Optimal. Leaf size=180 \[ \frac {b x (4 b c-9 a d)}{5 a^2 \sqrt [4]{a+b x^4} (b c-a d)^2}+\frac {d^2 \tan ^{-1}\left (\frac {x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} (b c-a d)^{9/4}}+\frac {d^2 \tanh ^{-1}\left (\frac {x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} (b c-a d)^{9/4}}+\frac {b x}{5 a \left (a+b x^4\right )^{5/4} (b c-a d)} \]
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Rubi [A] time = 0.20, antiderivative size = 180, 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} \[ \frac {b x (4 b c-9 a d)}{5 a^2 \sqrt [4]{a+b x^4} (b c-a d)^2}+\frac {d^2 \tan ^{-1}\left (\frac {x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} (b c-a d)^{9/4}}+\frac {d^2 \tanh ^{-1}\left (\frac {x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} (b c-a d)^{9/4}}+\frac {b x}{5 a \left (a+b x^4\right )^{5/4} (b c-a d)} \]
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 )^{9/4} \left (c+d x^4\right )} \, dx &=\frac {b x}{5 a (b c-a d) \left (a+b x^4\right )^{5/4}}-\frac {\int \frac {-4 b c+5 a d-4 b d x^4}{\left (a+b x^4\right )^{5/4} \left (c+d x^4\right )} \, dx}{5 a (b c-a d)}\\ &=\frac {b x}{5 a (b c-a d) \left (a+b x^4\right )^{5/4}}+\frac {b (4 b c-9 a d) x}{5 a^2 (b c-a d)^2 \sqrt [4]{a+b x^4}}+\frac {\int \frac {5 a^2 d^2}{\sqrt [4]{a+b x^4} \left (c+d x^4\right )} \, dx}{5 a^2 (b c-a d)^2}\\ &=\frac {b x}{5 a (b c-a d) \left (a+b x^4\right )^{5/4}}+\frac {b (4 b c-9 a d) x}{5 a^2 (b c-a d)^2 \sqrt [4]{a+b x^4}}+\frac {d^2 \int \frac {1}{\sqrt [4]{a+b x^4} \left (c+d x^4\right )} \, dx}{(b c-a d)^2}\\ &=\frac {b x}{5 a (b c-a d) \left (a+b x^4\right )^{5/4}}+\frac {b (4 b c-9 a d) x}{5 a^2 (b c-a d)^2 \sqrt [4]{a+b x^4}}+\frac {d^2 \operatorname {Subst}\left (\int \frac {1}{c-(b c-a d) x^4} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{(b c-a d)^2}\\ &=\frac {b x}{5 a (b c-a d) \left (a+b x^4\right )^{5/4}}+\frac {b (4 b c-9 a d) x}{5 a^2 (b c-a d)^2 \sqrt [4]{a+b x^4}}+\frac {d^2 \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 )}{2 \sqrt {c} (b c-a d)^2}+\frac {d^2 \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 )}{2 \sqrt {c} (b c-a d)^2}\\ &=\frac {b x}{5 a (b c-a d) \left (a+b x^4\right )^{5/4}}+\frac {b (4 b c-9 a d) x}{5 a^2 (b c-a d)^2 \sqrt [4]{a+b x^4}}+\frac {d^2 \tan ^{-1}\left (\frac {\sqrt [4]{b c-a d} x}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} (b c-a d)^{9/4}}+\frac {d^2 \tanh ^{-1}\left (\frac {\sqrt [4]{b c-a d} x}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} (b c-a d)^{9/4}}\\ \end {align*}
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Mathematica [C] time = 2.37, size = 621, normalized size = 3.45 \[ \frac {80 c^2 x^{12} (b c-a d)^3 \, _3F_2\left (2,2,\frac {13}{4};1,\frac {17}{4};\frac {(b c-a d) x^4}{c \left (b x^4+a\right )}\right )+80 d^2 x^{20} (b c-a d)^3 \, _3F_2\left (2,2,\frac {13}{4};1,\frac {17}{4};\frac {(b c-a d) x^4}{c \left (b x^4+a\right )}\right )+160 c d x^{16} (b c-a d)^3 \, _3F_2\left (2,2,\frac {13}{4};1,\frac {17}{4};\frac {(b c-a d) x^4}{c \left (b x^4+a\right )}\right )+2925 c^5 \left (a+b x^4\right )^3 \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {(b c-a d) x^4}{c \left (b x^4+a\right )}\right )-2925 c^5 \left (a+b x^4\right )^3+4680 c^4 d x^4 \left (a+b x^4\right )^3 \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {(b c-a d) x^4}{c \left (b x^4+a\right )}\right )-4680 c^4 d x^4 \left (a+b x^4\right )^3-585 c^4 x^4 \left (a+b x^4\right )^2 (b c-a d)+2080 c^3 d^2 x^8 \left (a+b x^4\right )^3 \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {(b c-a d) x^4}{c \left (b x^4+a\right )}\right )-2080 c^3 d^2 x^8 \left (a+b x^4\right )^3-936 c^3 d x^8 \left (a+b x^4\right )^2 (b c-a d)-416 c^2 d^2 x^{12} \left (a+b x^4\right )^2 (b c-a d)+280 c^2 x^{12} (b c-a d)^3 \, _2F_1\left (2,\frac {13}{4};\frac {17}{4};\frac {(b c-a d) x^4}{c \left (b x^4+a\right )}\right )+240 d^2 x^{20} (b c-a d)^3 \, _2F_1\left (2,\frac {13}{4};\frac {17}{4};\frac {(b c-a d) x^4}{c \left (b x^4+a\right )}\right )+520 c d x^{16} (b c-a d)^3 \, _2F_1\left (2,\frac {13}{4};\frac {17}{4};\frac {(b c-a d) x^4}{c \left (b x^4+a\right )}\right )}{325 c^4 x^7 \left (a+b x^4\right )^{13/4} (b c-a d)^2} \]
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 \[ \int \frac {1}{{\left (b x^{4} + a\right )}^{\frac {9}{4}} {\left (d x^{4} + c\right )}}\,{d x} \]
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 \[ \int \frac {1}{\left (b \,x^{4}+a \right )^{\frac {9}{4}} \left (d \,x^{4}+c \right )}\, dx \]
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 \[ \int \frac {1}{{\left (b x^{4} + a\right )}^{\frac {9}{4}} {\left (d x^{4} + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (b\,x^4+a\right )}^{9/4}\,\left (d\,x^4+c\right )} \,d x \]
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 \[ \int \frac {1}{\left (a + b x^{4}\right )^{\frac {9}{4}} \left (c + d x^{4}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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