Optimal. Leaf size=362 \[ \frac {b^{3/4} \sqrt {1-\frac {b x^4}{a}} (a d+2 b c) F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{4 a^{3/4} c \sqrt {a-b x^4} (b c-a d)^2}-\frac {3 \sqrt [4]{a} d \sqrt {1-\frac {b x^4}{a}} (3 b c-a d) \Pi \left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 \sqrt {a-b x^4} (b c-a d)^2}-\frac {3 \sqrt [4]{a} d \sqrt {1-\frac {b x^4}{a}} (3 b c-a d) \Pi \left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 \sqrt {a-b x^4} (b c-a d)^2}+\frac {b x (a d+2 b c)}{4 a c \sqrt {a-b x^4} (b c-a d)^2}-\frac {d x}{4 c \sqrt {a-b x^4} \left (c-d x^4\right ) (b c-a d)} \]
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Rubi [A] time = 0.40, antiderivative size = 362, 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 = {414, 527, 523, 224, 221, 409, 1219, 1218} \[ \frac {b^{3/4} \sqrt {1-\frac {b x^4}{a}} (a d+2 b c) F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{4 a^{3/4} c \sqrt {a-b x^4} (b c-a d)^2}-\frac {3 \sqrt [4]{a} d \sqrt {1-\frac {b x^4}{a}} (3 b c-a d) \Pi \left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 \sqrt {a-b x^4} (b c-a d)^2}-\frac {3 \sqrt [4]{a} d \sqrt {1-\frac {b x^4}{a}} (3 b c-a d) \Pi \left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 \sqrt {a-b x^4} (b c-a d)^2}+\frac {b x (a d+2 b c)}{4 a c \sqrt {a-b x^4} (b c-a d)^2}-\frac {d x}{4 c \sqrt {a-b x^4} \left (c-d x^4\right ) (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 221
Rule 224
Rule 409
Rule 414
Rule 523
Rule 527
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 )^2} \, dx &=-\frac {d x}{4 c (b c-a d) \sqrt {a-b x^4} \left (c-d x^4\right )}-\frac {\int \frac {-4 b c+3 a d-5 b d x^4}{\left (a-b x^4\right )^{3/2} \left (c-d x^4\right )} \, dx}{4 c (b c-a d)}\\ &=\frac {b (2 b c+a d) x}{4 a c (b c-a d)^2 \sqrt {a-b x^4}}-\frac {d x}{4 c (b c-a d) \sqrt {a-b x^4} \left (c-d x^4\right )}-\frac {\int \frac {-2 \left (2 b^2 c^2-8 a b c d+3 a^2 d^2\right )+2 b d (2 b c+a d) x^4}{\sqrt {a-b x^4} \left (c-d x^4\right )} \, dx}{8 a c (b c-a d)^2}\\ &=\frac {b (2 b c+a d) x}{4 a c (b c-a d)^2 \sqrt {a-b x^4}}-\frac {d x}{4 c (b c-a d) \sqrt {a-b x^4} \left (c-d x^4\right )}-\frac {(3 d (3 b c-a d)) \int \frac {1}{\sqrt {a-b x^4} \left (c-d x^4\right )} \, dx}{4 c (b c-a d)^2}+\frac {(b (2 b c+a d)) \int \frac {1}{\sqrt {a-b x^4}} \, dx}{4 a c (b c-a d)^2}\\ &=\frac {b (2 b c+a d) x}{4 a c (b c-a d)^2 \sqrt {a-b x^4}}-\frac {d x}{4 c (b c-a d) \sqrt {a-b x^4} \left (c-d x^4\right )}-\frac {(3 d (3 b c-a d)) \int \frac {1}{\left (1-\frac {\sqrt {d} x^2}{\sqrt {c}}\right ) \sqrt {a-b x^4}} \, dx}{8 c^2 (b c-a d)^2}-\frac {(3 d (3 b c-a d)) \int \frac {1}{\left (1+\frac {\sqrt {d} x^2}{\sqrt {c}}\right ) \sqrt {a-b x^4}} \, dx}{8 c^2 (b c-a d)^2}+\frac {\left (b (2 b c+a d) \sqrt {1-\frac {b x^4}{a}}\right ) \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}} \, dx}{4 a c (b c-a d)^2 \sqrt {a-b x^4}}\\ &=\frac {b (2 b c+a d) x}{4 a c (b c-a d)^2 \sqrt {a-b x^4}}-\frac {d x}{4 c (b c-a d) \sqrt {a-b x^4} \left (c-d x^4\right )}+\frac {b^{3/4} (2 b c+a d) \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 )}{4 a^{3/4} c (b c-a d)^2 \sqrt {a-b x^4}}-\frac {\left (3 d (3 b c-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 (b c-a d)^2 \sqrt {a-b x^4}}-\frac {\left (3 d (3 b c-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 (b c-a d)^2 \sqrt {a-b x^4}}\\ &=\frac {b (2 b c+a d) x}{4 a c (b c-a d)^2 \sqrt {a-b x^4}}-\frac {d x}{4 c (b c-a d) \sqrt {a-b x^4} \left (c-d x^4\right )}+\frac {b^{3/4} (2 b c+a d) \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 )}{4 a^{3/4} c (b c-a d)^2 \sqrt {a-b x^4}}-\frac {3 \sqrt [4]{a} d (3 b c-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 (b c-a d)^2 \sqrt {a-b x^4}}-\frac {3 \sqrt [4]{a} d (3 b c-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 (b c-a d)^2 \sqrt {a-b x^4}}\\ \end {align*}
