Optimal. Leaf size=973 \[ \text{result too large to display} \]
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Rubi [A] time = 3.70666, antiderivative size = 973, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{-\frac{\sqrt{-c} \left (b-\frac{a d}{c}\right )}{\sqrt{d}}} x}{\sqrt{b x^4+a}}\right ) (b c-a d)^2}{4 c d^2 \sqrt{\frac{b c-a d}{\sqrt{-c} \sqrt{d}}}}+\frac{\tan ^{-1}\left (\frac{\sqrt{\frac{\sqrt{-c} \left (b-\frac{a d}{c}\right )}{\sqrt{d}}} x}{\sqrt{b x^4+a}}\right ) (b c-a d)^2}{4 c d^2 \sqrt{-\frac{b c-a d}{\sqrt{-c} \sqrt{d}}}}+\frac{\sqrt [4]{b} \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 ) (b c-a d)^2}{4 \sqrt [4]{a} \left (\sqrt{b} c-\sqrt{a} \sqrt{-c} \sqrt{d}\right ) d^2 \sqrt{b x^4+a}}+\frac{\sqrt [4]{b} \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 ) (b c-a d)^2}{4 \sqrt [4]{a} \left (\sqrt{b} c+\sqrt{a} \sqrt{-c} \sqrt{d}\right ) d^2 \sqrt{b x^4+a}}-\frac{\left (\sqrt{b} \sqrt{-c}+\sqrt{a} \sqrt{d}\right ) \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 ) (b c-a d)^2}{8 \sqrt [4]{a} \sqrt [4]{b} c \left (\sqrt{b} \sqrt{-c}-\sqrt{a} \sqrt{d}\right ) d^2 \sqrt{b x^4+a}}-\frac{\left (\sqrt{b} \sqrt{-c}-\sqrt{a} \sqrt{d}\right ) \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 ) (b c-a d)^2}{8 \sqrt [4]{a} \sqrt [4]{b} c \left (\sqrt{b} \sqrt{-c}+\sqrt{a} \sqrt{d}\right ) d^2 \sqrt{b x^4+a}}-\frac{b^{3/4} (3 b c-5 a 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 )}{6 \sqrt [4]{a} d^2 \sqrt{b x^4+a}}+\frac{b x \sqrt{b x^4+a}}{3 d} \]
Warning: Unable to verify antiderivative.
[In] Int[(a + b*x^4)^(3/2)/(c + d*x^4),x]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**4+a)**(3/2)/(d*x**4+c),x)
[Out]
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Mathematica [C] time = 1.14671, size = 435, normalized size = 0.45 \[ \frac{x \left (\frac{25 a^2 c (3 a d-b c) F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{5 a c F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )-2 x^4 \left (2 a d F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )+b c F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )}+\frac{b \left (10 x^4 \left (a+b x^4\right ) \left (c+d x^4\right ) \left (2 a d F_1\left (\frac{9}{4};\frac{1}{2},2;\frac{13}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )+b c F_1\left (\frac{9}{4};\frac{3}{2},1;\frac{13}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )-9 a c \left (5 a \left (c+2 d x^4\right )+b x^4 \left (2 c+5 d x^4\right )\right ) F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )}{2 x^4 \left (2 a d F_1\left (\frac{9}{4};\frac{1}{2},2;\frac{13}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )+b c F_1\left (\frac{9}{4};\frac{3}{2},1;\frac{13}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )-9 a c F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}\right )}{15 d \sqrt{a+b x^4} \left (c+d x^4\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(a + b*x^4)^(3/2)/(c + d*x^4),x]
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Maple [C] time = 0.036, size = 322, normalized size = 0.3 \[{\frac{bx}{3\,d}\sqrt{b{x}^{4}+a}}+{1 \left ({\frac{b \left ( 2\,ad-bc \right ) }{{d}^{2}}}-{\frac{ab}{3\,d}} \right ) \sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}-{\frac{1}{8\,{d}^{3}}\sum _{{\it \_alpha}={\it RootOf} \left ( d{{\it \_Z}}^{4}+c \right ) }{\frac{-{a}^{2}{d}^{2}+2\,cabd-{b}^{2}{c}^{2}}{{{\it \_alpha}}^{3}} \left ( -{1{\it Artanh} \left ({\frac{2\,{{\it \_alpha}}^{2}b{x}^{2}+2\,a}{2}{\frac{1}{\sqrt{{\frac{ad-bc}{d}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}} \right ){\frac{1}{\sqrt{{\frac{ad-bc}{d}}}}}}+2\,{\frac{{{\it \_alpha}}^{3}d}{c\sqrt{b{x}^{4}+a}}\sqrt{1-{\frac{i\sqrt{b}{x}^{2}}{\sqrt{a}}}}\sqrt{1+{\frac{i\sqrt{b}{x}^{2}}{\sqrt{a}}}}{\it EllipticPi} \left ( x\sqrt{{\frac{i\sqrt{b}}{\sqrt{a}}}},{\frac{i\sqrt{a}{{\it \_alpha}}^{2}d}{c\sqrt{b}}},{1\sqrt{{\frac{-i\sqrt{b}}{\sqrt{a}}}}{\frac{1}{\sqrt{{\frac{i\sqrt{b}}{\sqrt{a}}}}}}} \right ){\frac{1}{\sqrt{{\frac{i\sqrt{b}}{\sqrt{a}}}}}}} \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^4+a)^(3/2)/(d*x^4+c),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{3}{2}}}{d x^{4} + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^(3/2)/(d*x^4 + c),x, algorithm="maxima")
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^(3/2)/(d*x^4 + c),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x^{4}\right )^{\frac{3}{2}}}{c + d x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**4+a)**(3/2)/(d*x**4+c),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{3}{2}}}{d x^{4} + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^(3/2)/(d*x^4 + c),x, algorithm="giac")
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