Optimal. Leaf size=913 \[ \text{result too large to display} \]
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Rubi [A] time = 1.13196, antiderivative size = 913, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {414, 523, 220, 409, 1217, 1707} \[ -\frac{d \left (\sqrt{b} x^2+\sqrt{a}\right ) \sqrt{\frac{b x^4+a}{\left (\sqrt{b} x^2+\sqrt{a}\right )^2}} \Pi \left (\frac{\left (\sqrt{b} \sqrt{-c}+\sqrt{a} \sqrt{d}\right )^2}{4 \sqrt{a} \sqrt{b} \sqrt{-c} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right ) \left (\sqrt{b} \sqrt{-c}-\sqrt{a} \sqrt{d}\right )^2}{8 \sqrt [4]{a} \sqrt [4]{b} c (b c-a d) (b c+a d) \sqrt{b x^4+a}}+\frac{d^{5/4} \tan ^{-1}\left (\frac{\sqrt{b c-a d} x}{\sqrt [4]{-c} \sqrt [4]{d} \sqrt{b x^4+a}}\right )}{4 (-c)^{3/4} (b c-a d)^{3/2}}-\frac{d^{5/4} \tan ^{-1}\left (\frac{\sqrt{a d-b c} x}{\sqrt [4]{-c} \sqrt [4]{d} \sqrt{b x^4+a}}\right )}{4 (-c)^{3/4} (a d-b c)^{3/2}}+\frac{b^{3/4} \left (\sqrt{b} x^2+\sqrt{a}\right ) \sqrt{\frac{b x^4+a}{\left (\sqrt{b} x^2+\sqrt{a}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 a^{5/4} (b c-a d) \sqrt{b x^4+a}}-\frac{\sqrt [4]{b} \left (\sqrt{b}+\frac{\sqrt{a} \sqrt{d}}{\sqrt{-c}}\right ) d \left (\sqrt{b} x^2+\sqrt{a}\right ) \sqrt{\frac{b x^4+a}{\left (\sqrt{b} x^2+\sqrt{a}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{a} (b c-a d) (b c+a d) \sqrt{b x^4+a}}-\frac{\sqrt [4]{b} \left (\sqrt{b} c+\sqrt{a} \sqrt{-c} \sqrt{d}\right ) d \left (\sqrt{b} x^2+\sqrt{a}\right ) \sqrt{\frac{b x^4+a}{\left (\sqrt{b} x^2+\sqrt{a}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{a} c \left (b^2 c^2-a^2 d^2\right ) \sqrt{b x^4+a}}-\frac{\left (\sqrt{b} \sqrt{-c}+\sqrt{a} \sqrt{d}\right )^2 d \left (\sqrt{b} x^2+\sqrt{a}\right ) \sqrt{\frac{b x^4+a}{\left (\sqrt{b} x^2+\sqrt{a}\right )^2}} \Pi \left (-\frac{\left (\sqrt{b} \sqrt{-c}-\sqrt{a} \sqrt{d}\right )^2}{4 \sqrt{a} \sqrt{b} \sqrt{-c} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{8 \sqrt [4]{a} \sqrt [4]{b} c (b c-a d) (b c+a d) \sqrt{b x^4+a}}+\frac{b x}{2 a (b c-a d) \sqrt{b x^4+a}} \]
Antiderivative was successfully verified.
