Optimal. Leaf size=113 \[ \frac {2 a+b x^2}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {3 b \left (b+2 c x^2\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {3 b c \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.06, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {1128, 652, 628,
632, 212} \begin {gather*} \frac {3 b c \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}+\frac {2 a+b x^2}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {3 b \left (b+2 c x^2\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 212
Rule 628
Rule 632
Rule 652
Rule 1128
Rubi steps
\begin {align*} \int \frac {x^3}{\left (a+b x^2+c x^4\right )^3} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x}{\left (a+b x+c x^2\right )^3} \, dx,x,x^2\right )\\ &=\frac {2 a+b x^2}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {(3 b) \text {Subst}\left (\int \frac {1}{\left (a+b x+c x^2\right )^2} \, dx,x,x^2\right )}{4 \left (b^2-4 a c\right )}\\ &=\frac {2 a+b x^2}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {3 b \left (b+2 c x^2\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {(3 b c) \text {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^2\right )}{2 \left (b^2-4 a c\right )^2}\\ &=\frac {2 a+b x^2}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {3 b \left (b+2 c x^2\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {(3 b c) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{\left (b^2-4 a c\right )^2}\\ &=\frac {2 a+b x^2}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {3 b \left (b+2 c x^2\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {3 b c \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.07, size = 114, normalized size = 1.01 \begin {gather*} \frac {\frac {\left (b^2-4 a c\right ) \left (2 a+b x^2\right )}{\left (a+b x^2+c x^4\right )^2}-\frac {3 b \left (b+2 c x^2\right )}{a+b x^2+c x^4}-\frac {12 b c \tan ^{-1}\left (\frac {b+2 c x^2}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}}{4 \left (b^2-4 a c\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.06, size = 128, normalized size = 1.13
method | result | size |
default | \(\frac {-b \,x^{2}-2 a}{4 \left (4 a c -b^{2}\right ) \left (c \,x^{4}+b \,x^{2}+a \right )^{2}}-\frac {3 b \left (\frac {2 c \,x^{2}+b}{\left (4 a c -b^{2}\right ) \left (c \,x^{4}+b \,x^{2}+a \right )}+\frac {4 c \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}}}\right )}{4 \left (4 a c -b^{2}\right )}\) | \(128\) |
risch | \(\frac {-\frac {3 b \,c^{2} x^{6}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {9 b^{2} c \,x^{4}}{4 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {\left (5 a c +b^{2}\right ) b \,x^{2}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {a \left (8 a c +b^{2}\right )}{4 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}}-\frac {3 c b \ln \left (\left (-\left (-4 a c +b^{2}\right )^{\frac {5}{2}}-16 a^{2} b \,c^{2}+8 a \,b^{3} c -b^{5}\right ) x^{2}-32 a^{3} c^{2}+16 a^{2} b^{2} c -2 b^{4} a \right )}{2 \left (-4 a c +b^{2}\right )^{\frac {5}{2}}}+\frac {3 c b \ln \left (\left (-\left (-4 a c +b^{2}\right )^{\frac {5}{2}}+16 a^{2} b \,c^{2}-8 a \,b^{3} c +b^{5}\right ) x^{2}+32 a^{3} c^{2}-16 a^{2} b^{2} c +2 b^{4} a \right )}{2 \left (-4 a c +b^{2}\right )^{\frac {5}{2}}}\) | \(297\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 392 vs.
\(2 (105) = 210\).
time = 0.36, size = 808, normalized size = 7.15 \begin {gather*} \left [-\frac {6 \, {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{6} + a b^{4} + 4 \, a^{2} b^{2} c - 32 \, a^{3} c^{2} + 9 \, {\left (b^{4} c - 4 \, a b^{2} c^{2}\right )} x^{4} + 2 \, {\left (b^{5} + a b^{3} c - 20 \, a^{2} b c^{2}\right )} x^{2} - 6 \, {\left (b c^{3} x^{8} + 2 \, b^{2} c^{2} x^{6} + 2 \, a b^{2} c x^{2} + {\left (b^{3} c + 2 \, a b c^{2}\right )} x^{4} + a^{2} b c\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c + {\left (2 \, c x^{2} + b\right )} \sqrt {b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right )}{4 \, {\left ({\left (b^{6} c^{2} - 12 \, a b^{4} c^{3} + 48 \, a^{2} b^{2} c^{4} - 64 \, a^{3} c^{5}\right )} x^{8} + a^{2} b^{6} - 12 \, a^{3} b^{4} c + 48 \, a^{4} b^{2} c^{2} - 64 \, a^{5} c^{3} + 2 \, {\left (b^{7} c - 12 \, a b^{5} c^{2} + 48 \, a^{2} b^{3} c^{3} - 