Optimal. Leaf size=353 \[ \frac {b (3 b c-a d) x \sqrt [4]{a+b x^4}}{4 c d^2}-\frac {(b c-a d) x \left (a+b x^4\right )^{5/4}}{4 c d \left (c+d x^4\right )}-\frac {\sqrt {a} b^{3/2} (3 b c-a d) \left (1+\frac {a}{b x^4}\right )^{3/4} x^3 F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{4 c d^2 \left (a+b x^4\right )^{3/4}}-\frac {3 (b c-a d) (2 b c+a d) \sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} \Pi \left (-\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d^2}-\frac {3 (b c-a d) (2 b c+a d) \sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} \Pi \left (\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d^2} \]
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Rubi [A]
time = 0.23, antiderivative size = 353, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 10, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {424, 542,
543, 243, 342, 281, 237, 416, 418, 1232} \begin {gather*} -\frac {3 \sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} (b c-a d) (a d+2 b c) \Pi \left (-\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}};\left .\text {ArcSin}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d^2}-\frac {3 \sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} (b c-a d) (a d+2 b c) \Pi \left (\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}};\left .\text {ArcSin}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d^2}-\frac {\sqrt {a} b^{3/2} x^3 \left (\frac {a}{b x^4}+1\right )^{3/4} (3 b c-a d) F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{4 c d^2 \left (a+b x^4\right )^{3/4}}+\frac {b x \sqrt [4]{a+b x^4} (3 b c-a d)}{4 c d^2}-\frac {x \left (a+b x^4\right )^{5/4} (b c-a d)}{4 c d \left (c+d x^4\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 237
Rule 243
Rule 281
Rule 342
Rule 416
Rule 418
Rule 424
Rule 542
Rule 543
Rule 1232
Rubi steps
\begin {align*} \int \frac {\left (a+b x^4\right )^{9/4}}{\left (c+d x^4\right )^2} \, dx &=-\frac {(b c-a d) x \left (a+b x^4\right )^{5/4}}{4 c d \left (c+d x^4\right )}+\frac {\int \frac {\sqrt [4]{a+b x^4} \left (a (b c+3 a d)+2 b (3 b c-a d) x^4\right )}{c+d x^4} \, dx}{4 c d}\\ &=\frac {b (3 b c-a d) x \sqrt [4]{a+b x^4}}{4 c d^2}-\frac {(b c-a d) x \left (a+b x^4\right )^{5/4}}{4 c d \left (c+d x^4\right )}+\frac {\int \frac {-2 a \left (3 b^2 c^2-2 a b c d-3 a^2 d^2\right )-4 b \left (3 b^2 c^2-3 a b c d-a^2 d^2\right ) x^4}{\left (a+b x^4\right )^{3/4} \left (c+d x^4\right )} \, dx}{8 c d^2}\\ &=\frac {b (3 b c-a d) x \sqrt [4]{a+b x^4}}{4 c d^2}-\frac {(b c-a d) x \left (a+b x^4\right )^{5/4}}{4 c d \left (c+d x^4\right )}+\frac {(a b (3 b c-a d)) \int \frac {1}{\left (a+b x^4\right )^{3/4}} \, dx}{4 c d^2}-\frac {(3 (b c-a d) (2 b c+a d)) \int \frac {\sqrt [4]{a+b x^4}}{c+d x^4} \, dx}{4 c d^2}\\ &=\frac {b (3 b c-a d) x \sqrt [4]{a+b x^4}}{4 c d^2}-\frac {(b c-a d) x \left (a+b x^4\right )^{5/4}}{4 c d \left (c+d x^4\right )}+\frac {\left (a b (3 b c-a d) \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 d^2 \left (a+b x^4\right )^{3/4}}-\frac {\left (3 (b c-a d) (2 b c+a d) \sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4}\right ) \text {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 d^2}\\ &=\frac {b (3 b c-a d) x \sqrt [4]{a+b x^4}}{4 c d^2}-\frac {(b c-a d) x \left (a+b x^4\right )^{5/4}}{4 c d \left (c+d x^4\right )}-\frac {\left (a b (3 b c-a d) \left (1+\frac {a}{b x^4}\right )^{3/4} x^3\right ) \text {Subst}\left (\int \frac {x}{\left (1+\frac {a x^4}{b}\right )^{3/4}} \, dx,x,\frac {1}{x}\right )}{4 c d^2 \left (a+b x^4\right )^{3/4}}-\frac {\left (3 (b c-a d) (2 b c+a d) \sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4}\right ) \text {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 d^2}-\frac {\left (3 (b c-a d) (2 b c+a d) \sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4}\right ) \text {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 d^2}\\ &=\frac {b (3 b c-a d) x \sqrt [4]{a+b x^4}}{4 c d^2}-\frac {(b c-a d) x \left (a+b x^4\right )^{5/4}}{4 c d \left (c+d x^4\right )}-\frac {3 (b c-a d) (2 b c+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 d^2}-\frac {3 (b c-a d) (2 b c+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 d^2}-\frac {\left (a b (3 b c-a d) \left (1+\frac {a}{b x^4}\right )^{3/4} x^3\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {a x^2}{b}\right )^{3/4}} \, dx,x,\frac {1}{x^2}\right )}{8 c d^2 \left (a+b x^4\right )^{3/4}}\\ &=\frac {b (3 b c-a d) x \sqrt [4]{a+b x^4}}{4 c d^2}-\frac {(b c-a d) x \left (a+b x^4\right )^{5/4}}{4 c d \left (c+d x^4\right )}-\frac {\sqrt {a} b^{3/2} (3 b c-a d) \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 d^2 \left (a+b x^4\right )^{3/4}}-\frac {3 (b c-a d) (2 b c+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 d^2}-\frac {3 (b c-a d) (2 b c+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 d^2}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in
optimal.
time = 10.35, size = 392, normalized size = 1.11 \begin {gather*} \frac {2 b \left (-3 b^2 c^2+3 a b c d+a^2 d^2\right ) x^5 \left (1+\frac {b x^4}{a}\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 )+\frac {5 c \left (-5 a c x \left (4 a^3 d^2+a^2 b d^2 x^4+b^3 c x^4 \left (3 c+2 d x^4\right )\right ) 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^5 \left (a+b x^4\right ) \left (-2 a b c d+a^2 d^2+b^2 c \left (3 c+2 d x^4\right )\right ) \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 )\right )}{\left (c+d x^4\right ) \left (-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 )+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 )\right )}}{20 c^2 d^2 \left (a+b x^4\right )^{3/4}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (b \,x^{4}+a \right )^{\frac {9}{4}}}{\left (d \,x^{4}+c \right )^{2}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \int \frac {\left (a + b x^{4}\right )^{\frac {9}{4}}}{\left (c + d x^{4}\right )^{2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (b\,x^4+a\right )}^{9/4}}{{\left (d\,x^4+c\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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