Optimal. Leaf size=426 \[ -\frac {b x \sqrt {a-b x^4} \left (21 a^2 d^2-122 a b c d+77 b^2 c^2\right )}{84 c d^3}+\frac {\sqrt [4]{a} b^{3/4} \sqrt {1-\frac {b x^4}{a}} \left (21 a^3 d^3+349 a^2 b c d^2-553 a b^2 c^2 d+231 b^3 c^3\right ) F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{84 c d^4 \sqrt {a-b x^4}}-\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} (3 a d+11 b c) (b c-a d)^3 \Pi \left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d^4 \sqrt {a-b x^4}}-\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} (3 a d+11 b c) (b c-a d)^3 \Pi \left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d^4 \sqrt {a-b x^4}}+\frac {b x \left (a-b x^4\right )^{3/2} (11 b c-7 a d)}{28 c d^2}-\frac {x \left (a-b x^4\right )^{5/2} (b c-a d)}{4 c d \left (c-d x^4\right )} \]
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Rubi [A] time = 0.54, antiderivative size = 426, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {413, 528, 523, 224, 221, 409, 1219, 1218} \[ -\frac {b x \sqrt {a-b x^4} \left (21 a^2 d^2-122 a b c d+77 b^2 c^2\right )}{84 c d^3}+\frac {\sqrt [4]{a} b^{3/4} \sqrt {1-\frac {b x^4}{a}} \left (349 a^2 b c d^2+21 a^3 d^3-553 a b^2 c^2 d+231 b^3 c^3\right ) F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{84 c d^4 \sqrt {a-b x^4}}-\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} (3 a d+11 b c) (b c-a d)^3 \Pi \left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d^4 \sqrt {a-b x^4}}-\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} (3 a d+11 b c) (b c-a d)^3 \Pi \left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d^4 \sqrt {a-b x^4}}+\frac {b x \left (a-b x^4\right )^{3/2} (11 b c-7 a d)}{28 c d^2}-\frac {x \left (a-b x^4\right )^{5/2} (b c-a d)}{4 c d \left (c-d x^4\right )} \]
Antiderivative was successfully verified.
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Rule 221
Rule 224
Rule 409
Rule 413
Rule 523
Rule 528
Rule 1218
Rule 1219
Rubi steps
\begin {align*} \int \frac {\left (a-b x^4\right )^{7/2}}{\left (c-d x^4\right )^2} \, dx &=-\frac {(b c-a d) x \left (a-b x^4\right )^{5/2}}{4 c d \left (c-d x^4\right )}-\frac {\int \frac {\left (a-b x^4\right )^{3/2} \left (-a (b c+3 a d)+b (11 b c-7 a d) x^4\right )}{c-d x^4} \, dx}{4 c d}\\ &=\frac {b (11 b c-7 a d) x \left (a-b x^4\right )^{3/2}}{28 c d^2}-\frac {(b c-a d) x \left (a-b x^4\right )^{5/2}}{4 c d \left (c-d x^4\right )}+\frac {\int \frac {\sqrt {a-b x^4} \left (-a \left (11 b^2 c^2-14 a b c d-21 a^2 d^2\right )+b \left (77 b^2 c^2-122 a b c d+21 a^2 d^2\right ) x^4\right )}{c-d x^4} \, dx}{28 c d^2}\\ &=-\frac {b \left (77 b^2 c^2-122 a b c d+21 a^2 d^2\right ) x \sqrt {a-b x^4}}{84 c d^3}+\frac {b (11 b c-7 a d) x \left (a-b x^4\right )^{3/2}}{28 c d^2}-\frac {(b c-a d) x \left (a-b x^4\right )^{5/2}}{4 c d \left (c-d x^4\right )}-\frac {\int \frac {-a \left (77 b^3 c^3-155 a b^2 c^2 d+63 a^2 b c d^2+63 a^3 d^3\right )+b \left (231 b^3 c^3-553 a b^2 c^2 d+349 a^2 b c d^2+21 a^3 d^3\right ) x^4}{\sqrt {a-b x^4} \left (c-d x^4\right )} \, dx}{84 c d^3}\\ &=-\frac {b \left (77 b^2 c^2-122 a b c d+21 a^2 d^2\right ) x \sqrt {a-b x^4}}{84 c d^3}+\frac {b (11 b c-7 a d) x \left (a-b x^4\right )^{3/2}}{28 c d^2}-\frac {(b c-a d) x \left (a-b x^4\right )^{5/2}}{4 c d \left (c-d x^4\right )}-\frac {\left ((b c-a d)^3 (11 b c+3 a d)\right ) \int \frac {1}{\sqrt {a-b x^4} \left (c-d x^4\right )} \, dx}{4 c d^4}+\frac {\left (b \left (231 b^3 c^3-553 a b^2 c^2 d+349 a^2 b c d^2+21 a^3 d^3\right )\right ) \int \frac {1}{\sqrt {a-b x^4}} \, dx}{84 c d^4}\\ &=-\frac {b \left (77 b^2 c^2-122 a b c d+21 a^2 d^2\right ) x \sqrt {a-b x^4}}{84 c d^3}+\frac {b (11 b c-7 a d) x \left (a-b x^4\right )^{3/2}}{28 c d^2}-\frac {(b c-a d) x \left (a-b x^4\right )^{5/2}}{4 c d \left (c-d x^4\right )}-\frac {\left ((b c-a d)^3 (11 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^4}-\frac {\left ((b c-a d)^3 (11 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^4}+\frac {\left (b \left (231 b^3 c^3-553 a b^2 c^2 d+349 a^2 b c d^2+21 a^3 d^3\right ) \sqrt {1-\frac {b x^4}{a}}\right ) \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}} \, dx}{84 c d^4 \sqrt {a-b x^4}}\\ &=-\frac {b \left (77 b^2 c^2-122 a b c d+21 a^2 d^2\right ) x \sqrt {a-b x^4}}{84 c d^3}+\frac {b (11 b c-7 a d) x \left (a-b x^4\right )^{3/2}}{28 c d^2}-\frac {(b c-a d) x \left (a-b x^4\right )^{5/2}}{4 c d \left (c-d x^4\right )}+\frac {\sqrt [4]{a} b^{3/4} \left (231 b^3 c^3-553 a b^2 c^2 d+349 a^2 b c d^2+21 a^3 d^3\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 )}{84 c d^4 \sqrt {a-b x^4}}-\frac {\left ((b c-a d)^3 (11 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^4 \sqrt {a-b x^4}}-\frac {\left ((b c-a d)^3 (11 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^4 \sqrt {a-b x^4}}\\ &=-\frac {b \left (77 b^2 c^2-122 a b c d+21 a^2 d^2\right ) x \sqrt {a-b x^4}}{84 c d^3}+\frac {b (11 b c-7 a d) x \left (a-b x^4\right )^{3/2}}{28 c d^2}-\frac {(b c-a d) x \left (a-b x^4\right )^{5/2}}{4 c d \left (c-d x^4\right )}+\frac {\sqrt [4]{a} b^{3/4} \left (231 b^3 c^3-553 a b^2 c^2 d+349 a^2 b c d^2+21 a^3 d^3\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 )}{84 c d^4 \sqrt {a-b x^4}}-\frac {\sqrt [4]{a} (b c-a d)^3 (11 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^4 \sqrt {a-b x^4}}-\frac {\sqrt [4]{a} (b c-a d)^3 (11 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^4 \sqrt {a-b x^4}}\\ \end {align*}
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Mathematica [C] time = 0.92, size = 477, normalized size = 1.12 \[ -\frac {b x^5 \sqrt {1-\frac {b x^4}{a}} \left (21 a^3 d^3+349 a^2 b c d^2-553 a b^2 c^2 d+231 b^3 c^3\right ) F_1\left (\frac {5}{4};\frac {1}{2},1;\frac {9}{4};\frac {b x^4}{a},\frac {d x^4}{c}\right )+\frac {5 c \left (2 x^5 \left (b x^4-a\right ) \left (21 a^3 d^3-63 a^2 b c d^2+a b^2 c d \left (155 c-92 d x^4\right )+b^3 c \left (-77 c^2+44 c d x^4+12 d^2 x^8\right )\right ) \left (2 a d F_1\left (\frac {5}{4};\frac {1}{2},2;\frac {9}{4};\frac {b x^4}{a},\frac {d x^4}{c}\right )+b c F_1\left (\frac {5}{4};\frac {3}{2},1;\frac {9}{4};\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )+5 a c x \left (-84 a^4 d^3+21 a^3 b d^3 x^4+29 a^2 b^2 c d^2 x^4+a b^3 c d x^4 \left (111 c-104 d x^4\right )+b^4 c x^4 \left (-77 c^2+44 c d x^4+12 d^2 x^8\right )\right ) F_1\left (\frac {1}{4};\frac {1}{2},1;\frac {5}{4};\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )}{\left (c-d x^4\right ) \left (2 x^4 \left (2 a d F_1\left (\frac {5}{4};\frac {1}{2},2;\frac {9}{4};\frac {b x^4}{a},\frac {d x^4}{c}\right )+b c F_1\left (\frac {5}{4};\frac {3}{2},1;\frac {9}{4};\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )+5 a c F_1\left (\frac {1}{4};\frac {1}{2},1;\frac {5}{4};\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )}}{420 c^2 d^3 \sqrt {a-b x^4}} \]
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 \[ \int \frac {{\left (-b x^{4} + a\right )}^{\frac {7}{2}}}{{\left (d x^{4} - c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.29, size = 540, normalized size = 1.27 \[ -\frac {\sqrt {-b \,x^{4}+a}\, b^{3} x^{5}}{7 d^{2}}+\frac {\left (\frac {\left (\frac {5 a \,b^{3}}{7 d^{2}}-\frac {2 \left (2 a d -b c \right ) b^{3}}{d^{3}}\right ) a}{3 b}+\frac {\left (6 a^{2} d^{2}-8 a b c d +3 b^{2} c^{2}\right ) b^{2}}{d^{4}}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b}{4 c \,d^{4}}\right ) \sqrt {-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \EllipticF \left (\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, x , i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {\left (\frac {5 a \,b^{3}}{7 d^{2}}-\frac {2 \left (2 a d -b c \right ) b^{3}}{d^{3}}\right ) \sqrt {-b \,x^{4}+a}\, x}{3 b}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \sqrt {-b \,x^{4}+a}\, x}{4 \left (d \,x^{4}-c \right ) c \,d^{3}}-\frac {\left (3 a^{4} d^{4}+2 a^{3} b \,d^{3} c -24 a^{2} b^{2} c^{2} d^{2}+30 c^{3} a \,b^{3} d -11 b^{4} c^{4}\right ) \left (-\frac {2 \sqrt {-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \RootOf \left (d \,\textit {\_Z}^{4}-c \right )^{3} d \EllipticPi \left (\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, x , \frac {\RootOf \left (d \,\textit {\_Z}^{4}-c \right )^{2} \sqrt {a}\, d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}\, c}-\frac {\arctanh \left (\frac {-2 \RootOf \left (d \,\textit {\_Z}^{4}-c \right )^{2} b \,x^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {-b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}\right )}{32 c \,d^{5} \RootOf \left (d \,\textit {\_Z}^{4}-c \right )^{3}} \]
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 \[ \int \frac {{\left (-b x^{4} + a\right )}^{\frac {7}{2}}}{{\left (d x^{4} - c\right )}^{2}}\,{d x} \]
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 \[ \int \frac {{\left (a-b\,x^4\right )}^{7/2}}{{\left (c-d\,x^4\right )}^2} \,d x \]
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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