Optimal. Leaf size=110 \[ \frac {b \sqrt {b c-a d} \tan ^{-1}\left (\frac {x^2 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^4}}\right )}{2 a^{5/2}}+\frac {\sqrt {c+d x^4} (3 b c-a d)}{6 a^2 c x^2}-\frac {\sqrt {c+d x^4}}{6 a x^6} \]
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Rubi [A] time = 0.16, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {465, 475, 583, 12, 377, 205} \begin {gather*} \frac {\sqrt {c+d x^4} (3 b c-a d)}{6 a^2 c x^2}+\frac {b \sqrt {b c-a d} \tan ^{-1}\left (\frac {x^2 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^4}}\right )}{2 a^{5/2}}-\frac {\sqrt {c+d x^4}}{6 a x^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 205
Rule 377
Rule 465
Rule 475
Rule 583
Rubi steps
\begin {align*} \int \frac {\sqrt {c+d x^4}}{x^7 \left (a+b x^4\right )} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt {c+d x^2}}{x^4 \left (a+b x^2\right )} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {c+d x^4}}{6 a x^6}+\frac {\operatorname {Subst}\left (\int \frac {-3 b c+a d-2 b d x^2}{x^2 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx,x,x^2\right )}{6 a}\\ &=-\frac {\sqrt {c+d x^4}}{6 a x^6}+\frac {(3 b c-a d) \sqrt {c+d x^4}}{6 a^2 c x^2}-\frac {\operatorname {Subst}\left (\int -\frac {3 b c (b c-a d)}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx,x,x^2\right )}{6 a^2 c}\\ &=-\frac {\sqrt {c+d x^4}}{6 a x^6}+\frac {(3 b c-a d) \sqrt {c+d x^4}}{6 a^2 c x^2}+\frac {(b (b c-a d)) \operatorname {Subst}\left (\int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx,x,x^2\right )}{2 a^2}\\ &=-\frac {\sqrt {c+d x^4}}{6 a x^6}+\frac {(3 b c-a d) \sqrt {c+d x^4}}{6 a^2 c x^2}+\frac {(b (b c-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 )}{2 a^2}\\ &=-\frac {\sqrt {c+d x^4}}{6 a x^6}+\frac {(3 b c-a d) \sqrt {c+d x^4}}{6 a^2 c x^2}+\frac {b \sqrt {b c-a d} \tan ^{-1}\left (\frac {\sqrt {b c-a d} x^2}{\sqrt {a} \sqrt {c+d x^4}}\right )}{2 a^{5/2}}\\ \end {align*}
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Mathematica [A] time = 5.33, size = 130, normalized size = 1.18 \begin {gather*} \frac {3 b c^2 x^4 \sqrt {\frac {d x^4}{c}+1} \sqrt {x^4 \left (\frac {b}{a}-\frac {d}{c}\right )} \sin ^{-1}\left (\frac {\sqrt {x^4 \left (\frac {b}{a}-\frac {d}{c}\right )}}{\sqrt {\frac {b x^4}{a}+1}}\right )+\left (c+d x^4\right ) \left (3 b c x^4-a \left (c+d x^4\right )\right )}{6 a^2 c x^6 \sqrt {c+d x^4}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.53, size = 151, normalized size = 1.37 \begin {gather*} \frac {b \sqrt {b c-a d} \tan ^{-1}\left (\frac {b \sqrt {d} x^4}{\sqrt {a} \sqrt {b c-a d}}+\frac {b x^2 \sqrt {c+d x^4}}{\sqrt {a} \sqrt {b c-a d}}+\frac {\sqrt {a} \sqrt {d}}{\sqrt {b c-a d}}\right )}{2 a^{5/2}}+\frac {\sqrt {c+d x^4} \left (-a c-a d x^4+3 b c x^4\right )}{6 a^2 c x^6} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 329, normalized size = 2.99 \begin {gather*} \left [\frac {3 \, b c x^{6} \sqrt {-\frac {b c - a d}{a}} \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 (a b c - 2 \, a^{2} d\right )} x^{6} - a^{2} c x^{2}\right )} \sqrt {d x^{4} + c} \sqrt {-\frac {b c - a d}{a}}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}\right ) + 4 \, {\left ({\left (3 \, b c - a d\right )} x^{4} - a c\right )} \sqrt {d x^{4} + c}}{24 \, a^{2} c x^{6}}, \frac {3 \, b c x^{6} \sqrt {\frac {b c - a d}{a}} \arctan \left (\frac {{\left ({\left (b c - 2 \, a d\right )} x^{4} - a c\right )} \sqrt {d x^{4} + c} \sqrt {\frac {b c - a d}{a}}}{2 \, {\left ({\left (b c d - a d^{2}\right )} x^{6} + {\left (b c^{2} - a c d\right )} x^{2}\right )}}\right ) + 2 \, {\left ({\left (3 \, b c - a d\right )} x^{4} - a c\right )} \sqrt {d x^{4} + c}}{12 \, a^{2} c x^{6}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.75, size = 225, normalized size = 2.05 \begin {gather*} -\frac {{\left (b^{2} c \sqrt {d} - a b d^{\frac {3}{2}}\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 )}{2 \, \sqrt {a b c d - a^{2} d^{2}} a^{2}} - \frac {3 \, {\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{4} b c \sqrt {d} - 3 \, {\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{4} a d^{\frac {3}{2}} - 6 \, {\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} b c^{2} \sqrt {d} + 3 \, b c^{3} \sqrt {d} - a c^{2} d^{\frac {3}{2}}}{3 \, {\left ({\left (\sqrt {d} x^{2} - \sqrt {d x^{4} + c}\right )}^{2} - c\right )}^{3} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.26, size = 1116, normalized size = 10.15 \begin {gather*} \frac {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 )}{4 \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}\, a}-\frac {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 )}{4 \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}\, a}-\frac {b^{2} c \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 {-a b}\, \sqrt {-\frac {a d -b c}{b}}\, a^{2}}+\frac {b^{2} c \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 {-a b}\, \sqrt {-\frac {a d -b c}{b}}\, a^{2}}-\frac {\sqrt {d \,x^{4}+c}\, b d \,x^{2}}{2 a^{2} c}-\frac {b \sqrt {d}\, \ln \left (\sqrt {d}\, x^{2}+\sqrt {d \,x^{4}+c}\right )}{2 a^{2}}+\frac {b \sqrt {d}\, \ln \left (\frac {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right ) d -\frac {\sqrt {-a b}\, d}{b}}{\sqrt {d}}+\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}}\right )}{4 a^{2}}+\frac {b \sqrt {d}\, \ln \left (\frac {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right ) d +\frac {\sqrt {-a b}\, d}{b}}{\sqrt {d}}+\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}}\right )}{4 a^{2}}-\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}}\, b^{2}}{4 \sqrt {-a b}\, a^{2}}+\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}}\, b^{2}}{4 \sqrt {-a b}\, a^{2}}+\frac {\left (d \,x^{4}+c \right )^{\frac {3}{2}} b}{2 a^{2} c \,x^{2}}-\frac {\left (d \,x^{4}+c \right )^{\frac {3}{2}}}{6 a c \,x^{6}} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {d x^{4} + c}}{{\left (b x^{4} + a\right )} x^{7}}\,{d x} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {d\,x^4+c}}{x^7\,\left (b\,x^4+a\right )} \,d x \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {c + d x^{4}}}{x^{7} \left (a + b x^{4}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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