Optimal. Leaf size=216 \[ \frac {\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{1024 a^4 x^4}-\frac {\left (7 b^2-4 a c\right ) \left (2 a+b x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{384 a^3 x^8}-\frac {\left (a+b x^2+c x^4\right )^{5/2}}{12 a x^{12}}+\frac {7 b \left (a+b x^2+c x^4\right )^{5/2}}{120 a^2 x^{10}}-\frac {\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) \tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{2048 a^{9/2}} \]
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Rubi [A]
time = 0.14, antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1128, 758, 820,
734, 738, 212} \begin {gather*} -\frac {\left (b^2-4 a c\right )^2 \left (7 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 )}{2048 a^{9/2}}+\frac {\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{1024 a^4 x^4}-\frac {\left (7 b^2-4 a c\right ) \left (2 a+b x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{384 a^3 x^8}+\frac {7 b \left (a+b x^2+c x^4\right )^{5/2}}{120 a^2 x^{10}}-\frac {\left (a+b x^2+c x^4\right )^{5/2}}{12 a x^{12}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 734
Rule 738
Rule 758
Rule 820
Rule 1128
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^{13}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^7} \, dx,x,x^2\right )\\ &=-\frac {\left (a+b x^2+c x^4\right )^{5/2}}{12 a x^{12}}-\frac {\text {Subst}\left (\int \frac {\left (\frac {7 b}{2}+c x\right ) \left (a+b x+c x^2\right )^{3/2}}{x^6} \, dx,x,x^2\right )}{12 a}\\ &=-\frac {\left (a+b x^2+c x^4\right )^{5/2}}{12 a x^{12}}+\frac {7 b \left (a+b x^2+c x^4\right )^{5/2}}{120 a^2 x^{10}}+\frac {\left (7 b^2-4 a c\right ) \text {Subst}\left (\int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^5} \, dx,x,x^2\right )}{48 a^2}\\ &=-\frac {\left (7 b^2-4 a c\right ) \left (2 a+b x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{384 a^3 x^8}-\frac {\left (a+b x^2+c x^4\right )^{5/2}}{12 a x^{12}}+\frac {7 b \left (a+b x^2+c x^4\right )^{5/2}}{120 a^2 x^{10}}-\frac {\left (\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right )\right ) \text {Subst}\left (\int \frac {\sqrt {a+b x+c x^2}}{x^3} \, dx,x,x^2\right )}{256 a^3}\\ &=\frac {\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{1024 a^4 x^4}-\frac {\left (7 b^2-4 a c\right ) \left (2 a+b x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{384 a^3 x^8}-\frac {\left (a+b x^2+c x^4\right )^{5/2}}{12 a x^{12}}+\frac {7 b \left (a+b x^2+c x^4\right )^{5/2}}{120 a^2 x^{10}}+\frac {\left (\left (b^2-4 a c\right )^2 \left (7 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 )}{2048 a^4}\\ &=\frac {\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{1024 a^4 x^4}-\frac {\left (7 b^2-4 a c\right ) \left (2 a+b x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{384 a^3 x^8}-\frac {\left (a+b x^2+c x^4\right )^{5/2}}{12 a x^{12}}+\frac {7 b \left (a+b x^2+c x^4\right )^{5/2}}{120 a^2 x^{10}}-\frac {\left (\left (b^2-4 a c\right )^2 \left (7 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 )}{1024 a^4}\\ &=\frac {\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{1024 a^4 x^4}-\frac {\left (7 b^2-4 a c\right ) \left (2 a+b x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{384 a^3 x^8}-\frac {\left (a+b x^2+c x^4\right )^{5/2}}{12 a x^{12}}+\frac {7 b \left (a+b x^2+c x^4\right )^{5/2}}{120 a^2 x^{10}}-\frac {\left (b^2-4 a c\right )^2 \left (7 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 )}{2048 a^{9/2}}\\ \end {align*}
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Mathematica [A]
time = 1.35, size = 201, normalized size = 0.93 \begin {gather*} \frac {-\frac {\sqrt {a} \sqrt {a+b x^2+c x^4} \left (1280 a^5-105 b^5 x^{10}+10 a b^3 x^8 \left (7 b+76 c x^2\right )+64 a^4 \left (26 b x^2+35 c x^4\right )+48 a^3 x^4 \left (b^2+6 b c x^2+10 c^2 x^4\right )-8 a^2 b x^6 \left (7 b^2+54 b c x^2+162 c^2 x^4\right )\right )}{x^{12}}+15 \left (b^2-4 a c\right )^2 \left (7 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 )}{15360 a^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(456\) vs.
