Optimal. Leaf size=281 \[ \frac {b^{3/4} \sqrt {1-\frac {b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 a^{3/4} \sqrt {a-b x^4} (b c-a d)}+\frac {b x}{2 a \sqrt {a-b x^4} (b c-a d)}-\frac {\sqrt [4]{a} d \sqrt {1-\frac {b x^4}{a}} \Pi \left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c \sqrt {a-b x^4} (b c-a d)}-\frac {\sqrt [4]{a} d \sqrt {1-\frac {b x^4}{a}} \Pi \left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c \sqrt {a-b x^4} (b c-a d)} \]
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Rubi [A] time = 0.22, antiderivative size = 281, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {414, 523, 224, 221, 409, 1219, 1218} \[ \frac {b^{3/4} \sqrt {1-\frac {b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 a^{3/4} \sqrt {a-b x^4} (b c-a d)}+\frac {b x}{2 a \sqrt {a-b x^4} (b c-a d)}-\frac {\sqrt [4]{a} d \sqrt {1-\frac {b x^4}{a}} \Pi \left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c \sqrt {a-b x^4} (b c-a d)}-\frac {\sqrt [4]{a} d \sqrt {1-\frac {b x^4}{a}} \Pi \left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c \sqrt {a-b x^4} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 221
Rule 224
Rule 409
Rule 414
Rule 523
Rule 1218
Rule 1219
Rubi steps
\begin {align*} \int \frac {1}{\left (a-b x^4\right )^{3/2} \left (c-d x^4\right )} \, dx &=\frac {b x}{2 a (b c-a d) \sqrt {a-b x^4}}+\frac {\int \frac {b c-2 a d-b d x^4}{\sqrt {a-b x^4} \left (c-d x^4\right )} \, dx}{2 a (b c-a d)}\\ &=\frac {b x}{2 a (b c-a d) \sqrt {a-b x^4}}+\frac {b \int \frac {1}{\sqrt {a-b x^4}} \, dx}{2 a (b c-a d)}-\frac {d \int \frac {1}{\sqrt {a-b x^4} \left (c-d x^4\right )} \, dx}{b c-a d}\\ &=\frac {b x}{2 a (b c-a d) \sqrt {a-b x^4}}-\frac {d \int \frac {1}{\left (1-\frac {\sqrt {d} x^2}{\sqrt {c}}\right ) \sqrt {a-b x^4}} \, dx}{2 c (b c-a d)}-\frac {d \int \frac {1}{\left (1+\frac {\sqrt {d} x^2}{\sqrt {c}}\right ) \sqrt {a-b x^4}} \, dx}{2 c (b c-a d)}+\frac {\left (b \sqrt {1-\frac {b x^4}{a}}\right ) \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}} \, dx}{2 a (b c-a d) \sqrt {a-b x^4}}\\ &=\frac {b x}{2 a (b c-a d) \sqrt {a-b x^4}}+\frac {b^{3/4} \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 )}{2 a^{3/4} (b c-a d) \sqrt {a-b x^4}}-\frac {\left (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}{2 c (b c-a d) \sqrt {a-b x^4}}-\frac {\left (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}{2 c (b c-a d) \sqrt {a-b x^4}}\\ &=\frac {b x}{2 a (b c-a d) \sqrt {a-b x^4}}+\frac {b^{3/4} \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 )}{2 a^{3/4} (b c-a d) \sqrt {a-b x^4}}-\frac {\sqrt [4]{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 )}{2 \sqrt [4]{b} c (b c-a d) \sqrt {a-b x^4}}-\frac {\sqrt [4]{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 )}{2 \sqrt [4]{b} c (b c-a d) \sqrt {a-b x^4}}\\ \end {align*}
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Mathematica [C] time = 0.32, size = 381, normalized size = 1.36 \[ \frac {5 a c x F_1\left (\frac {1}{4};\frac {1}{2},1;\frac {5}{4};\frac {b x^4}{a},\frac {d x^4}{c}\right ) \left (b d x^4 \sqrt {1-\frac {b x^4}{a}} \left (d x^4-c\right ) F_1\left (\frac {5}{4};\frac {1}{2},1;\frac {9}{4};\frac {b x^4}{a},\frac {d x^4}{c}\right )-5 c \left (2 a d-2 b c+b d x^4\right )\right )+2 b x^5 \left (c-d x^4\right ) \left (5 c-d x^4 \sqrt {1-\frac {b x^4}{a}} F_1\left (\frac {5}{4};\frac {1}{2},1;\frac {9}{4};\frac {b x^4}{a},\frac {d x^4}{c}\right )\right ) \left (2 a d F_1\left (\frac {5}{4};\frac {1}{2},2;\frac {9}{4};\frac {b x^4}{a},\frac {d x^4}{c}\right )+b c F_1\left (\frac {5}{4};\frac {3}{2},1;\frac {9}{4};\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )}{10 a c \sqrt {a-b x^4} \left (d x^4-c\right ) (a d-b c) \left (2 x^4 \left (2 a d F_1\left (\frac {5}{4};\frac {1}{2},2;\frac {9}{4};\frac {b x^4}{a},\frac {d x^4}{c}\right )+b c F_1\left (\frac {5}{4};\frac {3}{2},1;\frac {9}{4};\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )+5 a c F_1\left (\frac {1}{4};\frac {1}{2},1;\frac {5}{4};\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )} \]
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 \[ \int -\frac {1}{{\left (-b x^{4} + a\right )}^{\frac {3}{2}} {\left (d x^{4} - c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.36, size = 301, normalized size = 1.07 \[ -\frac {b x}{2 \left (a d -b c \right ) \sqrt {-\left (x^{4}-\frac {a}{b}\right ) b}\, a}-\frac {\sqrt {-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, b \EllipticF \left (\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, x , i\right )}{2 \left (a d -b c \right ) \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}\, a}-\frac {-\frac {2 \sqrt {-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \RootOf \left (d \,\textit {\_Z}^{4}-c \right )^{3} d \EllipticPi \left (\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, x , \frac {\RootOf \left (d \,\textit {\_Z}^{4}-c \right )^{2} \sqrt {a}\, d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}\, c}-\frac {\arctanh \left (\frac {-2 \RootOf \left (d \,\textit {\_Z}^{4}-c \right )^{2} b \,x^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {-b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}}{8 \left (a d -b c \right ) \RootOf \left (d \,\textit {\_Z}^{4}-c \right )^{3}} \]
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 \[ -\int \frac {1}{{\left (-b x^{4} + a\right )}^{\frac {3}{2}} {\left (d x^{4} - c\right )}}\,{d x} \]
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 \[ \int \frac {1}{{\left (a-b\,x^4\right )}^{3/2}\,\left (c-d\,x^4\right )} \,d x \]
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 \[ - \int \frac {1}{- a c \sqrt {a - b x^{4}} + a d x^{4} \sqrt {a - b x^{4}} + b c x^{4} \sqrt {a - b x^{4}} - b d x^{8} \sqrt {a - b x^{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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