Optimal. Leaf size=443 \[ \frac{x \left (3 a^2 c^2+13 a b c+8 b^2\right ) \left (a c+a d x^2+b\right )}{15 a^3 d^2 \left (c+d x^2\right ) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}-\frac{\sqrt{c} \left (3 a^2 c^2+13 a b c+8 b^2\right ) \left (a c+a d x^2+b\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b}{b+a c}\right )}{15 a^3 d^{5/2} \left (c+d x^2\right ) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \sqrt{\frac{c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}+\frac{c^{3/2} (3 a c+4 b) \left (a c+a d x^2+b\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b}{b+a c}\right )}{15 a^2 d^{5/2} \left (c+d x^2\right ) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \sqrt{\frac{c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}-\frac{x (3 a c+4 b) \left (a c+a d x^2+b\right )}{15 a^2 d^2 \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}+\frac{x^3 \left (a c+a d x^2+b\right )}{5 a d \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}} \]
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Rubi [A] time = 0.708877, antiderivative size = 498, normalized size of antiderivative = 1.12, number of steps used = 8, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381, Rules used = {6722, 1975, 478, 582, 531, 418, 492, 411} \[ \frac{x \left (3 a^2 c^2+13 a b c+8 b^2\right ) \sqrt{a c+a d x^2+b} \sqrt{a \left (c+d x^2\right )+b}}{15 a^3 d^2 \left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}}}-\frac{\sqrt{c} \left (3 a^2 c^2+13 a b c+8 b^2\right ) \sqrt{a c+a d x^2+b} \sqrt{a \left (c+d x^2\right )+b} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b}{b+a c}\right )}{15 a^3 d^{5/2} \left (c+d x^2\right ) \sqrt{\frac{c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}} \sqrt{a+\frac{b}{c+d x^2}}}+\frac{c^{3/2} (3 a c+4 b) \sqrt{a c+a d x^2+b} \sqrt{a \left (c+d x^2\right )+b} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b}{b+a c}\right )}{15 a^2 d^{5/2} \left (c+d x^2\right ) \sqrt{\frac{c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}} \sqrt{a+\frac{b}{c+d x^2}}}-\frac{x (3 a c+4 b) \sqrt{a c+a d x^2+b} \sqrt{a \left (c+d x^2\right )+b}}{15 a^2 d^2 \sqrt{a+\frac{b}{c+d x^2}}}+\frac{x^3 \sqrt{a c+a d x^2+b} \sqrt{a \left (c+d x^2\right )+b}}{5 a d \sqrt{a+\frac{b}{c+d x^2}}} \]
Antiderivative was successfully verified.
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Rule 6722
Rule 1975
Rule 478
Rule 582
Rule 531
Rule 418
Rule 492
Rule 411
Rubi steps
\begin{align*} \int \frac{x^4}{\sqrt{a+\frac{b}{c+d x^2}}} \, dx &=\frac{\sqrt{b+a \left (c+d x^2\right )} \int \frac{x^4 \sqrt{c+d x^2}}{\sqrt{b+a \left (c+d x^2\right )}} \, dx}{\sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}}\\ &=\frac{\sqrt{b+a \left (c+d x^2\right )} \int \frac{x^4 \sqrt{c+d x^2}}{\sqrt{b+a c+a d x^2}} \, dx}{\sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}}\\ &=\frac{x^3 \sqrt{b+a c+a d x^2} \sqrt{b+a \left (c+d x^2\right )}}{5 a d \sqrt{a+\frac{b}{c+d x^2}}}-\frac{\sqrt{b+a \left (c+d x^2\right )} \int \frac{x^2 \left (3 c (b+a c)+(4 b+3 a