Optimal. Leaf size=261 \[ \frac{c \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt{a}+\sqrt{b} (c+d x)^2\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b} d^2}-\frac{c \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt{a}+\sqrt{b} (c+d x)^2\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b} d^2}+\frac{c \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{b} d^2}-\frac{c \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{b} d^2}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} (c+d x)^2}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{b} d^2} \]
[Out]
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Rubi [A] time = 0.543396, 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 \[ \frac{c \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt{a}+\sqrt{b} (c+d x)^2\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b} d^2}-\frac{c \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt{a}+\sqrt{b} (c+d x)^2\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b} d^2}+\frac{c \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{b} d^2}-\frac{c \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{b} d^2}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} (c+d x)^2}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{b} d^2} \]
Antiderivative was successfully verified.
[In] Int[x/(a + b*(c + d*x)^4),x]
[Out]
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Rubi in Sympy [A] time = 67.3942, size = 250, normalized size = 0.96 \[ \frac{\operatorname{atan}{\left (\frac{\sqrt{b} \left (c + d x\right )^{2}}{\sqrt{a}} \right )}}{2 \sqrt{a} \sqrt{b} d^{2}} + \frac{\sqrt{2} c \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \left (- c - d x\right ) + \sqrt{a} + \sqrt{b} \left (c + d x\right )^{2} \right )}}{8 a^{\frac{3}{4}} \sqrt [4]{b} d^{2}} - \frac{\sqrt{2} c \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \left (c + d x\right ) + \sqrt{a} + \sqrt{b} \left (c + d x\right )^{2} \right )}}{8 a^{\frac{3}{4}} \sqrt [4]{b} d^{2}} + \frac{\sqrt{2} c \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \left (- c - d x\right )}{\sqrt [4]{a}} \right )}}{4 a^{\frac{3}{4}} \sqrt [4]{b} d^{2}} - \frac{\sqrt{2} c \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \left (c + d x\right )}{\sqrt [4]{a}} \right )}}{4 a^{\frac{3}{4}} \sqrt [4]{b} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(a+b*(d*x+c)**4),x)
[Out]
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Mathematica [C] time = 0.0430668, size = 104, normalized size = 0.4 \[ \frac{\text{RootSum}\left [\text{$\#$1}^4 b d^4+4 \text{$\#$1}^3 b c d^3+6 \text{$\#$1}^2 b c^2 d^2+4 \text{$\#$1} b c^3 d+a+b c^4\&,\frac{\text{$\#$1} \log (x-\text{$\#$1})}{\text{$\#$1}^3 d^3+3 \text{$\#$1}^2 c d^2+3 \text{$\#$1} c^2 d+c^3}\&\right ]}{4 b d} \]
Antiderivative was successfully verified.
[In] Integrate[x/(a + b*(c + d*x)^4),x]
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Maple [C] time = 0.006, size = 95, normalized size = 0.4 \[{\frac{1}{4\,bd}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{4}b{d}^{4}+4\,{{\it \_Z}}^{3}bc{d}^{3}+6\,{{\it \_Z}}^{2}b{c}^{2}{d}^{2}+4\,{\it \_Z}\,b{c}^{3}d+b{c}^{4}+a \right ) }{\frac{{\it \_R}\,\ln \left ( x-{\it \_R} \right ) }{{d}^{3}{{\it \_R}}^{3}+3\,c{d}^{2}{{\it \_R}}^{2}+3\,{c}^{2}d{\it \_R}+{c}^{3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(a+b*(d*x+c)^4),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{{\left (d x + c\right )}^{4} b + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((d*x + c)^4*b + a),x, algorithm="maxima")
[Out]
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((d*x + c)^4*b + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.92449, size = 131, normalized size = 0.5 \[ \operatorname{RootSum}{\left (256 t^{4} a^{3} b^{2} d^{8} + 32 t^{2} a^{2} b d^{4} - 16 t a b c^{2} d^{2} + a + b c^{4}, \left ( t \mapsto t \log{\left (x + \frac{128 t^{3} a^{3} b d^{6} + 16 t^{2} a^{2} b c^{2} d^{4} + 8 t a^{2} d^{2} + 4 t a b c^{4} d^{2} - a c^{2} - b c^{6}}{4 a c d - b c^{5} d} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(a+b*(d*x+c)**4),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{{\left (d x + c\right )}^{4} b + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((d*x + c)^4*b + a),x, algorithm="giac")
[Out]