Optimal. Leaf size=281 \[ \frac{b^{3/4} \sqrt{1-\frac{b x^4}{a}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{2 a^{3/4} \sqrt{a-b x^4} (b c-a d)}+\frac{b x}{2 a \sqrt{a-b x^4} (b c-a d)}-\frac{\sqrt [4]{a} d \sqrt{1-\frac{b x^4}{a}} \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c \sqrt{a-b x^4} (b c-a d)}-\frac{\sqrt [4]{a} d \sqrt{1-\frac{b x^4}{a}} \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c \sqrt{a-b x^4} (b c-a d)} \]
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Rubi [A] time = 0.222203, antiderivative size = 281, 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 = {414, 523, 224, 221, 409, 1219, 1218} \[ \frac{b^{3/4} \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 a^{3/4} \sqrt{a-b x^4} (b c-a d)}+\frac{b x}{2 a \sqrt{a-b x^4} (b c-a d)}-\frac{\sqrt [4]{a} d \sqrt{1-\frac{b x^4}{a}} \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c \sqrt{a-b x^4} (b c-a d)}-\frac{\sqrt [4]{a} d \sqrt{1-\frac{b x^4}{a}} \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c \sqrt{a-b x^4} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 414
Rule 523
Rule 224
Rule 221
Rule 409
Rule 1219
Rule 1218
Rubi steps
\begin{align*} \int \frac{1}{\left (a-b x^4\right )^{3/2} \left (c-d x^4\right )} \, dx &=\frac{b x}{2 a (b c-a d) \sqrt{a-b x^4}}+\frac{\int \frac{b c-2 a d-b d x^4}{\sqrt{a-b x^4} \left (c-d x^4\right )} \, dx}{2 a (b c-a d)}\\ &=\frac{b x}{2 a (b c-a d) \sqrt{a-b x^4}}+\frac{b \int \frac{1}{\sqrt{a-b x^4}} \, dx}{2 a (b c-a d)}-\frac{d \int \frac{1}{\sqrt{a-b x^4} \left (c-d x^4\right )} \, dx}{b c-a d}\\ &=\frac{b x}{2 a (b c-a d) \sqrt{a-b x^4}}-\frac{d \int \frac{1}{\left (1-\frac{\sqrt{d} x^2}{\sqrt{c}}\right ) \sqrt{a-b x^4}} \, dx}{2 c (b c-a d)}-\frac{d \int \frac{1}{\left (1+\frac{\sqrt{d} x^2}{\sqrt{c}}\right ) \sqrt{a-b x^4}} \, dx}{2 c (b c-a d)}+\frac{\left (b \sqrt{1-\frac{b x^4}{a}}\right ) \int \frac{1}{\sqrt{1-\frac{b x^4}{a}}} \, dx}{2 a (b c-a d) \sqrt{a-b x^4}}\\ &=\frac{b x}{2 a (b c-a d) \sqrt{a-b x^4}}+\frac{b^{3/4} \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 )}{2 a^{3/4} (b c-a d) \sqrt{a-b x^4}}-\frac{\left (d \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 (b c-a d) \sqrt{a-b x^4}}-\frac{\left (d \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 (b c-a d) \sqrt{a-b x^4}}\\ &=\frac{b x}{2 a (b c-a d) \sqrt{a-b x^4}}+\frac{b^{3/4} \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 )}{2 a^{3/4} (b c-a d) \sqrt{a-b x^4}}-\frac{\sqrt [4]{a} d \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 (b c-a d) \sqrt{a-b x^4}}-\frac{\sqrt [4]{a} d \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 (b c-a d) \sqrt{a-b x^4}}\\ \end{align*}
Mathematica [C] time = 0.230621, size = 381, normalized size = 1.36 \[ \frac{2 b x^5 \left (c-d x^4\right ) \left (5 c-d x^4 \sqrt{1-\frac{b x^4}{a}} F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right ) \left (2 a d F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )+b c F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )+5 a c x F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right ) \left (b d x^4 \sqrt{1-\frac{b x^4}{a}} \left (d x^4-c\right ) F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )-5 c \left (2 a d-2 b c+b d x^4\right )\right )}{10 a c \sqrt{a-b x^4} \left (d x^4-c\right ) (a d-b c) \left (2 x^4 \left (2 a d F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )+b c F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )+5 a c F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.052, size = 301, normalized size = 1.1 \begin{align*} -{\frac{bx}{2\,a \left ( ad-bc \right ) }{\frac{1}{\sqrt{- \left ({x}^{4}-{\frac{a}{b}} \right ) b}}}}-{\frac{b}{2\,a \left ( ad-bc \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}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{4}d-c \right ) }{\frac{1}{ \left ( ad-bc \right ){{\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{1}{{\left (-b x^{4} + a\right )}^{\frac{3}{2}}{\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}{- a c \sqrt{a - b x^{4}} + a d x^{4} \sqrt{a - b x^{4}} + b c x^{4} \sqrt{a - b x^{4}} - b d x^{8} \sqrt{a - b 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{1}{{\left (-b x^{4} + a\right )}^{\frac{3}{2}}{\left (d x^{4} - c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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