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 = 0.23, antiderivative size = 184, normalized size of antiderivative = 1.00, 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 206
Rule 217
Rule 220
Rule 275
Rule 1680
Rule 1885
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*}
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Mathematica [C] time = 0.58, 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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fricas [F] time = 1.29, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x}{\sqrt {b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \, b c^{3} d x + b c^{4} + a}}, x\right ) \]
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 \[ \int \frac {x}{\sqrt {b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \, b c^{3} d x + b c^{4} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.42, size = 1528, normalized size = 8.30 \[ \text {result too large to display} \]
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 \[ \int \frac {x}{\sqrt {b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \, b c^{3} d x + b c^{4} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x}{\sqrt {b\,c^4+4\,b\,c^3\,d\,x+6\,b\,c^2\,d^2\,x^2+4\,b\,c\,d^3\,x^3+b\,d^4\,x^4+a}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\sqrt {a + b c^{4} + 4 b c^{3} d x + 6 b c^{2} d^{2} x^{2} + 4 b c d^{3} x^{3} + b d^{4} x^{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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