64 \, a^{3} b c^{4}\right )} x^{6} + {\left (b^{8} - 10 \, a b^{6} c + 24 \, a^{2} b^{4} c^{2} + 32 \, a^{3} b^{2} c^{3} - 128 \, a^{4} c^{4}\right )} x^{4} + 2 \, {\left (a b^{7} - 12 \, a^{2} b^{5} c + 48 \, a^{3} b^{3} c^{2} - 64 \, a^{4} b c^{3}\right )} x^{2}\right )}}, -\frac {6 \, {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{6} + a b^{4} + 4 \, a^{2} b^{2} c - 32 \, a^{3} c^{2} + 9 \, {\left (b^{4} c - 4 \, a b^{2} c^{2}\right )} x^{4} + 2 \, {\left (b^{5} + a b^{3} c - 20 \, a^{2} b c^{2}\right )} x^{2} - 12 \, {\left (b c^{3} x^{8} + 2 \, b^{2} c^{2} x^{6} + 2 \, a b^{2} c x^{2} + {\left (b^{3} c + 2 \, a b c^{2}\right )} x^{4} + a^{2} b c\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {{\left (2 \, c x^{2} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right )}{4 \, {\left ({\left (b^{6} c^{2} - 12 \, a b^{4} c^{3} + 48 \, a^{2} b^{2} c^{4} - 64 \, a^{3} c^{5}\right )} x^{8} + a^{2} b^{6} - 12 \, a^{3} b^{4} c + 48 \, a^{4} b^{2} c^{2} - 64 \, a^{5} c^{3} + 2 \, {\left (b^{7} c - 12 \, a b^{5} c^{2} + 48 \, a^{2} b^{3} c^{3} - 64 \, a^{3} b c^{4}\right )} x^{6} + {\left (b^{8} - 10 \, a b^{6} c + 24 \, a^{2} b^{4} c^{2} + 32 \, a^{3} b^{2} c^{3} - 128 \, a^{4} c^{4}\right )} x^{4} + 2 \, {\left (a b^{7} - 12 \, a^{2} b^{5} c + 48 \, a^{3} b^{3} c^{2} - 64 \, a^{4} b c^{3}\right )} x^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 491 vs.
\(2 (105) = 210\).
time = 1.40, size = 491, normalized size = 4.35 \begin {gather*} \frac {3 b c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \log {\left (x^{2} + \frac {- 192 a^{3} b c^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 144 a^{2} b^{3} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} - 36 a b^{5} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 3 b^{7} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 3 b^{2} c}{6 b c^{2}} \right )}}{2} - \frac {3 b c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \log {\left (x^{2} + \frac {192 a^{3} b c^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} - 144 a^{2} b^{3} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 36 a b^{5} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} - 3 b^{7} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 3 b^{2} c}{6 b c^{2}} \right )}}{2} + \frac {- 8 a^{2} c - a b^{2} - 9 b^{2} c x^{4} - 6 b c^{2} x^{6} + x^{2} \left (- 10 a b c - 2 b^{3}\right )}{64 a^{4} c^{2} - 32 a^{3} b^{2} c + 4 a^{2} b^{4} + x^{8} \cdot \left (64 a^{2} c^{4} - 32 a b^{2} c^{3} + 4 b^{4} c^{2}\right ) + x^{6} \cdot \left (128 a^{2} b c^{3} - 64 a b^{3} c^{2} + 8 b^{5} c\right ) + x^{4} \cdot \left (128 a^{3} c^{3} - 24 a b^{4} c + 4 b^{6}\right ) + x^{2} \cdot \left (128 a^{3} b c^{2} - 64 a^{2} b^{3} c + 8 a b^{5}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 6.00, size = 143, normalized size = 1.27 \begin {gather*} -\frac {3 \, b c \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {6 \, b c^{2} x^{6} + 9 \, b^{2} c x^{4} + 2 \, b^{3} x^{2} + 10 \, a b c x^{2} + a b^{2} + 8 \, a^{2} c}{4 \, {\left (c x^{4} + b x^{2} + a\right )}^{2} {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 4.39, size = 400, normalized size = 3.54 \begin {gather*} -\frac {\frac {8\,c\,a^2+a\,b^2}{4\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {x^2\,\left (b^3+5\,a\,c\,b\right )}{2\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {9\,b^2\,c\,x^4}{4\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {3\,b\,c^2\,x^6}{2\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}}{x^4\,\left (b^2+2\,a\,c\right )+a^2+c^2\,x^8+2\,a\,b\,x^2+2\,b\,c\,x^6}-\frac {3\,b\,c\,\mathrm {atan}\left (\frac {\left (x^2\,\left (\frac {9\,b^2\,c^4}{a\,{\left (4\,a\,c-b^2\right )}^{9/2}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {b^3\,c^2\,\left (144\,a^2\,b\,c^4-72\,a\,b^3\,c^3+9\,b^5\,c^2\right )}{a\,{\left (4\,a\,c-b^2\right )}^{15/2}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}\right )+\frac {18\,b^3\,c^4}{{\left (4\,a\,c-b^2\right )}^{15/2}}\right )\,\left (b^4\,{\left (4\,a\,c-b^2\right )}^5+16\,a^2\,c^2\,{\left (4\,a\,c-b^2\right )}^5-8\,a\,b^2\,c\,{\left (4\,a\,c-b^2\right )}^5\right )}{18\,b^2\,c^4}\right )}{{\left (4\,a\,c-b^2\right )}^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________