Optimal. Leaf size=154 \[ -\frac{8 b^{7/2} (c x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} \text{EllipticF}\left (\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ),2\right )}{77 a^{5/2} c^8 \left (a+b x^2\right )^{3/4}}+\frac{4 b^2 \sqrt [4]{a+b x^2}}{77 a^2 c^5 (c x)^{3/2}}-\frac{2 b \sqrt [4]{a+b x^2}}{77 a c^3 (c x)^{7/2}}-\frac{2 \sqrt [4]{a+b x^2}}{11 c (c x)^{11/2}} \]
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Rubi [A] time = 0.110039, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {277, 325, 329, 237, 335, 275, 231} \[ \frac{4 b^2 \sqrt [4]{a+b x^2}}{77 a^2 c^5 (c x)^{3/2}}-\frac{8 b^{7/2} (c x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{77 a^{5/2} c^8 \left (a+b x^2\right )^{3/4}}-\frac{2 b \sqrt [4]{a+b x^2}}{77 a c^3 (c x)^{7/2}}-\frac{2 \sqrt [4]{a+b x^2}}{11 c (c x)^{11/2}} \]
Antiderivative was successfully verified.
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Rule 277
Rule 325
Rule 329
Rule 237
Rule 335
Rule 275
Rule 231
Rubi steps
\begin{align*} \int \frac{\sqrt [4]{a+b x^2}}{(c x)^{13/2}} \, dx &=-\frac{2 \sqrt [4]{a+b x^2}}{11 c (c x)^{11/2}}+\frac{b \int \frac{1}{(c x)^{9/2} \left (a+b x^2\right )^{3/4}} \, dx}{11 c^2}\\ &=-\frac{2 \sqrt [4]{a+b x^2}}{11 c (c x)^{11/2}}-\frac{2 b \sqrt [4]{a+b x^2}}{77 a c^3 (c x)^{7/2}}-\frac{\left (6 b^2\right ) \int \frac{1}{(c x)^{5/2} \left (a+b x^2\right )^{3/4}} \, dx}{77 a c^4}\\ &=-\frac{2 \sqrt [4]{a+b x^2}}{11 c (c x)^{11/2}}-\frac{2 b \sqrt [4]{a+b x^2}}{77 a c^3 (c x)^{7/2}}+\frac{4 b^2 \sqrt [4]{a+b x^2}}{77 a^2 c^5 (c x)^{3/2}}+\frac{\left (4 b^3\right ) \int \frac{1}{\sqrt{c x} \left (a+b x^2\right )^{3/4}} \, dx}{77 a^2 c^6}\\ &=-\frac{2 \sqrt [4]{a+b x^2}}{11 c (c x)^{11/2}}-\frac{2 b \sqrt [4]{a+b x^2}}{77 a c^3 (c x)^{7/2}}+\frac{4 b^2 \sqrt [4]{a+b x^2}}{77 a^2 c^5 (c x)^{3/2}}+\frac{\left (8 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{\left (a+\frac{b x^4}{c^2}\right )^{3/4}} \, dx,x,\sqrt{c x}\right )}{77 a^2 c^7}\\ &=-\frac{2 \sqrt [4]{a+b x^2}}{11 c (c x)^{11/2}}-\frac{2 b \sqrt [4]{a+b x^2}}{77 a c^3 (c x)^{7/2}}+\frac{4 b^2 \sqrt [4]{a+b x^2}}{77 a^2 c^5 (c x)^{3/2}}+\frac{\left (8 b^3 \left (1+\frac{a}{b x^2}\right )^{3/4} (c x)^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{a c^2}{b x^4}\right )^{3/4} x^3} \, dx,x,\sqrt{c x}\right )}{77 a^2 c^7 \left (a+b x^2\right )^{3/4}}\\ &=-\frac{2 \sqrt [4]{a+b x^2}}{11 c (c x)^{11/2}}-\frac{2 b \sqrt [4]{a+b x^2}}{77 a c^3 (c x)^{7/2}}+\frac{4 b^2 \sqrt [4]{a+b x^2}}{77 a^2 c^5 (c x)^{3/2}}-\frac{\left (8 b^3 \left (1+\frac{a}{b x^2}\right )^{3/4} (c x)^{3/2}\right ) \operatorname{Subst}\left (\int \frac{x}{\left (1+\frac{a c^2 x^4}{b}\right )^{3/4}} \, dx,x,\frac{1}{\sqrt{c x}}\right )}{77 a^2 c^7 \left (a+b x^2\right )^{3/4}}\\ &=-\frac{2 \sqrt [4]{a+b x^2}}{11 c (c x)^{11/2}}-\frac{2 b \sqrt [4]{a+b x^2}}{77 a c^3 (c x)^{7/2}}+\frac{4 b^2 \sqrt [4]{a+b x^2}}{77 a^2 c^5 (c x)^{3/2}}-\frac{\left (4 b^3 \left (1+\frac{a}{b x^2}\right )^{3/4} (c x)^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{a c^2 x^2}{b}\right )^{3/4}} \, dx,x,\frac{1}{c x}\right )}{77 a^2 c^7 \left (a+b x^2\right )^{3/4}}\\ &=-\frac{2 \sqrt [4]{a+b x^2}}{11 c (c x)^{11/2}}-\frac{2 b \sqrt [4]{a+b x^2}}{77 a c^3 (c x)^{7/2}}+\frac{4 b^2 \sqrt [4]{a+b x^2}}{77 a^2 c^5 (c x)^{3/2}}-\frac{8 b^{7/2} \left (1+\frac{a}{b x^2}\right )^{3/4} (c x)^{3/2} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{77 a^{5/2} c^8 \left (a+b x^2\right )^{3/4}}\\ \end{align*}
Mathematica [C] time = 0.012955, size = 56, normalized size = 0.36 \[ -\frac{2 x \sqrt [4]{a+b x^2} \, _2F_1\left (-\frac{11}{4},-\frac{1}{4};-\frac{7}{4};-\frac{b x^2}{a}\right )}{11 (c x)^{13/2} \sqrt [4]{\frac{b x^2}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.026, size = 0, normalized size = 0. \begin{align*} \int{\sqrt [4]{b{x}^{2}+a} \left ( cx \right ) ^{-{\frac{13}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{2} + a\right )}^{\frac{1}{4}}}{\left (c x\right )^{\frac{13}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{c x}}{c^{7} x^{7}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{2} + a\right )}^{\frac{1}{4}}}{\left (c x\right )^{\frac{13}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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