Optimal. Leaf size=353 \[ -\frac{\sqrt{a} b^{3/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} (3 b c-a d) \text{EllipticF}\left (\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right ),2\right )}{4 c d^2 \left (a+b x^4\right )^{3/4}}-\frac{3 \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} (b c-a d) (a d+2 b c) \Pi \left (-\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d^2}-\frac{3 \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} (b c-a d) (a d+2 b c) \Pi \left (\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d^2}+\frac{b x \sqrt [4]{a+b x^4} (3 b c-a d)}{4 c d^2}-\frac{x \left (a+b x^4\right )^{5/4} (b c-a d)}{4 c d \left (c+d x^4\right )} \]
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Rubi [A] time = 0.347931, antiderivative size = 353, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.476, Rules used = {413, 528, 529, 237, 335, 275, 231, 407, 409, 1218} \[ -\frac{\sqrt{a} b^{3/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} (3 b c-a d) F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{4 c d^2 \left (a+b x^4\right )^{3/4}}-\frac{3 \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} (b c-a d) (a d+2 b c) \Pi \left (-\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d^2}-\frac{3 \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} (b c-a d) (a d+2 b c) \Pi \left (\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d^2}+\frac{b x \sqrt [4]{a+b x^4} (3 b c-a d)}{4 c d^2}-\frac{x \left (a+b x^4\right )^{5/4} (b c-a d)}{4 c d \left (c+d x^4\right )} \]
Antiderivative was successfully verified.
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Rule 413
Rule 528
Rule 529
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 )^{9/4}}{\left (c+d x^4\right )^2} \, dx &=-\frac{(b c-a d) x \left (a+b x^4\right )^{5/4}}{4 c d \left (c+d x^4\right )}+\frac{\int \frac{\sqrt [4]{a+b x^4} \left (a (b c+3 a d)+2 b (3 b c-a d) x^4\right )}{c+d x^4} \, dx}{4 c d}\\ &=\frac{b (3 b c-a d) x \sqrt [4]{a+b x^4}}{4 c d^2}-\frac{(b c-a d) x \left (a+b x^4\right )^{5/4}}{4 c d \left (c+d x^4\right )}+\frac{\int \frac{-2 a \left (3 b^2 c^2-2 a b c d-3 a^2 d^2\right )-4 b \left (3 b^2 c^2-3 a b c d-a^2 d^2\right ) x^4}{\left (a+b x^4\right )^{3/4} \left (c+d x^4\right )} \, dx}{8 c d^2}\\ &=\frac{b (3 b c-a d) x \sqrt [4]{a+b x^4}}{4 c d^2}-\frac{(b c-a d) x \left (a+b x^4\right )^{5/4}}{4 c d \left (c+d x^4\right )}+\frac{(a b (3 b c-a d)) \int \frac{1}{\left (a+b x^4\right )^{3/4}} \, dx}{4 c d^2}-\frac{(3 (b c-a d) (2 b c+a d)) \int \frac{\sqrt [4]{a+b x^4}}{c+d x^4} \, dx}{4 c d^2}\\ &=\frac{b (3 b c-a d) x \sqrt [4]{a+b x^4}}{4 c d^2}-\frac{(b c-a d) x \left (a+b x^4\right )^{5/4}}{4 c d \left (c+d x^4\right )}+\frac{\left (a b (3 b c-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}{4 c d^2 \left (a+b x^4\right )^{3/4}}-\frac{\left (3 (b c-a d) (2 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 )}{4 c d^2}\\ &=\frac{b (3 b c-a d) x \sqrt [4]{a+b x^4}}{4 c d^2}-\frac{(b c-a d) x \left (a+b x^4\right )^{5/4}}{4 c d \left (c+d x^4\right )}-\frac{\left (a b (3 b c-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 )}{4 c d^2 \left (a+b x^4\right )^{3/4}}-\frac{\left (3 (b c-a d) (2 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 )}{8 c^2 d^2}-\frac{\left (3 (b c-a d) (2 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 )}{8 c^2 d^2}\\ &=\frac{b (3 b c-a d) x \sqrt [4]{a+b x^4}}{4 c d^2}-\frac{(b c-a d) x \left (a+b x^4\right )^{5/4}}{4 c d \left (c+d x^4\right )}-\frac{3 (b c-a d) (2 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 )}{8 \sqrt [4]{b} c^2 d^2}-\frac{3 (b c-a d) (2 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 )}{8 \sqrt [4]{b} c^2 d^2}-\frac{\left (a b (3 b c-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 )}{8 c d^2 \left (a+b x^4\right )^{3/4}}\\ &=\frac{b (3 b c-a d) x \sqrt [4]{a+b x^4}}{4 c d^2}-\frac{(b c-a d) x \left (a+b x^4\right )^{5/4}}{4 c d \left (c+d x^4\right )}-\frac{\sqrt{a} b^{3/2} (3 b c-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 )}{4 c d^2 \left (a+b x^4\right )^{3/4}}-\frac{3 (b c-a d) (2 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 )}{8 \sqrt [4]{b} c^2 d^2}-\frac{3 (b c-a d) (2 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 )}{8 \sqrt [4]{b} c^2 d^2}\\ \end{align*}
Mathematica [C] time = 0.5029, size = 392, normalized size = 1.11 \[ \frac{2 b x^5 \left (\frac{b x^4}{a}+1\right )^{3/4} \left (a^2 d^2+3 a b c d-3 b^2 c^2\right ) F_1\left (\frac{5}{4};\frac{3}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )+\frac{5 c \left (x^5 \left (a+b x^4\right ) \left (a^2 d^2-2 a b c d+b^2 c \left (3 c+2 d x^4\right )\right ) \left (4 a d F_1\left (\frac{5}{4};\frac{3}{4},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )+3 b c F_1\left (\frac{5}{4};\frac{7}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )-5 a c x \left (a^2 b d^2 x^4+4 a^3 d^2+b^3 c x^4 \left (3 c+2 d x^4\right )\right ) F_1\left (\frac{1}{4};\frac{3}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )}{\left (c+d x^4\right ) \left (x^4 \left (4 a d F_1\left (\frac{5}{4};\frac{3}{4},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )+3 b c F_1\left (\frac{5}{4};\frac{7}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )-5 a c F_1\left (\frac{1}{4};\frac{3}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )}}{20 c^2 d^2 \left (a+b x^4\right )^{3/4}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.253, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( d{x}^{4}+c \right ) ^{2}} \left ( b{x}^{4}+a \right ) ^{{\frac{9}{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{9}{4}}}{{\left (d x^{4} + c\right )}^{2}}\,{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(-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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{4} + a\right )}^{\frac{9}{4}}}{{\left (d x^{4} + c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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