Optimal. Leaf size=700 \[ -\frac {(b c-a d) \tan ^{-1}\left (\frac {x \sqrt {\frac {\sqrt {-a} \left (\frac {b c}{a}-d\right )}{\sqrt {b}}}}{\sqrt {c+d x^4}}\right )}{4 b^2 \sqrt {-\frac {b c-a d}{\sqrt {-a} \sqrt {b}}}}-\frac {(b c-a d) \tan ^{-1}\left (\frac {x \sqrt {\frac {b c-a d}{\sqrt {-a} \sqrt {b}}}}{\sqrt {c+d x^4}}\right )}{4 b^2 \sqrt {\frac {b c-a d}{\sqrt {-a} \sqrt {b}}}}-\frac {\left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \left (\sqrt {-a} \sqrt {d}+\sqrt {b} \sqrt {c}\right ) (b c-a d) \Pi \left (-\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}};2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{8 b^2 \sqrt [4]{c} \sqrt [4]{d} \sqrt {c+d x^4} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right )}-\frac {\left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) (b c-a d) \Pi \left (\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}};2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{8 b^2 \sqrt [4]{c} \sqrt [4]{d} \sqrt {c+d x^4} \left (\sqrt {-a} \sqrt {d}+\sqrt {b} \sqrt {c}\right )}+\frac {c^{3/4} \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} (b c-2 a d) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{3 b \sqrt [4]{d} \sqrt {c+d x^4} (a d+b c)}+\frac {x \sqrt {c+d x^4}}{3 b} \]
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Rubi [A] time = 1.25, antiderivative size = 904, normalized size of antiderivative = 1.29, number of steps used = 10, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {478, 523, 220, 409, 1217, 1707} \[ -\frac {(b c-a d) \left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\sqrt {c}\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]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right ) \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right )^2}{8 b^2 \sqrt [4]{c} \sqrt [4]{d} (b c+a d) \sqrt {d x^4+c}}-\frac {\sqrt {-a} \sqrt [4]{d} (b c-a d) \left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\sqrt {c}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right ) \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right )}{4 b^2 \sqrt [4]{c} (b c+a d) \sqrt {d x^4+c}}-\frac {\sqrt [4]{-a} \sqrt {b c-a d} \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {d x^4+c}}\right )}{4 b^{7/4}}+\frac {\sqrt [4]{-a} \sqrt {a d-b c} \tan ^{-1}\left (\frac {\sqrt {a d-b c} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {d x^4+c}}\right )}{4 b^{7/4}}+\frac {(2 b c-3 a d) \left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\sqrt {c}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{6 b^2 \sqrt [4]{c} \sqrt [4]{d} \sqrt {d x^4+c}}-\frac {a \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {-a}}+\sqrt {d}\right ) \sqrt [4]{d} (b c-a d) \left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\sqrt {c}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{4 b^2 \sqrt [4]{c} (b c+a d) \sqrt {d x^4+c}}-\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2 (b c-a d) \left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\sqrt {c}\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]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{8 b^2 \sqrt [4]{c} \sqrt [4]{d} (b c+a d) \sqrt {d x^4+c}}+\frac {x \sqrt {d x^4+c}}{3 b} \]
Warning: Unable to verify antiderivative.
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Rule 220
Rule 409
Rule 478
Rule 523
Rule 1217
Rule 1707
Rubi steps
\begin {align*} \int \frac {x^4 \sqrt {c+d x^4}}{a+b x^4} \, dx &=\frac {x \sqrt {c+d x^4}}{3 b}-\frac {\int \frac {a c+(-2 b c+3 a d) x^4}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx}{3 b}\\ &=\frac {x \sqrt {c+d x^4}}{3 b}+\frac {(2 b c-3 a d) \int \frac {1}{\sqrt {c+d x^4}} \, dx}{3 b^2}-\frac {(a (b c-a d)) \int \frac {1}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx}{b^2}\\ &=\frac {x \sqrt {c+d x^4}}{3 b}+\frac {(2 b c-3 a d) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{6 b^2 \sqrt [4]{c} \sqrt [4]{d} \sqrt {c+d x^4}}-\frac {(b c-a d) \int \frac {1}{\left (1-\frac {\sqrt {b} x^2}{\sqrt {-a}}\right ) \sqrt {c+d x^4}} \, dx}{2 b^2}-\frac {(b c-a d) \int \frac {1}{\left (1+\frac {\sqrt {b} x^2}{\sqrt {-a}}\right ) \sqrt {c+d x^4}} \, dx}{2 b^2}\\ &=\frac {x \sqrt {c+d x^4}}{3 b}+\frac {(2 b c-3 a d) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{6 b^2 \sqrt [4]{c} \sqrt [4]{d} \sqrt {c+d x^4}}-\frac {\left (\sqrt {c} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) (b c-a d)\right ) \int \frac {1+\frac {\sqrt {d} x^2}{\sqrt {c}}}{\left (1-\frac {\sqrt {b} x^2}{\sqrt {-a}}\right ) \sqrt {c+d x^4}} \, dx}{2 b^{3/2} (b c+a d)}-\frac {\left (\sqrt {c} \left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right ) (b c-a d)\right ) \int \frac {1+\frac {\sqrt {d} x^2}{\sqrt {c}}}{\left (1+\frac {\sqrt {b} x^2}{\sqrt {-a}}\right ) \sqrt {c+d x^4}} \, dx}{2 b^{3/2} (b c+a d)}-\frac {\left (a \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {-a}}+\sqrt {d}\right ) \sqrt {d} (b c-a d)\right ) \int \frac {1}{\sqrt {c+d x^4}} \, dx}{2 b^2 (b c+a d)}-\frac {\left (\sqrt {-a} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \sqrt {d} (b c-a d)\right ) \int \frac {1}{\sqrt {c+d x^4}} \, dx}{2 b^2 (b c+a d)}\\ &=\frac {x \sqrt {c+d x^4}}{3 b}-\frac {\sqrt [4]{-a} \sqrt {b c-a d} \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {c+d x^4}}\right )}{4 b^{7/4}}+\frac {\sqrt [4]{-a} \sqrt {-b c+a d} \tan ^{-1}\left (\frac {\sqrt {-b c+a d} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {c+d x^4}}\right )}{4 b^{7/4}}+\frac {(2 b c-3 a d) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{6 b^2 \sqrt [4]{c} \sqrt [4]{d} \sqrt {c+d x^4}}-\frac {a \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {-a}}+\sqrt {d}\right ) \sqrt [4]{d} (b c-a d) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{4 b^2 \sqrt [4]{c} (b c+a d) \sqrt {c+d x^4}}-\frac {\sqrt {-a} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \sqrt [4]{d} (b c-a d) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{4 b^2 \sqrt [4]{c} (b c+a d) \sqrt {c+d x^4}}-\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2 (b c-a d) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} 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]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{8 b^2 \sqrt [4]{c} \sqrt [4]{d} (b c+a d) \sqrt {c+d x^4}}-\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right )^2 (b c-a d) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} 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]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{8 b^2 \sqrt [4]{c} \sqrt [4]{d} (b c+a d) \sqrt {c+d x^4}}\\ \end {align*}
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Mathematica [C] time = 0.48, size = 241, normalized size = 0.34 \[ \frac {x \left (5 \left (\frac {5 a^2 c^2 F_1\left (\frac {1}{4};\frac {1}{2},1;\frac {5}{4};-\frac {d x^4}{c},-\frac {b x^4}{a}\right )}{\left (a+b x^4\right ) \left (2 x^4 \left (2 b c F_1\left (\frac {5}{4};\frac {1}{2},2;\frac {9}{4};-\frac {d x^4}{c},-\frac {b x^4}{a}\right )+a d F_1\left (\frac {5}{4};\frac {3}{2},1;\frac {9}{4};-\frac {d x^4}{c},-\frac {b x^4}{a}\right )\right )-5 a c F_1\left (\frac {1}{4};\frac {1}{2},1;\frac {5}{4};-\frac {d x^4}{c},-\frac {b x^4}{a}\right )\right )}+c+d x^4\right )+\frac {x^4 \sqrt {\frac {d x^4}{c}+1} (2 b c-3 a d) F_1\left (\frac {5}{4};\frac {1}{2},1;\frac {9}{4};-\frac {d x^4}{c},-\frac {b x^4}{a}\right )}{a}\right )}{15 b \sqrt {c+d x^4}} \]
Warning: Unable to verify antiderivative.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d x^{4} + c} x^{4}}{b x^{4} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.35, size = 368, normalized size = 0.53 \[ -\frac {\left (\frac {\sqrt {-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}+1}\, \sqrt {\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}+1}\, d \EllipticF \left (\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, x , i\right )}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, \sqrt {d \,x^{4}+c}\, b}-\frac {\left (a d -b c \right ) \left (\frac {2 \sqrt {-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}+1}\, \sqrt {\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}+1}\, \RootOf \left (b \,\textit {\_Z}^{4}+a \right )^{3} b \EllipticPi \left (\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, x , \frac {i \RootOf \left (b \,\textit {\_Z}^{4}+a \right )^{2} b \sqrt {c}}{a \sqrt {d}}, \frac {\sqrt {-\frac {i \sqrt {d}}{\sqrt {c}}}}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}}\right )}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, \sqrt {d \,x^{4}+c}\, a}-\frac {\arctanh \left (\frac {2 \RootOf \left (b \,\textit {\_Z}^{4}+a \right )^{2} d \,x^{2}+2 c}{2 \sqrt {\frac {-a d +b c}{b}}\, \sqrt {d \,x^{4}+c}}\right )}{\sqrt {\frac {-a d +b c}{b}}}\right )}{8 b^{2} \RootOf \left (b \,\textit {\_Z}^{4}+a \right )^{3}}\right ) a}{b}+\frac {\frac {2 \sqrt {-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}+1}\, \sqrt {\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}+1}\, c \EllipticF \left (\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, x , i\right )}{3 \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, \sqrt {d \,x^{4}+c}}+\frac {\sqrt {d \,x^{4}+c}\, x}{3}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d x^{4} + c} x^{4}}{b x^{4} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^4\,\sqrt {d\,x^4+c}}{b\,x^4+a} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4} \sqrt {c + d x^{4}}}{a + b x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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