Optimal. Leaf size=200 \[ \frac{\sqrt{\frac{d x^4}{c}+1} (e x)^{m+1} \left (a^2 d^2 (m+3) (m+7)+b c (m+1) (b c (m+5)-2 a d (m+7))\right ) \, _2F_1\left (\frac{1}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right )}{d^2 e (m+1) (m+3) (m+7) \sqrt{c+d x^4}}-\frac{b \sqrt{c+d x^4} (e x)^{m+1} (b c (m+5)-2 a d (m+7))}{d^2 e (m+3) (m+7)}+\frac{b^2 \sqrt{c+d x^4} (e x)^{m+5}}{d e^5 (m+7)} \]
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Rubi [A] time = 0.212091, antiderivative size = 194, normalized size of antiderivative = 0.97, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {464, 459, 365, 364} \[ \frac{\sqrt{\frac{d x^4}{c}+1} (e x)^{m+1} \left (\frac{a^2 d^2 (m+7)}{m+1}+\frac{b c (b c (m+5)-2 a d (m+7))}{m+3}\right ) \, _2F_1\left (\frac{1}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right )}{d^2 e (m+7) \sqrt{c+d x^4}}-\frac{b \sqrt{c+d x^4} (e x)^{m+1} (b c (m+5)-2 a d (m+7))}{d^2 e (m+3) (m+7)}+\frac{b^2 \sqrt{c+d x^4} (e x)^{m+5}}{d e^5 (m+7)} \]
Antiderivative was successfully verified.
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Rule 464
Rule 459
Rule 365
Rule 364
Rubi steps
\begin{align*} \int \frac{(e x)^m \left (a+b x^4\right )^2}{\sqrt{c+d x^4}} \, dx &=\frac{b^2 (e x)^{5+m} \sqrt{c+d x^4}}{d e^5 (7+m)}+\frac{\int \frac{(e x)^m \left (a^2 d (7+m)-b (b c (5+m)-2 a d (7+m)) x^4\right )}{\sqrt{c+d x^4}} \, dx}{d (7+m)}\\ &=-\frac{b (b c (5+m)-2 a d (7+m)) (e x)^{1+m} \sqrt{c+d x^4}}{d^2 e (3+m) (7+m)}+\frac{b^2 (e x)^{5+m} \sqrt{c+d x^4}}{d e^5 (7+m)}-\left (-a^2-\frac{b c (1+m) (b c (5+m)-2 a d (7+m))}{d^2 (3+m) (7+m)}\right ) \int \frac{(e x)^m}{\sqrt{c+d x^4}} \, dx\\ &=-\frac{b (b c (5+m)-2 a d (7+m)) (e x)^{1+m} \sqrt{c+d x^4}}{d^2 e (3+m) (7+m)}+\frac{b^2 (e x)^{5+m} \sqrt{c+d x^4}}{d e^5 (7+m)}-\frac{\left (\left (-a^2-\frac{b c (1+m) (b c (5+m)-2 a d (7+m))}{d^2 (3+m) (7+m)}\right ) \sqrt{1+\frac{d x^4}{c}}\right ) \int \frac{(e x)^m}{\sqrt{1+\frac{d x^4}{c}}} \, dx}{\sqrt{c+d x^4}}\\ &=-\frac{b (b c (5+m)-2 a d (7+m)) (e x)^{1+m} \sqrt{c+d x^4}}{d^2 e (3+m) (7+m)}+\frac{b^2 (e x)^{5+m} \sqrt{c+d x^4}}{d e^5 (7+m)}+\frac{\left (a^2+\frac{b c (1+m) (b c (5+m)-2 a d (7+m))}{d^2 (3+m) (7+m)}\right ) (e x)^{1+m} \sqrt{1+\frac{d x^4}{c}} \, _2F_1\left (\frac{1}{2},\frac{1+m}{4};\frac{5+m}{4};-\frac{d x^4}{c}\right )}{e (1+m) \sqrt{c+d x^4}}\\ \end{align*}
Mathematica [A] time = 0.137103, size = 164, normalized size = 0.82 \[ \frac{x \sqrt{\frac{d x^4}{c}+1} (e x)^m \left (a^2 \left (m^2+14 m+45\right ) \, _2F_1\left (\frac{1}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right )+b (m+1) x^4 \left (2 a (m+9) \, _2F_1\left (\frac{1}{2},\frac{m+5}{4};\frac{m+9}{4};-\frac{d x^4}{c}\right )+b (m+5) x^4 \, _2F_1\left (\frac{1}{2},\frac{m+9}{4};\frac{m+13}{4};-\frac{d x^4}{c}\right )\right )\right )}{(m+1) (m+5) (m+9) \sqrt{c+d x^4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.029, size = 0, normalized size = 0. \begin{align*} \int{ \left ( ex \right ) ^{m} \left ( b{x}^{4}+a \right ) ^{2}{\frac{1}{\sqrt{d{x}^{4}+c}}}}\, dx \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 (b x^{4} + a\right )}^{2} \left (e x\right )^{m}}{\sqrt{d x^{4} + 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 (b^{2} x^{8} + 2 \, a b x^{4} + a^{2}\right )} \left (e x\right )^{m}}{\sqrt{d x^{4} + c}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 55.6202, size = 185, normalized size = 0.92 \begin{align*} \frac{a^{2} e^{m} x x^{m} \Gamma \left (\frac{m}{4} + \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m}{4} + \frac{1}{4} \\ \frac{m}{4} + \frac{5}{4} \end{matrix}\middle |{\frac{d x^{4} e^{i \pi }}{c}} \right )}}{4 \sqrt{c} \Gamma \left (\frac{m}{4} + \frac{5}{4}\right )} + \frac{a b e^{m} x^{5} x^{m} \Gamma \left (\frac{m}{4} + \frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m}{4} + \frac{5}{4} \\ \frac{m}{4} + \frac{9}{4} \end{matrix}\middle |{\frac{d x^{4} e^{i \pi }}{c}} \right )}}{2 \sqrt{c} \Gamma \left (\frac{m}{4} + \frac{9}{4}\right )} + \frac{b^{2} e^{m} x^{9} x^{m} \Gamma \left (\frac{m}{4} + \frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m}{4} + \frac{9}{4} \\ \frac{m}{4} + \frac{13}{4} \end{matrix}\middle |{\frac{d x^{4} e^{i \pi }}{c}} \right )}}{4 \sqrt{c} \Gamma \left (\frac{m}{4} + \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 (b x^{4} + a\right )}^{2} \left (e x\right )^{m}}{\sqrt{d x^{4} + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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