Optimal. Leaf size=274 \[ -\frac{\sqrt{a} b^{3/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} \text{EllipticF}\left (\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right ),2\right )}{2 d \left (a+b x^4\right )^{3/4}}-\frac{\sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} (b c-a d) \Pi \left (-\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d}-\frac{\sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} (b c-a d) \Pi \left (\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d}+\frac{b x \sqrt [4]{a+b x^4}}{2 d} \]
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Rubi [A] time = 0.155669, antiderivative size = 274, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {408, 195, 237, 335, 275, 231, 407, 409, 1218} \[ -\frac{\sqrt{a} b^{3/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{2 d \left (a+b x^4\right )^{3/4}}-\frac{\sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} (b c-a d) \Pi \left (-\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d}-\frac{\sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} (b c-a d) \Pi \left (\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d}+\frac{b x \sqrt [4]{a+b x^4}}{2 d} \]
Antiderivative was successfully verified.
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Rule 408
Rule 195
Rule 237
Rule 335
Rule 275
Rule 231
Rule 407
Rule 409
Rule 1218
Rubi steps
\begin{align*} \int \frac{\left (a+b x^4\right )^{5/4}}{c+d x^4} \, dx &=\frac{b \int \sqrt [4]{a+b x^4} \, dx}{d}-\frac{(b c-a d) \int \frac{\sqrt [4]{a+b x^4}}{c+d x^4} \, dx}{d}\\ &=\frac{b x \sqrt [4]{a+b x^4}}{2 d}+\frac{(a b) \int \frac{1}{\left (a+b x^4\right )^{3/4}} \, dx}{2 d}-\frac{\left ((b c-a d) \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-b x^4} \left (c-(b c-a d) x^4\right )} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )}{d}\\ &=\frac{b x \sqrt [4]{a+b x^4}}{2 d}+\frac{\left (a b \left (1+\frac{a}{b x^4}\right )^{3/4} x^3\right ) \int \frac{1}{\left (1+\frac{a}{b x^4}\right )^{3/4} x^3} \, dx}{2 d \left (a+b x^4\right )^{3/4}}-\frac{\left ((b c-a d) \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{\sqrt{b c-a d} x^2}{\sqrt{c}}\right ) \sqrt{1-b x^4}} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )}{2 c d}-\frac{\left ((b c-a d) \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{\sqrt{b c-a d} x^2}{\sqrt{c}}\right ) \sqrt{1-b x^4}} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )}{2 c d}\\ &=\frac{b x \sqrt [4]{a+b x^4}}{2 d}-\frac{(b c-a d) \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} \Pi \left (-\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d}-\frac{(b c-a d) \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} \Pi \left (\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d}-\frac{\left (a b \left (1+\frac{a}{b x^4}\right )^{3/4} x^3\right ) \operatorname{Subst}\left (\int \frac{x}{\left (1+\frac{a x^4}{b}\right )^{3/4}} \, dx,x,\frac{1}{x}\right )}{2 d \left (a+b x^4\right )^{3/4}}\\ &=\frac{b x \sqrt [4]{a+b x^4}}{2 d}-\frac{(b c-a d) \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} \Pi \left (-\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d}-\frac{(b c-a d) \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} \Pi \left (\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d}-\frac{\left (a b \left (1+\frac{a}{b x^4}\right )^{3/4} x^3\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{a x^2}{b}\right )^{3/4}} \, dx,x,\frac{1}{x^2}\right )}{4 d \left (a+b x^4\right )^{3/4}}\\ &=\frac{b x \sqrt [4]{a+b x^4}}{2 d}-\frac{\sqrt{a} b^{3/2} \left (1+\frac{a}{b x^4}\right )^{3/4} x^3 F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{2 d \left (a+b x^4\right )^{3/4}}-\frac{(b c-a d) \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} \Pi \left (-\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d}-\frac{(b c-a d) \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} \Pi \left (\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d}\\ \end{align*}
Mathematica [C] time = 0.321675, size = 346, normalized size = 1.26 \[ \frac{x \left (\frac{5 \left (b x^4 \left (a+b x^4\right ) \left (c+d x^4\right ) \left (4 a d F_1\left (\frac{5}{4};\frac{3}{4},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )+3 b c F_1\left (\frac{5}{4};\frac{7}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )-5 a c \left (2 a^2 d+a b d x^4+b^2 x^4 \left (c+d x^4\right )\right ) F_1\left (\frac{1}{4};\frac{3}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )}{\left (c+d x^4\right ) \left (x^4 \left (4 a d F_1\left (\frac{5}{4};\frac{3}{4},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )+3 b c F_1\left (\frac{5}{4};\frac{7}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )-5 a c F_1\left (\frac{1}{4};\frac{3}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )}+\frac{b x^4 \left (\frac{b x^4}{a}+1\right )^{3/4} (3 a d-2 b c) F_1\left (\frac{5}{4};\frac{3}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{c}\right )}{10 d \left (a+b x^4\right )^{3/4}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.416, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{d{x}^{4}+c} \left ( b{x}^{4}+a \right ) ^{{\frac{5}{4}}}}\, 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 )}^{\frac{5}{4}}}{d x^{4} + c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b x^{4}\right )^{\frac{5}{4}}}{c + d x^{4}}\, 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{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}{d x^{4} + c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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