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 = 0.26, antiderivative size = 261, normalized size of antiderivative = 1.00, 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 204
Rule 205
Rule 211
Rule 275
Rule 371
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1876
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*}
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Mathematica [C] time = 0.03, size = 104, normalized size = 0.40 \[ \frac {\text {RootSum}\left [\text {$\#$1}^4 b d^4+4 \text {$\#$1}^3 b c d^3+6 \text {$\#$1}^2 b c^2 d^2+4 \text {$\#$1} b c^3 d+a+b c^4\& ,\frac {\text {$\#$1} \log (x-\text {$\#$1})}{\text {$\#$1}^3 d^3+3 \text {$\#$1}^2 c d^2+3 \text {$\#$1} c^2 d+c^3}\& \right ]}{4 b d} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{{\left (d x + c\right )}^{4} b + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 95, normalized size = 0.36 \[ \frac {\RootOf \left (b \,d^{4} \textit {\_Z}^{4}+4 b \,d^{3} c \,\textit {\_Z}^{3}+6 b \,d^{2} c^{2} \textit {\_Z}^{2}+4 b \,c^{3} d \textit {\_Z} +b \,c^{4}+a \right ) \ln \left (-\RootOf \left (b \,d^{4} \textit {\_Z}^{4}+4 b \,d^{3} c \,\textit {\_Z}^{3}+6 b \,d^{2} c^{2} \textit {\_Z}^{2}+4 b \,c^{3} d \textit {\_Z} +b \,c^{4}+a \right )+x \right )}{4 b d \left (d^{3} \RootOf \left (b \,d^{4} \textit {\_Z}^{4}+4 b \,d^{3} c \,\textit {\_Z}^{3}+6 b \,d^{2} c^{2} \textit {\_Z}^{2}+4 b \,c^{3} d \textit {\_Z} +b \,c^{4}+a \right )^{3}+3 \RootOf \left (b \,d^{4} \textit {\_Z}^{4}+4 b \,d^{3} c \,\textit {\_Z}^{3}+6 b \,d^{2} c^{2} \textit {\_Z}^{2}+4 b \,c^{3} d \textit {\_Z} +b \,c^{4}+a \right )^{2} c \,d^{2}+3 \RootOf \left (b \,d^{4} \textit {\_Z}^{4}+4 b \,d^{3} c \,\textit {\_Z}^{3}+6 b \,d^{2} c^{2} \textit {\_Z}^{2}+4 b \,c^{3} d \textit {\_Z} +b \,c^{4}+a \right ) c^{2} d +c^{3}\right )} \]
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 \[ \int \frac {x}{{\left (d x + c\right )}^{4} b + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.36, size = 205, normalized size = 0.79 \[ \sum _{k=1}^4\ln \left (-\mathrm {root}\left (256\,a^3\,b^2\,d^8\,z^4+32\,a^2\,b\,d^4\,z^2-16\,a\,b\,c^2\,d^2\,z+b\,c^4+a,z,k\right )\,\left (-\mathrm {root}\left (256\,a^3\,b^2\,d^8\,z^4+32\,a^2\,b\,d^4\,z^2-16\,a\,b\,c^2\,d^2\,z+b\,c^4+a,z,k\right )\,\left (16\,a\,x\,b^3\,d^{12}+32\,a\,c\,b^3\,d^{11}\right )+4\,b^3\,c^3\,d^9+4\,b^3\,c^2\,d^{10}\,x\right )+b^2\,d^8\,x\right )\,\mathrm {root}\left (256\,a^3\,b^2\,d^8\,z^4+32\,a^2\,b\,d^4\,z^2-16\,a\,b\,c^2\,d^2\,z+b\,c^4+a,z,k\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.88, size = 131, normalized size = 0.50 \[ \operatorname {RootSum} {\left (256 t^{4} a^{3} b^{2} d^{8} + 32 t^{2} a^{2} b d^{4} - 16 t a b c^{2} d^{2} + a + b c^{4}, \left (t \mapsto t \log {\left (x + \frac {128 t^{3} a^{3} b d^{6} + 16 t^{2} a^{2} b c^{2} d^{4} + 8 t a^{2} d^{2} + 4 t a b c^{4} d^{2} - a c^{2} - b c^{6}}{4 a c d - b c^{5} d} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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