Optimal. Leaf size=266 \[ \frac{b x \left (-5 a^2 d^2-56 a b c d+16 b^2 c^2\right )}{20 a^2 c \sqrt [4]{a+b x^4} (b c-a d)^3}+\frac{3 d^2 (4 b c-a d) \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} (b c-a d)^{13/4}}+\frac{3 d^2 (4 b c-a d) \tanh ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} (b c-a d)^{13/4}}-\frac{d x}{4 c \left (a+b x^4\right )^{5/4} \left (c+d x^4\right ) (b c-a d)}+\frac{b x (5 a d+4 b c)}{20 a c \left (a+b x^4\right )^{5/4} (b c-a d)^2} \]
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Rubi [A] time = 0.292855, antiderivative size = 266, normalized size of antiderivative = 1., 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 = {414, 527, 12, 377, 212, 208, 205} \[ \frac{b x \left (-5 a^2 d^2-56 a b c d+16 b^2 c^2\right )}{20 a^2 c \sqrt [4]{a+b x^4} (b c-a d)^3}+\frac{3 d^2 (4 b c-a d) \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} (b c-a d)^{13/4}}+\frac{3 d^2 (4 b c-a d) \tanh ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} (b c-a d)^{13/4}}-\frac{d x}{4 c \left (a+b x^4\right )^{5/4} \left (c+d x^4\right ) (b c-a d)}+\frac{b x (5 a d+4 b c)}{20 a c \left (a+b x^4\right )^{5/4} (b c-a d)^2} \]
Antiderivative was successfully verified.
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Rule 414
Rule 527
Rule 12
Rule 377
Rule 212
Rule 208
Rule 205
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 ) \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)^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 ) \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)^3}+\frac{\left (3 d^2 (4 b c-a d)\right ) \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)^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*}
Mathematica [C] time = 3.81365, size = 1216, normalized size = 4.57 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.425, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( d{x}^{4}+c \right ) ^{2}} \left ( b{x}^{4}+a \right ) ^{-{\frac{9}{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{9}{4}}{\left (d x^{4} + c\right )}^{2}}\,{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(-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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{9}{4}}{\left (d x^{4} + c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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