Optimal. Leaf size=277 \[ -\frac{\sqrt [4]{a} b^{3/4} \sqrt{1-\frac{b x^4}{a}} (3 b c-5 a d) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{3 d^2 \sqrt{a-b x^4}}+\frac{\sqrt [4]{a} \sqrt{1-\frac{b x^4}{a}} (b c-a d)^2 \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d^2 \sqrt{a-b x^4}}+\frac{\sqrt [4]{a} \sqrt{1-\frac{b x^4}{a}} (b c-a d)^2 \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d^2 \sqrt{a-b x^4}}+\frac{b x \sqrt{a-b x^4}}{3 d} \]
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Rubi [A] time = 0.262202, antiderivative size = 277, 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 = {416, 523, 224, 221, 409, 1219, 1218} \[ -\frac{\sqrt [4]{a} b^{3/4} \sqrt{1-\frac{b x^4}{a}} (3 b c-5 a d) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{3 d^2 \sqrt{a-b x^4}}+\frac{\sqrt [4]{a} \sqrt{1-\frac{b x^4}{a}} (b c-a d)^2 \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d^2 \sqrt{a-b x^4}}+\frac{\sqrt [4]{a} \sqrt{1-\frac{b x^4}{a}} (b c-a d)^2 \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d^2 \sqrt{a-b x^4}}+\frac{b x \sqrt{a-b x^4}}{3 d} \]
Antiderivative was successfully verified.
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Rule 416
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}}{c-d x^4} \, dx &=\frac{b x \sqrt{a-b x^4}}{3 d}-\frac{\int \frac{a (b c-3 a d)-b (3 b c-5 a d) x^4}{\sqrt{a-b x^4} \left (c-d x^4\right )} \, dx}{3 d}\\ &=\frac{b x \sqrt{a-b x^4}}{3 d}-\frac{(b (3 b c-5 a d)) \int \frac{1}{\sqrt{a-b x^4}} \, dx}{3 d^2}+\frac{(b c-a d)^2 \int \frac{1}{\sqrt{a-b x^4} \left (c-d x^4\right )} \, dx}{d^2}\\ &=\frac{b x \sqrt{a-b x^4}}{3 d}+\frac{(b c-a d)^2 \int \frac{1}{\left (1-\frac{\sqrt{d} x^2}{\sqrt{c}}\right ) \sqrt{a-b x^4}} \, dx}{2 c d^2}+\frac{(b c-a d)^2 \int \frac{1}{\left (1+\frac{\sqrt{d} x^2}{\sqrt{c}}\right ) \sqrt{a-b x^4}} \, dx}{2 c d^2}-\frac{\left (b (3 b c-5 a d) \sqrt{1-\frac{b x^4}{a}}\right ) \int \frac{1}{\sqrt{1-\frac{b x^4}{a}}} \, dx}{3 d^2 \sqrt{a-b x^4}}\\ &=\frac{b x \sqrt{a-b x^4}}{3 d}-\frac{\sqrt [4]{a} b^{3/4} (3 b c-5 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 )}{3 d^2 \sqrt{a-b x^4}}+\frac{\left ((b c-a d)^2 \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}{2 c d^2 \sqrt{a-b x^4}}+\frac{\left ((b c-a d)^2 \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}{2 c d^2 \sqrt{a-b x^4}}\\ &=\frac{b x \sqrt{a-b x^4}}{3 d}-\frac{\sqrt [4]{a} b^{3/4} (3 b c-5 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 )}{3 d^2 \sqrt{a-b x^4}}+\frac{\sqrt [4]{a} (b c-a d)^2 \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 )}{2 \sqrt [4]{b} c d^2 \sqrt{a-b x^4}}+\frac{\sqrt [4]{a} (b c-a d)^2 \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 )}{2 \sqrt [4]{b} c d^2 \sqrt{a-b x^4}}\\ \end{align*}
Mathematica [C] time = 0.35143, size = 341, normalized size = 1.23 \[ -\frac{x \left (\frac{5 \left (5 a c \left (3 a^2 d-a b d x^4+b^2 x^4 \left (d x^4-c\right )\right ) F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )+2 b x^4 \left (a-b x^4\right ) \left (c-d x^4\right ) \left (2 a d F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )+b c F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )\right )}{\left (d x^4-c\right ) \left (2 x^4 \left (2 a d F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )+b c F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )+5 a c F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )}+\frac{b x^4 \sqrt{1-\frac{b x^4}{a}} (5 a d-3 b c) F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )}{c}\right )}{15 d \sqrt{a-b x^4}} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.019, size = 311, normalized size = 1.1 \begin{align*}{\frac{bx}{3\,d}\sqrt{-b{x}^{4}+a}}-{ \left ( -{\frac{b \left ( 2\,ad-bc \right ) }{{d}^{2}}}+{\frac{ab}{3\,d}} \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{1}{8\,{d}^{3}}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{4}d-c \right ) }{\frac{-{a}^{2}{d}^{2}+2\,cabd-{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}}}{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{a \sqrt{a - b x^{4}}}{- c + d x^{4}}\, dx - \int - \frac{b x^{4} \sqrt{a - b x^{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{3}{2}}}{d x^{4} - c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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