Optimal. Leaf size=150 \[ -\frac {2 \left (6 a^2 c^2-6 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^3 \left (b^2-4 a c\right )^{3/2}}+\frac {2 x \left (b^2-3 a c\right )}{c^2 \left (b^2-4 a c\right )}-\frac {b x^2}{c \left (b^2-4 a c\right )}+\frac {x^3 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {b \log \left (a+b x+c x^2\right )}{c^3} \]
________________________________________________________________________________________
Rubi [A] time = 0.16, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {1585, 738, 800, 634, 618, 206, 628} \begin {gather*} -\frac {2 \left (6 a^2 c^2-6 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^3 \left (b^2-4 a c\right )^{3/2}}+\frac {2 x \left (b^2-3 a c\right )}{c^2 \left (b^2-4 a c\right )}+\frac {x^3 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {b x^2}{c \left (b^2-4 a c\right )}-\frac {b \log \left (a+b x+c x^2\right )}{c^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 618
Rule 628
Rule 634
Rule 738
Rule 800
Rule 1585
Rubi steps
\begin {align*} \int \frac {x^8}{\left (a x^2+b x^3+c x^4\right )^2} \, dx &=\int \frac {x^4}{\left (a+b x+c x^2\right )^2} \, dx\\ &=\frac {x^3 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\int \frac {x^2 (6 a+2 b x)}{a+b x+c x^2} \, dx}{-b^2+4 a c}\\ &=\frac {x^3 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\int \left (-\frac {2 \left (b^2-3 a c\right )}{c^2}+\frac {2 b x}{c}+\frac {2 \left (a \left (b^2-3 a c\right )+b \left (b^2-4 a c\right ) x\right )}{c^2 \left (a+b x+c x^2\right )}\right ) \, dx}{-b^2+4 a c}\\ &=\frac {2 \left (b^2-3 a c\right ) x}{c^2 \left (b^2-4 a c\right )}-\frac {b x^2}{c \left (b^2-4 a c\right )}+\frac {x^3 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {2 \int \frac {a \left (b^2-3 a c\right )+b \left (b^2-4 a c\right ) x}{a+b x+c x^2} \, dx}{c^2 \left (b^2-4 a c\right )}\\ &=\frac {2 \left (b^2-3 a c\right ) x}{c^2 \left (b^2-4 a c\right )}-\frac {b x^2}{c \left (b^2-4 a c\right )}+\frac {x^3 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {b \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{c^3}+\frac {\left (b^4-6 a b^2 c+6 a^2 c^2\right ) \int \frac {1}{a+b x+c x^2} \, dx}{c^3 \left (b^2-4 a c\right )}\\ &=\frac {2 \left (b^2-3 a c\right ) x}{c^2 \left (b^2-4 a c\right )}-\frac {b x^2}{c \left (b^2-4 a c\right )}+\frac {x^3 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {b \log \left (a+b x+c x^2\right )}{c^3}-\frac {\left (2 \left (b^4-6 a b^2 c+6 a^2 c^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{c^3 \left (b^2-4 a c\right )}\\ &=\frac {2 \left (b^2-3 a c\right ) x}{c^2 \left (b^2-4 a c\right )}-\frac {b x^2}{c \left (b^2-4 a c\right )}+\frac {x^3 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {2 \left (b^4-6 a b^2 c+6 a^2 c^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^3 \left (b^2-4 a c\right )^{3/2}}-\frac {b \log \left (a+b x+c x^2\right )}{c^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.20, size = 132, normalized size = 0.88 \begin {gather*} \frac {-\frac {2 \left (6 a^2 c^2-6 a b^2 c+b^4\right ) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}}+\frac {a^2 c (3 b-2 c x)-a b^2 (b-4 c x)+b^4 (-x)}{\left (b^2-4 a c\right ) (a+x (b+c x))}-b \log (a+x (b+c x))+c x}{c^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^8}{\left (a x^2+b x^3+c x^4\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.89, size = 837, normalized size = 5.58 \begin {gather*} \left [-\frac {a b^{5} - 7 \, a^{2} b^{3} c + 12 \, a^{3} b c^{2} - {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{3} - {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{2} + {\left (a b^{4} - 6 \, a^{2} b^{2} c + 6 \, a^{3} c^{2} + {\left (b^{4} c - 6 \, a b^{2} c^{2} + 6 \, a^{2} c^{3}\right )} x^{2} + {\left (b^{5} - 6 \, a b^{3} c + 6 \, a^{2} b c^{2}\right )} x\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c + \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) + {\left (b^{6} - 9 \, a b^{4} c + 26 \, a^{2} b^{2} c^{2} - 24 \, a^{3} c^{3}\right )} x + {\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2} + {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{2} + {\left (b^{6} - 8 \, a b^{4} c + 16 \, a^{2} b^{2} c^{2}\right )} x\right )} \log \left (c x^{2} + b x + a\right )}{a b^{4} c^{3} - 8 \, a^{2} b^{2} c^{4} + 16 \, a^{3} c^{5} + {\left (b^{4} c^{4} - 8 \, a b^{2} c^{5} + 16 \, a^{2} c^{6}\right )} x^{2} + {\left (b^{5} c^{3} - 8 \, a b^{3} c^{4} + 16 \, a^{2} b c^{5}\right )} x}, -\frac {a b^{5} - 7 \, a^{2} b^{3} c + 12 \, a^{3} b c^{2} - {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{3} - {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{2} + 2 \, {\left (a b^{4} - 6 \, a^{2} b^{2} c + 6 \, a^{3} c^{2} + {\left (b^{4} c - 6 \, a b^{2} c^{2} + 6 \, a^{2} c^{3}\right )} x^{2} + {\left (b^{5} - 6 \, a b^{3} c + 6 \, a^{2} b c^{2}\right )} x\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + {\left (b^{6} - 9 \, a b^{4} c + 26 \, a^{2} b^{2} c^{2} - 24 \, a^{3} c^{3}\right )} x + {\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2} + {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{2} + {\left (b^{6} - 8 \, a b^{4} c + 16 \, a^{2} b^{2} c^{2}\right )} x\right )} \log \left (c x^{2} + b x + a\right )}{a b^{4} c^{3} - 8 \, a^{2} b^{2} c^{4} + 16 \, a^{3} c^{5} + {\left (b^{4} c^{4} - 8 \, a b^{2} c^{5} + 16 \, a^{2} c^{6}\right )} x^{2} + {\left (b^{5} c^{3} - 8 \, a b^{3} c^{4} + 16 \, a^{2} b c^{5}\right )} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.57, size = 161, normalized size = 1.07 \begin {gather*} \frac {2 \, {\left (b^{4} - 6 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {x}{c^{2}} - \frac {b \log \left (c x^{2} + b x + a\right )}{c^{3}} - \frac {\frac {{\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} x}{c} + \frac {a b^{3} - 3 \, a^{2} b c}{c}}{{\left (c x^{2} + b x + a\right )} {\left (b^{2} - 4 \, a c\right )} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.01, size = 352, normalized size = 2.35 \begin {gather*} \frac {2 a^{2} x}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) c}-\frac {12 a^{2} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}} c}-\frac {4 a \,b^{2} x}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) c^{2}}+\frac {12 a \,b^{2} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}} c^{2}}+\frac {b^{4} x}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) c^{3}}-\frac {2 b^{4} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}} c^{3}}-\frac {3 a^{2} b}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) c^{2}}+\frac {a \,b^{3}}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) c^{3}}-\frac {4 a b \ln \left (c \,x^{2}+b x +a \right )}{\left (4 a c -b^{2}\right ) c^{2}}+\frac {b^{3} \ln \left (c \,x^{2}+b x +a \right )}{\left (4 a c -b^{2}\right ) c^{3}}+\frac {x}{c^{2}} \end {gather*}
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]
________________________________________________________________________________________
mupad [B] time = 2.46, size = 261, normalized size = 1.74 \begin {gather*} \frac {x}{c^2}+\frac {\frac {a\,\left (b^3-3\,a\,b\,c\right )}{c\,\left (4\,a\,c-b^2\right )}+\frac {x\,\left (2\,a^2\,c^2-4\,a\,b^2\,c+b^4\right )}{c\,\left (4\,a\,c-b^2\right )}}{c^3\,x^2+b\,c^2\,x+a\,c^2}+\frac {\ln \left (c\,x^2+b\,x+a\right )\,\left (-128\,a^3\,b\,c^3+96\,a^2\,b^3\,c^2-24\,a\,b^5\,c+2\,b^7\right )}{2\,\left (64\,a^3\,c^6-48\,a^2\,b^2\,c^5+12\,a\,b^4\,c^4-b^6\,c^3\right )}-\frac {2\,\mathrm {atan}\left (\frac {2\,c\,x}{\sqrt {4\,a\,c-b^2}}-\frac {b^3\,c^2-4\,a\,b\,c^3}{c^2\,{\left (4\,a\,c-b^2\right )}^{3/2}}\right )\,\left (6\,a^2\,c^2-6\,a\,b^2\,c+b^4\right )}{c^3\,{\left (4\,a\,c-b^2\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 1.79, size = 842, normalized size = 5.61 \begin {gather*} \left (- \frac {b}{c^{3}} - \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (6 a^{2} c^{2} - 6 a b^{2} c + b^{4}\right )}{c^{3} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}\right ) \log {\left (x + \frac {- 10 a^{2} b c - 16 a^{2} c^{4} \left (- \frac {b}{c^{3}} - \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (6 a^{2} c^{2} - 6 a b^{2} c + b^{4}\right )}{c^{3} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}\right ) + 2 a b^{3} + 8 a b^{2} c^{3} \left (- \frac {b}{c^{3}} - \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (6 a^{2} c^{2} - 6 a b^{2} c + b^{4}\right )}{c^{3} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}\right ) - b^{4} c^{2} \left (- \frac {b}{c^{3}} - \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (6 a^{2} c^{2} - 6 a b^{2} c + b^{4}\right )}{c^{3} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}\right )}{12 a^{2} c^{2} - 12 a b^{2} c + 2 b^{4}} \right )} + \left (- \frac {b}{c^{3}} + \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (6 a^{2} c^{2} - 6 a b^{2} c + b^{4}\right )}{c^{3} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}\right ) \log {\left (x + \frac {- 10 a^{2} b c - 16 a^{2} c^{4} \left (- \frac {b}{c^{3}} + \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (6 a^{2} c^{2} - 6 a b^{2} c + b^{4}\right )}{c^{3} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}\right ) + 2 a b^{3} + 8 a b^{2} c^{3} \left (- \frac {b}{c^{3}} + \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (6 a^{2} c^{2} - 6 a b^{2} c + b^{4}\right )}{c^{3} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}\right ) - b^{4} c^{2} \left (- \frac {b}{c^{3}} + \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (6 a^{2} c^{2} - 6 a b^{2} c + b^{4}\right )}{c^{3} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}\right )}{12 a^{2} c^{2} - 12 a b^{2} c + 2 b^{4}} \right )} + \frac {- 3 a^{2} b c + a b^{3} + x \left (2 a^{2} c^{2} - 4 a b^{2} c + b^{4}\right )}{4 a^{2} c^{4} - a b^{2} c^{3} + x^{2} \left (4 a c^{5} - b^{2} c^{4}\right ) + x \left (4 a b c^{4} - b^{3} c^{3}\right )} + \frac {x}{c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________