Optimal. Leaf size=195 \[ \frac {b^2-2 a c+b c x^2}{a \left (b^2-4 a c\right ) x^4 \sqrt {a+b x^2+c x^4}}-\frac {\left (5 b^2-12 a c\right ) \sqrt {a+b x^2+c x^4}}{4 a^2 \left (b^2-4 a c\right ) x^4}+\frac {b \left (15 b^2-52 a c\right ) \sqrt {a+b x^2+c x^4}}{8 a^3 \left (b^2-4 a c\right ) x^2}-\frac {3 \left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{16 a^{7/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.15, antiderivative size = 195, 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 = {1128, 754, 848,
820, 738, 212} \begin {gather*} -\frac {3 \left (5 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 )}{16 a^{7/2}}+\frac {b \left (15 b^2-52 a c\right ) \sqrt {a+b x^2+c x^4}}{8 a^3 x^2 \left (b^2-4 a c\right )}-\frac {\left (5 b^2-12 a c\right ) \sqrt {a+b x^2+c x^4}}{4 a^2 x^4 \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x^2}{a x^4 \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 212
Rule 738
Rule 754
Rule 820
Rule 848
Rule 1128
Rubi steps
\begin {align*} \int \frac {1}{x^5 \left (a+b x^2+c x^4\right )^{3/2}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^3 \left (a+b x+c x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=\frac {b^2-2 a c+b c x^2}{a \left (b^2-4 a c\right ) x^4 \sqrt {a+b x^2+c x^4}}-\frac {\text {Subst}\left (\int \frac {\frac {1}{2} \left (-5 b^2+12 a c\right )-2 b c x}{x^3 \sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{a \left (b^2-4 a c\right )}\\ &=\frac {b^2-2 a c+b c x^2}{a \left (b^2-4 a c\right ) x^4 \sqrt {a+b x^2+c x^4}}-\frac {\left (5 b^2-12 a c\right ) \sqrt {a+b x^2+c x^4}}{4 a^2 \left (b^2-4 a c\right ) x^4}+\frac {\text {Subst}\left (\int \frac {-\frac {1}{4} b \left (15 b^2-52 a c\right )-\frac {1}{2} c \left (5 b^2-12 a c\right ) x}{x^2 \sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{2 a^2 \left (b^2-4 a c\right )}\\ &=\frac {b^2-2 a c+b c x^2}{a \left (b^2-4 a c\right ) x^4 \sqrt {a+b x^2+c x^4}}-\frac {\left (5 b^2-12 a c\right ) \sqrt {a+b x^2+c x^4}}{4 a^2 \left (b^2-4 a c\right ) x^4}+\frac {b \left (15 b^2-52 a c\right ) \sqrt {a+b x^2+c x^4}}{8 a^3 \left (b^2-4 a c\right ) x^2}+\frac {\left (3 \left (5 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 )}{16 a^3}\\ &=\frac {b^2-2 a c+b c x^2}{a \left (b^2-4 a c\right ) x^4 \sqrt {a+b x^2+c x^4}}-\frac {\left (5 b^2-12 a c\right ) \sqrt {a+b x^2+c x^4}}{4 a^2 \left (b^2-4 a c\right ) x^4}+\frac {b \left (15 b^2-52 a c\right ) \sqrt {a+b x^2+c x^4}}{8 a^3 \left (b^2-4 a c\right ) x^2}-\frac {\left (3 \left (5 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 )}{8 a^3}\\ &=\frac {b^2-2 a c+b c x^2}{a \left (b^2-4 a c\right ) x^4 \sqrt {a+b x^2+c x^4}}-\frac {\left (5 b^2-12 a c\right ) \sqrt {a+b x^2+c x^4}}{4 a^2 \left (b^2-4 a c\right ) x^4}+\frac {b \left (15 b^2-52 a c\right ) \sqrt {a+b x^2+c x^4}}{8 a^3 \left (b^2-4 a c\right ) x^2}-\frac {3 \left (5 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 )}{16 a^{7/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.69, size = 166, normalized size = 0.85 \begin {gather*} \frac {-8 a^3 c-15 b^3 x^4 \left (b+c x^2\right )+2 a^2 \left (b^2+10 b c x^2-12 c^2 x^4\right )+a b x^2 \left (-5 b^2+62 b c x^2+52 c^2 x^4\right )}{8 a^3 \left (-b^2+4 a c\right ) x^4 \sqrt {a+b x^2+c x^4}}+\frac {3 \left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x^2-\sqrt {a+b x^2+c x^4}}{\sqrt {a}}\right )}{8 a^{7/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.09, size = 314, normalized size = 1.61
method | result | size |
default | \(-\frac {1}{4 a \,x^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {5 b}{8 a^{2} x^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {15 b^{2}}{16 a^{3} \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {15 b^{3} x^{2} c}{8 a^{3} \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {15 b^{4}}{16 a^{3} \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {15 b^{2} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{16 a^{\frac {7}{2}}}+\frac {13 b \,c^{2} x^{2}}{2 a^{2} \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {13 b^{2} c}{4 a^{2} \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {3 c}{4 a^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {3 c \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{4 a^{\frac {5}{2}}}\) | \(314\) |
