Optimal. Leaf size=131 \[ \frac{\left (\sqrt{a}+\sqrt{b} d^2 \left (\frac{c}{d}+x\right )^2\right ) \sqrt{\frac{a+b d^4 \left (\frac{c}{d}+x\right )^4}{\left (\sqrt{a}+\sqrt{b} d^2 \left (\frac{c}{d}+x\right )^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} d \sqrt{a+b d^4 \left (\frac{c}{d}+x\right )^4}} \]
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Rubi [A] time = 0.100435, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 49, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.041, Rules used = {1106, 220} \[ \frac{\left (\sqrt{a}+\sqrt{b} d^2 \left (\frac{c}{d}+x\right )^2\right ) \sqrt{\frac{a+b d^4 \left (\frac{c}{d}+x\right )^4}{\left (\sqrt{a}+\sqrt{b} d^2 \left (\frac{c}{d}+x\right )^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} d \sqrt{a+b d^4 \left (\frac{c}{d}+x\right )^4}} \]
Antiderivative was successfully verified.
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Rule 1106
Rule 220
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a+b c^4+4 b c^3 d x+6 b c^2 d^2 x^2+4 b c d^3 x^3+b d^4 x^4}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b d^4 x^4}} \, dx,x,\frac{c}{d}+x\right )\\ &=\frac{\left (\sqrt{a}+\sqrt{b} d^2 \left (\frac{c}{d}+x\right )^2\right ) \sqrt{\frac{a+b d^4 \left (\frac{c}{d}+x\right )^4}{\left (\sqrt{a}+\sqrt{b} d^2 \left (\frac{c}{d}+x\right )^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} d \sqrt{a+b d^4 \left (\frac{c}{d}+x\right )^4}}\\ \end{align*}
Mathematica [C] time = 0.0573804, size = 90, normalized size = 0.69 \[ -\frac{i \sqrt{\frac{a+b (c+d x)^4}{a}} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} (c+d x)\right )\right |-1\right )}{d \sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \sqrt{a+b (c+d x)^4}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.019, size = 1036, normalized size = 7.9 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \, b c^{3} d x + b c^{4} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{\sqrt{b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \, b c^{3} d x + b c^{4} + a}}, x\right ) \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}{\sqrt{a + b c^{4} + 4 b c^{3} d x + 6 b c^{2} d^{2} x^{2} + 4 b c d^{3} x^{3} + b d^{4} x^{4}}}\, 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}{\sqrt{b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \, b c^{3} d x + b c^{4} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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