Optimal. Leaf size=343 \[ \frac {b \left (21 b^2-68 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{8 a^3 x^4 \left (b^2-4 a c\right )}-\frac {\left (9 b^2-20 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{4 a^2 x^5 \left (b^2-4 a c\right )}-\frac {15 \left (16 a^2 c^2-56 a b^2 c+21 b^4\right ) \tanh ^{-1}\left (\frac {x (2 a+b x)}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}\right )}{128 a^{11/2}}+\frac {b \left (1808 a^2 c^2-1680 a b^2 c+315 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{64 a^5 x^2 \left (b^2-4 a c\right )}-\frac {\left (240 a^2 c^2-448 a b^2 c+105 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{32 a^4 x^3 \left (b^2-4 a c\right )}+\frac {2 \left (-2 a c+b^2+b c x\right )}{a x^3 \left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}} \]
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Rubi [A] time = 0.62, antiderivative size = 343, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1924, 1951, 12, 1904, 206} \begin {gather*} \frac {b \left (1808 a^2 c^2-1680 a b^2 c+315 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{64 a^5 x^2 \left (b^2-4 a c\right )}-\frac {\left (240 a^2 c^2-448 a b^2 c+105 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{32 a^4 x^3 \left (b^2-4 a c\right )}-\frac {15 \left (16 a^2 c^2-56 a b^2 c+21 b^4\right ) \tanh ^{-1}\left (\frac {x (2 a+b x)}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}\right )}{128 a^{11/2}}+\frac {b \left (21 b^2-68 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{8 a^3 x^4 \left (b^2-4 a c\right )}-\frac {\left (9 b^2-20 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{4 a^2 x^5 \left (b^2-4 a c\right )}+\frac {2 \left (-2 a c+b^2+b c x\right )}{a x^3 \left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 1904
Rule 1924
Rule 1951
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx &=\frac {2 \left (b^2-2 a c+b c x\right )}{a \left (b^2-4 a c\right ) x^3 \sqrt {a x^2+b x^3+c x^4}}-\frac {2 \int \frac {-\frac {9 b^2}{2}+10 a c-4 b c x}{x^4 \sqrt {a x^2+b x^3+c x^4}} \, dx}{a \left (b^2-4 a c\right )}\\ &=\frac {2 \left (b^2-2 a c+b c x\right )}{a \left (b^2-4 a c\right ) x^3 \sqrt {a x^2+b x^3+c x^4}}-\frac {\left (9 b^2-20 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{4 a^2 \left (b^2-4 a c\right ) x^5}+\frac {\int \frac {-\frac {3}{4} b \left (21 b^2-68 a c\right )-\frac {3}{2} c \left (9 b^2-20 a c\right ) x}{x^3 \sqrt {a x^2+b x^3+c x^4}} \, dx}{2 a^2 \left (b^2-4 a c\right )}\\ &=\frac {2 \left (b^2-2 a c+b c x\right )}{a \left (b^2-4 a c\right ) x^3 \sqrt {a x^2+b x^3+c x^4}}-\frac {\left (9 b^2-20 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{4 a^2 \left (b^2-4 a c\right ) x^5}+\frac {b \left (21 b^2-68 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{8 a^3 \left (b^2-4 a c\right ) x^4}-\frac {\int \frac {-\frac {3}{8} \left (105 b^4-448 a b^2 c+240 a^2 c^2\right )-\frac {3}{2} b c \left (21 b^2-68 a c\right ) x}{x^2 \sqrt {a x^2+b x^3+c x^4}} \, dx}{6 a^3 \left (b^2-4 a c\right )}\\ &=\frac {2 \left (b^2-2 a c+b c x\right )}{a \left (b^2-4 a c\right ) x^3 \sqrt {a x^2+b