Optimal. Leaf size=321 \[ \frac {\sqrt [4]{a} b^{3/4} \sqrt {1-\frac {b x^4}{a}} \left (47 a^2 d^2-56 a b c d+21 b^2 c^2\right ) F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{21 d^3 \sqrt {a-b x^4}}-\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} (b c-a d)^3 \Pi \left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d^3 \sqrt {a-b x^4}}-\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} (b c-a d)^3 \Pi \left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d^3 \sqrt {a-b x^4}}-\frac {b x \sqrt {a-b x^4} (7 b c-13 a d)}{21 d^2}+\frac {b x \left (a-b x^4\right )^{3/2}}{7 d} \]
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Rubi [A] time = 0.38, antiderivative size = 321, 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 = {416, 528, 523, 224, 221, 409, 1219, 1218} \[ \frac {\sqrt [4]{a} b^{3/4} \sqrt {1-\frac {b x^4}{a}} \left (47 a^2 d^2-56 a b c d+21 b^2 c^2\right ) F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{21 d^3 \sqrt {a-b x^4}}-\frac {b x \sqrt {a-b x^4} (7 b c-13 a d)}{21 d^2}-\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} (b c-a d)^3 \Pi \left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d^3 \sqrt {a-b x^4}}-\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} (b c-a d)^3 \Pi \left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d^3 \sqrt {a-b x^4}}+\frac {b x \left (a-b x^4\right )^{3/2}}{7 d} \]
Antiderivative was successfully verified.
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Rule 221
Rule 224
Rule 409
Rule 416
Rule 523
Rule 528
Rule 1218
Rule 1219
Rubi steps
\begin {align*} \int \frac {\left (a-b x^4\right )^{5/2}}{c-d x^4} \, dx &=\frac {b x \left (a-b x^4\right )^{3/2}}{7 d}-\frac {\int \frac {\sqrt {a-b x^4} \left (a (b c-7 a d)-b (7 b c-13 a d) x^4\right )}{c-d x^4} \, dx}{7 d}\\ &=-\frac {b (7 b c-13 a d) x \sqrt {a-b x^4}}{21 d^2}+\frac {b x \left (a-b x^4\right )^{3/2}}{7 d}+\frac {\int \frac {a \left (7 b^2 c^2-16 a b c d+21 a^2 d^2\right )-b \left (21 b^2 c^2-56 a b c d+47 a^2 d^2\right ) x^4}{\sqrt {a-b x^4} \left (c-d x^4\right )} \, dx}{21 d^2}\\ &=-\frac {b (7 b c-13 a d) x \sqrt {a-b x^4}}{21 d^2}+\frac {b x \left (a-b x^4\right )^{3/2}}{7 d}-\frac {(b c-a d)^3 \int \frac {1}{\sqrt {a-b x^4} \left (c-d x^4\right )} \, dx}{d^3}+\frac {\left (b \left (21 b^2 c^2-56 a b c d+47 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a-b x^4}} \, dx}{21 d^3}\\ &=-\frac {b (7 b c-13 a d) x \sqrt {a-b x^4}}{21 d^2}+\frac {b x \left (a-b x^4\right )^{3/2}}{7 d}-\frac {(b c-a d)^3 \int \frac {1}{\left (1-\frac {\sqrt {d} x^2}{\sqrt {c}}\right ) \sqrt {a-b x^4}} \, dx}{2 c d^3}-\frac {(b c-a d)^3 \int \frac {1}{\left (1+\frac {\sqrt {d} x^2}{\sqrt {c}}\right ) \sqrt {a-b x^4}} \, dx}{2 c d^3}+\frac {\left (b \left (21 b^2 c^2-56 a b c d+47 a^2 d^2\right ) \sqrt {1-\frac {b x^4}{a}}\right ) \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}} \, dx}{21 d^3 \sqrt {a-b x^4}}\\ &=-\frac {b (7 b c-13 a d) x \sqrt {a-b x^4}}{21 d^2}+\frac {b x \left (a-b x^4\right )^{3/2}}{7 d}+\frac {\sqrt [4]{a} b^{3/4} \left (21 b^2 c^2-56 a b c d+47 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 )}{21 d^3 \sqrt {a-b x^4}}-\frac {\left ((b c-a d)^3 \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 d^3 \sqrt {a-b x^4}}-\frac {\left ((b c-a d)^3 \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 d^3 \sqrt {a-b x^4}}\\ &=-\frac {b (7 b c-13 a d) x \sqrt {a-b x^4}}{21 d^2}+\frac {b x \left (a-b x^4\right )^{3/2}}{7 d}+\frac {\sqrt [4]{a} b^{3/4} \left (21 b^2 c^2-56 a b c d+47 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 )}{21 d^3 \sqrt {a-b x^4}}-\frac {\sqrt [4]{a} (b c-a d)^3 \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 d^3 \sqrt {a-b x^4}}-\frac {\sqrt [4]{a} (b c-a d)^3 \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 d^3 \sqrt {a-b x^4}}\\ \end {align*}
