Optimal. Leaf size=266 \[ \frac {b (4 b c+5 a d) x}{20 a c (b c-a d)^2 \left (a+b x^4\right )^{5/4}}+\frac {b \left (16 b^2 c^2-56 a b c d-5 a^2 d^2\right ) x}{20 a^2 c (b c-a d)^3 \sqrt [4]{a+b x^4}}-\frac {d x}{4 c (b c-a d) \left (a+b x^4\right )^{5/4} \left (c+d x^4\right )}+\frac {3 d^2 (4 b c-a d) \tan ^{-1}\left (\frac {\sqrt [4]{b c-a d} x}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} (b c-a d)^{13/4}}+\frac {3 d^2 (4 b c-a d) \tanh ^{-1}\left (\frac {\sqrt [4]{b c-a d} x}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} (b c-a d)^{13/4}} \]
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Rubi [A]
time = 0.22, antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {425, 541, 12,
385, 218, 214, 211} \begin {gather*} \frac {b x \left (-5 a^2 d^2-56 a b c d+16 b^2 c^2\right )}{20 a^2 c \sqrt [4]{a+b x^4} (b c-a d)^3}+\frac {3 d^2 (4 b c-a d) \text {ArcTan}\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)^{13/4}}+\frac {3 d^2 (4 b c-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)^{13/4}}-\frac {d x}{4 c \left (a+b x^4\right )^{5/4} \left (c+d x^4\right ) (b c-a d)}+\frac {b x (5 a d+4 b c)}{20 a c \left (a+b x^4\right )^{5/4} (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 211
Rule 214
Rule 218
Rule 385
Rule 425
Rule 541
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b x^4\right )^{9/4} \left (c+d x^4\right )^2} \, dx &=-\frac {d x}{4 c (b c-a d) \left (a+b x^4\right )^{5/4} \left (c+d x^4\right )}+\frac {\int \frac {4 b c-3 a d-8 b d x^4}{\left (a+b x^4\right )^{9/4} \left (c+d x^4\right )} \, dx}{4 c (b c-a d)}\\ &=\frac {b (4 b c+5 a d) x}{20 a c (b c-a d)^2 \left (a+b x^4\right )^{5/4}}-\frac {d x}{4 c (b c-a d) \left (a+b x^4\right )^{5/4} \left (c+d x^4\right )}-\frac {\int \frac {-16 b^2 c^2+40 a b c d-15 a^2 d^2-4 b d (4 b c+5 a d) x^4}{\left (a+b x^4\right )^{5/4} \left (c+d x^4\right )} \, dx}{20 a c (b c-a d)^2}\\ &=\frac {b (4 b c+5 a d) x}{20 a c (b c-a d)^2 \left (a+b x^4\right )^{5/4}}+\frac {b \left (16 b^2 c^2-56 a b c d-5 a^2 d^2\right ) x}{20 a^2 c (b c-a d)^3 \sqrt [4]{a+b x^4}}-\frac {d x}{4 c (b c-a d) \left (a+b x^4\right )^{5/4} \left (c+d x^4\right )}+\frac {\int \frac {15 a^2 d^2 (4 b c-a d)}{\sqrt [4]{a+b x^4} \left (c+d x^4\right )} \, dx}{20 a^2 c (b c-a d)^3}\\ &=\frac {b (4 b c+5 a d) x}{20 a c (b c-a d)^2 \left (a+b x^4\right )^{5/4}}+\frac {b \left (16 b^2 c^2-56 a b c d-5 a^2 d^2\right ) x}{20 a^2 c (b c-a d)^3 \sqrt [4]{a+b x^4}}-\frac {d x}{4 c (b c-a d) \left (a+b x^4\right )^{5/4} \left (c+d x^4\right )}+\frac {\left (3 d^2 (4 b c-a d)\right ) \int \frac {1}{\sqrt [4]{a+b x^4} \left (c+d x^4\right )} \, dx}{4 c (b c-a d)^3}\\ &=\frac {b (4 b c+5 a d) x}{20 a c (b c-a d)^2 \left (a+b x^4\right )^{5/4}}+\frac {b \left (16 b^2 c^2-56 a b c d-5 a^2 d^2\right ) x}{20 a^2 c (b c-a d)^3 \sqrt [4]{a+b x^4}}-\frac {d x}{4 c (b c-a d) \left (a+b x^4\right )^{5/4} \left (c+d x^4\right )}+\frac {\left (3 d^2 (4 b c-a d)\right ) \text {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)^3}\\ &=\frac {b (4 b c+5 a d) x}{20 a c (b c-a d)^2 \left (a+b x^4\right )^{5/4}}+\frac {b \left (16 b^2 c^2-56 a b c d-5 a^2 d^2\right ) x}{20 a^2 c (b c-a d)^3 \sqrt [4]{a+b x^4}}-\frac {d x}{4 c (b c-a d) \left (a+b x^4\right )^{5/4} \left (c+d x^4\right )}+\frac {\left (3 d^2 (4 b c-a d)\right ) \text {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)^3}+\frac {\left (3 d^2 (4 b c-a d)\right ) \text {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)^3}\\ &=\frac {b (4 b c+5 a d) x}{20 a c (b c-a d)^2 \left (a+b x^4\right )^{5/4}}+\frac {b \left (16 b^2 c^2-56 a b c d-5 a^2 d^2\right ) x}{20 a^2 c (b c-a d)^3 \sqrt [4]{a+b x^4}}-\frac {d x}{4 c (b c-a d) \left (a+b x^4\right )^{5/4} \left (c+d x^4\right )}+\frac {3 d^2 (4 b c-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)^{13/4}}+\frac {3 d^2 (4 b c-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)^{13/4}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 4.68, size = 357, normalized size = 1.34 \begin {gather*} \frac {\left (\frac {1}{80}+\frac {i}{80}\right ) \left (\frac {(2-2 i) c^{3/4} x \left (5 a^4 d^3+10 a^3 b d^3 x^4-16 b^4 c^2 x^4 \left (c+d x^4\right )+5 a^2 b^2 d \left (12 c^2+12 c d x^4+d^2 x^8\right )+4 a b^3 c \left (-5 c^2+9 c d x^4+14 d^2 x^8\right )\right )}{a^2 (-b c+a d)^3 \left (a+b x^4\right )^{5/4} \left (c+d x^4\right )}+\frac {15 d^2 (4 b c-a d) \tan ^{-1}\left (\frac {\frac {(1-i) \sqrt [4]{b c-a d} x^2}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}-\frac {(1+i) \sqrt [4]{c} \sqrt [4]{a+b x^4}}{\sqrt [4]{b c-a d}}}{2 x}\right )}{(b c-a d)^{13/4}}+\frac {15 d^2 (4 b c-a d) \tanh ^{-1}\left (\frac {\frac {(1-i) \sqrt [4]{b c-a d} x^2}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}+\frac {(1+i) \sqrt [4]{c} \sqrt [4]{a+b x^4}}{\sqrt [4]{b c-a d}}}{2 x}\right )}{(b c-a d)^{13/4}}\right )}{c^{7/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (b \,x^{4}+a \right )^{\frac {9}{4}} \left (d \,x^{4}+c \right )^{2}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \int \frac {1}{\left (a + b x^{4}\right )^{\frac {9}{4}} \left (c + d x^{4}\right )^{2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{{\left (b\,x^4+a\right )}^{9/4}\,{\left (d\,x^4+c\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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