c) d x^2\right )}{\sqrt{c+d x^2} \sqrt{b+a c+a d x^2}} \, dx}{5 a d \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}}\\ &=-\frac{(4 b+3 a c) x \sqrt{b+a c+a d x^2} \sqrt{b+a \left (c+d x^2\right )}}{15 a^2 d^2 \sqrt{a+\frac{b}{c+d x^2}}}+\frac{x^3 \sqrt{b+a c+a d x^2} \sqrt{b+a \left (c+d x^2\right )}}{5 a d \sqrt{a+\frac{b}{c+d x^2}}}+\frac{\sqrt{b+a \left (c+d x^2\right )} \int \frac{c (b+a c) (4 b+3 a c) d+\left (8 b^2+13 a b c+3 a^2 c^2\right ) d^2 x^2}{\sqrt{c+d x^2} \sqrt{b+a c+a d x^2}} \, dx}{15 a^2 d^3 \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}}\\ &=-\frac{(4 b+3 a c) x \sqrt{b+a c+a d x^2} \sqrt{b+a \left (c+d x^2\right )}}{15 a^2 d^2 \sqrt{a+\frac{b}{c+d x^2}}}+\frac{x^3 \sqrt{b+a c+a d x^2} \sqrt{b+a \left (c+d x^2\right )}}{5 a d \sqrt{a+\frac{b}{c+d x^2}}}+\frac{\left (c (b+a c) (4 b+3 a c) \sqrt{b+a \left (c+d x^2\right )}\right ) \int \frac{1}{\sqrt{c+d x^2} \sqrt{b+a c+a d x^2}} \, dx}{15 a^2 d^2 \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}}+\frac{\left (\left (8 b^2+13 a b c+3 a^2 c^2\right ) \sqrt{b+a \left (c+d x^2\right )}\right ) \int \frac{x^2}{\sqrt{c+d x^2} \sqrt{b+a c+a d x^2}} \, dx}{15 a^2 d \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}}\\ &=-\frac{(4 b+3 a c) x \sqrt{b+a c+a d x^2} \sqrt{b+a \left (c+d x^2\right )}}{15 a^2 d^2 \sqrt{a+\frac{b}{c+d x^2}}}+\frac{x^3 \sqrt{b+a c+a d x^2} \sqrt{b+a \left (c+d x^2\right )}}{5 a d \sqrt{a+\frac{b}{c+d x^2}}}+\frac{\left (8 b^2+13 a b c+3 a^2 c^2\right ) x \sqrt{b+a c+a d x^2} \sqrt{b+a \left (c+d x^2\right )}}{15 a^3 d^2 \left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}}}+\frac{c^{3/2} (4 b+3 a c) \sqrt{b+a c+a d x^2} \sqrt{b+a \left (c+d x^2\right )} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b}{b+a c}\right )}{15 a^2 d^{5/2} \left (c+d x^2\right ) \sqrt{\frac{c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}} \sqrt{a+\frac{b}{c+d x^2}}}-\frac{\left (c \left (8 b^2+13 a b c+3 a^2 c^2\right ) \sqrt{b+a \left (c+d x^2\right )}\right ) \int \frac{\sqrt{b+a c+a d x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{15 a^3 d^2 \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}}\\ &=-\frac{(4 b+3 a c) x \sqrt{b+a c+a d x^2} \sqrt{b+a \left (c+d x^2\right )}}{15 a^2 d^2 \sqrt{a+\frac{b}{c+d x^2}}}+\frac{x^3 \sqrt{b+a c+a d x^2} \sqrt{b+a \left (c+d x^2\right )}}{5 a d \sqrt{a+\frac{b}{c+d x^2}}}+\frac{\left (8 b^2+13 a b c+3 a^2 c^2\right ) x \sqrt{b+a c+a d x^2} \sqrt{b+a \left (c+d x^2\right )}}{15 a^3 d^2 \left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}}}-\frac{\sqrt{c} \left (8 b^2+13 a b c+3 a^2 c^2\right ) \sqrt{b+a c+a d x^2} \sqrt{b+a \left (c+d x^2\right )} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b}{b+a c}\right )}{15 a^3 d^{5/2} \left (c+d x^2\right ) \sqrt{\frac{c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}} \sqrt{a+\frac{b}{c+d x^2}}}+\frac{c^{3/2} (4 b+3 a c) \sqrt{b+a c+a d x^2} \sqrt{b+a \left (c+d x^2\right )} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b}{b+a c}\right )}{15 a^2 d^{5/2} \left (c+d x^2\right ) \sqrt{\frac{c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}} \sqrt{a+\frac{b}{c+d x^2}}}\\ \end{align*}
