Optimal. Leaf size=123 \[ -\frac{a^2 \sqrt{c+d x^4}}{4 b^2 \left (a+b x^4\right ) (b c-a d)}+\frac{a (4 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{4 b^{5/2} (b c-a d)^{3/2}}+\frac{\sqrt{c+d x^4}}{2 b^2 d} \]
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Rubi [A] time = 0.146328, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {446, 89, 80, 63, 208} \[ -\frac{a^2 \sqrt{c+d x^4}}{4 b^2 \left (a+b x^4\right ) (b c-a d)}+\frac{a (4 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{4 b^{5/2} (b c-a d)^{3/2}}+\frac{\sqrt{c+d x^4}}{2 b^2 d} \]
Antiderivative was successfully verified.
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Rule 446
Rule 89
Rule 80
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{x^{11}}{\left (a+b x^4\right )^2 \sqrt{c+d x^4}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{x^2}{(a+b x)^2 \sqrt{c+d x}} \, dx,x,x^4\right )\\ &=-\frac{a^2 \sqrt{c+d x^4}}{4 b^2 (b c-a d) \left (a+b x^4\right )}+\frac{\operatorname{Subst}\left (\int \frac{-\frac{1}{2} a (2 b c-a d)+b (b c-a d) x}{(a+b x) \sqrt{c+d x}} \, dx,x,x^4\right )}{4 b^2 (b c-a d)}\\ &=\frac{\sqrt{c+d x^4}}{2 b^2 d}-\frac{a^2 \sqrt{c+d x^4}}{4 b^2 (b c-a d) \left (a+b x^4\right )}-\frac{(a (4 b c-3 a d)) \operatorname{Subst}\left (\int \frac{1}{(a+b x) \sqrt{c+d x}} \, dx,x,x^4\right )}{8 b^2 (b c-a d)}\\ &=\frac{\sqrt{c+d x^4}}{2 b^2 d}-\frac{a^2 \sqrt{c+d x^4}}{4 b^2 (b c-a d) \left (a+b x^4\right )}-\frac{(a (4 b c-3 a d)) \operatorname{Subst}\left (\int \frac{1}{a-\frac{b c}{d}+\frac{b x^2}{d}} \, dx,x,\sqrt{c+d x^4}\right )}{4 b^2 d (b c-a d)}\\ &=\frac{\sqrt{c+d x^4}}{2 b^2 d}-\frac{a^2 \sqrt{c+d x^4}}{4 b^2 (b c-a d) \left (a+b x^4\right )}+\frac{a (4 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{4 b^{5/2} (b c-a d)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.228681, size = 107, normalized size = 0.87 \[ \frac{1}{4} \left (\frac{\sqrt{c+d x^4} \left (\frac{a^2}{\left (a+b x^4\right ) (a d-b c)}+\frac{2}{d}\right )}{b^2}+\frac{a (4 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{b^{5/2} (b c-a d)^{3/2}}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.014, size = 876, normalized size = 7.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.58557, size = 983, normalized size = 7.99 \begin{align*} \left [\frac{{\left (4 \, a^{2} b c d - 3 \, a^{3} d^{2} +{\left (4 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} x^{4}\right )} \sqrt{b^{2} c - a b d} \log \left (\frac{b d x^{4} + 2 \, b c - a d + 2 \, \sqrt{d x^{4} + c} \sqrt{b^{2} c - a b d}}{b x^{4} + a}\right ) + 2 \,{\left (2 \, a b^{3} c^{2} - 5 \, a^{2} b^{2} c d + 3 \, a^{3} b d^{2} + 2 \,{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{4}\right )} \sqrt{d x^{4} + c}}{8 \,{\left (a b^{5} c^{2} d - 2 \, a^{2} b^{4} c d^{2} + a^{3} b^{3} d^{3} +{\left (b^{6} c^{2} d - 2 \, a b^{5} c d^{2} + a^{2} b^{4} d^{3}\right )} x^{4}\right )}}, -\frac{{\left (4 \, a^{2} b c d - 3 \, a^{3} d^{2} +{\left (4 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} x^{4}\right )} \sqrt{-b^{2} c + a b d} \arctan \left (\frac{\sqrt{d x^{4} + c} \sqrt{-b^{2} c + a b d}}{b d x^{4} + b c}\right ) -{\left (2 \, a b^{3} c^{2} - 5 \, a^{2} b^{2} c d + 3 \, a^{3} b d^{2} + 2 \,{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{4}\right )} \sqrt{d x^{4} + c}}{4 \,{\left (a b^{5} c^{2} d - 2 \, a^{2} b^{4} c d^{2} + a^{3} b^{3} d^{3} +{\left (b^{6} c^{2} d - 2 \, a b^{5} c d^{2} + a^{2} b^{4} d^{3}\right )} x^{4}\right )}}\right ] \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 [A] time = 1.55646, size = 181, normalized size = 1.47 \begin{align*} -\frac{\sqrt{d x^{4} + c} a^{2} d}{4 \,{\left (b^{3} c - a b^{2} d\right )}{\left ({\left (d x^{4} + c\right )} b - b c + a d\right )}} - \frac{{\left (4 \, a b c - 3 \, a^{2} d\right )} \arctan \left (\frac{\sqrt{d x^{4} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{4 \,{\left (b^{3} c - a b^{2} d\right )} \sqrt{-b^{2} c + a b d}} + \frac{\sqrt{d x^{4} + c}}{2 \, b^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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