elliptic | \(-\frac {1}{4 a \,x^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {5 b}{8 a^{2} x^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {15 b^{2}}{16 a^{3} \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {15 b^{3} x^{2} c}{8 a^{3} \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {15 b^{4}}{16 a^{3} \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {15 b^{2} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{16 a^{\frac {7}{2}}}+\frac {13 b \,c^{2} x^{2}}{2 a^{2} \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {13 b^{2} c}{4 a^{2} \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {3 c}{4 a^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {3 c \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{4 a^{\frac {5}{2}}}\) | \(314\) |
risch | \(-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (-7 b \,x^{2}+2 a \right )}{8 a^{3} x^{4}}+\frac {3 b \,c^{2} x^{2}}{a^{2} \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {c^{2}}{a \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (4 a c -b^{2}\right )}-\frac {b^{3} x^{2} c}{a^{3} \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {b^{2} c}{2 a^{2} \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {b^{4}}{16 a^{3} \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {3 c}{4 a^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {15 b^{2}}{16 a^{3} \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {3 c \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{4 a^{\frac {5}{2}}}-\frac {15 b^{2} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{16 a^{\frac {7}{2}}}\) | \(334\) |
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 = 0.46, size = 615, normalized size = 3.15 \begin {gather*} \left [-\frac {3 \, {\left ({\left (5 \, b^{4} c - 24 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{8} + {\left (5 \, b^{5} - 24 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x^{6} + {\left (5 \, a b^{4} - 24 \, a^{2} b^{2} c + 16 \, a^{3} c^{2}\right )} x^{4}\right )} \sqrt {a} \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 ({\left (15 \, a b^{3} c - 52 \, a^{2} b c^{2}\right )} x^{6} - 2 \, a^{3} b^{2} + 8 \, a^{4} c + {\left (15 \, a b^{4} - 62 \, a^{2} b^{2} c + 24 \, a^{3} c^{2}\right )} x^{4} + 5 \, {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{32 \, {\left ({\left (a^{4} b^{2} c - 4 \, a^{5} c^{2}\right )} x^{8} + {\left (a^{4} b^{3} - 4 \, a^{5} b c\right )} x^{6} + {\left (a^{5} b^{2} - 4 \, a^{6} c\right )} x^{4}\right )}}, \frac {3 \, {\left ({\left (5 \, b^{4} c - 24 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{8} + {\left (5 \, b^{5} - 24 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x^{6} + {\left (5 \, a b^{4} - 24 \, a^{2} b^{2} c + 16 \, a^{3} c^{2}\right )} x^{4}\right )} \sqrt {-a} \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 ({\left (15 \, a b^{3} c - 52 \, a^{2} b c^{2}\right )} x^{6} - 2 \, a^{3} b^{2} + 8 \, a^{4} c + {\left (15 \, a b^{4} - 62 \, a^{2} b^{2} c + 24 \, a^{3} c^{2}\right )} x^{4} + 5 \, {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{16 \, {\left ({\left (a^{4} b^{2} c - 4 \, a^{5} c^{2}\right )} x^{8} + {\left (a^{4} b^{3} - 4 \, a^{5} b c\right )} x^{6} + {\left (a^{5} b^{2} - 4 \, a^{6} c\right )} x^{4}\right )}}\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 {1}{x^{5} \left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 5.01, size = 350, normalized size = 1.79 \begin {gather*} \frac {\frac {{\left (a^{3} b^{3} c - 3 \, a^{4} b c^{2}\right )} x^{2}}{a^{6} b^{2} - 4 \, a^{7} c} + \frac {a^{3} b^{4} - 4 \, a^{4} b^{2} c + 2 \, a^{5} c^{2}}{a^{6} b^{2} - 4 \, a^{7} c}}{\sqrt {c x^{4} + b x^{2} + a}} + \frac {3 \, {\left (5 \, b^{2} - 4 \, a c\right )} \arctan \left (-\frac {\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}}{\sqrt {-a}}\right )}{8 \, \sqrt {-a} a^{3}} - \frac {7 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{3} b^{2} - 4 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{3} a c + 8 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{2} a b \sqrt {c} - 9 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} a b^{2} - 4 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} a^{2} c - 16 \, a^{2} b \sqrt {c}}{8 \, {\left ({\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{2} - a\right )}^{2} a^{3}} \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 {1}{x^5\,{\left (c\,x^4+b\,x^2+a\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________