Optimal. Leaf size=262 \[ \frac {3 x \left (5 b^2-4 a c\right ) \sqrt {a+b x+c x^2} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{7/2} \sqrt {a x^2+b x^3+c x^4}}-\frac {b \left (15 b^2-52 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{4 c^3 x \left (b^2-4 a c\right )}+\frac {\left (5 b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{2 c^2 \left (b^2-4 a c\right )}+\frac {2 x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}-\frac {2 b x \sqrt {a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right )} \]
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Rubi [A] time = 0.51, antiderivative size = 262, 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 = {1923, 1949, 12, 1914, 621, 206} \begin {gather*} \frac {\left (5 b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{2 c^2 \left (b^2-4 a c\right )}-\frac {b \left (15 b^2-52 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{4 c^3 x \left (b^2-4 a c\right )}+\frac {3 x \left (5 b^2-4 a c\right ) \sqrt {a+b x+c x^2} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{7/2} \sqrt {a x^2+b x^3+c x^4}}+\frac {2 x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}-\frac {2 b x \sqrt {a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 621
Rule 1914
Rule 1923
Rule 1949
Rubi steps
\begin {align*} \int \frac {x^7}{\left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx &=\frac {2 x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}-\frac {2 \int \frac {x^3 (6 a+3 b x)}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{b^2-4 a c}\\ &=\frac {2 x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}-\frac {2 b x \sqrt {a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right )}+\frac {2 \int \frac {x^2 \left (6 a b+\frac {3}{2} \left (5 b^2-12 a c\right ) x\right )}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{3 c \left (b^2-4 a c\right )}\\ &=\frac {2 x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}+\frac {\left (5 b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{2 c^2 \left (b^2-4 a c\right )}-\frac {2 b x \sqrt {a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right )}-\frac {\int \frac {x \left (\frac {3}{2} a \left (5 b^2-12 a c\right )+\frac {3}{4} b \left (15 b^2-52 a c\right ) x\right )}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{3 c^2 \left (b^2-4 a c\right )}\\ &=\frac {2 x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}+\frac {\left (5 b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{2 c^2 \left (b^2-4 a c\right )}-\frac {b \left (15 b^2-52 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{4 c^3 \left (b^2-4 a c\right ) x}-\frac {2 b x \sqrt {a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right )}+\frac {\int \frac {9 \left (b^2-4 a c\right ) \left (5 b^2-4 a c\right ) x}{8 \sqrt {a x^2+b x^3+c x^4}} \, dx}{3 c^3 \left (b^2-4 a c\right )}\\ &=\frac {2 x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}+\frac {\left (5 b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{2 c^2 \left (b^2-4 a c\right )}-\frac {b \left (15 b^2-52 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{4 c^3 \left (b^2-4 a c\right ) x}-\frac {2 b x \sqrt {a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right )}+\frac {\left (3 \left (5 b^2-4 a c\right )\right ) \int \frac {x}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{8 c^3}\\ &=\frac {2 x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}+\frac {\left (5 b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{2 c^2 \left (b^2-4 a c\right )}-\frac {b \left (15 b^2-52 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{4 c^3 \left (b^2-4 a c\right ) x}-\frac {2 b x \sqrt {a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right )}+\frac {\left (3 \left (5 b^2-4 a c\right ) x \sqrt {a+b x+c x^2}\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{8 c^3 \sqrt {a x^2+b x^3+c x^4}}\\ &=\frac {2 x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}+\frac {\left (5 b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{2 c^2 \left (b^2-4 a c\right )}-\frac {b \left (15 b^2-52 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{4 c^3 \left (b^2-4 a c\right ) x}-\frac {2 b x \sqrt {a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right )}+\frac {\left (3 \left (5 b^2-4 a c\right ) x \sqrt {a+b x+c x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{4 c^3 \sqrt {a x^2+b x^3+c x^4}}\\ &=\frac {2 x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}+\frac {\left (5 b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{2 c^2 \left (b^2-4 