Optimal. Leaf size=221 \[ -\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {a}+\sqrt {b} (c+d x)^2\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b} d}+\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {a}+\sqrt {b} (c+d x)^2\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b} d}-\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b} d}+\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}+1\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b} d} \]
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Rubi [A] time = 0.19, antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {247, 211, 1165, 628, 1162, 617, 204} \[ -\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {a}+\sqrt {b} (c+d x)^2\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b} d}+\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {a}+\sqrt {b} (c+d x)^2\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b} d}-\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b} d}+\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}+1\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b} d} \]
Antiderivative was successfully verified.
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Rule 204
Rule 211
Rule 247
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {1}{a+b (c+d x)^4} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,c+d x\right )}{d}\\ &=\frac {\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}+\frac {\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}\\ &=\frac {\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}+\frac {\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}-\frac {\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}-\frac {\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}\\ &=-\frac {\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}+\frac {\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}+\frac {\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}-\frac {\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}\\ &=-\frac {\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}+\frac {\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}-\frac {\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}+\frac {\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}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 161, normalized size = 0.73 \[ \frac {-\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {a}+\sqrt {b} (c+d x)^2\right )+\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {a}+\sqrt {b} (c+d x)^2\right )-2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )+2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b} d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 189, normalized size = 0.86 \[ \left (-\frac {1}{a^{3} b d^{4}}\right )^{\frac {1}{4}} \arctan \left (a^{2} b d^{4} \sqrt {\frac {a^{2} d^{2} \sqrt {-\frac {1}{a^{3} b d^{4}}} + d^{2} x^{2} + 2 \, c d x + c^{2}}{d^{2}}} \left (-\frac {1}{a^{3} b d^{4}}\right )^{\frac {3}{4}} - {\left (a^{2} b d^{4} x + a^{2} b c d^{3}\right )} \left (-\frac {1}{a^{3} b d^{4}}\right )^{\frac {3}{4}}\right ) + \frac {1}{4} \, \left (-\frac {1}{a^{3} b d^{4}}\right )^{\frac {1}{4}} \log \left (a d \left (-\frac {1}{a^{3} b d^{4}}\right )^{\frac {1}{4}} + d x + c\right ) - \frac {1}{4} \, \left (-\frac {1}{a^{3} b d^{4}}\right )^{\frac {1}{4}} \log \left (-a d \left (-\frac {1}{a^{3} b d^{4}}\right )^{\frac {1}{4}} + d x + c\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.50, size = 103, normalized size = 0.47 \[ -\frac {1}{2} \, \left (-\frac {1}{a^{3} b d^{4}}\right )^{\frac {1}{4}} \arctan \left (-\frac {b d x + b c}{\left (-a b^{3}\right )^{\frac {1}{4}}}\right ) + \frac {1}{4} \, \left (-\frac {1}{a^{3} b d^{4}}\right )^{\frac {1}{4}} \log \left ({\left | b d x + b c + \left (-a b^{3}\right )^{\frac {1}{4}} \right |}\right ) - \frac {1}{4} \, \left (-\frac {1}{a^{3} b d^{4}}\right )^{\frac {1}{4}} \log \left ({\left | -b d x - b c + \left (-a b^{3}\right )^{\frac {1}{4}} \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.00, size = 94, normalized size = 0.43 \[ \frac {\ln \left (-\RootOf \left (b \,d^{4} \textit {\_Z}^{4}+4 b \,d^{3} c \,\textit {\_Z}^{3}+6 b \,d^{2} c^{2} \textit {\_Z}^{2}+4 b \,c^{3} d \textit {\_Z} +b \,c^{4}+a \right )+x \right )}{4 b d \left (d^{3} \RootOf \left (b \,d^{4} \textit {\_Z}^{4}+4 b \,d^{3} c \,\textit {\_Z}^{3}+6 b \,d^{2} c^{2} \textit {\_Z}^{2}+4 b \,c^{3} d \textit {\_Z} +b \,c^{4}+a \right )^{3}+3 \RootOf \left (b \,d^{4} \textit {\_Z}^{4}+4 b \,d^{3} c \,\textit {\_Z}^{3}+6 b \,d^{2} c^{2} \textit {\_Z}^{2}+4 b \,c^{3} d \textit {\_Z} +b \,c^{4}+a \right )^{2} c \,d^{2}+3 \RootOf \left (b \,d^{4} \textit {\_Z}^{4}+4 b \,d^{3} c \,\textit {\_Z}^{3}+6 b \,d^{2} c^{2} \textit {\_Z}^{2}+4 b \,c^{3} d \textit {\_Z} +b \,c^{4}+a \right ) c^{2} d +c^{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (d x + c\right )}^{4} b + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 60, normalized size = 0.27 \[ -\frac {\mathrm {atan}\left (\frac {b^{1/4}\,c}{{\left (-a\right )}^{1/4}}+\frac {b^{1/4}\,d\,x}{{\left (-a\right )}^{1/4}}\right )+\mathrm {atanh}\left (\frac {b^{1/4}\,c}{{\left (-a\right )}^{1/4}}+\frac {b^{1/4}\,d\,x}{{\left (-a\right )}^{1/4}}\right )}{2\,{\left (-a\right )}^{3/4}\,b^{1/4}\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.30, size = 26, normalized size = 0.12 \[ \frac {\operatorname {RootSum} {\left (256 t^{4} a^{3} b + 1, \left (t \mapsto t \log {\left (x + \frac {4 t a + c}{d} \right )} \right )\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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