Optimal. Leaf size=308 \[ -\frac {\sqrt {a} b^{3/2} x^3 \left (\frac {a}{b x^4}+1\right )^{3/4} F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{4 c \left (a+b x^4\right )^{3/4} (b c-a d)}+\frac {\sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} (2 b c-3 a d) \Pi \left (-\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 (b c-a d)}+\frac {\sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} (2 b c-3 a d) \Pi \left (\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 (b c-a d)}+\frac {x \sqrt [4]{a+b x^4}}{4 c \left (c+d x^4\right )} \]
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Rubi [A] time = 0.20, antiderivative size = 308, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {412, 529, 237, 335, 275, 231, 407, 409, 1218} \[ -\frac {\sqrt {a} b^{3/2} x^3 \left (\frac {a}{b x^4}+1\right )^{3/4} F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{4 c \left (a+b x^4\right )^{3/4} (b c-a d)}+\frac {\sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} (2 b c-3 a d) \Pi \left (-\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 (b c-a d)}+\frac {\sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} (2 b c-3 a d) \Pi \left (\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 (b c-a d)}+\frac {x \sqrt [4]{a+b x^4}}{4 c \left (c+d x^4\right )} \]
Antiderivative was successfully verified.
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Rule 231
Rule 237
Rule 275
Rule 335
Rule 407
Rule 409
Rule 412
Rule 529
Rule 1218
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{a+b x^4}}{\left (c+d x^4\right )^2} \, dx &=\frac {x \sqrt [4]{a+b x^4}}{4 c \left (c+d x^4\right )}-\frac {\int \frac {-3 a-2 b x^4}{\left (a+b x^4\right )^{3/4} \left (c+d x^4\right )} \, dx}{4 c}\\ &=\frac {x \sqrt [4]{a+b x^4}}{4 c \left (c+d x^4\right )}+\frac {(a b) \int \frac {1}{\left (a+b x^4\right )^{3/4}} \, dx}{4 c (b c-a d)}+\frac {(2 b c-3 a d) \int \frac {\sqrt [4]{a+b x^4}}{c+d x^4} \, dx}{4 c (b c-a d)}\\ &=\frac {x \sqrt [4]{a+b x^4}}{4 c \left (c+d x^4\right )}+\frac {\left (a b \left (1+\frac {a}{b x^4}\right )^{3/4} x^3\right ) \int \frac {1}{\left (1+\frac {a}{b x^4}\right )^{3/4} x^3} \, dx}{4 c (b c-a d) \left (a+b x^4\right )^{3/4}}+\frac {\left ((2 b c-3 a d) \sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-b x^4} \left (c-(b c-a d) x^4\right )} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{4 c (b c-a d)}\\ &=\frac {x \sqrt [4]{a+b x^4}}{4 c \left (c+d x^4\right )}-\frac {\left (a b \left (1+\frac {a}{b x^4}\right )^{3/4} x^3\right ) \operatorname {Subst}\left (\int \frac {x}{\left (1+\frac {a x^4}{b}\right )^{3/4}} \, dx,x,\frac {1}{x}\right )}{4 c (b c-a d) \left (a+b x^4\right )^{3/4}}+\frac {\left ((2 b c-3 a d) \sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {\sqrt {b c-a d} x^2}{\sqrt {c}}\right ) \sqrt {1-b x^4}} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{8 c^2 (b c-a d)}+\frac {\left ((2 b c-3 a d) \sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {\sqrt {b c-a d} x^2}{\sqrt {c}}\right ) \sqrt {1-b x^4}} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{8 c^2 (b c-a d)}\\ &=\frac {x \sqrt [4]{a+b x^4}}{4 c \left (c+d x^4\right )}+\frac {(2 b c-3 a d) \sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} \Pi \left (-\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 (b c-a d)}+\frac {(2 b c-3 a d) \sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} \Pi \left (\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 (b c-a d)}-\frac {\left (a b \left (1+\frac {a}{b x^4}\right )^{3/4} x^3\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {a x^2}{b}\right )^{3/4}} \, dx,x,\frac {1}{x^2}\right )}{8 c (b c-a d) \left (a+b x^4\right )^{3/4}}\\ &=\frac {x \sqrt [4]{a+b x^4}}{4 c \left (c+d x^4\right )}-\frac {\sqrt {a} b^{3/2} \left (1+\frac {a}{b x^4}\right )^{3/4} x^3 F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{4 c (b c-a d) \left (a+b x^4\right )^{3/4}}+\frac {(2 b c-3 a d) \sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} \Pi \left (-\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 (b c-a d)}+\frac {(2 b c-3 a d) \sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} \Pi \left (\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 (b c-a d)}\\ \end {align*}
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Mathematica [C] time = 0.31, size = 233, normalized size = 0.76 \[ \frac {x \left (\frac {5 \left (\frac {a+b x^4}{c}-\frac {15 a^2 F_1\left (\frac {1}{4};\frac {3}{4},1;\frac {5}{4};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{x^4 \left (4 a d F_1\left (\frac {5}{4};\frac {3}{4},2;\frac {9}{4};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+3 b c F_1\left (\frac {5}{4};\frac {7}{4},1;\frac {9}{4};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )\right )-5 a c F_1\left (\frac {1}{4};\frac {3}{4},1;\frac {5}{4};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}\right )}{c+d x^4}+\frac {2 b x^4 \left (\frac {b x^4}{a}+1\right )^{3/4} F_1\left (\frac {5}{4};\frac {3}{4},1;\frac {9}{4};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{c^2}\right )}{20 \left (a+b x^4\right )^{3/4}} \]
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 \[ \int \frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{{\left (d x^{4} + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.66, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{4}+a \right )^{\frac {1}{4}}}{\left (d \,x^{4}+c \right )^{2}}\, dx \]
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 \[ \int \frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{{\left (d x^{4} + c\right )}^{2}}\,{d x} \]
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 \[ \int \frac {{\left (b\,x^4+a\right )}^{1/4}}{{\left (d\,x^4+c\right )}^2} \,d x \]
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 \[ \int \frac {\sqrt [4]{a + b x^{4}}}{\left (c + d x^{4}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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