Optimal. Leaf size=365 \[ \frac {b (7 b c-3 a d) x \sqrt {a-b x^4}}{12 c d^2}-\frac {(b c-a d) x \left (a-b x^4\right )^{3/2}}{4 c d \left (c-d x^4\right )}-\frac {\sqrt [4]{a} b^{3/4} \left (21 b^2 c^2-26 a b c d-3 a^2 d^2\right ) \sqrt {1-\frac {b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{12 c d^3 \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} (b c-a d)^2 (7 b c+3 a d) \sqrt {1-\frac {b x^4}{a}} \Pi \left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d^3 \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} (b c-a d)^2 (7 b c+3 a d) \sqrt {1-\frac {b x^4}{a}} \Pi \left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d^3 \sqrt {a-b x^4}} \]
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Rubi [A]
time = 0.25, antiderivative size = 365, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {424, 542, 537,
230, 227, 418, 1233, 1232} \begin {gather*} -\frac {\sqrt [4]{a} b^{3/4} \sqrt {1-\frac {b x^4}{a}} \left (-3 a^2 d^2-26 a b c d+21 b^2 c^2\right ) F\left (\left .\text {ArcSin}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{12 c d^3 \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} (3 a d+7 b c) (b c-a d)^2 \Pi \left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}};\left .\text {ArcSin}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d^3 \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} (3 a d+7 b c) (b c-a d)^2 \Pi \left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}};\left .\text {ArcSin}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d^3 \sqrt {a-b x^4}}+\frac {b x \sqrt {a-b x^4} (7 b c-3 a d)}{12 c d^2}-\frac {x \left (a-b x^4\right )^{3/2} (b c-a d)}{4 c d \left (c-d x^4\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 227
Rule 230
Rule 418
Rule 424
Rule 537
Rule 542
Rule 1232
Rule 1233
Rubi steps
\begin {align*} \int \frac {\left (a-b x^4\right )^{5/2}}{\left (c-d x^4\right )^2} \, dx &=-\frac {(b c-a d) x \left (a-b x^4\right )^{3/2}}{4 c d \left (c-d x^4\right )}-\frac {\int \frac {\sqrt {a-b x^4} \left (-a (b c+3 a d)+b (7 b c-3 a d) x^4\right )}{c-d x^4} \, dx}{4 c d}\\ &=\frac {b (7 b c-3 a d) x \sqrt {a-b x^4}}{12 c d^2}-\frac {(b c-a d) x \left (a-b x^4\right )^{3/2}}{4 c d \left (c-d x^4\right )}+\frac {\int \frac {-a \left (7 b^2 c^2-6 a b c d-9 a^2 d^2\right )+b \left (21 b^2 c^2-26 a b c d-3 a^2 d^2\right ) x^4}{\sqrt {a-b x^4} \left (c-d x^4\right )} \, dx}{12 c d^2}\\ &=\frac {b (7 b c-3 a d) x \sqrt {a-b x^4}}{12 c d^2}-\frac {(b c-a d) x \left (a-b x^4\right )^{3/2}}{4 c d \left (c-d x^4\right )}+\frac {\left ((b c-a d)^2 (7 b c+3 a d)\right ) \int \frac {1}{\sqrt {a-b x^4} \left (c-d x^4\right )} \, dx}{4 c d^3}-\frac {\left (b \left (21 b^2 c^2-26 a b c d-3 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a-b x^4}} \, dx}{12 c d^3}\\ &=\frac {b (7 b c-3 a d) x \sqrt {a-b x^4}}{12 c d^2}-\frac {(b c-a d) x \left (a-b x^4\right )^{3/2}}{4 c d \left (c-d x^4\right )}+\frac {\left ((b c-a d)^2 (7 b c+3 a d)\right ) \int \frac {1}{\left (1-\frac {\sqrt {d} x^2}{\sqrt {c}}\right ) \sqrt {a-b x^4}} \, dx}{8 c^2 d^3}+\frac {\left ((b c-a d)^2 (7 b c+3 a d)\right ) \int \frac {1}{\left (1+\frac {\sqrt {d} x^2}{\sqrt {c}}\right ) \sqrt {a-b x^4}} \, dx}{8 c^2 d^3}-\frac {\left (b \left (21 b^2 c^2-26 a b c d-3 a^2 d^2\right ) \sqrt {1-\frac {b x^4}{a}}\right ) \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}} \, dx}{12 c d^3 \sqrt {a-b x^4}}\\ &=\frac {b (7 b c-3 a d) x \sqrt {a-b x^4}}{12 c d^2}-\frac {(b c-a d) x \left (a-b x^4\right )^{3/2}}{4 c d \left (c-d x^4\right )}-\frac {\sqrt [4]{a} b^{3/4} \left (21 b^2 c^2-26 a b c d-3 a^2 d^2\right ) \sqrt {1-\frac {b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{12 c d^3 \sqrt {a-b x^4}}+\frac {\left ((b c-a d)^2 (7 b c+3 a d) \sqrt {1-\frac {b x^4}{a}}\right ) \int \frac {1}{\left (1-\frac {\sqrt {d} x^2}{\sqrt {c}}\right ) \sqrt {1-\frac {b x^4}{a}}} \, dx}{8 c^2 d^3 \sqrt {a-b x^4}}+\frac {\left ((b c-a d)^2 (7 b c+3 a d) \sqrt {1-\frac {b x^4}{a}}\right ) \int \frac {1}{\left (1+\frac {\sqrt {d} x^2}{\sqrt {c}}\right ) \sqrt {1-\frac {b x^4}{a}}} \, dx}{8 c^2 d^3 \sqrt {a-b x^4}}\\ &=\frac {b (7 b c-3 a d) x \sqrt {a-b x^4}}{12 c d^2}-\frac {(b c-a d) x \left (a-b x^4\right )^{3/2}}{4 c d \left (c-d x^4\right )}-\frac {\sqrt [4]{a} b^{3/4} \left (21 b^2 c^2-26 a b c d-3 a^2 d^2\right ) \sqrt {1-\frac {b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{12 c d^3 \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} (b c-a d)^2 (7 b c+3 a d) \sqrt {1-\frac {b x^4}{a}} \Pi \left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d^3 \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} (b c-a d)^2 (7 b c+3 a d) \sqrt {1-\frac {b x^4}{a}} \Pi \left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d^3 \sqrt {a-b x^4}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in
optimal.
time = 10.42, size = 396, normalized size = 1.08 \begin {gather*} -\frac {b \left (-21 b^2 c^2+26 a b c d+3 a^2 d^2\right ) x^5 \sqrt {1-\frac {b x^4}{a}} F_1\left (\frac {5}{4};\frac {1}{2},1;\frac {9}{4};\frac {b x^4}{a},\frac {d x^4}{c}\right )+\frac {5 c \left (5 a c x \left (12 a^3 d^2+2 a b^2 c d x^4-3 a^2 b d^2 x^4+b^3 c x^4 \left (-7 c+4 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 )+2 x^5 \left (a-b x^4\right ) \left (-6 a b c d+3 a^2 d^2+b^2 c \left (7 c-4 d x^4\right )\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 )\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 )}}{60 c^2 d^2 \sqrt {a-b x^4}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.32, size = 411, normalized size = 1.13 Too large to display
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 {5}{2}}}{\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 (a-b\,x^4\right )}^{5/2}}{{\left (c-d\,x^4\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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