Optimal. Leaf size=205 \[ -\frac {d (8 b c-3 a d) \tan ^{-1}\left (\frac {x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} (b c-a d)^{9/4}}-\frac {d (8 b c-3 a d) \tanh ^{-1}\left (\frac {x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} (b c-a d)^{9/4}}+\frac {b x (a d+4 b c)}{4 a c \sqrt [4]{a+b x^4} (b c-a d)^2}-\frac {d x}{4 c \sqrt [4]{a+b x^4} \left (c+d x^4\right ) (b c-a d)} \]
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Rubi [A] time = 0.18, antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {414, 527, 12, 377, 212, 208, 205} \[ -\frac {d (8 b c-3 a d) \tan ^{-1}\left (\frac {x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} (b c-a d)^{9/4}}-\frac {d (8 b c-3 a d) \tanh ^{-1}\left (\frac {x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} (b c-a d)^{9/4}}+\frac {b x (a d+4 b c)}{4 a c \sqrt [4]{a+b x^4} (b c-a d)^2}-\frac {d x}{4 c \sqrt [4]{a+b x^4} \left (c+d x^4\right ) (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 205
Rule 208
Rule 212
Rule 377
Rule 414
Rule 527
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b x^4\right )^{5/4} \left (c+d x^4\right )^2} \, dx &=-\frac {d x}{4 c (b c-a d) \sqrt [4]{a+b x^4} \left (c+d x^4\right )}+\frac {\int \frac {4 b c-3 a d-4 b d x^4}{\left (a+b x^4\right )^{5/4} \left (c+d x^4\right )} \, dx}{4 c (b c-a d)}\\ &=\frac {b (4 b c+a d) x}{4 a c (b c-a d)^2 \sqrt [4]{a+b x^4}}-\frac {d x}{4 c (b c-a d) \sqrt [4]{a+b x^4} \left (c+d x^4\right )}-\frac {\int \frac {a d (8 b c-3 a d)}{\sqrt [4]{a+b x^4} \left (c+d x^4\right )} \, dx}{4 a c (b c-a d)^2}\\ &=\frac {b (4 b c+a d) x}{4 a c (b c-a d)^2 \sqrt [4]{a+b x^4}}-\frac {d x}{4 c (b c-a d) \sqrt [4]{a+b x^4} \left (c+d x^4\right )}-\frac {(d (8 b c-3 a d)) \int \frac {1}{\sqrt [4]{a+b x^4} \left (c+d x^4\right )} \, dx}{4 c (b c-a d)^2}\\ &=\frac {b (4 b c+a d) x}{4 a c (b c-a d)^2 \sqrt [4]{a+b x^4}}-\frac {d x}{4 c (b c-a d) \sqrt [4]{a+b x^4} \left (c+d x^4\right )}-\frac {(d (8 b c-3 a d)) \operatorname {Subst}\left (\int \frac {1}{c-(b c-a d) x^4} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{4 c (b c-a d)^2}\\ &=\frac {b (4 b c+a d) x}{4 a c (b c-a d)^2 \sqrt [4]{a+b x^4}}-\frac {d x}{4 c (b c-a d) \sqrt [4]{a+b x^4} \left (c+d x^4\right )}-\frac {(d (8 b c-3 a d)) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c}-\sqrt {b c-a d} x^2} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{8 c^{3/2} (b c-a d)^2}-\frac {(d (8 b c-3 a d)) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c}+\sqrt {b c-a d} x^2} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{8 c^{3/2} (b c-a d)^2}\\ &=\frac {b (4 b c+a d) x}{4 a c (b c-a d)^2 \sqrt [4]{a+b x^4}}-\frac {d x}{4 c (b c-a d) \sqrt [4]{a+b x^4} \left (c+d x^4\right )}-\frac {d (8 b c-3 a d) \tan ^{-1}\left (\frac {\sqrt [4]{b c-a d} x}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} (b c-a d)^{9/4}}-\frac {d (8 b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt [4]{b c-a d} x}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} (b c-a d)^{9/4}}\\ \end {align*}
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Mathematica [C] time = 2.27, size = 625, normalized size = 3.05 \[ \frac {c \left (a+b x^4\right )^{3/4} \left (\frac {320 d^2 x^{20} (b c-a d)^3 \, _3F_2\left (2,2,\frac {9}{4};1,\frac {17}{4};\frac {(b c-a d) x^4}{c \left (b x^4+a\right )}\right )}{c^5 \left (a+b x^4\right )^3}+\frac {640 d x^{16} (b c-a d)^3 \, _3F_2\left (2,2,\frac {9}{4};1,\frac {17}{4};\frac {(b c-a d) x^4}{c \left (b x^4+a\right )}\right )}{c^4 \left (a+b x^4\right )^3}+\frac {320 x^{12} (b c-a d)^3 \, _3F_2\left (2,2,\frac {9}{4};1,\frac {17}{4};\frac {(b c-a d) x^4}{c \left (b x^4+a\right )}\right )}{c^3 \left (a+b x^4\right )^3}+\frac {16380 d^2 x^{12} (a d-b c) \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {(b c-a d) x^4}{c \left (b x^4+a\right )}\right )}{c^3 \left (a+b x^4\right )}+\frac {7488 d^2 x^{12} (b c-a d)}{c^3 \left (a+b x^4\right )}+\frac {44460 d^2 x^8 \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {(b c-a d) x^4}{c \left (b x^4+a\right )}\right )}{c^2}+\frac {33930 d x^8 (a d-b c) \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {(b c-a d) x^4}{c \left (b x^4+a\right )}\right )}{c^2 \left (a+b x^4\right )}+\frac {14976 d x^8 (b c-a d)}{c^2 \left (a+b x^4\right )}+\frac {94770 d x^4 \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {(b c-a d) x^4}{c \left (b x^4+a\right )}\right )}{c}-\frac {14625 x^4 (b c-a d) \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {(b c-a d) x^4}{c \left (b x^4+a\right )}\right )}{c \left (a+b x^4\right )}+47385 \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {(b c-a d) x^4}{c \left (b x^4+a\right )}\right )+\frac {5148 x^4 (b c-a d)}{c \left (a+b x^4\right )}-\frac {44460 d^2 x^8}{c^2}-\frac {94770 d x^4}{c}-47385\right )}{2340 x^7 \left (c+d x^4\right ) (b c-a d)^2} \]
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 \[ \int \frac {1}{{\left (b x^{4} + a\right )}^{\frac {5}{4}} {\left (d x^{4} + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.60, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,x^{4}+a \right )^{\frac {5}{4}} \left (d \,x^{4}+c \right )^{2}}\, dx \]
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 \[ \int \frac {1}{{\left (b x^{4} + a\right )}^{\frac {5}{4}} {\left (d x^{4} + c\right )}^{2}}\,{d x} \]
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 \[ \int \frac {1}{{\left (b\,x^4+a\right )}^{5/4}\,{\left (d\,x^4+c\right )}^2} \,d x \]
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 \[ \int \frac {1}{\left (a + b x^{4}\right )^{\frac {5}{4}} \left (c + d x^{4}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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