Optimal. Leaf size=153 \[ -\frac{5 a^{3/4} c^{7/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right ),\frac{1}{2}\right )}{6 b^{9/4} \sqrt{a+b x^2}}+\frac{5 c^3 \sqrt{c x} \sqrt{a+b x^2}}{3 b^2}-\frac{c (c x)^{5/2}}{b \sqrt{a+b x^2}} \]
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Rubi [A] time = 0.0866138, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {288, 321, 329, 220} \[ -\frac{5 a^{3/4} c^{7/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{6 b^{9/4} \sqrt{a+b x^2}}+\frac{5 c^3 \sqrt{c x} \sqrt{a+b x^2}}{3 b^2}-\frac{c (c x)^{5/2}}{b \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
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Rule 288
Rule 321
Rule 329
Rule 220
Rubi steps
\begin{align*} \int \frac{(c x)^{7/2}}{\left (a+b x^2\right )^{3/2}} \, dx &=-\frac{c (c x)^{5/2}}{b \sqrt{a+b x^2}}+\frac{\left (5 c^2\right ) \int \frac{(c x)^{3/2}}{\sqrt{a+b x^2}} \, dx}{2 b}\\ &=-\frac{c (c x)^{5/2}}{b \sqrt{a+b x^2}}+\frac{5 c^3 \sqrt{c x} \sqrt{a+b x^2}}{3 b^2}-\frac{\left (5 a c^4\right ) \int \frac{1}{\sqrt{c x} \sqrt{a+b x^2}} \, dx}{6 b^2}\\ &=-\frac{c (c x)^{5/2}}{b \sqrt{a+b x^2}}+\frac{5 c^3 \sqrt{c x} \sqrt{a+b x^2}}{3 b^2}-\frac{\left (5 a c^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+\frac{b x^4}{c^2}}} \, dx,x,\sqrt{c x}\right )}{3 b^2}\\ &=-\frac{c (c x)^{5/2}}{b \sqrt{a+b x^2}}+\frac{5 c^3 \sqrt{c x} \sqrt{a+b x^2}}{3 b^2}-\frac{5 a^{3/4} 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 )}{6 b^{9/4} \sqrt{a+b x^2}}\\ \end{align*}
Mathematica [C] time = 0.0313815, size = 74, normalized size = 0.48 \[ \frac{c^3 \sqrt{c x} \left (-5 a \sqrt{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^2}{a}\right )+5 a+2 b x^2\right )}{3 b^2 \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.031, size = 128, normalized size = 0.8 \begin{align*} -{\frac{{c}^{3}}{6\,{b}^{3}x}\sqrt{cx} \left ( 5\,\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\,{b}^{2}{x}^{3}-10\,abx \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 \frac{\left (c x\right )^{\frac{7}{2}}}{{\left (b x^{2} + a\right )}^{\frac{3}{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{\sqrt{b x^{2} + a} \sqrt{c x} c^{3} x^{3}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 159.68, size = 44, normalized size = 0.29 \begin{align*} \frac{c^{\frac{7}{2}} x^{\frac{9}{2}} \Gamma \left (\frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{9}{4} \\ \frac{13}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac{3}{2}} \Gamma \left (\frac{13}{4}\right )} \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 (c x\right )^{\frac{7}{2}}}{{\left (b x^{2} + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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