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Mathematica [C] time = 0.52, size = 374, normalized size = 1.03 \[ \frac {x \left (\frac {c \left (25 a c \left (4 a^2 d^2-a b d \left (8 c+d x^4\right )+2 b^2 c \left (2 c-d x^4\right )\right ) F_1\left (\frac {1}{4};\frac {1}{2},1;\frac {5}{4};\frac {b x^4}{a},\frac {d x^4}{c}\right )-10 x^4 \left (-a^2 d^2+a b d^2 x^4-2 b^2 c \left (c-d x^4\right )\right ) \left (2 a d F_1\left (\frac {5}{4};\frac {1}{2},2;\frac {9}{4};\frac {b x^4}{a},\frac {d x^4}{c}\right )+b c F_1\left (\frac {5}{4};\frac {3}{2},1;\frac {9}{4};\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )\right )}{\left (c-d x^4\right ) \left (2 x^4 \left (2 a d F_1\left (\frac {5}{4};\frac {1}{2},2;\frac {9}{4};\frac {b x^4}{a},\frac {d x^4}{c}\right )+b c F_1\left (\frac {5}{4};\frac {3}{2},1;\frac {9}{4};\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )+5 a c F_1\left (\frac {1}{4};\frac {1}{2},1;\frac {5}{4};\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )}-b d x^4 \sqrt {1-\frac {b x^4}{a}} (a d+2 b c) F_1\left (\frac {5}{4};\frac {1}{2},1;\frac {9}{4};\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )}{20 a c^2 \sqrt {a-b x^4} (b c-a d)^2} \]
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 \[ \int \frac {1}{{\left (-b x^{4} + a\right )}^{\frac {3}{2}} {\left (d x^{4} - c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.35, size = 375, normalized size = 1.04 \[ \frac {b^{2} x}{2 \left (a d -b c \right )^{2} \sqrt {-\left (x^{4}-\frac {a}{b}\right ) b}\, a}-\frac {\sqrt {-b \,x^{4}+a}\, d^{2} x}{4 \left (a d -b c \right )^{2} \left (d \,x^{4}-c \right ) c}+\frac {\left (\frac {b^{2}}{2 \left (a d -b c \right )^{2} a}+\frac {b d}{4 \left (a d -b c \right )^{2} c}\right ) \sqrt {-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \EllipticF \left (\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, x , i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {3 \left (a d -3 b c \right ) \left (-\frac {2 \sqrt {-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \RootOf \left (d \,\textit {\_Z}^{4}-c \right )^{3} d \EllipticPi \left (\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, x , \frac {\RootOf \left (d \,\textit {\_Z}^{4}-c \right )^{2} \sqrt {a}\, d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}\, c}-\frac {\arctanh \left (\frac {-2 \RootOf \left (d \,\textit {\_Z}^{4}-c \right )^{2} b \,x^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {-b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}\right )}{32 c \left (a d -b c \right )^{2} \RootOf \left (d \,\textit {\_Z}^{4}-c \right )^{3}} \]
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 \[ \int \frac {1}{{\left (-b x^{4} + a\right )}^{\frac {3}{2}} {\left (d x^{4} - c\right )}^{2}}\,{d x} \]
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 \[ \int \frac {1}{{\left (a-b\,x^4\right )}^{3/2}\,{\left (c-d\,x^4\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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