Optimal. Leaf size=104 \[ \frac {(b c-2 a d) \tan ^{-1}\left (\frac {x^2 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^4}}\right )}{4 a^{3/2} (b c-a d)^{3/2}}+\frac {b x^2 \sqrt {c+d x^4}}{4 a \left (a+b x^4\right ) (b c-a d)} \]
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Rubi [A] time = 0.09, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {465, 382, 377, 205} \[ \frac {(b c-2 a d) \tan ^{-1}\left (\frac {x^2 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^4}}\right )}{4 a^{3/2} (b c-a d)^{3/2}}+\frac {b x^2 \sqrt {c+d x^4}}{4 a \left (a+b x^4\right ) (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 205
Rule 377
Rule 382
Rule 465
Rubi steps
\begin {align*} \int \frac {x}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx,x,x^2\right )\\ &=\frac {b x^2 \sqrt {c+d x^4}}{4 a (b c-a d) \left (a+b x^4\right )}+\frac {(b c-2 a d) \operatorname {Subst}\left (\int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx,x,x^2\right )}{4 a (b c-a d)}\\ &=\frac {b x^2 \sqrt {c+d x^4}}{4 a (b c-a d) \left (a+b x^4\right )}+\frac {(b c-2 a d) \operatorname {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x^2}{\sqrt {c+d x^4}}\right )}{4 a (b c-a d)}\\ &=\frac {b x^2 \sqrt {c+d x^4}}{4 a (b c-a d) \left (a+b x^4\right )}+\frac {(b c-2 a d) \tan ^{-1}\left (\frac {\sqrt {b c-a d} x^2}{\sqrt {a} \sqrt {c+d x^4}}\right )}{4 a^{3/2} (b c-a d)^{3/2}}\\ \end {align*}
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Mathematica [C] time = 6.01, size = 407, normalized size = 3.91 \[ \frac {x^2 \sqrt {c+d x^4} \left (-30 d x^4 \sqrt {\frac {a x^4 \left (c+d x^4\right ) (b c-a d)}{c^2 \left (a+b x^4\right )^2}}-45 c \sqrt {\frac {a x^4 \left (c+d x^4\right ) (b c-a d)}{c^2 \left (a+b x^4\right )^2}}+16 d x^4 \left (\frac {x^4 (b c-a d)}{c \left (a+b x^4\right )}\right )^{5/2} \sqrt {\frac {a \left (c+d x^4\right )}{c \left (a+b x^4\right )}} \, _2F_1\left (2,3;\frac {7}{2};\frac {(b c-a d) x^4}{c \left (b x^4+a\right )}\right )+16 c \left (\frac {x^4 (b c-a d)}{c \left (a+b x^4\right )}\right )^{5/2} \sqrt {\frac {a \left (c+d x^4\right )}{c \left (a+b x^4\right )}} \, _2F_1\left (2,3;\frac {7}{2};\frac {(b c-a d) x^4}{c \left (b x^4+a\right )}\right )+30 d x^4 \sin ^{-1}\left (\sqrt {\frac {x^4 (b c-a d)}{c \left (a+b x^4\right )}}\right )+45 c \sin ^{-1}\left (\sqrt {\frac {x^4 (b c-a d)}{c \left (a+b x^4\right )}}\right )\right )}{60 c^2 \left (a+b x^4\right )^2 \left (\frac {x^4 (b c-a d)}{c \left (a+b x^4\right )}\right )^{3/2} \sqrt {\frac {a \left (c+d x^4\right )}{c \left (a+b x^4\right )}}} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.92, size = 467, normalized size = 4.49 \[ \left [\frac {4 \, \sqrt {d x^{4} + c} {\left (a b^{2} c - a^{2} b d\right )} x^{2} - {\left ({\left (b^{2} c - 2 \, a b d\right )} x^{4} + a b c - 2 \, a^{2} d\right )} \sqrt {-a b c + a^{2} d} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{8} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{4} + a^{2} c^{2} - 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{6} - a c x^{2}\right )} \sqrt {d x^{4} + c} \sqrt {-a b c + a^{2} d}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}\right )}{16 \, {\left (a^{3} b^{2} c^{2} - 2 \, a^{4} b c d + a^{5} d^{2} + {\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2}\right )} x^{4}\right )}}, \frac {2 \, \sqrt {d x^{4} + c} {\left (a b^{2} c - a^{2} b d\right )} x^{2} + {\left ({\left (b^{2} c - 2 \, a b d\right )} x^{4} + a b c - 2 \, a^{2} d\right )} \sqrt {a b c - a^{2} d} \arctan \left (\frac {{\left ({\left (b c - 2 \, a d\right )} x^{4} - a c\right )} \sqrt {d x^{4} + c} \sqrt {a b c - a^{2} d}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{6} + {\left (a b c^{2} - a^{2} c d\right )} x^{2}\right )}}\right )}{8 \, {\left (a^{3} b^{2} c^{2} - 2 \, a^{4} b c d + a^{5} d^{2} + {\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2}\right )} x^{4}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.38, size = 237, normalized size = 2.28 \[ -\frac {1}{4} \, d^{\frac {3}{2}} {\left (\frac {{\left (b c - 2 \, a d\right )} \arctan \left (\frac {{\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{{\left (a b c d - a^{2} d^{2}\right )}^{\frac {3}{2}}} + \frac {2 \, {\left ({\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} b c - 2 \, {\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} a d - b c^{2}\right )}}{{\left ({\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{4} b - 2 \, {\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} b c + 4 \, {\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} a d + b c^{2}\right )} {\left (a b c d - a^{2} d^{2}\right )}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.19, size = 867, normalized size = 8.34 \[ \frac {\sqrt {-a b}\, d \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 )}{8 \left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}\, a b}-\frac {\sqrt {-a b}\, d \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 )}{8 \left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}\, a b}-\frac {\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 )}{8 \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}\, a}+\frac {\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 )}{8 \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}\, a}-\frac {\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}}}{8 \left (a d -b c \right ) \left (x^{2}-\frac {\sqrt {-a b}}{b}\right ) a}-\frac {\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}}}{8 \left (a d -b c \right ) \left (x^{2}+\frac {\sqrt {-a b}}{b}\right ) a} \]
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 \[ \int \frac {x}{{\left (b x^{4} + a\right )}^{2} \sqrt {d x^{4} + c}}\,{d x} \]
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 \[ \int \frac {x}{{\left (b\,x^4+a\right )}^2\,\sqrt {d\,x^4+c}} \,d x \]
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 \[ \int \frac {x}{\left (a + b x^{4}\right )^{2} \sqrt {c + d x^{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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