Mathematica [C] time = 0.792588, size = 297, normalized size = 0.67 \[ -\frac{\sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \left (x \left (c+d x^2\right ) \sqrt{\frac{a d}{a c+b}} \left (3 a^2 \left (c^2-d^2 x^4\right )+a b \left (7 c+d x^2\right )+4 b^2\right )+i c \left (3 a^2 c^2+13 a b c+8 b^2\right ) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{a c+a d x^2+b}{a c+b}} E\left (i \sinh ^{-1}\left (\sqrt{\frac{a d}{b+a c}} x\right )|\frac{b}{a c}+1\right )-2 i b c (3 a c+2 b) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{a c+a d x^2+b}{a c+b}} F\left (i \sinh ^{-1}\left (\sqrt{\frac{a d}{b+a c}} x\right )|\frac{b}{a c}+1\right )\right )}{15 a^2 d^2 \sqrt{\frac{a d}{a c+b}} \left (a \left (c+d x^2\right )+b\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 665, normalized size = 1.5 \begin{align*}{\frac{d{x}^{2}+c}{15\,{a}^{2}{d}^{2}} \left ( 3\,\sqrt{-{\frac{ad}{ac+b}}}{x}^{7}{a}^{2}{d}^{3}+3\,\sqrt{-{\frac{ad}{ac+b}}}{x}^{5}{a}^{2}c{d}^{2}-\sqrt{-{\frac{ad}{ac+b}}}{x}^{5}ab{d}^{2}-3\,\sqrt{-{\frac{ad}{ac+b}}}{x}^{3}{a}^{2}{c}^{2}d-8\,\sqrt{-{\frac{ad}{ac+b}}}{x}^{3}abcd+3\,\sqrt{{\frac{ad{x}^{2}+ac+b}{ac+b}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticE} \left ( x\sqrt{-{\frac{ad}{ac+b}}},\sqrt{{\frac{ac+b}{ac}}} \right ){a}^{2}{c}^{3}-4\,\sqrt{-{\frac{ad}{ac+b}}}{x}^{3}{b}^{2}d-3\,\sqrt{-{\frac{ad}{ac+b}}}x{a}^{2}{c}^{3}-6\,\sqrt{{\frac{ad{x}^{2}+ac+b}{ac+b}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ( x\sqrt{-{\frac{ad}{ac+b}}},\sqrt{{\frac{ac+b}{ac}}} \right ) ab{c}^{2}+13\,\sqrt{{\frac{ad{x}^{2}+ac+b}{ac+b}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticE} \left ( x\sqrt{-{\frac{ad}{ac+b}}},\sqrt{{\frac{ac+b}{ac}}} \right ) ab{c}^{2}-7\,\sqrt{-{\frac{ad}{ac+b}}}xab{c}^{2}-4\,\sqrt{{\frac{ad{x}^{2}+ac+b}{ac+b}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ( x\sqrt{-{\frac{ad}{ac+b}}},\sqrt{{\frac{ac+b}{ac}}} \right ){b}^{2}c+8\,\sqrt{{\frac{ad{x}^{2}+ac+b}{ac+b}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticE} \left ( x\sqrt{-{\frac{ad}{ac+b}}},\sqrt{{\frac{ac+b}{ac}}} \right ){b}^{2}c-4\,\sqrt{-{\frac{ad}{ac+b}}}x{b}^{2}c \right ) \sqrt{{\frac{ad{x}^{2}+ac+b}{d{x}^{2}+c}}}{\frac{1}{\sqrt{a{d}^{2}{x}^{4}+2\,acd{x}^{2}+bd{x}^{2}+{c}^{2}a+bc}}}{\frac{1}{\sqrt{-{\frac{ad}{ac+b}}}}}{\frac{1}{\sqrt{ \left ( d{x}^{2}+c \right ) \left ( ad{x}^{2}+ac+b \right ) }}}} \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{x^{4}}{\sqrt{a + \frac{b}{d x^{2} + c}}}\,{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{{\left (d x^{6} + c x^{4}\right )} \sqrt{\frac{a d x^{2} + a c + b}{d x^{2} + c}}}{a d x^{2} + a c + b}, 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{x^{4}}{\sqrt{\frac{a c + a d x^{2} + b}{c + d x^{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{x^{4}}{\sqrt{a + \frac{b}{d x^{2} + c}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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