Optimal. Leaf size=468 \[ \frac{\sqrt [4]{c} \sqrt{x} \left (\sqrt{a} b \sqrt{c}-6 a c+2 b^2\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{2 a^{7/4} \left (b^2-4 a c\right ) \sqrt{a x+b x^3+c x^5}}+\frac{2 \sqrt{c} x^{3/2} \left (b^2-3 a c\right ) \left (a+b x^2+c x^4\right )}{a^2 \left (b^2-4 a c\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{a x+b x^3+c x^5}}-\frac{2 \left (b^2-3 a c\right ) \sqrt{a x+b x^3+c x^5}}{a^2 x^{3/2} \left (b^2-4 a c\right )}-\frac{2 \sqrt [4]{c} \sqrt{x} \left (b^2-3 a c\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{a^{7/4} \left (b^2-4 a c\right ) \sqrt{a x+b x^3+c x^5}}+\frac{-2 a c+b^2+b c x^2}{a \sqrt{x} \left (b^2-4 a c\right ) \sqrt{a x+b x^3+c x^5}} \]
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Rubi [A] time = 0.408659, antiderivative size = 468, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1924, 1951, 1953, 1197, 1103, 1195} \[ \frac{2 \sqrt{c} x^{3/2} \left (b^2-3 a c\right ) \left (a+b x^2+c x^4\right )}{a^2 \left (b^2-4 a c\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{a x+b x^3+c x^5}}-\frac{2 \left (b^2-3 a c\right ) \sqrt{a x+b x^3+c x^5}}{a^2 x^{3/2} \left (b^2-4 a c\right )}+\frac{\sqrt [4]{c} \sqrt{x} \left (\sqrt{a} b \sqrt{c}-6 a c+2 b^2\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{2 a^{7/4} \left (b^2-4 a c\right ) \sqrt{a x+b x^3+c x^5}}-\frac{2 \sqrt [4]{c} \sqrt{x} \left (b^2-3 a c\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{a^{7/4} \left (b^2-4 a c\right ) \sqrt{a x+b x^3+c x^5}}+\frac{-2 a c+b^2+b c x^2}{a \sqrt{x} \left (b^2-4 a c\right ) \sqrt{a x+b x^3+c x^5}} \]
Antiderivative was successfully verified.
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Rule 1924
Rule 1951
Rule 1953
Rule 1197
Rule 1103
Rule 1195
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{x} \left (a x+b x^3+c x^5\right )^{3/2}} \, dx &=\frac{b^2-2 a c+b c x^2}{a \left (b^2-4 a c\right ) \sqrt{x} \sqrt{a x+b x^3+c x^5}}-\frac{\int \frac{-2 b^2+6 a c-b c x^2}{x^{3/2} \sqrt{a x+b x^3+c x^5}} \, dx}{a \left (b^2-4 a c\right )}\\ &=\frac{b^2-2 a c+b c x^2}{a \left (b^2-4 a c\right ) \sqrt{x} \sqrt{a x+b x^3+c x^5}}-\frac{2 \left (b^2-3 a c\right ) \sqrt{a x+b x^3+c x^5}}{a^2 \left (b^2-4 a c\right ) x^{3/2}}+\frac{\int \frac{\sqrt{x} \left (a b c+2 c \left (b^2-3 a c\right ) x^2\right )}{\sqrt{a x+b x^3+c x^5}} \, dx}{a^2 \left (b^2-4 a c\right )}\\ &=\frac{b^2-2 a c+b c x^2}{a \left (b^2-4 a c\right ) \sqrt{x} \sqrt{a x+b x^3+c x^5}}-\frac{2 \left (b^2-3 a c\right ) \sqrt{a x+b x^3+c x^5}}{a^2 \left (b^2-4 a c\right ) x^{3/2}}+\frac{\left (\sqrt{x} \sqrt{a+b x^2+c x^4}\right ) \int \frac{a b c+2 c \left (b^2-3 a c\right ) x^2}{\sqrt{a+b x^2+c x^4}} \, dx}{a^2 \left (b^2-4 a c\right ) \sqrt{a x+b x^3+c x^5}}\\ &=\frac{b^2-2 a c+b c x^2}{a \left (b^2-4 a c\right ) \sqrt{x} \sqrt{a x+b x^3+c x^5}}-\frac{2 \left (b^2-3 a c\right ) \sqrt{a x+b x^3+c x^5}}{a^2 \left (b^2-4 a