Optimal. Leaf size=331 \[ -\frac{21 b^{5/4} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right ),\frac{1}{2}\right )}{10 a^{11/4} c^{7/2} \sqrt{a+b x^2}}-\frac{21 b^{3/2} \sqrt{c x} \sqrt{a+b x^2}}{5 a^3 c^4 \left (\sqrt{a}+\sqrt{b} x\right )}+\frac{21 b^{5/4} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{5 a^{11/4} c^{7/2} \sqrt{a+b x^2}}+\frac{21 b \sqrt{a+b x^2}}{5 a^3 c^3 \sqrt{c x}}-\frac{7 \sqrt{a+b x^2}}{5 a^2 c (c x)^{5/2}}+\frac{1}{a c (c x)^{5/2} \sqrt{a+b x^2}} \]
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Rubi [A] time = 0.254267, antiderivative size = 331, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {290, 325, 329, 305, 220, 1196} \[ -\frac{21 b^{3/2} \sqrt{c x} \sqrt{a+b x^2}}{5 a^3 c^4 \left (\sqrt{a}+\sqrt{b} x\right )}-\frac{21 b^{5/4} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{10 a^{11/4} c^{7/2} \sqrt{a+b x^2}}+\frac{21 b^{5/4} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{5 a^{11/4} c^{7/2} \sqrt{a+b x^2}}+\frac{21 b \sqrt{a+b x^2}}{5 a^3 c^3 \sqrt{c x}}-\frac{7 \sqrt{a+b x^2}}{5 a^2 c (c x)^{5/2}}+\frac{1}{a c (c x)^{5/2} \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
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Rule 290
Rule 325
Rule 329
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{1}{(c x)^{7/2} \left (a+b x^2\right )^{3/2}} \, dx &=\frac{1}{a c (c x)^{5/2} \sqrt{a+b x^2}}+\frac{7 \int \frac{1}{(c x)^{7/2} \sqrt{a+b x^2}} \, dx}{2 a}\\ &=\frac{1}{a c (c x)^{5/2} \sqrt{a+b x^2}}-\frac{7 \sqrt{a+b x^2}}{5 a^2 c (c x)^{5/2}}-\frac{(21 b) \int \frac{1}{(c x)^{3/2} \sqrt{a+b x^2}} \, dx}{10 a^2 c^2}\\ &=\frac{1}{a c (c x)^{5/2} \sqrt{a+b x^2}}-\frac{7 \sqrt{a+b x^2}}{5 a^2 c (c x)^{5/2}}+\frac{21 b \sqrt{a+b x^2}}{5 a^3 c^3 \sqrt{c x}}-\frac{\left (21 b^2\right ) \int \frac{\sqrt{c x}}{\sqrt{a+b x^2}} \, dx}{10 a^3 c^4}\\ &=\frac{1}{a c (c x)^{5/2} \sqrt{a+b x^2}}-\frac{7 \sqrt{a+b x^2}}{5 a^2 c (c x)^{5/2}}+\frac{21 b \sqrt{a+b x^2}}{5 a^3 c^3 \sqrt{c x}}-\frac{\left (21 b^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+\frac{b x^4}{c^2}}} \, dx,x,\sqrt{c x}\right )}{5 a^3 c^5}\\ &=\frac{1}{a c (c x)^{5/2} \sqrt{a+b x^2}}-\frac{7 \sqrt{a+b x^2}}{5 a^2 c (c x)^{5/2}}+\frac{21 b \sqrt{a+b x^2}}{5 a^3 c^3 \sqrt{c x}}-\frac{\left (21 b^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+\frac{b x^4}{c^2}}} \, dx,x,\sqrt{c x}\right )}{5 a^{5/2} c^4}+\frac{\left (21 b^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1-\frac{\sqrt{b} x^2}{\sqrt{a} c}}{\sqrt{a+\frac{b x^4}{c^2}}} \, dx,x,\sqrt{c x}\right )}{5 a^{5/2} c^4}\\ &=\frac{1}{a c (c x)^{5/2} \sqrt{a+b x^2}}-\frac{7 \sqrt{a+b x^2}}{5 a^2 c (c x)^{5/2}}+\frac{21 b \sqrt{a+b x^2}}{5 a^3 c^3 \sqrt{c x}}-\frac{21 b^{3/2} \sqrt{c x} \sqrt{a+b x^2}}{5 a^3 c^4 \left (\sqrt{a}+\sqrt{b} x\right )}+\frac{21 b^{5/4} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{5 a^{11/4} c^{7/2} \sqrt{a+b x^2}}-\frac{21 b^{5/4} \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 )}{10 a^{11/4} c^{7/2} \sqrt{a+b x^2}}\\ \end{align*}
Mathematica [C] time = 0.0128172, size = 59, normalized size = 0.18 \[ -\frac{2 x \sqrt{\frac{b x^2}{a}+1} \, _2F_1\left (-\frac{5}{4},\frac{3}{2};-\frac{1}{4};-\frac{b x^2}{a}\right )}{5 a (c x)^{7/2} \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 219, normalized size = 0.7 \begin{align*} -{\frac{1}{10\,{x}^{2}{c}^{3}{a}^{3}} \left ( 42\,\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticE} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ){x}^{2}ab-21\,\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 ){x}^{2}ab-42\,{b}^{2}{x}^{4}-28\,ab{x}^{2}+4\,{a}^{2} \right ){\frac{1}{\sqrt{cx}}}{\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{1}{{\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 (\frac{\sqrt{b x^{2} + a} \sqrt{c x}}{b^{2} c^{4} x^{8} + 2 \, a b c^{4} x^{6} + a^{2} c^{4} x^{4}}, 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{1}{{\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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