Optimal. Leaf size=249 \[ \frac {3 b \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {x (2 a+b x)}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}\right )}{256 a^{7/2}}+\frac {b \left (5 b^2-28 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{320 a^2 x^3}-\frac {\left (128 a^2 c^2-100 a b^2 c+15 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{640 a^3 x^2}-\frac {\left (b^2-8 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{80 a x^4}-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{5 x^8}-\frac {3 (b+4 c x) \sqrt {a x^2+b x^3+c x^4}}{40 x^5} \]
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Rubi [A] time = 0.50, antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1920, 1941, 1951, 12, 1904, 206} \begin {gather*} -\frac {\left (128 a^2 c^2-100 a b^2 c+15 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{640 a^3 x^2}+\frac {b \left (5 b^2-28 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{320 a^2 x^3}+\frac {3 b \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {x (2 a+b x)}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}\right )}{256 a^{7/2}}-\frac {\left (b^2-8 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{80 a x^4}-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{5 x^8}-\frac {3 (b+4 c x) \sqrt {a x^2+b x^3+c x^4}}{40 x^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 1904
Rule 1920
Rule 1941
Rule 1951
Rubi steps
\begin {align*} \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^9} \, dx &=-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{5 x^8}+\frac {3}{10} \int \frac {(b+2 c x) \sqrt {a x^2+b x^3+c x^4}}{x^6} \, dx\\ &=-\frac {3 (b+4 c x) \sqrt {a x^2+b x^3+c x^4}}{40 x^5}-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{5 x^8}+\frac {3}{160} \int \frac {2 \left (b^2-8 a c\right )-4 b c x}{x^3 \sqrt {a x^2+b x^3+c x^4}} \, dx\\ &=-\frac {\left (b^2-8 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{80 a x^4}-\frac {3 (b+4 c x) \sqrt {a x^2+b x^3+c x^4}}{40 x^5}-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{5 x^8}-\frac {\int \frac {b \left (5 b^2-28 a c\right )+4 c \left (b^2-8 a c\right ) x}{x^2 \sqrt {a x^2+b x^3+c x^4}} \, dx}{160 a}\\ &=-\frac {\left (b^2-8 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{80 a x^4}+\frac {b \left (5 b^2-28 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{320 a^2 x^3}-\frac {3 (b+4 c x) \sqrt {a x^2+b x^3+c x^4}}{40 x^5}-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{5 x^8}+\frac {\int \frac {\frac {1}{2} \left (15 b^4-100 a b^2 c+128 a^2 c^2\right )+b c \left (5 b^2-28 a c\right ) x}{x \sqrt {a x^2+b x^3+c x^4}} \, dx}{320 a^2}\\ &=-\frac {\left (b^2-8 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{80 a x^4}+\frac {b \left (5 b^2-28 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{320 a^2 x^3}-\frac {\left (15 b^4-100 a b^2 c+128 a^2 c^2\right ) \sqrt {a x^2+b x^3+c x^4}}{640 a^3 x^2}-\frac {3 (b+4 c x) \sqrt {a x^2+b x^3+c x^4}}{40 x^5}-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{5 x^8}-\frac {\int \frac {15 b \left (b^2-4 a c\right )^2}{4 \sqrt {a x^2+b x^3+c x^4}} \, dx}{320 a^3}\\ &=-\frac {\left (b^2-8 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{80 a x^4}+\frac {b \left (5 b^2-28 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{320 a^2 x^3}-\frac {\left (15 b^4-100 a b^2 c+128 a^2 c^2\right ) \sqrt {a x^2+b x^3+c x^4}}{640 a^3 x^2}-\frac {3 (b+4 c x) \sqrt {a x^2+b x^3+c x^4}}{40 x^5}-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{5 x^8}-\frac {\left (3 b \left (b^2-4 a c\right )^2\right ) \int \frac {1}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{256 a^3}\\ &=-\frac {\left (b^2-8 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{80 a x^4}+\frac {b \left (5 b^2-28 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{320 a^2 x^3}-\frac {\left (15 b^4-100 a b^2 c+128 a^2 c^2\right ) \sqrt {a x^2+b x^3+c x^4}}{640 a^3 x^2}-\frac {3 (b+4 c x) \sqrt {a x^2+b x^3+c x^4}}{40 x^5}-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{5 x^8}+\frac {\left (3 b \left (b^2-4 a c\right )^2\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 )}{128 a^3}\\ &=-\frac {\left (b^2-8 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{80 a x^4}+\frac {b \left (5 b^2-28 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{320 a^2 x^3}-\frac {\left (15 b^4-100 a b^2 c+128 a^2 c^2\right ) \sqrt {a x^2+b x^3+c