c\right ) x^{3/2}}-\frac{\left (2 \sqrt{c} \left (b^2-3 a c\right ) \sqrt{x} \sqrt{a+b x^2+c x^4}\right ) \int \frac{1-\frac{\sqrt{c} x^2}{\sqrt{a}}}{\sqrt{a+b x^2+c x^4}} \, dx}{a^{3/2} \left (b^2-4 a c\right ) \sqrt{a x+b x^3+c x^5}}+\frac{\left (\left (\sqrt{a} b c^{3/2}+2 c \left (b^2-3 a c\right )\right ) \sqrt{x} \sqrt{a+b x^2+c x^4}\right ) \int \frac{1}{\sqrt{a+b x^2+c x^4}} \, dx}{a^{3/2} \sqrt{c} \left (b^2-4 a c\right ) \sqrt{a x+b x^3+c x^5}}\\ &=\frac{b^2-2 a c+b c x^2}{a \left (b^2-4 a c\right ) \sqrt{x} \sqrt{a x+b x^3+c x^5}}+\frac{2 \sqrt{c} \left (b^2-3 a c\right ) x^{3/2} \left (a+b x^2+c x^4\right )}{a^2 \left (b^2-4 a c\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{a x+b x^3+c x^5}}-\frac{2 \left (b^2-3 a c\right ) \sqrt{a x+b x^3+c x^5}}{a^2 \left (b^2-4 a c\right ) x^{3/2}}-\frac{2 \sqrt [4]{c} \left (b^2-3 a c\right ) \sqrt{x} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{a^{7/4} \left (b^2-4 a c\right ) \sqrt{a x+b x^3+c x^5}}+\frac{\sqrt [4]{c} \left (2 b^2+\sqrt{a} b \sqrt{c}-6 a c\right ) \sqrt{x} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{2 a^{7/4} \left (b^2-4 a c\right ) \sqrt{a x+b x^3+c x^5}}\\ \end{align*}
Mathematica [C] time = 1.35104, size = 519, normalized size = 1.11 \[ -\frac{i x \left (b^2 \sqrt{b^2-4 a c}-3 a c \sqrt{b^2-4 a c}+4 a b c-b^3\right ) \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{-2 \sqrt{b^2-4 a c}+2 b+4 c x^2}{b-\sqrt{b^2-4 a c}}} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{2} x \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}}\right ),\frac{\sqrt{b^2-4 a c}+b}{b-\sqrt{b^2-4 a c}}\right )+2 \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \left (-4 a^2 c+a \left (b^2-7 b c x^2-6 c^2 x^4\right )+2 b^2 x^2 \left (b+c x^2\right )\right )-i x \left (b^2-3 a c\right ) \left (\sqrt{b^2-4 a c}-b\right ) \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{-2 \sqrt{b^2-4 a c}+2 b+4 c x^2}{b-\sqrt{b^2-4 a c}}} E\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )}{2 a^2 \sqrt{x} \left (b^2-4 a c\right ) \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{x \left (a+b x^2+c x^4\right )}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.029, size = 1136, normalized size = 2.4 \begin{align*} \text{result too large to display} \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 (c x^{5} + b x^{3} + a x\right )}^{\frac{3}{2}} \sqrt{x}}\,{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{c x^{5} + b x^{3} + a x} \sqrt{x}}{c^{2} x^{11} + 2 \, b c x^{9} +{\left (b^{2} + 2 \, a c\right )} x^{7} + 2 \, a b x^{5} + a^{2} x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x} \left (x \left (a + b x^{2} + c x^{4}\right )\right )^{\frac{3}{2}}}\, dx \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 (c x^{5} + b x^{3} + a x\right )}^{\frac{3}{2}} \sqrt{x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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