Optimal. Leaf size=117 \[ -\frac {b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {b c-a d}}\right )}{2 a^2 \sqrt {b c-a d}}+\frac {(a d+2 b c) \tanh ^{-1}\left (\frac {\sqrt {c+d x^4}}{\sqrt {c}}\right )}{4 a^2 c^{3/2}}-\frac {\sqrt {c+d x^4}}{4 a c x^4} \]
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Rubi [A] time = 0.12, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {446, 103, 156, 63, 208} \[ -\frac {b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {b c-a d}}\right )}{2 a^2 \sqrt {b c-a d}}+\frac {(a d+2 b c) \tanh ^{-1}\left (\frac {\sqrt {c+d x^4}}{\sqrt {c}}\right )}{4 a^2 c^{3/2}}-\frac {\sqrt {c+d x^4}}{4 a c x^4} \]
Antiderivative was successfully verified.
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Rule 63
Rule 103
Rule 156
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {1}{x^5 \left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{x^2 (a+b x) \sqrt {c+d x}} \, dx,x,x^4\right )\\ &=-\frac {\sqrt {c+d x^4}}{4 a c x^4}-\frac {\operatorname {Subst}\left (\int \frac {\frac {1}{2} (2 b c+a d)+\frac {b d x}{2}}{x (a+b x) \sqrt {c+d x}} \, dx,x,x^4\right )}{4 a c}\\ &=-\frac {\sqrt {c+d x^4}}{4 a c x^4}+\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^4\right )}{4 a^2}-\frac {(2 b c+a d) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^4\right )}{8 a^2 c}\\ &=-\frac {\sqrt {c+d x^4}}{4 a c x^4}+\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x^4}\right )}{2 a^2 d}-\frac {(2 b c+a d) \operatorname {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^4}\right )}{4 a^2 c d}\\ &=-\frac {\sqrt {c+d x^4}}{4 a c x^4}+\frac {(2 b c+a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x^4}}{\sqrt {c}}\right )}{4 a^2 c^{3/2}}-\frac {b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {b c-a d}}\right )}{2 a^2 \sqrt {b c-a d}}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 151, normalized size = 1.29 \[ \frac {b^{3/2} \sqrt {b c-a d} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {b c-a d}}\right )}{2 a^2 (a d-b c)}+\frac {b \tanh ^{-1}\left (\frac {\sqrt {c+d x^4}}{\sqrt {c}}\right )}{2 a^2 \sqrt {c}}+\frac {d \tanh ^{-1}\left (\frac {\sqrt {c+d x^4}}{\sqrt {c}}\right )}{4 a c^{3/2}}-\frac {\sqrt {c+d x^4}}{4 a c x^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.03, size = 565, normalized size = 4.83 \[ \left [\frac {2 \, b c^{2} x^{4} \sqrt {\frac {b}{b c - a d}} \log \left (\frac {b d x^{4} + 2 \, b c - a d - 2 \, \sqrt {d x^{4} + c} {\left (b c - a d\right )} \sqrt {\frac {b}{b c - a d}}}{b x^{4} + a}\right ) + {\left (2 \, b c + a d\right )} \sqrt {c} x^{4} \log \left (\frac {d x^{4} + 2 \, \sqrt {d x^{4} + c} \sqrt {c} + 2 \, c}{x^{4}}\right ) - 2 \, \sqrt {d x^{4} + c} a c}{8 \, a^{2} c^{2} x^{4}}, -\frac {4 \, b c^{2} x^{4} \sqrt {-\frac {b}{b c - a d}} \arctan \left (-\frac {\sqrt {d x^{4} + c} {\left (b c - a d\right )} \sqrt {-\frac {b}{b c - a d}}}{b d x^{4} + b c}\right ) - {\left (2 \, b c + a d\right )} \sqrt {c} x^{4} \log \left (\frac {d x^{4} + 2 \, \sqrt {d x^{4} + c} \sqrt {c} + 2 \, c}{x^{4}}\right ) + 2 \, \sqrt {d x^{4} + c} a c}{8 \, a^{2} c^{2} x^{4}}, \frac {b c^{2} x^{4} \sqrt {\frac {b}{b c - a d}} \log \left (\frac {b d x^{4} + 2 \, b c - a d - 2 \, \sqrt {d x^{4} + c} {\left (b c - a d\right )} \sqrt {\frac {b}{b c - a d}}}{b x^{4} + a}\right ) - {\left (2 \, b c + a d\right )} \sqrt {-c} x^{4} \arctan \left (\frac {\sqrt {d x^{4} + c} \sqrt {-c}}{c}\right ) - \sqrt {d x^{4} + c} a c}{4 \, a^{2} c^{2} x^{4}}, -\frac {2 \, b c^{2} x^{4} \sqrt {-\frac {b}{b c - a d}} \arctan \left (-\frac {\sqrt {d x^{4} + c} {\left (b c - a d\right )} \sqrt {-\frac {b}{b c - a d}}}{b d x^{4} + b c}\right ) + {\left (2 \, b c + a d\right )} \sqrt {-c} x^{4} \arctan \left (\frac {\sqrt {d x^{4} + c} \sqrt {-c}}{c}\right ) + \sqrt {d x^{4} + c} a c}{4 \, a^{2} c^{2} x^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 104, normalized size = 0.89 \[ \frac {b^{2} \arctan \left (\frac {\sqrt {d x^{4} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{2 \, \sqrt {-b^{2} c + a b d} a^{2}} - \frac {{\left (2 \, b c + a d\right )} \arctan \left (\frac {\sqrt {d x^{4} + c}}{\sqrt {-c}}\right )}{4 \, a^{2} \sqrt {-c} c} - \frac {\sqrt {d x^{4} + c}}{4 \, a c x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.29, size = 402, normalized size = 3.44 \[ -\frac {b \ln \left (\frac {\frac {2 \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {2 \left (a d -b c \right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-\frac {a d -b c}{b}}\, a^{2}}-\frac {b \ln \left (\frac {-\frac {2 \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {2 \left (a d -b c \right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-\frac {a d -b c}{b}}\, a^{2}}+\frac {d \ln \left (\frac {2 c +2 \sqrt {d \,x^{4}+c}\, \sqrt {c}}{x^{2}}\right )}{4 a \,c^{\frac {3}{2}}}+\frac {b \ln \left (\frac {2 c +2 \sqrt {d \,x^{4}+c}\, \sqrt {c}}{x^{2}}\right )}{2 a^{2} \sqrt {c}}-\frac {\sqrt {d \,x^{4}+c}}{4 a c \,x^{4}} \]
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 \[ \int \frac {1}{{\left (b x^{4} + a\right )} \sqrt {d x^{4} + c} x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.35, size = 396, normalized size = 3.38 \[ \frac {\ln \left (\sqrt {d\,x^4+c}\,{\left (b^4\,c-a\,b^3\,d\right )}^{3/2}+b^6\,c^2+a^2\,b^4\,d^2-2\,a\,b^5\,c\,d\right )\,\sqrt {b^4\,c-a\,b^3\,d}}{4\,a^3\,d-4\,a^2\,b\,c}-\frac {\ln \left (\sqrt {d\,x^4+c}\,{\left (b^4\,c-a\,b^3\,d\right )}^{3/2}-b^6\,c^2-a^2\,b^4\,d^2+2\,a\,b^5\,c\,d\right )\,\sqrt {b^4\,c-a\,b^3\,d}}{4\,\left (a^3\,d-a^2\,b\,c\right )}-\frac {\sqrt {d\,x^4+c}}{4\,a\,c\,x^4}-\frac {\mathrm {atan}\left (\frac {b^4\,d^4\,\sqrt {d\,x^4+c}\,3{}\mathrm {i}}{16\,\sqrt {c^3}\,\left (\frac {3\,b^4\,d^4}{16\,c}+\frac {5\,a\,b^3\,d^5}{32\,c^2}+\frac {a^2\,b^2\,d^6}{32\,c^3}\right )}+\frac {b^2\,d^6\,\sqrt {d\,x^4+c}\,1{}\mathrm {i}}{32\,\sqrt {c^3}\,\left (\frac {5\,b^3\,d^5}{32\,a}+\frac {b^2\,d^6}{32\,c}+\frac {3\,b^4\,c\,d^4}{16\,a^2}\right )}+\frac {b^3\,d^5\,\sqrt {d\,x^4+c}\,5{}\mathrm {i}}{32\,\sqrt {c^3}\,\left (\frac {3\,b^4\,d^4}{16\,a}+\frac {5\,b^3\,d^5}{32\,c}+\frac {a\,b^2\,d^6}{32\,c^2}\right )}\right )\,\left (a\,d+2\,b\,c\right )\,1{}\mathrm {i}}{4\,a^2\,\sqrt {c^3}} \]
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 \[ \int \frac {1}{x^{5} \left (a + b x^{4}\right ) \sqrt {c + d x^{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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