Optimal. Leaf size=116 \[ \frac {b \left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{16 a^2 x^4}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{6 a x^6}-\frac {b \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{32 a^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.07, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1128, 744, 734,
738, 212} \begin {gather*} -\frac {b \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{32 a^{5/2}}+\frac {b \left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{16 a^2 x^4}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{6 a x^6} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 212
Rule 734
Rule 738
Rule 744
Rule 1128
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x^2+c x^4}}{x^7} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {\sqrt {a+b x+c x^2}}{x^4} \, dx,x,x^2\right )\\ &=-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{6 a x^6}-\frac {b \text {Subst}\left (\int \frac {\sqrt {a+b x+c x^2}}{x^3} \, dx,x,x^2\right )}{4 a}\\ &=\frac {b \left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{16 a^2 x^4}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{6 a x^6}+\frac {\left (b \left (b^2-4 a c\right )\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{32 a^2}\\ &=\frac {b \left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{16 a^2 x^4}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{6 a x^6}-\frac {\left (b \left (b^2-4 a c\right )\right ) \text {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b x^2}{\sqrt {a+b x^2+c x^4}}\right )}{16 a^2}\\ &=\frac {b \left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{16 a^2 x^4}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{6 a x^6}-\frac {b \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{32 a^{5/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.41, size = 108, normalized size = 0.93 \begin {gather*} \frac {\sqrt {a+b x^2+c x^4} \left (-8 a^2-2 a b x^2+3 b^2 x^4-8 a c x^4\right )}{48 a^2 x^6}+\frac {\left (b^3-4 a b c\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x^2-\sqrt {a+b x^2+c x^4}}{\sqrt {a}}\right )}{16 a^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(221\) vs.
\(2(98)=196\).
time = 0.05, size = 222, normalized size = 1.91
method | result | size |
risch | \(-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (8 c \,x^{4} a -3 b^{2} x^{4}+2 a b \,x^{2}+8 a^{2}\right )}{48 x^{6} a^{2}}+\frac {b c \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{8 a^{\frac {3}{2}}}-\frac {b^{3} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{32 a^{\frac {5}{2}}}\) | \(133\) |
default | \(-\frac {\left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{6 a \,x^{6}}+\frac {b \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{8 a^{2} x^{4}}-\frac {b^{2} \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{16 a^{3} x^{2}}+\frac {b^{3} \sqrt {c \,x^{4}+b \,x^{2}+a}}{16 a^{3}}-\frac {b^{3} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{32 a^{\frac {5}{2}}}+\frac {b^{2} c \sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{16 a^{3}}-\frac {b c \sqrt {c \,x^{4}+b \,x^{2}+a}}{8 a^{2}}+\frac {b c \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{8 a^{\frac {3}{2}}}\) | \(222\) |
elliptic | \(-\frac {\left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{6 a \,x^{6}}+\frac {b \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{8 a^{2} x^{4}}-\frac {b^{2} \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{16 a^{3} x^{2}}+\frac {b^{3} \sqrt {c \,x^{4}+b \,x^{2}+a}}{16 a^{3}}-\frac {b^{3} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{32 a^{\frac {5}{2}}}+\frac {b^{2} c \sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{16 a^{3}}-\frac {b c \sqrt {c \,x^{4}+b \,x^{2}+a}}{8 a^{2}}+\frac {b c \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{8 a^{\frac {3}{2}}}\) | \(222\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.38, size = 261, normalized size = 2.25 \begin {gather*} \left [-\frac {3 \, {\left (b^{3} - 4 \, a b c\right )} \sqrt {a} x^{6} \log \left (-\frac {{\left (b^{2} + 4 \, a c\right )} x^{4} + 8 \, a b x^{2} + 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (b x^{2} + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{4}}\right ) + 4 \, {\left (2 \, a^{2} b x^{2} - {\left (3 \, a b^{2} - 8 \, a^{2} c\right )} x^{4} + 8 \, a^{3}\right )} \sqrt {c x^{4} + b x^{2} + a}}{192 \, a^{3} x^{6}}, \frac {3 \, {\left (b^{3} - 4 \, a b c\right )} \sqrt {-a} x^{6} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (b x^{2} + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{4} + a b x^{2} + a^{2}\right )}}\right ) - 2 \, {\left (2 \, a^{2} b x^{2} - {\left (3 \, a b^{2} - 8 \, a^{2} c\right )} x^{4} + 8 \, a^{3}\right )} \sqrt {c x^{4} + b x^{2} + a}}{96 \, a^{3} x^{6}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a + b x^{2} + c x^{4}}}{x^{7}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 359 vs.
\(2 (98) = 196\).
time = 4.36, size = 359, normalized size = 3.09 \begin {gather*} \frac {{\left (b^{3} - 4 \, a b c\right )} \arctan \left (-\frac {\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}}{\sqrt {-a}}\right )}{16 \, \sqrt {-a} a^{2}} - \frac {3 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{5} b^{3} - 12 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{5} a b c - 48 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{4} a^{2} c^{\frac {3}{2}} - 8 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{3} a b^{3} - 48 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{3} a^{2} b c - 48 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{2} a^{2} b^{2} \sqrt {c} - 3 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} a^{2} b^{3} - 36 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} a^{3} b c - 16 \, a^{4} c^{\frac {3}{2}}}{48 \, {\left ({\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{2} - a\right )}^{3} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {c\,x^4+b\,x^2+a}}{x^7} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________