Optimal. Leaf size=145 \[ \frac {b \left (5 b^2-12 a c\right ) \tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{32 a^{7/2}}-\frac {\left (15 b^2-16 a c\right ) \sqrt {a+b x^2+c x^4}}{48 a^3 x^2}+\frac {5 b \sqrt {a+b x^2+c x^4}}{24 a^2 x^4}-\frac {\sqrt {a+b x^2+c x^4}}{6 a x^6} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.17, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1114, 744, 834, 806, 724, 206} \[ -\frac {\left (15 b^2-16 a c\right ) \sqrt {a+b x^2+c x^4}}{48 a^3 x^2}+\frac {b \left (5 b^2-12 a c\right ) \tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{32 a^{7/2}}+\frac {5 b \sqrt {a+b x^2+c x^4}}{24 a^2 x^4}-\frac {\sqrt {a+b x^2+c x^4}}{6 a x^6} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 724
Rule 744
Rule 806
Rule 834
Rule 1114
Rubi steps
\begin {align*} \int \frac {1}{x^7 \sqrt {a+b x^2+c x^4}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^4 \sqrt {a+b x+c x^2}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {a+b x^2+c x^4}}{6 a x^6}-\frac {\operatorname {Subst}\left (\int \frac {\frac {5 b}{2}+2 c x}{x^3 \sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{6 a}\\ &=-\frac {\sqrt {a+b x^2+c x^4}}{6 a x^6}+\frac {5 b \sqrt {a+b x^2+c x^4}}{24 a^2 x^4}+\frac {\operatorname {Subst}\left (\int \frac {\frac {1}{4} \left (15 b^2-16 a c\right )+\frac {5 b c x}{2}}{x^2 \sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{12 a^2}\\ &=-\frac {\sqrt {a+b x^2+c x^4}}{6 a x^6}+\frac {5 b \sqrt {a+b x^2+c x^4}}{24 a^2 x^4}-\frac {\left (15 b^2-16 a c\right ) \sqrt {a+b x^2+c x^4}}{48 a^3 x^2}-\frac {\left (b \left (5 b^2-12 a c\right )\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{32 a^3}\\ &=-\frac {\sqrt {a+b x^2+c x^4}}{6 a x^6}+\frac {5 b \sqrt {a+b x^2+c x^4}}{24 a^2 x^4}-\frac {\left (15 b^2-16 a c\right ) \sqrt {a+b x^2+c x^4}}{48 a^3 x^2}+\frac {\left (b \left (5 b^2-12 a c\right )\right ) \operatorname {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^3}\\ &=-\frac {\sqrt {a+b x^2+c x^4}}{6 a x^6}+\frac {5 b \sqrt {a+b x^2+c x^4}}{24 a^2 x^4}-\frac {\left (15 b^2-16 a c\right ) \sqrt {a+b x^2+c x^4}}{48 a^3 x^2}+\frac {b \left (5 b^2-12 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^{7/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.08, size = 112, normalized size = 0.77 \[ \frac {b \left (5 b^2-12 a c\right ) \tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{32 a^{7/2}}+\frac {\sqrt {a+b x^2+c x^4} \left (-8 a^2+2 a \left (5 b x^2+8 c x^4\right )-15 b^2 x^4\right )}{48 a^3 x^6} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.92, size = 265, normalized size = 1.83 \[ \left [-\frac {3 \, {\left (5 \, b^{3} - 12 \, a b c\right )} \sqrt {a} x^{6} \log \left (-\frac {{\left (b^{2} + 4 \, a c\right )} x^{4} + 8 \, a b x^{2} - 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (b x^{2} + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{4}}\right ) - 4 \, {\left (10 \, a^{2} b x^{2} - {\left (15 \, a b^{2} - 16 \, a^{2} c\right )} x^{4} - 8 \, a^{3}\right )} \sqrt {c x^{4} + b x^{2} + a}}{192 \, a^{4} x^{6}}, -\frac {3 \, {\left (5 \, b^{3} - 12 \, a b c\right )} \sqrt {-a} x^{6} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (b x^{2} + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{4} + a b x^{2} + a^{2}\right )}}\right ) - 2 \, {\left (10 \, a^{2} b x^{2} - {\left (15 \, a b^{2} - 16 \, a^{2} c\right )} x^{4} - 8 \, a^{3}\right )} \sqrt {c x^{4} + b x^{2} + a}}{96 \, a^{4} x^{6}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.26, size = 335, normalized size = 2.31 \[ -\frac {{\left (5 \, b^{3} - 12 \, a b c\right )} \arctan \left (-\frac {\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}}{\sqrt {-a}}\right )}{16 \, \sqrt {-a} a^{3}} + \frac {15 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{5} b^{3} - 36 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{5} a b c - 40 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{3} a b^{3} + 96 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{3} a^{2} b c + 96 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{2} a^{3} c^{\frac {3}{2}} + 33 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} a^{2} b^{3} + 36 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} a^{3} b c + 48 \, a^{3} b^{2} \sqrt {c} - 32 \, a^{4} c^{\frac {3}{2}}}{48 \, {\left ({\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{2} - a\right )}^{3} a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 176, normalized size = 1.21 \[ -\frac {3 b c \ln \left (\frac {b \,x^{2}+2 a +2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {a}}{x^{2}}\right )}{8 a^{\frac {5}{2}}}+\frac {5 b^{3} \ln \left (\frac {b \,x^{2}+2 a +2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {a}}{x^{2}}\right )}{32 a^{\frac {7}{2}}}+\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, c}{3 a^{2} x^{2}}-\frac {5 \sqrt {c \,x^{4}+b \,x^{2}+a}\, b^{2}}{16 a^{3} x^{2}}+\frac {5 \sqrt {c \,x^{4}+b \,x^{2}+a}\, b}{24 a^{2} x^{4}}-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{6 a \,x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x^7\,\sqrt {c\,x^4+b\,x^2+a}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{7} \sqrt {a + b x^{2} + c x^{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________