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Mathematica [C] time = 0.82, size = 290, normalized size = 0.90 \[ \frac {x \left (-\frac {b x^4 \sqrt {1-\frac {b x^4}{a}} \left (47 a^2 d^2-56 a b c d+21 b^2 c^2\right ) F_1\left (\frac {5}{4};\frac {1}{2},1;\frac {9}{4};\frac {b x^4}{a},\frac {d x^4}{c}\right )}{c}+\frac {25 a^2 c \left (21 a^2 d^2-16 a b c d+7 b^2 c^2\right ) F_1\left (\frac {1}{4};\frac {1}{2},1;\frac {5}{4};\frac {b x^4}{a},\frac {d x^4}{c}\right )}{\left (c-d x^4\right ) \left (2 x^4 \left (2 a d F_1\left (\frac {5}{4};\frac {1}{2},2;\frac {9}{4};\frac {b x^4}{a},\frac {d x^4}{c}\right )+b c F_1\left (\frac {5}{4};\frac {3}{2},1;\frac {9}{4};\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )+5 a c F_1\left (\frac {1}{4};\frac {1}{2},1;\frac {5}{4};\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )}+5 b \left (b x^4-a\right ) \left (-16 a d+7 b c+3 b d x^4\right )\right )}{105 d^2 \sqrt {a-b x^4}} \]
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 \[ \int -\frac {{\left (-b x^{4} + a\right )}^{\frac {5}{2}}}{d x^{4} - c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.31, size = 408, normalized size = 1.27 \[ -\frac {\sqrt {-b \,x^{4}+a}\, b^{2} x^{5}}{7 d}-\frac {\left (\frac {\left (-\frac {5 a \,b^{2}}{7 d}+\frac {\left (3 a d -b c \right ) b^{2}}{d^{2}}\right ) a}{3 b}-\frac {\left (3 a^{2} d^{2}-3 a b c d +b^{2} c^{2}\right ) b}{d^{3}}\right ) \sqrt {-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \EllipticF \left (\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, x , i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}+\frac {\left (-\frac {5 a \,b^{2}}{7 d}+\frac {\left (3 a d -b c \right ) b^{2}}{d^{2}}\right ) \sqrt {-b \,x^{4}+a}\, x}{3 b}+\frac {\left (-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \left (-\frac {2 \sqrt {-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \RootOf \left (d \,\textit {\_Z}^{4}-c \right )^{3} d \EllipticPi \left (\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, x , \frac {\RootOf \left (d \,\textit {\_Z}^{4}-c \right )^{2} \sqrt {a}\, d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}\, c}-\frac {\arctanh \left (\frac {-2 \RootOf \left (d \,\textit {\_Z}^{4}-c \right )^{2} b \,x^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {-b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}\right )}{8 d^{4} \RootOf \left (d \,\textit {\_Z}^{4}-c \right )^{3}} \]
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 \[ -\int \frac {{\left (-b x^{4} + a\right )}^{\frac {5}{2}}}{d x^{4} - c}\,{d x} \]
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 \[ \int \frac {{\left (a-b\,x^4\right )}^{5/2}}{c-d\,x^4} \,d x \]
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 \[ - \int \frac {a^{2} \sqrt {a - b x^{4}}}{- c + d x^{4}}\, dx - \int \frac {b^{2} x^{8} \sqrt {a - b x^{4}}}{- c + d x^{4}}\, dx - \int \left (- \frac {2 a b x^{4} \sqrt {a - b x^{4}}}{- c + d x^{4}}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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