x^4}}{640 a^3 x^2}-\frac {3 (b+4 c x) \sqrt {a x^2+b x^3+c x^4}}{40 x^5}-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{5 x^8}+\frac {3 b \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {x (2 a+b x)}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}\right )}{256 a^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 177, normalized size = 0.71 \begin {gather*} \frac {\sqrt {x^2 (a+x (b+c x))} \left (15 b x^5 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+x (b+c x)}}\right )-2 \sqrt {a} \sqrt {a+x (b+c x)} \left (128 a^4+16 a^3 x (11 b+16 c x)+8 a^2 x^2 \left (b^2+7 b c x+16 c^2 x^2\right )-10 a b^2 x^3 (b+10 c x)+15 b^4 x^4\right )\right )}{1280 a^{7/2} x^6 \sqrt {a+x (b+c x)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 2.67, size = 211, normalized size = 0.85 \begin {gather*} -\frac {3 \left (16 a^2 b c^2-8 a b^3 c+b^5\right ) \log \left (2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}-2 a x-b x^2\right )}{256 a^{7/2}}+\frac {3 \log (x) \left (16 a^2 b c^2-8 a b^3 c+b^5\right )}{128 a^{7/2}}+\frac {\sqrt {a x^2+b x^3+c x^4} \left (-128 a^4-176 a^3 b x-256 a^3 c x^2-8 a^2 b^2 x^2-56 a^2 b c x^3-128 a^2 c^2 x^4+10 a b^3 x^3+100 a b^2 c x^4-15 b^4 x^4\right )}{640 a^3 x^6} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.77, size = 394, normalized size = 1.58 \begin {gather*} \left [\frac {15 \, {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} \sqrt {a} x^{6} \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 (176 \, a^{4} b x + 128 \, a^{5} + {\left (15 \, a b^{4} - 100 \, a^{2} b^{2} c + 128 \, a^{3} c^{2}\right )} x^{4} - 2 \, {\left (5 \, a^{2} b^{3} - 28 \, a^{3} b c\right )} x^{3} + 8 \, {\left (a^{3} b^{2} + 32 \, a^{4} c\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{3} + a x^{2}}}{2560 \, a^{4} x^{6}}, -\frac {15 \, {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} \sqrt {-a} x^{6} \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 (176 \, a^{4} b x + 128 \, a^{5} + {\left (15 \, a b^{4} - 100 \, a^{2} b^{2} c + 128 \, a^{3} c^{2}\right )} x^{4} - 2 \, {\left (5 \, a^{2} b^{3} - 28 \, a^{3} b c\right )} x^{3} + 8 \, {\left (a^{3} b^{2} + 32 \, a^{4} c\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{3} + a x^{2}}}{1280 \, a^{4} x^{6}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 534, normalized size = 2.14 \begin {gather*} \frac {\left (c \,x^{4}+b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}} \left (240 a^{\frac {7}{2}} b \,c^{2} x^{5} \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )-120 a^{\frac {5}{2}} b^{3} c \,x^{5} \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )+15 a^{\frac {3}{2}} b^{5} x^{5} \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )+120 \sqrt {c \,x^{2}+b x +a}\, a^{2} b^{2} c^{2} x^{6}-30 \sqrt {c \,x^{2}+b x +a}\, a \,b^{4} c \,x^{6}-240 \sqrt {c \,x^{2}+b x +a}\, a^{3} b \,c^{2} x^{5}+180 \sqrt {c \,x^{2}+b x +a}\, a^{2} b^{3} c \,x^{5}-30 \sqrt {c \,x^{2}+b x +a}\, a \,b^{5} x^{5}+120 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a \,b^{2} c^{2} x^{6}-10 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b^{4} c \,x^{6}-80 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a^{2} b \,c^{2} x^{5}+100 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a \,b^{3} c \,x^{5}-10 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b^{5} x^{5}-120 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} a \,b^{2} c \,x^{4}+10 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} b^{4} x^{4}+80 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} a^{2} b c \,x^{3}+20 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} a \,b^{3} x^{3}-80 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} a^{2} b^{2} x^{2}+160 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} a^{3} b x -256 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} a^{4}\right )}{1280 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a^{5} x^{8}} \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 {{\left (c x^{4} + b x^{3} + a x^{2}\right )}^{\frac {3}{2}}}{x^{9}}\,{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 {{\left (c\,x^4+b\,x^3+a\,x^2\right )}^{3/2}}{x^9} \,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 {\left (x^{2} \left (a + b x + c x^{2}\right )\right )^{\frac {3}{2}}}{x^{9}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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