x^3+c x^4}}-\frac {\left (9 b^2-20 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{4 a^2 \left (b^2-4 a c\right ) x^5}+\frac {b \left (21 b^2-68 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{8 a^3 \left (b^2-4 a c\right ) x^4}-\frac {\left (105 b^4-448 a b^2 c+240 a^2 c^2\right ) \sqrt {a x^2+b x^3+c x^4}}{32 a^4 \left (b^2-4 a c\right ) x^3}+\frac {\int \frac {-\frac {3}{16} b \left (315 b^4-1680 a b^2 c+1808 a^2 c^2\right )-\frac {3}{8} c \left (105 b^4-448 a b^2 c+240 a^2 c^2\right ) x}{x \sqrt {a x^2+b x^3+c x^4}} \, dx}{12 a^4 \left (b^2-4 a c\right )}\\ &=\frac {2 \left (b^2-2 a c+b c x\right )}{a \left (b^2-4 a c\right ) x^3 \sqrt {a x^2+b x^3+c x^4}}-\frac {\left (9 b^2-20 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{4 a^2 \left (b^2-4 a c\right ) x^5}+\frac {b \left (21 b^2-68 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{8 a^3 \left (b^2-4 a c\right ) x^4}-\frac {\left (105 b^4-448 a b^2 c+240 a^2 c^2\right ) \sqrt {a x^2+b x^3+c x^4}}{32 a^4 \left (b^2-4 a c\right ) x^3}+\frac {b \left (315 b^4-1680 a b^2 c+1808 a^2 c^2\right ) \sqrt {a x^2+b x^3+c x^4}}{64 a^5 \left (b^2-4 a c\right ) x^2}-\frac {\int -\frac {45 \left (b^2-4 a c\right ) \left (21 b^4-56 a b^2 c+16 a^2 c^2\right )}{32 \sqrt {a x^2+b x^3+c x^4}} \, dx}{12 a^5 \left (b^2-4 a c\right )}\\ &=\frac {2 \left (b^2-2 a c+b c x\right )}{a \left (b^2-4 a c\right ) x^3 \sqrt {a x^2+b x^3+c x^4}}-\frac {\left (9 b^2-20 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{4 a^2 \left (b^2-4 a c\right ) x^5}+\frac {b \left (21 b^2-68 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{8 a^3 \left (b^2-4 a c\right ) x^4}-\frac {\left (105 b^4-448 a b^2 c+240 a^2 c^2\right ) \sqrt {a x^2+b x^3+c x^4}}{32 a^4 \left (b^2-4 a c\right ) x^3}+\frac {b \left (315 b^4-1680 a b^2 c+1808 a^2 c^2\right ) \sqrt {a x^2+b x^3+c x^4}}{64 a^5 \left (b^2-4 a c\right ) x^2}+\frac {\left (15 \left (21 b^4-56 a b^2 c+16 a^2 c^2\right )\right ) \int \frac {1}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{128 a^5}\\ &=\frac {2 \left (b^2-2 a c+b c x\right )}{a \left (b^2-4 a c\right ) x^3 \sqrt {a x^2+b x^3+c x^4}}-\frac {\left (9 b^2-20 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{4 a^2 \left (b^2-4 a c\right ) x^5}+\frac {b \left (21 b^2-68 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{8 a^3 \left (b^2-4 a c\right ) x^4}-\frac {\left (105 b^4-448 a b^2 c+240 a^2 c^2\right ) \sqrt {a x^2+b x^3+c x^4}}{32 a^4 \left (b^2-4 a c\right ) x^3}+\frac {b \left (315 b^4-1680 a b^2 c+1808 a^2 c^2\right ) \sqrt {a x^2+b x^3+c x^4}}{64 a^5 \left (b^2-4 a c\right ) x^2}-\frac {\left (15 \left (21 b^4-56 a b^2 c+16 a^2 c^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {x (2 a+b x)}{\sqrt {a x^2+b x^3+c x^4}}\right )}{64 a^5}\\ &=\frac {2 \left (b^2-2 a c+b c x\right )}{a \left (b^2-4 a c\right ) x^3 \sqrt {a x^2+b x^3+c x^4}}-\frac {\left (9 b^2-20 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{4 a^2 \left (b^2-4 a c\right ) x^5}+\frac {b \left (21 b^2-68 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{8 a^3 \left (b^2-4 a c\right ) x^4}-\frac {\left (105 b^4-448 a b^2 c+240 a^2 c^2\right ) \sqrt {a x^2+b x^3+c x^4}}{32 a^4 \left (b^2-4 a c\right ) x^3}+\frac {b \left (315 b^4-1680 a b^2 c+1808 a^2 c^2\right ) \sqrt {a x^2+b x^3+c x^4}}{64 a^5 \left (b^2-4 a c\right ) x^2}-\frac {15 \left (21 b^4-56 a b^2 c+16 a^2 c^2\right ) \tanh ^{-1}\left (\frac {x (2 a+b x)}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}\right )}{128 a^{11/2}}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 272, normalized size = 0.79 \begin {gather*} \frac {15 x^4 \left (-64 a^3 c^3+240 a^2 b^2 c^2-140 a b^4 c+21 b^6\right ) \sqrt {a+x (b+c x)} \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+x (b+c x)}}\right )-2 \sqrt {a} \left (64 a^5 c-16 a^4 \left (b^2+6 b c x+10 c^2 x^2\right )+8 a^3 x \left (3 b^3+26 b^2 c x+98 b c^2 x^2-60 c^3 x^3\right )+2 a^2 b x^2 \left (-21 b^3-308 b^2 c x+1352 b c^2 x^2+904 c^3 x^3\right )+105 a b^3 x^3 \left (b^2-18 b c x-16 c^2 x^2\right )+315 b^5 x^4 (b+c x)\right )}{128 a^{11/2} x^3 \left (4 a c-b^2\right ) \sqrt {x^2 (a+x (b+c x))}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 5.24, size = 331, normalized size = 0.97 \begin {gather*} \frac {15 \left (16 a^2 c^2-56 a b^2 c+21 b^4\right ) \log \left (2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}-2 a x-b x^2\right )}{128 a^{11/2}}-\frac {15 \log (x) \left (16 a^2 c^2-56 a b^2 c+21 b^4\right )}{64 a^{11/2}}+\frac {\sqrt {a x^2+b x^3+c x^4} \left (-64 a^5 c+16 a^4 b^2+96 a^4 b c x+160 a^4 c^2 x^2-24 a^3 b^3 x-208 a^3 b^2 c x^2-784 a^3 b c^2 x^3+480 a^3 c^3 x^4+42 a^2 b^4 x^2+616 a^2 b^3 c x^3-2704 a^2 b^2 c^2 x^4-1808 a^2 b c^3 x^5-105 a b^5 x^3+1890 a b^4 c x^4+1680 a b^3 c^2 x^5-315 b^6 x^4-315 b^5 c x^5\right )}{64 a^5 x^5 \left (4 a c-b^2\right ) \left (a+b x+c x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 2.35, size = 866, normalized size = 2.52 \begin {gather*} \left [\frac {15 \, {\left ({\left (21 \, b^{6} c - 140 \, a b^{4} c^{2} + 240 \, a^{2} b^{2} c^{3} - 64 \, a^{3} c^{4}\right )} x^{7} + {\left (21 \, b^{7} - 140 \, a b^{5} c + 240 \, a^{2} b^{3} c^{2} - 64 \, a^{3} b c^{3}\right )} x^{6} + {\left (21 \, a b^{6} - 140 \, a^{2} b^{4} c + 240 \, a^{3} b^{2} c^{2} - 64 \, a^{4} c^{3}\right )} x^{5}\right )} \sqrt {a} \log \left (-\frac {8 \, a b x^{2} + {\left (b^{2} + 4 \, a c\right )} x^{3} + 8 \, a^{2} x - 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (b x + 2 \, a\right )} \sqrt {a}}{x^{3}}\right ) - 4 \, {\left (16 \, a^{5} b^{2} - 64 \, a^{6} c - {\left (315 \, a b^{5} c - 1680 \, a^{2} b^{3} c^{2} + 1808 \, a^{3} b c^{3}\right )} x^{5} - {\left (315 \, a b^{6} - 1890 \, a^{2} b^{4} c + 2704 \, a^{3} b^{2} c^{2} - 480 \, a^{4} c^{3}\right )} x^{4} - 7 \, {\left (15 \, a^{2} b^{5} - 88 \, a^{3} b^{3} c + 112 \, a^{4} b c^{2}\right )} x^{3} + 2 \, {\left (21 \, a^{3} b^{4} - 104 \, a^{4} b^{2} c + 80 \, a^{5} c^{2}\right )} x^{2} - 24 \, {\left (a^{4} b^{3} - 4 \, a^{5} b c\right )} x\right )} \sqrt {c x^{4} + b x^{3} + a x^{2}}}{256 \, {\left ({\left (a^{6} b^{2} c - 4 \, a^{7} c^{2}\right )} x^{7} + {\left (a^{6} b^{3} - 4 \, a^{7} b c\right )} x^{6} + {\left (a^{7} b^{2} - 4 \, a^{8} c\right )} x^{5}\right )}}, \frac {15 \, {\left ({\left (21 \, b^{6} c - 140 \, a b^{4} c^{2} + 240 \, a^{2} b^{2} c^{3} - 64 \, a^{3} c^{4}\right )} x^{7} + {\left (21 \, b^{7} - 140 \, a b^{5} c + 240 \, a^{2} b^{3} c^{2} - 64 \, a^{3} b c^{3}\right )} x^{6} + {\left (21 \, a b^{6} - 140 \, a^{2} b^{4} c + 240 \, a^{3} b^{2} c^{2} - 64 \, a^{4} c^{3}\right )} x^{5}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (b x + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{3} + a b x^{2} + a^{2} x\right )}}\right ) - 2 \, {\left (16 \, a^{5} b^{2} - 64 \, a^{6} c - {\left (315 \, a b^{5} c - 1680 \, a^{2} b^{3} c^{2} + 1808 \, a^{3} b c^{3}\right )} x^{5} - {\left (315 \, a b^{6} - 1890 \, a^{2} b^{4} c + 2704 \, a^{3} b^{2} c^{2} - 480 \, a^{4} c^{3}\right )} x^{4} - 7 \, {\left (15 \, a^{2} b^{5} - 88 \, a^{3} b^{3} c + 112 \, a^{4} b c^{2}\right )} x^{3} + 2 \, {\left (21 \, a^{3} b^{4} - 104 \, a^{4} b^{2} c + 80 \, a^{5} c^{2}\right )} x^{2} - 24 \, {\left (a^{4} b^{3} - 4 \, a^{5} b c\right )} x\right )} \sqrt {c x^{4} + b x^{3} + a x^{2}}}{128 \, {\left ({\left (a^{6} b^{2} c - 4 \, a^{7} c^{2}\right )} x^{7} + {\left (a^{6} b^{3} - 4 \, a^{7} b c\right )} x^{6} + {\left (a^{7} b^{2} - 4 \, a^{8} c\right )} x^{5}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (c x^{4} + b x^{3} + a x^{2}\right )}^{\frac {3}{2}} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 446, normalized size = 1.30 \begin {gather*} -\frac {\left (c \,x^{2}+b x +a \right ) \left (3616 a^{\frac {7}{2}} b \,c^{3} x^{5}-3360 a^{\frac {5}{2}} b^{3} c^{2} x^{5}+630 a^{\frac {3}{2}} b^{5} c \,x^{5}+960 \sqrt {c \,x^{2}+b x +a}\, a^{4} c^{3} x^{4} \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )-3600 \sqrt {c \,x^{2}+b x +a}\, a^{3} b^{2} c^{2} x^{4} \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )+2100 \sqrt {c \,x^{2}+b x +a}\, a^{2} b^{4} c \,x^{4} \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )-315 \sqrt {c \,x^{2}+b x +a}\, a \,b^{6} x^{4} \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )-960 a^{\frac {9}{2}} c^{3} x^{4}+5408 a^{\frac {7}{2}} b^{2} c^{2} x^{4}-3780 a^{\frac {5}{2}} b^{4} c \,x^{4}+630 a^{\frac {3}{2}} b^{6} x^{4}+1568 a^{\frac {9}{2}} b \,c^{2} x^{3}-1232 a^{\frac {7}{2}} b^{3} c \,x^{3}+210 a^{\frac {5}{2}} b^{5} x^{3}-320 a^{\frac {11}{2}} c^{2} x^{2}+416 a^{\frac {9}{2}} b^{2} c \,x^{2}-84 a^{\frac {7}{2}} b^{4} x^{2}-192 a^{\frac {11}{2}} b c x +48 a^{\frac {9}{2}} b^{3} x +128 a^{\frac {13}{2}} c -32 a^{\frac {11}{2}} b^{2}\right )}{128 \left (c \,x^{4}+b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}} \left (4 a c -b^{2}\right ) a^{\frac {13}{2}} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (c x^{4} + b x^{3} + a x^{2}\right )}^{\frac {3}{2}} x^{2}}\,{d x} \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 {1}{x^2\,{\left (c\,x^4+b\,x^3+a\,x^2\right )}^{3/2}} \,d x \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 {1}{x^{2} \left (x^{2} \left (a + b x + c x^{2}\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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