Optimal. Leaf size=152 \[ -\frac{a^{5/2} c^2 (c x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} \text{EllipticF}\left (\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ),2\right )}{12 b^{3/2} \left (a+b x^2\right )^{3/4}}-\frac{a^2 c^3 \sqrt{c x} \sqrt [4]{a+b x^2}}{12 b^2}+\frac{(c x)^{9/2} \sqrt [4]{a+b x^2}}{5 c}+\frac{a c (c x)^{5/2} \sqrt [4]{a+b x^2}}{30 b} \]
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Rubi [A] time = 0.107769, antiderivative size = 152, 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 = {279, 321, 329, 237, 335, 275, 231} \[ -\frac{a^2 c^3 \sqrt{c x} \sqrt [4]{a+b x^2}}{12 b^2}-\frac{a^{5/2} c^2 (c x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{12 b^{3/2} \left (a+b x^2\right )^{3/4}}+\frac{(c x)^{9/2} \sqrt [4]{a+b x^2}}{5 c}+\frac{a c (c x)^{5/2} \sqrt [4]{a+b x^2}}{30 b} \]
Antiderivative was successfully verified.
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Rule 279
Rule 321
Rule 329
Rule 237
Rule 335
Rule 275
Rule 231
Rubi steps
\begin{align*} \int (c x)^{7/2} \sqrt [4]{a+b x^2} \, dx &=\frac{(c x)^{9/2} \sqrt [4]{a+b x^2}}{5 c}+\frac{1}{10} a \int \frac{(c x)^{7/2}}{\left (a+b x^2\right )^{3/4}} \, dx\\ &=\frac{a c (c x)^{5/2} \sqrt [4]{a+b x^2}}{30 b}+\frac{(c x)^{9/2} \sqrt [4]{a+b x^2}}{5 c}-\frac{\left (a^2 c^2\right ) \int \frac{(c x)^{3/2}}{\left (a+b x^2\right )^{3/4}} \, dx}{12 b}\\ &=-\frac{a^2 c^3 \sqrt{c x} \sqrt [4]{a+b x^2}}{12 b^2}+\frac{a c (c x)^{5/2} \sqrt [4]{a+b x^2}}{30 b}+\frac{(c x)^{9/2} \sqrt [4]{a+b x^2}}{5 c}+\frac{\left (a^3 c^4\right ) \int \frac{1}{\sqrt{c x} \left (a+b x^2\right )^{3/4}} \, dx}{24 b^2}\\ &=-\frac{a^2 c^3 \sqrt{c x} \sqrt [4]{a+b x^2}}{12 b^2}+\frac{a c (c x)^{5/2} \sqrt [4]{a+b x^2}}{30 b}+\frac{(c x)^{9/2} \sqrt [4]{a+b x^2}}{5 c}+\frac{\left (a^3 c^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 )}{12 b^2}\\ &=-\frac{a^2 c^3 \sqrt{c x} \sqrt [4]{a+b x^2}}{12 b^2}+\frac{a c (c x)^{5/2} \sqrt [4]{a+b x^2}}{30 b}+\frac{(c x)^{9/2} \sqrt [4]{a+b x^2}}{5 c}+\frac{\left (a^3 c^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 )}{12 b^2 \left (a+b x^2\right )^{3/4}}\\ &=-\frac{a^2 c^3 \sqrt{c x} \sqrt [4]{a+b x^2}}{12 b^2}+\frac{a c (c x)^{5/2} \sqrt [4]{a+b x^2}}{30 b}+\frac{(c x)^{9/2} \sqrt [4]{a+b x^2}}{5 c}-\frac{\left (a^3 c^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 )}{12 b^2 \left (a+b x^2\right )^{3/4}}\\ &=-\frac{a^2 c^3 \sqrt{c x} \sqrt [4]{a+b x^2}}{12 b^2}+\frac{a c (c x)^{5/2} \sqrt [4]{a+b x^2}}{30 b}+\frac{(c x)^{9/2} \sqrt [4]{a+b x^2}}{5 c}-\frac{\left (a^3 c^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 )}{24 b^2 \left (a+b x^2\right )^{3/4}}\\ &=-\frac{a^2 c^3 \sqrt{c x} \sqrt [4]{a+b x^2}}{12 b^2}+\frac{a c (c x)^{5/2} \sqrt [4]{a+b x^2}}{30 b}+\frac{(c x)^{9/2} \sqrt [4]{a+b x^2}}{5 c}-\frac{a^{5/2} c^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 )}{12 b^{3/2} \left (a+b x^2\right )^{3/4}}\\ \end{align*}
Mathematica [C] time = 0.0653525, size = 102, normalized size = 0.67 \[ \frac{c^3 \sqrt{c x} \sqrt [4]{a+b x^2} \left (\sqrt [4]{\frac{b x^2}{a}+1} \left (-5 a^2+a b x^2+6 b^2 x^4\right )+5 a^2 \, _2F_1\left (-\frac{1}{4},\frac{1}{4};\frac{5}{4};-\frac{b x^2}{a}\right )\right )}{30 b^2 \sqrt [4]{\frac{b x^2}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.024, size = 0, normalized size = 0. \begin{align*} \int \left ( cx \right ) ^{{\frac{7}{2}}}\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{\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 ({\left (b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{c x} c^{3} x^{3}, 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{\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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