Optimal. Leaf size=212 \[ \frac{4 a^{15/4} c^{7/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right ),\frac{1}{2}\right )}{231 b^{9/4} \sqrt{a+b x^2}}-\frac{8 a^3 c^3 \sqrt{c x} \sqrt{a+b x^2}}{231 b^2}+\frac{8 a^2 c (c x)^{5/2} \sqrt{a+b x^2}}{385 b}+\frac{2 (c x)^{9/2} \left (a+b x^2\right )^{3/2}}{15 c}+\frac{4 a (c x)^{9/2} \sqrt{a+b x^2}}{55 c} \]
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Rubi [A] time = 0.130183, antiderivative size = 212, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {279, 321, 329, 220} \[ -\frac{8 a^3 c^3 \sqrt{c x} \sqrt{a+b x^2}}{231 b^2}+\frac{4 a^{15/4} c^{7/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{231 b^{9/4} \sqrt{a+b x^2}}+\frac{8 a^2 c (c x)^{5/2} \sqrt{a+b x^2}}{385 b}+\frac{2 (c x)^{9/2} \left (a+b x^2\right )^{3/2}}{15 c}+\frac{4 a (c x)^{9/2} \sqrt{a+b x^2}}{55 c} \]
Antiderivative was successfully verified.
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Rule 279
Rule 321
Rule 329
Rule 220
Rubi steps
\begin{align*} \int (c x)^{7/2} \left (a+b x^2\right )^{3/2} \, dx &=\frac{2 (c x)^{9/2} \left (a+b x^2\right )^{3/2}}{15 c}+\frac{1}{5} (2 a) \int (c x)^{7/2} \sqrt{a+b x^2} \, dx\\ &=\frac{4 a (c x)^{9/2} \sqrt{a+b x^2}}{55 c}+\frac{2 (c x)^{9/2} \left (a+b x^2\right )^{3/2}}{15 c}+\frac{1}{55} \left (4 a^2\right ) \int \frac{(c x)^{7/2}}{\sqrt{a+b x^2}} \, dx\\ &=\frac{8 a^2 c (c x)^{5/2} \sqrt{a+b x^2}}{385 b}+\frac{4 a (c x)^{9/2} \sqrt{a+b x^2}}{55 c}+\frac{2 (c x)^{9/2} \left (a+b x^2\right )^{3/2}}{15 c}-\frac{\left (4 a^3 c^2\right ) \int \frac{(c x)^{3/2}}{\sqrt{a+b x^2}} \, dx}{77 b}\\ &=-\frac{8 a^3 c^3 \sqrt{c x} \sqrt{a+b x^2}}{231 b^2}+\frac{8 a^2 c (c x)^{5/2} \sqrt{a+b x^2}}{385 b}+\frac{4 a (c x)^{9/2} \sqrt{a+b x^2}}{55 c}+\frac{2 (c x)^{9/2} \left (a+b x^2\right )^{3/2}}{15 c}+\frac{\left (4 a^4 c^4\right ) \int \frac{1}{\sqrt{c x} \sqrt{a+b x^2}} \, dx}{231 b^2}\\ &=-\frac{8 a^3 c^3 \sqrt{c x} \sqrt{a+b x^2}}{231 b^2}+\frac{8 a^2 c (c x)^{5/2} \sqrt{a+b x^2}}{385 b}+\frac{4 a (c x)^{9/2} \sqrt{a+b x^2}}{55 c}+\frac{2 (c x)^{9/2} \left (a+b x^2\right )^{3/2}}{15 c}+\frac{\left (8 a^4 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+\frac{b x^4}{c^2}}} \, dx,x,\sqrt{c x}\right )}{231 b^2}\\ &=-\frac{8 a^3 c^3 \sqrt{c x} \sqrt{a+b x^2}}{231 b^2}+\frac{8 a^2 c (c x)^{5/2} \sqrt{a+b x^2}}{385 b}+\frac{4 a (c x)^{9/2} \sqrt{a+b x^2}}{55 c}+\frac{2 (c x)^{9/2} \left (a+b x^2\right )^{3/2}}{15 c}+\frac{4 a^{15/4} c^{7/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{231 b^{9/4} \sqrt{a+b x^2}}\\ \end{align*}
Mathematica [C] time = 0.0772995, size = 102, normalized size = 0.48 \[ \frac{2 c^3 \sqrt{c x} \sqrt{a+b x^2} \left (5 a^3 \, _2F_1\left (-\frac{3}{2},\frac{1}{4};\frac{5}{4};-\frac{b x^2}{a}\right )-\left (5 a-11 b x^2\right ) \left (a+b x^2\right )^2 \sqrt{\frac{b x^2}{a}+1}\right )}{165 b^2 \sqrt{\frac{b x^2}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 163, normalized size = 0.8 \begin{align*}{\frac{2\,{c}^{3}}{1155\,{b}^{3}x}\sqrt{cx} \left ( 77\,{b}^{5}{x}^{9}+196\,a{b}^{4}{x}^{7}+10\,\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{-ab}{a}^{4}+131\,{a}^{2}{b}^{3}{x}^{5}-8\,{a}^{3}{b}^{2}{x}^{3}-20\,{a}^{4}bx \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}} \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{3}{2}} \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 c^{3} x^{5} + a c^{3} x^{3}\right )} \sqrt{b x^{2} + a} \sqrt{c x}, 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{3}{2}} \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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