\(2(190)=380\).
time = 0.09, size = 457, normalized size = 2.12
method | result | size |
risch | \(-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (-1296 a^{2} b \,c^{2} x^{10}+760 a \,b^{3} c \,x^{10}-105 b^{5} x^{10}+480 a^{3} c^{2} x^{8}-432 a^{2} b^{2} c \,x^{8}+70 b^{4} a \,x^{8}+288 a^{3} b c \,x^{6}-56 a^{2} b^{3} x^{6}+2240 a^{4} c \,x^{4}+48 b^{2} a^{3} x^{4}+1664 b \,a^{4} x^{2}+1280 a^{5}\right )}{15360 x^{12} a^{4}}+\frac {c^{3} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{32 a^{\frac {3}{2}}}-\frac {9 c^{2} b^{2} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{128 a^{\frac {5}{2}}}+\frac {15 b^{4} c \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{512 a^{\frac {7}{2}}}-\frac {7 b^{6} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{2048 a^{\frac {9}{2}}}\) | \(310\) |
default | \(-\frac {7 b^{6} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{2048 a^{\frac {9}{2}}}-\frac {b^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{320 a \,x^{8}}+\frac {7 b^{3} \sqrt {c \,x^{4}+b \,x^{2}+a}}{1920 a^{2} x^{6}}-\frac {7 b^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}}{1536 a^{3} x^{4}}+\frac {7 b^{5} \sqrt {c \,x^{4}+b \,x^{2}+a}}{1024 a^{4} x^{2}}-\frac {9 c^{2} b^{2} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{128 a^{\frac {5}{2}}}-\frac {c^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{32 a \,x^{4}}-\frac {a \sqrt {c \,x^{4}+b \,x^{2}+a}}{12 x^{12}}-\frac {7 c \sqrt {c \,x^{4}+b \,x^{2}+a}}{48 x^{8}}+\frac {27 c^{2} b \sqrt {c \,x^{4}+b \,x^{2}+a}}{320 a^{2} x^{2}}+\frac {15 b^{4} c \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{512 a^{\frac {7}{2}}}-\frac {19 b^{3} c \sqrt {c \,x^{4}+b \,x^{2}+a}}{384 a^{3} x^{2}}+\frac {9 b^{2} c \sqrt {c \,x^{4}+b \,x^{2}+a}}{320 a^{2} x^{4}}-\frac {3 b c \sqrt {c \,x^{4}+b \,x^{2}+a}}{160 a \,x^{6}}-\frac {13 b \sqrt {c \,x^{4}+b \,x^{2}+a}}{120 x^{10}}+\frac {c^{3} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{32 a^{\frac {3}{2}}}\) | \(457\) |
elliptic | \(-\frac {7 b^{6} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{2048 a^{\frac {9}{2}}}-\frac {b^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{320 a \,x^{8}}+\frac {7 b^{3} \sqrt {c \,x^{4}+b \,x^{2}+a}}{1920 a^{2} x^{6}}-\frac {7 b^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}}{1536 a^{3} x^{4}}+\frac {7 b^{5} \sqrt {c \,x^{4}+b \,x^{2}+a}}{1024 a^{4} x^{2}}-\frac {9 c^{2} b^{2} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{128 a^{\frac {5}{2}}}-\frac {c^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{32 a \,x^{4}}-\frac {a \sqrt {c \,x^{4}+b \,x^{2}+a}}{12 x^{12}}-\frac {7 c \sqrt {c \,x^{4}+b \,x^{2}+a}}{48 x^{8}}+\frac {27 c^{2} b \sqrt {c \,x^{4}+b \,x^{2}+a}}{320 a^{2} x^{2}}+\frac {15 b^{4} c \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{512 a^{\frac {7}{2}}}-\frac {19 b^{3} c \sqrt {c \,x^{4}+b \,x^{2}+a}}{384 a^{3} x^{2}}+\frac {9 b^{2} c \sqrt {c \,x^{4}+b \,x^{2}+a}}{320 a^{2} x^{4}}-\frac {3 b c \sqrt {c \,x^{4}+b \,x^{2}+a}}{160 a \,x^{6}}-\frac {13 b \sqrt {c \,x^{4}+b \,x^{2}+a}}{120 x^{10}}+\frac {c^{3} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{32 a^{\frac {3}{2}}}\) | \(457\) |
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [A]
time = 0.54, size = 473, normalized size = 2.19 \begin {gather*} \left [-\frac {15 \, {\left (7 \, b^{6} - 60 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {a} x^{12} \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 (105 \, a b^{5} - 760 \, a^{2} b^{3} c + 1296 \, a^{3} b c^{2}\right )} x^{10} - 2 \, {\left (35 \, a^{2} b^{4} - 216 \, a^{3} b^{2} c + 240 \, a^{4} c^{2}\right )} x^{8} - 1664 \, a^{5} b x^{2} + 8 \, {\left (7 \, a^{3} b^{3} - 36 \, a^{4} b c\right )} x^{6} - 1280 \, a^{6} - 16 \, {\left (3 \, a^{4} b^{2} + 140 \, a^{5} c\right )} x^{4}\right )} \sqrt {c x^{4} + b x^{2} + a}}{61440 \, a^{5} x^{12}}, \frac {15 \, {\left (7 \, b^{6} - 60 