Optimal. Leaf size=309 \[ \frac{\sqrt [4]{a} b^{3/4} \sqrt{1-\frac{b x^4}{a}} (a d+3 b c) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{4 c d^2 \sqrt{a-b x^4}}-\frac{3 \sqrt [4]{a} \sqrt{1-\frac{b x^4}{a}} (b c-a d) (a d+b c) \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d^2 \sqrt{a-b x^4}}-\frac{3 \sqrt [4]{a} \sqrt{1-\frac{b x^4}{a}} (b c-a d) (a d+b c) \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d^2 \sqrt{a-b x^4}}-\frac{x \sqrt{a-b x^4} (b c-a d)}{4 c d \left (c-d x^4\right )} \]
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Rubi [A] time = 0.277241, antiderivative size = 309, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {413, 523, 224, 221, 409, 1219, 1218} \[ \frac{\sqrt [4]{a} b^{3/4} \sqrt{1-\frac{b x^4}{a}} (a d+3 b c) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c d^2 \sqrt{a-b x^4}}-\frac{3 \sqrt [4]{a} \sqrt{1-\frac{b x^4}{a}} (b c-a d) (a d+b c) \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d^2 \sqrt{a-b x^4}}-\frac{3 \sqrt [4]{a} \sqrt{1-\frac{b x^4}{a}} (b c-a d) (a d+b c) \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d^2 \sqrt{a-b x^4}}-\frac{x \sqrt{a-b x^4} (b c-a d)}{4 c d \left (c-d x^4\right )} \]
Antiderivative was successfully verified.
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Rule 413
Rule 523
Rule 224
Rule 221
Rule 409
Rule 1219
Rule 1218
Rubi steps
\begin{align*} \int \frac{\left (a-b x^4\right )^{3/2}}{\left (c-d x^4\right )^2} \, dx &=-\frac{(b c-a d) x \sqrt{a-b x^4}}{4 c d \left (c-d x^4\right )}-\frac{\int \frac{-a (b c+3 a d)+b (3 b c+a d) x^4}{\sqrt{a-b x^4} \left (c-d x^4\right )} \, dx}{4 c d}\\ &=-\frac{(b c-a d) x \sqrt{a-b x^4}}{4 c d \left (c-d x^4\right )}+\frac{\left (3 \left (a^2-\frac{b^2 c^2}{d^2}\right )\right ) \int \frac{1}{\sqrt{a-b x^4} \left (c-d x^4\right )} \, dx}{4 c}+\frac{(b (3 b c+a d)) \int \frac{1}{\sqrt{a-b x^4}} \, dx}{4 c d^2}\\ &=-\frac{(b c-a d) x \sqrt{a-b x^4}}{4 c d \left (c-d x^4\right )}+\frac{\left (3 \left (a^2-\frac{b^2 c^2}{d^2}\right )\right ) \int \frac{1}{\left (1-\frac{\sqrt{d} x^2}{\sqrt{c}}\right ) \sqrt{a-b x^4}} \, dx}{8 c^2}+\frac{\left (3 \left (a^2-\frac{b^2 c^2}{d^2}\right )\right ) \int \frac{1}{\left (1+\frac{\sqrt{d} x^2}{\sqrt{c}}\right ) \sqrt{a-b x^4}} \, dx}{8 c^2}+\frac{\left (b (3 b c+a d) \sqrt{1-\frac{b x^4}{a}}\right ) \int \frac{1}{\sqrt{1-\frac{b x^4}{a}}} \, dx}{4 c d^2 \sqrt{a-b x^4}}\\ &=-\frac{(b c-a d) x \sqrt{a-b x^4}}{4 c d \left (c-d x^4\right )}+\frac{\sqrt [4]{a} b^{3/4} (3 b c+a d) \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c d^2 \sqrt{a-b x^4}}+\frac{\left (3 \left (a^2-\frac{b^2 c^2}{d^2}\right ) \sqrt{1-\frac{b x^4}{a}}\right ) \int \frac{1}{\left (1-\frac{\sqrt{d} x^2}{\sqrt{c}}\right ) \sqrt{1-\frac{b x^4}{a}}} \, dx}{8 c^2 \sqrt{a-b x^4}}+\frac{\left (3 \left (a^2-\frac{b^2 c^2}{d^2}\right ) \sqrt{1-\frac{b x^4}{a}}\right ) \int \frac{1}{\left (1+\frac{\sqrt{d} x^2}{\sqrt{c}}\right ) \sqrt{1-\frac{b x^4}{a}}} \, dx}{8 c^2 \sqrt{a-b x^4}}\\ &=-\frac{(b c-a d) x \sqrt{a-b x^4}}{4 c d \left (c-d x^4\right )}+\frac{\sqrt [4]{a} b^{3/4} (3 b c+a d) \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c d^2 \sqrt{a-b x^4}}+\frac{3 \sqrt [4]{a} \left (a^2-\frac{b^2 c^2}{d^2}\right ) \sqrt{1-\frac{b x^4}{a}} \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 \sqrt{a-b x^4}}+\frac{3 \sqrt [4]{a} \left (a^2-\frac{b^2 c^2}{d^2}\right ) \sqrt{1-\frac{b x^4}{a}} \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 \sqrt{a-b x^4}}\\ \end{align*}
Mathematica [C] time = 0.300378, size = 342, normalized size = 1.11 \[ \frac{x \left (\frac{5 c \left (-5 a c \left (4 a^2 d-a b d x^4+b^2 c x^4\right ) F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )-2 x^4 \left (a-b x^4\right ) (a d-b c) \left (2 a d F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )+b c F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )\right )}{2 x^4 \left (2 a d F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )+b c F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )+5 a c F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )}-b x^4 \sqrt{1-\frac{b x^4}{a}} \left (d x^4-c\right ) (a d+3 b c) F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )}{20 c^2 d \sqrt{a-b x^4} \left (d x^4-c\right )} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.026, size = 329, normalized size = 1.1 \begin{align*} -{\frac{ \left ( ad-bc \right ) x}{4\,cd \left ( d{x}^{4}-c \right ) }\sqrt{-b{x}^{4}+a}}+{ \left ({\frac{{b}^{2}}{{d}^{2}}}+{\frac{ \left ( ad-bc \right ) b}{4\,c{d}^{2}}} \right ) \sqrt{1-{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{-b{x}^{4}+a}}}}-{\frac{3}{32\,c{d}^{3}}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{4}d-c \right ) }{\frac{{a}^{2}{d}^{2}-{b}^{2}{c}^{2}}{{{\it \_alpha}}^{3}} \left ( -{{\it Artanh} \left ({\frac{-2\,{{\it \_alpha}}^{2}b{x}^{2}+2\,a}{2}{\frac{1}{\sqrt{{\frac{ad-bc}{d}}}}}{\frac{1}{\sqrt{-b{x}^{4}+a}}}} \right ){\frac{1}{\sqrt{{\frac{ad-bc}{d}}}}}}-2\,{\frac{{{\it \_alpha}}^{3}d}{c\sqrt{-b{x}^{4}+a}}\sqrt{1-{\frac{{x}^{2}\sqrt{b}}{\sqrt{a}}}}\sqrt{1+{\frac{{x}^{2}\sqrt{b}}{\sqrt{a}}}}{\it EllipticPi} \left ( x\sqrt{{\frac{\sqrt{b}}{\sqrt{a}}}},{\frac{\sqrt{a}{{\it \_alpha}}^{2}d}{c\sqrt{b}}},{\sqrt{-{\frac{\sqrt{b}}{\sqrt{a}}}}{\frac{1}{\sqrt{{\frac{\sqrt{b}}{\sqrt{a}}}}}}} \right ){\frac{1}{\sqrt{{\frac{\sqrt{b}}{\sqrt{a}}}}}}} \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{{\left (-b x^{4} + a\right )}^{\frac{3}{2}}}{{\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a - b x^{4}\right )^{\frac{3}{2}}}{\left (- c + d x^{4}\right )^{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{{\left (-b x^{4} + a\right )}^{\frac{3}{2}}}{{\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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