Optimal. Leaf size=87 \[ \frac{d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{4 \sqrt{b} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x^4}}{4 \left (a+b x^4\right ) (b c-a d)} \]
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Rubi [A] time = 0.0735342, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {444, 51, 63, 208} \[ \frac{d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{4 \sqrt{b} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x^4}}{4 \left (a+b x^4\right ) (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 444
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{x^3}{\left (a+b x^4\right )^2 \sqrt{c+d x^4}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{(a+b x)^2 \sqrt{c+d x}} \, dx,x,x^4\right )\\ &=-\frac{\sqrt{c+d x^4}}{4 (b c-a d) \left (a+b x^4\right )}-\frac{d \operatorname{Subst}\left (\int \frac{1}{(a+b x) \sqrt{c+d x}} \, dx,x,x^4\right )}{8 (b c-a d)}\\ &=-\frac{\sqrt{c+d x^4}}{4 (b c-a d) \left (a+b x^4\right )}-\frac{\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 c-a d)}\\ &=-\frac{\sqrt{c+d x^4}}{4 (b c-a d) \left (a+b x^4\right )}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{4 \sqrt{b} (b c-a d)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0865039, size = 85, normalized size = 0.98 \[ \frac{1}{4} \left (\frac{\sqrt{c+d x^4}}{\left (a+b x^4\right ) (a d-b c)}+\frac{d \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{a d-b c}}\right )}{\sqrt{b} (a d-b c)^{3/2}}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.006, size = 541, normalized size = 6.2 \begin{align*}{\frac{1}{8\,ab \left ( ad-bc \right ) }\sqrt{-ab}\sqrt{ \left ({x}^{2}+{\frac{1}{b}\sqrt{-ab}} \right ) ^{2}d-2\,{\frac{d\sqrt{-ab}}{b} \left ({x}^{2}+{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \left ({x}^{2}+{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}}-{\frac{d}{8\,b \left ( ad-bc \right ) }\ln \left ({ \left ( -2\,{\frac{ad-bc}{b}}-2\,{\frac{d\sqrt{-ab}}{b} \left ({x}^{2}+{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ({x}^{2}+{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d-2\,{\frac{d\sqrt{-ab}}{b} \left ({x}^{2}+{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ({x}^{2}+{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}}-{\frac{1}{8\,ab \left ( ad-bc \right ) }\sqrt{-ab}\sqrt{ \left ({x}^{2}-{\frac{1}{b}\sqrt{-ab}} \right ) ^{2}d+2\,{\frac{d\sqrt{-ab}}{b} \left ({x}^{2}-{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \left ({x}^{2}-{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}}-{\frac{d}{8\,b \left ( ad-bc \right ) }\ln \left ({ \left ( -2\,{\frac{ad-bc}{b}}+2\,{\frac{d\sqrt{-ab}}{b} \left ({x}^{2}-{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ({x}^{2}-{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d+2\,{\frac{d\sqrt{-ab}}{b} \left ({x}^{2}-{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ({x}^{2}-{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}} \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 = 1.57579, size = 639, normalized size = 7.34 \begin{align*} \left [-\frac{{\left (b d x^{4} + a d\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 \, \sqrt{d x^{4} + c}{\left (b^{2} c - a b d\right )}}{8 \,{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2} +{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{4}\right )}}, -\frac{{\left (b d x^{4} + a d\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 ) + \sqrt{d x^{4} + c}{\left (b^{2} c - a b d\right )}}{4 \,{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2} +{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{4}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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Giac [A] time = 1.51874, size = 124, normalized size = 1.43 \begin{align*} -\frac{1}{4} \, d{\left (\frac{\arctan \left (\frac{\sqrt{d x^{4} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d}{\left (b c - a d\right )}} + \frac{\sqrt{d x^{4} + c}}{{\left ({\left (d x^{4} + c\right )} b - b c + a d\right )}{\left (b c - a d\right )}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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