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 = 0.0961695, antiderivative size = 134, normalized size of antiderivative = 1., 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 382
Rule 377
Rule 212
Rule 208
Rule 205
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*}
Mathematica [C] time = 0.34564, size = 256, normalized size = 1.91 \[ -\frac{36 c^2 d x^4 \left (a+b x^4\right )^2 \, _2F_1\left (\frac{1}{4},1;\frac{5}{4};\frac{(b c-a d) x^4}{c \left (b x^4+a\right )}\right )+45 c^3 \left (a+b x^4\right )^2 \, _2F_1\left (\frac{1}{4},1;\frac{5}{4};\frac{(b c-a d) x^4}{c \left (b x^4+a\right )}\right )-36 c^2 d x^4 \left (a+b x^4\right )^2-45 c^3 \left (a+b x^4\right )^2+4 d x^{12} (b c-a d)^2 \, _2F_1\left (2,\frac{9}{4};\frac{13}{4};\frac{(b c-a d) x^4}{c \left (b x^4+a\right )}\right )+4 c x^8 (b c-a d)^2 \, _2F_1\left (2,\frac{9}{4};\frac{13}{4};\frac{(b c-a d) x^4}{c \left (b x^4+a\right )}\right )}{9 c^3 x^3 \left (a+b x^4\right )^{9/4} (a d-b c)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.411, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{d{x}^{4}+c} \left ( b{x}^{4}+a \right ) ^{-{\frac{5}{4}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{5}{4}}{\left (d x^{4} + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b x^{4}\right )^{\frac{5}{4}} \left (c + d x^{4}\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{5}{4}}{\left (d x^{4} + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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