a c\right )}-\frac {b \left (15 b^2-52 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{4 c^3 \left (b^2-4 a c\right ) x}-\frac {2 b x \sqrt {a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right )}+\frac {3 \left (5 b^2-4 a c\right ) x \sqrt {a+b x+c x^2} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{7/2} \sqrt {a x^2+b x^3+c x^4}}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 183, normalized size = 0.70 \begin {gather*} \frac {x \left (2 \sqrt {c} \left (4 a^2 c (6 c x-13 b)+a \left (15 b^3-62 b^2 c x-20 b c^2 x^2+8 c^3 x^3\right )+b^2 x \left (15 b^2+5 b c x-2 c^2 x^2\right )\right )-3 \left (16 a^2 c^2-24 a b^2 c+5 b^4\right ) \sqrt {a+x (b+c x)} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )}{8 c^{7/2} \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 = 2.80, size = 192, normalized size = 0.73 \begin {gather*} \frac {\left (-52 a^2 b c+24 a^2 c^2 x+15 a b^3-62 a b^2 c x-20 a b c^2 x^2+8 a c^3 x^3+15 b^4 x+5 b^3 c x^2-2 b^2 c^2 x^3\right ) \sqrt {a x^2+b x^3+c x^4}}{4 c^3 x \left (4 a c-b^2\right ) \left (a+b x+c x^2\right )}-\frac {3 \left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a} x-\sqrt {a x^2+b x^3+c x^4}}\right )}{4 c^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.55, size = 616, normalized size = 2.35 \begin {gather*} \left [-\frac {3 \, {\left ({\left (5 \, b^{4} c - 24 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{3} + {\left (5 \, b^{5} - 24 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x^{2} + {\left (5 \, a b^{4} - 24 \, a^{2} b^{2} c + 16 \, a^{3} c^{2}\right )} x\right )} \sqrt {c} \log \left (-\frac {8 \, c^{2} x^{3} + 8 \, b c x^{2} - 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {c} + {\left (b^{2} + 4 \, a c\right )} x}{x}\right ) + 4 \, {\left (15 \, a b^{3} c - 52 \, a^{2} b c^{2} - 2 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{3} + 5 \, {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{2} + {\left (15 \, b^{4} c - 62 \, a b^{2} c^{2} + 24 \, a^{2} c^{3}\right )} x\right )} \sqrt {c x^{4} + b x^{3} + a x^{2}}}{16 \, {\left ({\left (b^{2} c^{5} - 4 \, a c^{6}\right )} x^{3} + {\left (b^{3} c^{4} - 4 \, a b c^{5}\right )} x^{2} + {\left (a b^{2} c^{4} - 4 \, a^{2} c^{5}\right )} x\right )}}, -\frac {3 \, {\left ({\left (5 \, b^{4} c - 24 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{3} + {\left (5 \, b^{5} - 24 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x^{2} + {\left (5 \, a b^{4} - 24 \, a^{2} b^{2} c + 16 \, a^{3} c^{2}\right )} x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{3} + b c x^{2} + a c x\right )}}\right ) + 2 \, {\left (15 \, a b^{3} c - 52 \, a^{2} b c^{2} - 2 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{3} + 5 \, {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{2} + {\left (15 \, b^{4} c - 62 \, a b^{2} c^{2} + 24 \, a^{2} c^{3}\right )} x\right )} \sqrt {c x^{4} + b x^{3} + a x^{2}}}{8 \, {\left ({\left (b^{2} c^{5} - 4 \, a c^{6}\right )} x^{3} + {\left (b^{3} c^{4} - 4 \, a b c^{5}\right )} x^{2} + {\left (a b^{2} c^{4} - 4 \, a^{2} c^{5}\right )} x\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 {x^{7}}{{\left (c x^{4} + b x^{3} + a x^{2}\right )}^{\frac {3}{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 = 283, normalized size = 1.08 \begin {gather*} \frac {\left (c \,x^{2}+b x +a \right ) \left (16 a \,c^{\frac {9}{2}} x^{3}-4 b^{2} c^{\frac {7}{2}} x^{3}-40 a b \,c^{\frac {7}{2}} x^{2}+10 b^{3} c^{\frac {5}{2}} x^{2}+48 a^{2} c^{\frac {7}{2}} x -124 a \,b^{2} c^{\frac {5}{2}} x +30 b^{4} c^{\frac {3}{2}} x -48 \sqrt {c \,x^{2}+b x +a}\, a^{2} c^{3} \ln \left (\frac {2 c x +b +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}}{2 \sqrt {c}}\right )+72 \sqrt {c \,x^{2}+b x +a}\, a \,b^{2} c^{2} \ln \left (\frac {2 c x +b +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}}{2 \sqrt {c}}\right )-15 \sqrt {c \,x^{2}+b x +a}\, b^{4} c \ln \left (\frac {2 c x +b +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}}{2 \sqrt {c}}\right )-104 a^{2} b \,c^{\frac {5}{2}}+30 a \,b^{3} c^{\frac {3}{2}}\right ) x^{3}}{8 \left (c \,x^{4}+b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}} \left (4 a c -b^{2}\right ) c^{\frac {9}{2}}} \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 {x^{7}}{{\left (c x^{4} + b x^{3} + a x^{2}\right )}^{\frac {3}{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 {x^7}{{\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 {x^{7}}{\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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