{6}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}, x\right ) \]
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 \[ \int \frac {\left (c x\right )^{\frac {13}{2}}}{{\left (b x^{2} + a\right )}^{\frac {5}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.33, size = 0, normalized size = 0.00 \[ \int \frac {\left (c x \right )^{\frac {13}{2}}}{\left (b \,x^{2}+a \right )^{\frac {5}{4}}}\, dx \]
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 \[ \int \frac {\left (c x\right )^{\frac {13}{2}}}{{\left (b x^{2} + a\right )}^{\frac {5}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c\,x\right )}^{13/2}}{{\left (b\,x^2+a\right )}^{5/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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