Optimal. Leaf size=184 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b} d^2 \left (\frac{c}{d}+x\right )^2}{\sqrt{a+b d^4 \left (\frac{c}{d}+x\right )^4}}\right )}{2 \sqrt{b} d^2}-\frac{c \left (\sqrt{a}+\sqrt{b} d^2 \left (\frac{c}{d}+x\right )^2\right ) \sqrt{\frac{a+b d^4 \left (\frac{c}{d}+x\right )^4}{\left (\sqrt{a}+\sqrt{b} d^2 \left (\frac{c}{d}+x\right )^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} d^2 \sqrt{a+b d^4 \left (\frac{c}{d}+x\right )^4}} \]
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Rubi [A] time = 0.232949, antiderivative size = 184, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 51, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {1680, 1885, 220, 275, 217, 206} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b} d^2 \left (\frac{c}{d}+x\right )^2}{\sqrt{a+b d^4 \left (\frac{c}{d}+x\right )^4}}\right )}{2 \sqrt{b} d^2}-\frac{c \left (\sqrt{a}+\sqrt{b} d^2 \left (\frac{c}{d}+x\right )^2\right ) \sqrt{\frac{a+b d^4 \left (\frac{c}{d}+x\right )^4}{\left (\sqrt{a}+\sqrt{b} d^2 \left (\frac{c}{d}+x\right )^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} d^2 \sqrt{a+b d^4 \left (\frac{c}{d}+x\right )^4}} \]
Antiderivative was successfully verified.
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Rule 1680
Rule 1885
Rule 220
Rule 275
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{x}{\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{-\frac{c}{d}+x}{\sqrt{a+b d^4 x^4}} \, dx,x,\frac{c}{d}+x\right )\\ &=\operatorname{Subst}\left (\int \left (-\frac{c}{d \sqrt{a+b d^4 x^4}}+\frac{x}{\sqrt{a+b d^4 x^4}}\right ) \, dx,x,\frac{c}{d}+x\right )\\ &=-\frac{c \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b d^4 x^4}} \, dx,x,\frac{c}{d}+x\right )}{d}+\operatorname{Subst}\left (\int \frac{x}{\sqrt{a+b d^4 x^4}} \, dx,x,\frac{c}{d}+x\right )\\ &=-\frac{c \left (\sqrt{a}+\sqrt{b} (c+d x)^2\right ) \sqrt{\frac{a+b (c+d x)^4}{\left (\sqrt{a}+\sqrt{b} (c+d x)^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^2 \sqrt{a+b (c+d x)^4}}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b d^4 x^2}} \, dx,x,\left (\frac{c}{d}+x\right )^2\right )\\ &=-\frac{c \left (\sqrt{a}+\sqrt{b} (c+d x)^2\right ) \sqrt{\frac{a+b (c+d x)^4}{\left (\sqrt{a}+\sqrt{b} (c+d x)^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^2 \sqrt{a+b (c+d x)^4}}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1-b d^4 x^2} \, dx,x,\frac{\left (\frac{c}{d}+x\right )^2}{\sqrt{a+b (c+d x)^4}}\right )\\ &=\frac{\tanh ^{-1}\left (\frac{\sqrt{b} (c+d x)^2}{\sqrt{a+b (c+d x)^4}}\right )}{2 \sqrt{b} d^2}-\frac{c \left (\sqrt{a}+\sqrt{b} (c+d x)^2\right ) \sqrt{\frac{a+b (c+d x)^4}{\left (\sqrt{a}+\sqrt{b} (c+d x)^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^2 \sqrt{a+b (c+d x)^4}}\\ \end{align*}
Mathematica [C] time = 0.57034, size = 330, normalized size = 1.79 \[ \frac{\sqrt [4]{-1} \sqrt{2} \sqrt{-\frac{i \left (\sqrt [4]{-1} \sqrt [4]{a}+\sqrt [4]{b} (c+d x)\right )}{\sqrt [4]{-1} \sqrt [4]{a}-\sqrt [4]{b} (c+d x)}} \left (\sqrt{b} (c+d x)^2+i \sqrt{a}\right ) \left (\left (\sqrt [4]{-1} \sqrt [4]{a}-\sqrt [4]{b} c\right ) F\left (\left .\sin ^{-1}\left (\sqrt{-\frac{i \left (\sqrt [4]{b} (c+d x)+\sqrt [4]{-1} \sqrt [4]{a}\right )}{\sqrt [4]{-1} \sqrt [4]{a}-\sqrt [4]{b} (c+d x)}}\right )\right |-1\right )-2 \sqrt [4]{-1} \sqrt [4]{a} \Pi \left (-i;\left .\sin ^{-1}\left (\sqrt{-\frac{i \left (\sqrt [4]{b} (c+d x)+\sqrt [4]{-1} \sqrt [4]{a}\right )}{\sqrt [4]{-1} \sqrt [4]{a}-\sqrt [4]{b} (c+d x)}}\right )\right |-1\right )\right )}{\sqrt [4]{a} \sqrt{b} d^2 \sqrt{\frac{\sqrt{b} (c+d x)^2+i \sqrt{a}}{\left (\sqrt [4]{-1} \sqrt [4]{a}-\sqrt [4]{b} (c+d x)\right )^2}} \sqrt{a+b (c+d x)^4}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.382, size = 1528, normalized size = 8.3 \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{x}{\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{x}{\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{x}{\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{x}{\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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