Optimal. Leaf size=277 \[ -\frac {\sqrt [4]{a} b^{3/4} \sqrt {1-\frac {b x^4}{a}} (3 b c-5 a d) F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{3 d^2 \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} (b c-a d)^2 \Pi \left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d^2 \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} (b c-a d)^2 \Pi \left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d^2 \sqrt {a-b x^4}}+\frac {b x \sqrt {a-b x^4}}{3 d} \]
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Rubi [A] time = 0.26, antiderivative size = 277, 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 = {416, 523, 224, 221, 409, 1219, 1218} \[ -\frac {\sqrt [4]{a} b^{3/4} \sqrt {1-\frac {b x^4}{a}} (3 b c-5 a d) F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{3 d^2 \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} (b c-a d)^2 \Pi \left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d^2 \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} (b c-a d)^2 \Pi \left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d^2 \sqrt {a-b x^4}}+\frac {b x \sqrt {a-b x^4}}{3 d} \]
Antiderivative was successfully verified.
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Rule 221
Rule 224
Rule 409
Rule 416
Rule 523
Rule 1218
Rule 1219
Rubi steps
\begin {align*} \int \frac {\left (a-b x^4\right )^{3/2}}{c-d x^4} \, dx &=\frac {b x \sqrt {a-b x^4}}{3 d}-\frac {\int \frac {a (b c-3 a d)-b (3 b c-5 a d) x^4}{\sqrt {a-b x^4} \left (c-d x^4\right )} \, dx}{3 d}\\ &=\frac {b x \sqrt {a-b x^4}}{3 d}-\frac {(b (3 b c-5 a d)) \int \frac {1}{\sqrt {a-b x^4}} \, dx}{3 d^2}+\frac {(b c-a d)^2 \int \frac {1}{\sqrt {a-b x^4} \left (c-d x^4\right )} \, dx}{d^2}\\ &=\frac {b x \sqrt {a-b x^4}}{3 d}+\frac {(b c-a d)^2 \int \frac {1}{\left (1-\frac {\sqrt {d} x^2}{\sqrt {c}}\right ) \sqrt {a-b x^4}} \, dx}{2 c d^2}+\frac {(b c-a d)^2 \int \frac {1}{\left (1+\frac {\sqrt {d} x^2}{\sqrt {c}}\right ) \sqrt {a-b x^4}} \, dx}{2 c d^2}-\frac {\left (b (3 b c-5 a d) \sqrt {1-\frac {b x^4}{a}}\right ) \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}} \, dx}{3 d^2 \sqrt {a-b x^4}}\\ &=\frac {b x \sqrt {a-b x^4}}{3 d}-\frac {\sqrt [4]{a} b^{3/4} (3 b c-5 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 )}{3 d^2 \sqrt {a-b x^4}}+\frac {\left ((b c-a d)^2 \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^2 \sqrt {a-b x^4}}+\frac {\left ((b c-a d)^2 \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^2 \sqrt {a-b x^4}}\\ &=\frac {b x \sqrt {a-b x^4}}{3 d}-\frac {\sqrt [4]{a} b^{3/4} (3 b c-5 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 )}{3 d^2 \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} (b c-a d)^2 \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^2 \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} (b c-a d)^2 \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^2 \sqrt {a-b x^4}}\\ \end {align*}
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Mathematica [C] time = 0.46, size = 341, normalized size = 1.23 \[ -\frac {x \left (\frac {5 \left (5 a c \left (3 a^2 d-a b d x^4+b^2 x^4 \left (d x^4-c\right )\right ) F_1\left (\frac {1}{4};\frac {1}{2},1;\frac {5}{4};\frac {b x^4}{a},\frac {d x^4}{c}\right )+2 b x^4 \left (a-b x^4\right ) \left (c-d x^4\right ) \left (2 a d F_1\left (\frac {5}{4};\frac {1}{2},2;\frac {9}{4};\frac {b x^4}{a},\frac {d x^4}{c}\right )+b c F_1\left (\frac {5}{4};\frac {3}{2},1;\frac {9}{4};\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )\right )}{\left (d x^4-c\right ) \left (2 x^4 \left (2 a d F_1\left (\frac {5}{4};\frac {1}{2},2;\frac {9}{4};\frac {b x^4}{a},\frac {d x^4}{c}\right )+b c F_1\left (\frac {5}{4};\frac {3}{2},1;\frac {9}{4};\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )+5 a c F_1\left (\frac {1}{4};\frac {1}{2},1;\frac {5}{4};\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )}+\frac {b x^4 \sqrt {1-\frac {b x^4}{a}} (5 a d-3 b c) F_1\left (\frac {5}{4};\frac {1}{2},1;\frac {9}{4};\frac {b x^4}{a},\frac {d x^4}{c}\right )}{c}\right )}{15 d \sqrt {a-b x^4}} \]
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 \[ \int -\frac {{\left (-b x^{4} + a\right )}^{\frac {3}{2}}}{d x^{4} - c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.29, size = 311, normalized size = 1.12 \[ \frac {\sqrt {-b \,x^{4}+a}\, b x}{3 d}-\frac {\left (\frac {a b}{3 d}-\frac {\left (2 a d -b c \right ) b}{d^{2}}\right ) \sqrt {-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \EllipticF \left (\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, x , i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}+\frac {\left (-a^{2} d^{2}+2 a b c d -b^{2} c^{2}\right ) \left (-\frac {2 \sqrt {-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \RootOf \left (d \,\textit {\_Z}^{4}-c \right )^{3} d \EllipticPi \left (\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, x , \frac {\RootOf \left (d \,\textit {\_Z}^{4}-c \right )^{2} \sqrt {a}\, d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}\, c}-\frac {\arctanh \left (\frac {-2 \RootOf \left (d \,\textit {\_Z}^{4}-c \right )^{2} b \,x^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {-b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}\right )}{8 d^{3} \RootOf \left (d \,\textit {\_Z}^{4}-c \right )^{3}} \]
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 \[ -\int \frac {{\left (-b x^{4} + a\right )}^{\frac {3}{2}}}{d x^{4} - c}\,{d x} \]
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 \[ \int \frac {{\left (a-b\,x^4\right )}^{3/2}}{c-d\,x^4} \,d x \]
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 \[ - \int \frac {a \sqrt {a - b x^{4}}}{- c + d x^{4}}\, dx - \int \left (- \frac {b x^{4} \sqrt {a - b x^{4}}}{- c + d x^{4}}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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