Optimal. Leaf size=126 \[ -\frac{4 b^{3/2} \sqrt{c x} \sqrt [4]{\frac{a}{b x^2}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{5 a^{3/2} c^4 \sqrt [4]{a+b x^2}}+\frac{4 b}{5 a c^3 \sqrt{c x} \sqrt [4]{a+b x^2}}-\frac{2 \left (a+b x^2\right )^{3/4}}{5 a c (c x)^{5/2}} \]
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Rubi [A] time = 0.0490704, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {325, 316, 284, 335, 196} \[ -\frac{4 b^{3/2} \sqrt{c x} \sqrt [4]{\frac{a}{b x^2}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{5 a^{3/2} c^4 \sqrt [4]{a+b x^2}}+\frac{4 b}{5 a c^3 \sqrt{c x} \sqrt [4]{a+b x^2}}-\frac{2 \left (a+b x^2\right )^{3/4}}{5 a c (c x)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 325
Rule 316
Rule 284
Rule 335
Rule 196
Rubi steps
\begin{align*} \int \frac{1}{(c x)^{7/2} \sqrt [4]{a+b x^2}} \, dx &=-\frac{2 \left (a+b x^2\right )^{3/4}}{5 a c (c x)^{5/2}}-\frac{(2 b) \int \frac{1}{(c x)^{3/2} \sqrt [4]{a+b x^2}} \, dx}{5 a c^2}\\ &=\frac{4 b}{5 a c^3 \sqrt{c x} \sqrt [4]{a+b x^2}}-\frac{2 \left (a+b x^2\right )^{3/4}}{5 a c (c x)^{5/2}}+\frac{\left (2 b^2\right ) \int \frac{\sqrt{c x}}{\left (a+b x^2\right )^{5/4}} \, dx}{5 a c^4}\\ &=\frac{4 b}{5 a c^3 \sqrt{c x} \sqrt [4]{a+b x^2}}-\frac{2 \left (a+b x^2\right )^{3/4}}{5 a c (c x)^{5/2}}+\frac{\left (2 b \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}{5 a c^4 \sqrt [4]{a+b x^2}}\\ &=\frac{4 b}{5 a c^3 \sqrt{c x} \sqrt [4]{a+b x^2}}-\frac{2 \left (a+b x^2\right )^{3/4}}{5 a c (c x)^{5/2}}-\frac{\left (2 b \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 )}{5 a c^4 \sqrt [4]{a+b x^2}}\\ &=\frac{4 b}{5 a c^3 \sqrt{c x} \sqrt [4]{a+b x^2}}-\frac{2 \left (a+b x^2\right )^{3/4}}{5 a c (c x)^{5/2}}-\frac{4 b^{3/2} \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 )}{5 a^{3/2} c^4 \sqrt [4]{a+b x^2}}\\ \end{align*}
Mathematica [C] time = 0.0122788, size = 56, normalized size = 0.44 \[ -\frac{2 x \sqrt [4]{\frac{b x^2}{a}+1} \, _2F_1\left (-\frac{5}{4},\frac{1}{4};-\frac{1}{4};-\frac{b x^2}{a}\right )}{5 (c x)^{7/2} \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.036, size = 0, normalized size = 0. \begin{align*} \int{ \left ( cx \right ) ^{-{\frac{7}{2}}}{\frac{1}{\sqrt [4]{b{x}^{2}+a}}}}\, 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{1}{{\left (b x^{2} + a\right )}^{\frac{1}{4}} \left (c x\right )^{\frac{7}{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{3}{4}} \sqrt{c x}}{b c^{4} x^{6} + a c^{4} x^{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 96.643, size = 34, normalized size = 0.27 \begin{align*} - \frac{{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{2}}} \right )}}{3 \sqrt [4]{b} c^{\frac{7}{2}} x^{3}} \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{1}{{\left (b x^{2} + a\right )}^{\frac{1}{4}} \left (c x\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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