Optimal. Leaf size=261 \[ \frac{c \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt{a}+\sqrt{b} (c+d x)^2\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b} d^2}-\frac{c \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt{a}+\sqrt{b} (c+d x)^2\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b} d^2}+\frac{c \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{b} d^2}-\frac{c \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{b} d^2}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} (c+d x)^2}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{b} d^2} \]
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Rubi [A] time = 0.26372, antiderivative size = 261, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {371, 1876, 211, 1165, 628, 1162, 617, 204, 275, 205} \[ \frac{c \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt{a}+\sqrt{b} (c+d x)^2\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b} d^2}-\frac{c \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt{a}+\sqrt{b} (c+d x)^2\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b} d^2}+\frac{c \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{b} d^2}-\frac{c \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{b} d^2}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} (c+d x)^2}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{b} d^2} \]
Antiderivative was successfully verified.
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Rule 371
Rule 1876
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rule 275
Rule 205
Rubi steps
\begin{align*} \int \frac{x}{a+b (c+d x)^4} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{-c+x}{a+b x^4} \, dx,x,c+d x\right )}{d^2}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{c}{a+b x^4}+\frac{x}{a+b x^4}\right ) \, dx,x,c+d x\right )}{d^2}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x}{a+b x^4} \, dx,x,c+d x\right )}{d^2}-\frac{c \operatorname{Subst}\left (\int \frac{1}{a+b x^4} \, dx,x,c+d x\right )}{d^2}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,(c+d x)^2\right )}{2 d^2}-\frac{c \operatorname{Subst}\left (\int \frac{\sqrt{a}-\sqrt{b} x^2}{a+b x^4} \, dx,x,c+d x\right )}{2 \sqrt{a} d^2}-\frac{c \operatorname{Subst}\left (\int \frac{\sqrt{a}+\sqrt{b} x^2}{a+b x^4} \, dx,x,c+d x\right )}{2 \sqrt{a} d^2}\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{b} (c+d x)^2}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{b} d^2}-\frac{c \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,c+d x\right )}{4 \sqrt{a} \sqrt{b} d^2}-\frac{c \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,c+d x\right )}{4 \sqrt{a} \sqrt{b} d^2}+\frac{c \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,c+d x\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b} d^2}+\frac{c \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,c+d x\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b} d^2}\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{b} (c+d x)^2}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{b} d^2}+\frac{c \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt{b} (c+d x)^2\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b} d^2}-\frac{c \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt{b} (c+d x)^2\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b} d^2}-\frac{c \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{b} d^2}+\frac{c \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{b} d^2}\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{b} (c+d x)^2}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{b} d^2}+\frac{c \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{b} d^2}-\frac{c \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{b} d^2}+\frac{c \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt{b} (c+d x)^2\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b} d^2}-\frac{c \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt{b} (c+d x)^2\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b} d^2}\\ \end{align*}
Mathematica [C] time = 0.0272071, size = 104, normalized size = 0.4 \[ \frac{\text{RootSum}\left [6 \text{$\#$1}^2 b c^2 d^2+4 \text{$\#$1}^3 b c d^3+\text{$\#$1}^4 b d^4+4 \text{$\#$1} b c^3 d+a+b c^4\& ,\frac{\text{$\#$1} \log (x-\text{$\#$1})}{3 \text{$\#$1}^2 c d^2+\text{$\#$1}^3 d^3+3 \text{$\#$1} c^2 d+c^3}\& \right ]}{4 b d} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.003, size = 95, normalized size = 0.4 \begin{align*}{\frac{1}{4\,bd}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{4}b{d}^{4}+4\,{{\it \_Z}}^{3}bc{d}^{3}+6\,{{\it \_Z}}^{2}b{c}^{2}{d}^{2}+4\,{\it \_Z}\,b{c}^{3}d+b{c}^{4}+a \right ) }{\frac{{\it \_R}\,\ln \left ( x-{\it \_R} \right ) }{{d}^{3}{{\it \_R}}^{3}+3\,c{d}^{2}{{\it \_R}}^{2}+3\,{\it \_R}\,{c}^{2}d+{c}^{3}}}} \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}{{\left (d x + c\right )}^{4} b + a}\,{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 [A] time = 0.901808, size = 131, normalized size = 0.5 \begin{align*} \operatorname{RootSum}{\left (256 t^{4} a^{3} b^{2} d^{8} + 32 t^{2} a^{2} b d^{4} - 16 t a b c^{2} d^{2} + a + b c^{4}, \left ( t \mapsto t \log{\left (x + \frac{128 t^{3} a^{3} b d^{6} + 16 t^{2} a^{2} b c^{2} d^{4} + 8 t a^{2} d^{2} + 4 t a b c^{4} d^{2} - a c^{2} - b c^{6}}{4 a c d - b c^{5} d} \right )} \right )\right )} \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}{{\left (d x + c\right )}^{4} b + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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