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Rule 414
Rule 523
Rule 220
Rule 409
Rule 1217
Rule 1707
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{b^{3/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 a^{5/4} (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{b x}{2 a (b c-a d) \sqrt{a+b x^4}}+\frac{b^{3/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 a^{5/4} (b c-a d) \sqrt{a+b x^4}}-\frac{\left (\sqrt{b} \left (\sqrt{b}+\frac{\sqrt{a} \sqrt{d}}{\sqrt{-c}}\right ) d\right ) \int \frac{1}{\sqrt{a+b x^4}} \, dx}{2 (b c-a d) (b c+a d)}-\frac{\left (\sqrt{b} \left (\sqrt{b} c+\sqrt{a} \sqrt{-c} \sqrt{d}\right ) d\right ) \int \frac{1}{\sqrt{a+b x^4}} \, dx}{2 c \left (b^2 c^2-a^2 d^2\right )}+\frac{\left (\sqrt{a} \left (\sqrt{b} \sqrt{-c}-\sqrt{a} \sqrt{d}\right ) d^{3/2}\right ) \int \frac{1+\frac{\sqrt{b} x^2}{\sqrt{a}}}{\left (1-\frac{\sqrt{d} x^2}{\sqrt{-c}}\right ) \sqrt{a+b x^4}} \, dx}{2 c \left (b^2 c^2-a^2 d^2\right )}-\frac{\left (\sqrt{a} \left (\sqrt{b} \sqrt{-c}+\sqrt{a} \sqrt{d}\right ) d^{3/2}\right ) \int \frac{1+\frac{\sqrt{b} x^2}{\sqrt{a}}}{\left (1+\frac{\sqrt{d} x^2}{\sqrt{-c}}\right ) \sqrt{a+b x^4}} \, dx}{2 c \left (b^2 c^2-a^2 d^2\right )}\\ &=\frac{b x}{2 a (b c-a d) \sqrt{a+b x^4}}+\frac{d^{5/4} \tan ^{-1}\left (\frac{\sqrt{b c-a d} x}{\sqrt [4]{-c} \sqrt [4]{d} \sqrt{a+b x^4}}\right )}{4 (-c)^{3/4} (b c-a d)^{3/2}}-\frac{d^{5/4} \tan ^{-1}\left (\frac{\sqrt{-b c+a d} x}{\sqrt [4]{-c} \sqrt [4]{d} \sqrt{a+b x^4}}\right )}{4 (-c)^{3/4} (-b c+a d)^{3/2}}+\frac{b^{3/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 a^{5/4} (b c-a d) \sqrt{a+b x^4}}-\frac{\sqrt [4]{b} \left (\sqrt{b}+\frac{\sqrt{a} \sqrt{d}}{\sqrt{-c}}\right ) d \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{a} (b c-a d) (b c+a d) \sqrt{a+b x^4}}-\frac{\sqrt [4]{b} \left (\sqrt{b} c+\sqrt{a} \sqrt{-c} \sqrt{d}\right ) d \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{a} c \left (b^2 c^2-a^2 d^2\right ) \sqrt{a+b x^4}}-\frac{\left (\sqrt{b} \sqrt{-c}+\sqrt{a} \sqrt{d}\right )^2 d \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \Pi \left (-\frac{\left (\sqrt{b} \sqrt{-c}-\sqrt{a} \sqrt{d}\right )^2}{4 \sqrt{a} \sqrt{b} \sqrt{-c} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{8 \sqrt [4]{a} \sqrt [4]{b} c \left (b^2 c^2-a^2 d^2\right ) \sqrt{a+b x^4}}-\frac{\left (\sqrt{b} \sqrt{-c}-\sqrt{a} \sqrt{d}\right )^2 d \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \Pi \left (\frac{\left (\sqrt{b} \sqrt{-c}+\sqrt{a} \sqrt{d}\right )^2}{4 \sqrt{a} \sqrt{b} \sqrt{-c} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{8 \sqrt [4]{a} \sqrt [4]{b} c \left (b^2 c^2-a^2 d^2\right ) \sqrt{a+b x^4}}\\ \end{align*}
Mathematica [C] time = 0.245768, size = 331, normalized size = 0.36 \[ \frac{x \left (\frac{5 \left (2 b x^4 \left (c+d x^4\right ) \left (2 a d F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )+b c F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )+5 a c \left (2 a d-b \left (2 c+d x^4\right )\right ) F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )}{\left (c+d x^4\right ) \left (5 a c F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )-2 x^4 \left (2 a d F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )+b c F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )\right )}-\frac{b d x^4 \sqrt{\frac{b x^4}{a}+1} F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{c}\right )}{10 a \sqrt{a+b x^4} (a d-b c)} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.025, size = 313, normalized size = 0.3 \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-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\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{i\sqrt{b}{x}^{2}}{\sqrt{a}}}}\sqrt{1+{\frac{i\sqrt{b}{x}^{2}}{\sqrt{a}}}}{\it EllipticPi} \left ( x\sqrt{{\frac{i\sqrt{b}}{\sqrt{a}}}},{\frac{i\sqrt{a}{{\it \_alpha}}^{2}d}{c\sqrt{b}}},{\sqrt{{\frac{-i\sqrt{b}}{\sqrt{a}}}}{\frac{1}{\sqrt{{\frac{i\sqrt{b}}{\sqrt{a}}}}}}} \right ){\frac{1}{\sqrt{{\frac{i\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}{\left (a + b x^{4}\right )^{\frac{3}{2}} \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{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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