Optimal. Leaf size=134 \[ -\frac {d \tan ^{-1}\left (\frac {x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} (b c-a d)^{5/4}}-\frac {d \tanh ^{-1}\left (\frac {x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} (b c-a d)^{5/4}}+\frac {b x}{a \sqrt [4]{a+b x^4} (b c-a d)} \]
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Rubi [A] time = 0.10, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {382, 377, 212, 208, 205} \[ -\frac {d \tan ^{-1}\left (\frac {x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} (b c-a d)^{5/4}}-\frac {d \tanh ^{-1}\left (\frac {x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} (b c-a d)^{5/4}}+\frac {b x}{a \sqrt [4]{a+b x^4} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 205
Rule 208
Rule 212
Rule 377
Rule 382
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b x^4\right )^{5/4} \left (c+d x^4\right )} \, dx &=\frac {b x}{a (b c-a d) \sqrt [4]{a+b x^4}}-\frac {d \int \frac {1}{\sqrt [4]{a+b x^4} \left (c+d x^4\right )} \, dx}{b c-a d}\\ &=\frac {b x}{a (b c-a d) \sqrt [4]{a+b x^4}}-\frac {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 )}{b c-a d}\\ &=\frac {b x}{a (b c-a d) \sqrt [4]{a+b x^4}}-\frac {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 )}{2 \sqrt {c} (b c-a d)}-\frac {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 )}{2 \sqrt {c} (b c-a d)}\\ &=\frac {b x}{a (b c-a d) \sqrt [4]{a+b x^4}}-\frac {d \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)^{5/4}}-\frac {d \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)^{5/4}}\\ \end {align*}
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Mathematica [C] time = 0.58, size = 256, normalized size = 1.91 \[ -\frac {45 c^3 \left (a+b x^4\right )^2 \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {(b c-a d) x^4}{c \left (b x^4+a\right )}\right )-45 c^3 \left (a+b x^4\right )^2+36 c^2 d x^4 \left (a+b x^4\right )^2 \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {(b c-a d) x^4}{c \left (b x^4+a\right )}\right )-36 c^2 d x^4 \left (a+b x^4\right )^2+4 d x^{12} (b c-a d)^2 \, _2F_1\left (2,\frac {9}{4};\frac {13}{4};\frac {(b c-a d) x^4}{c \left (b x^4+a\right )}\right )+4 c x^8 (b c-a d)^2 \, _2F_1\left (2,\frac {9}{4};\frac {13}{4};\frac {(b c-a d) x^4}{c \left (b x^4+a\right )}\right )}{9 c^3 x^3 \left (a+b x^4\right )^{9/4} (a d-b c)} \]
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 \[ \int \frac {1}{{\left (b x^{4} + a\right )}^{\frac {5}{4}} {\left (d x^{4} + c\right )}}\,{d x} \]
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 \[ \int \frac {1}{\left (b \,x^{4}+a \right )^{\frac {5}{4}} \left (d \,x^{4}+c \right )}\, dx \]
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 \[ \int \frac {1}{{\left (b x^{4} + a\right )}^{\frac {5}{4}} {\left (d x^{4} + c\right )}}\,{d x} \]
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 \[ \int \frac {1}{{\left (b\,x^4+a\right )}^{5/4}\,\left (d\,x^4+c\right )} \,d x \]
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 \[ \int \frac {1}{\left (a + b x^{4}\right )^{\frac {5}{4}} \left (c + d x^{4}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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