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {-a} x^{12} \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 (105 \, a b^{5} - 760 \, a^{2} b^{3} c + 1296 \, a^{3} b c^{2}\right )} x^{10} - 2 \, {\left (35 \, a^{2} b^{4} - 216 \, a^{3} b^{2} c + 240 \, a^{4} c^{2}\right )} x^{8} - 1664 \, a^{5} b x^{2} + 8 \, {\left (7 \, a^{3} b^{3} - 36 \, a^{4} b c\right )} x^{6} - 1280 \, a^{6} - 16 \, {\left (3 \, a^{4} b^{2} + 140 \, a^{5} c\right )} x^{4}\right )} \sqrt {c x^{4} + b x^{2} + a}}{30720 \, a^{5} x^{12}}\right ] \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 {\left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}}{x^{13}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1235 vs.
\(2 (190) = 380\).
time = 3.41, size = 1235, normalized size = 5.72 \begin {gather*} \frac {{\left (7 \, b^{6} - 60 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \arctan \left (-\frac {\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}}{\sqrt {-a}}\right )}{1024 \, \sqrt {-a} a^{4}} - \frac {105 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{11} b^{6} - 900 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{11} a b^{4} c + 2160 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{11} a^{2} b^{2} c^{2} - 960 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{11} a^{3} c^{3} - 595 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{9} a b^{6} + 5100 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{9} a^{2} b^{4} c - 12240 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{9} a^{3} b^{2} c^{2} - 15040 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{9} a^{4} c^{3} - 76800 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{8} a^{4} b c^{\frac {5}{2}} + 1386 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{7} a^{2} b^{6} - 11880 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{7} a^{3} b^{4} c - 97440 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{7} a^{4} b^{2} c^{2} - 24960 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{7} a^{5} c^{3} - 112640 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{6} a^{4} b^{3} c^{\frac {3}{2}} - 61440 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{6} a^{5} b c^{\frac {5}{2}} - 1686 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{5} a^{3} b^{6} - 42600 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{5} a^{4} b^{4} c - 128160 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{5} a^{5} b^{2} c^{2} - 24960 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{5} a^{6} c^{3} - 15360 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{4} a^{4} b^{5} \sqrt {c} - 61440 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{4} a^{5} b^{3} c^{\frac {3}{2}} - 92160 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{4} a^{6} b c^{\frac {5}{2}} - 595 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{3} a^{4} b^{6} - 25620 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{3} a^{5} b^{4} c - 58320 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{3} a^{6} b^{2} c^{2} - 15040 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{3} a^{7} c^{3} - 30720 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{2} a^{6} b^{3} c^{\frac {3}{2}} - 12288 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{2} a^{7} b c^{\frac {5}{2}} + 105 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} a^{5} b^{6} - 900 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} a^{6} b^{4} c - 13200 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} a^{7} b^{2} c^{2} - 960 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} a^{8} c^{3} - 3072 \, a^{8} b c^{\frac {5}{2}}}{15360 \, {\left ({\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{2} - a\right )}^{6} a^{4}} \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 = -0.00 \begin {gather*} \int \frac {{\left (c\,x^4+b\,x^2+a\right )}^{3/2}}{x^{13}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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