Optimal. Leaf size=304 \[ -\frac{b^{3/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} (2 b c-5 a d) \text{EllipticF}\left (\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right ),2\right )}{3 a^{3/2} \left (a+b x^4\right )^{3/4} (b c-a d)^2}+\frac{d^2 \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} \Pi \left (-\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c (b c-a d)^2}+\frac{d^2 \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} \Pi \left (\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c (b c-a d)^2}+\frac{b x}{3 a \left (a+b x^4\right )^{3/4} (b c-a d)} \]
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Rubi [A] time = 0.238478, antiderivative size = 304, 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 = {414, 529, 237, 335, 275, 231, 407, 409, 1218} \[ -\frac{b^{3/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} (2 b c-5 a d) F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{3 a^{3/2} \left (a+b x^4\right )^{3/4} (b c-a d)^2}+\frac{d^2 \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} \Pi \left (-\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c (b c-a d)^2}+\frac{d^2 \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} \Pi \left (\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c (b c-a d)^2}+\frac{b x}{3 a \left (a+b x^4\right )^{3/4} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 414
Rule 529
Rule 237
Rule 335
Rule 275
Rule 231
Rule 407
Rule 409
Rule 1218
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b x^4\right )^{7/4} \left (c+d x^4\right )} \, dx &=\frac{b x}{3 a (b c-a d) \left (a+b x^4\right )^{3/4}}-\frac{\int \frac{-2 b c+3 a d-2 b d x^4}{\left (a+b x^4\right )^{3/4} \left (c+d x^4\right )} \, dx}{3 a (b c-a d)}\\ &=\frac{b x}{3 a (b c-a d) \left (a+b x^4\right )^{3/4}}+\frac{d^2 \int \frac{\sqrt [4]{a+b x^4}}{c+d x^4} \, dx}{(b c-a d)^2}+\frac{(b (2 b c-5 a d)) \int \frac{1}{\left (a+b x^4\right )^{3/4}} \, dx}{3 a (b c-a d)^2}\\ &=\frac{b x}{3 a (b c-a d) \left (a+b x^4\right )^{3/4}}+\frac{\left (b (2 b c-5 a d) \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}{3 a (b c-a d)^2 \left (a+b x^4\right )^{3/4}}+\frac{\left (d^2 \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 )}{(b c-a d)^2}\\ &=\frac{b x}{3 a (b c-a d) \left (a+b x^4\right )^{3/4}}-\frac{\left (b (2 b c-5 a d) \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 )}{3 a (b c-a d)^2 \left (a+b x^4\right )^{3/4}}+\frac{\left (d^2 \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 (b c-a d)^2}+\frac{\left (d^2 \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 (b c-a d)^2}\\ &=\frac{b x}{3 a (b c-a d) \left (a+b x^4\right )^{3/4}}+\frac{d^2 \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 (b c-a d)^2}+\frac{d^2 \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 (b c-a d)^2}-\frac{\left (b (2 b c-5 a d) \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 )}{6 a (b c-a d)^2 \left (a+b x^4\right )^{3/4}}\\ &=\frac{b x}{3 a (b c-a d) \left (a+b x^4\right )^{3/4}}-\frac{b^{3/2} (2 b c-5 a d) \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 )}{3 a^{3/2} (b c-a d)^2 \left (a+b x^4\right )^{3/4}}+\frac{d^2 \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 (b c-a d)^2}+\frac{d^2 \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 (b c-a d)^2}\\ \end{align*}
Mathematica [C] time = 0.255581, size = 332, normalized size = 1.09 \[ \frac{x \left (\frac{5 \left (b x^4 \left (c+d x^4\right ) \left (4 a d F_1\left (\frac{5}{4};\frac{3}{4},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )+3 b c F_1\left (\frac{5}{4};\frac{7}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )+5 a c \left (3 a d-b \left (3 c+d x^4\right )\right ) F_1\left (\frac{1}{4};\frac{3}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )}{\left (c+d x^4\right ) \left (5 a c F_1\left (\frac{1}{4};\frac{3}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )-x^4 \left (4 a d F_1\left (\frac{5}{4};\frac{3}{4},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )+3 b c F_1\left (\frac{5}{4};\frac{7}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )\right )}-\frac{2 b d x^4 \left (\frac{b x^4}{a}+1\right )^{3/4} F_1\left (\frac{5}{4};\frac{3}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{c}\right )}{15 a \left (a+b x^4\right )^{3/4} (a d-b c)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.403, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{d{x}^{4}+c} \left ( b{x}^{4}+a \right ) ^{-{\frac{7}{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{1}{{\left (b x^{4} + a\right )}^{\frac{7}{4}}{\left (d x^{4} + c\right )}}\,{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{1}{\left (a + b x^{4}\right )^{\frac{7}{4}} \left (c + d x^{4}\right )}\, 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 (b x^{4} + a\right )}^{\frac{7}{4}}{\left (d x^{4} + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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