Optimal. Leaf size=262 \[ -\frac {\left (-30 a^2 b c^2 e+12 a^2 c^3 d+20 a b^3 c e-12 a b^2 c^2 d-3 b^5 e+2 b^4 c d\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^4 \left (b^2-4 a c\right )^{3/2}}-\frac {\left (2 a c e-3 b^2 e+2 b c d\right ) \log \left (a+b x+c x^2\right )}{2 c^4}-\frac {x^2 \left (8 a c e-3 b^2 e+2 b c d\right )}{2 c^2 \left (b^2-4 a c\right )}+\frac {x^3 \left (x \left (2 a c e+b^2 (-e)+b c d\right )+a (2 c d-b e)\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {x \left (11 a b c e-6 a c^2 d-3 b^3 e+2 b^2 c d\right )}{c^3 \left (b^2-4 a c\right )} \]
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Rubi [A] time = 0.64, antiderivative size = 262, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {818, 800, 634, 618, 206, 628} \begin {gather*} -\frac {\left (-30 a^2 b c^2 e+12 a^2 c^3 d-12 a b^2 c^2 d+20 a b^3 c e+2 b^4 c d-3 b^5 e\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^4 \left (b^2-4 a c\right )^{3/2}}-\frac {x^2 \left (8 a c e-3 b^2 e+2 b c d\right )}{2 c^2 \left (b^2-4 a c\right )}-\frac {\left (2 a c e-3 b^2 e+2 b c d\right ) \log \left (a+b x+c x^2\right )}{2 c^4}+\frac {x \left (11 a b c e-6 a c^2 d+2 b^2 c d-3 b^3 e\right )}{c^3 \left (b^2-4 a c\right )}+\frac {x^3 \left (x \left (2 a c e+b^2 (-e)+b c d\right )+a (2 c d-b e)\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 628
Rule 634
Rule 800
Rule 818
Rubi steps
\begin {align*} \int \frac {x^4 (d+e x)}{\left (a+b x+c x^2\right )^2} \, dx &=\frac {x^3 \left (a (2 c d-b e)+\left (b c d-b^2 e+2 a c e\right ) x\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\int \frac {x^2 \left (-3 a (2 c d-b e)-\left (2 b c d-3 b^2 e+8 a c e\right ) x\right )}{a+b x+c x^2} \, dx}{c \left (b^2-4 a c\right )}\\ &=\frac {x^3 \left (a (2 c d-b e)+\left (b c d-b^2 e+2 a c e\right ) x\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\int \left (\frac {2 b^2 c d-6 a c^2 d-3 b^3 e+11 a b c e}{c^2}-\frac {\left (2 b c d-3 b^2 e+8 a c e\right ) x}{c}-\frac {a \left (2 b^2 c d-6 a c^2 d-3 b^3 e+11 a b c e\right )+\left (b^2-4 a c\right ) \left (2 b c d-3 b^2 e+2 a c e\right ) x}{c^2 \left (a+b x+c x^2\right )}\right ) \, dx}{c \left (b^2-4 a c\right )}\\ &=\frac {\left (2 b^2 c d-6 a c^2 d-3 b^3 e+11 a b c e\right ) x}{c^3 \left (b^2-4 a c\right )}-\frac {\left (2 b c d-3 b^2 e+8 a c e\right ) x^2}{2 c^2 \left (b^2-4 a c\right )}+\frac {x^3 \left (a (2 c d-b e)+\left (b c d-b^2 e+2 a c e\right ) x\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\int \frac {a \left (2 b^2 c d-6 a c^2 d-3 b^3 e+11 a b c e\right )+\left (b^2-4 a c\right ) \left (2 b c d-3 b^2 e+2 a c e\right ) x}{a+b x+c x^2} \, dx}{c^3 \left (b^2-4 a c\right )}\\ &=\frac {\left (2 b^2 c d-6 a c^2 d-3 b^3 e+11 a b c e\right ) x}{c^3 \left (b^2-4 a c\right )}-\frac {\left (2 b c d-3 b^2 e+8 a c e\right ) x^2}{2 c^2 \left (b^2-4 a c\right )}+\frac {x^3 \left (a (2 c d-b e)+\left (b c d-b^2 e+2 a c e\right ) x\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\left (2 b c d-3 b^2 e+2 a c e\right ) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 c^4}+\frac {\left (2 b^4 c d-12 a b^2 c^2 d+12 a^2 c^3 d-3 b^5 e+20 a b^3 c e-30 a^2 b c^2 e\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 c^4 \left (b^2-4 a c\right )}\\ &=\frac {\left (2 b^2 c d-6 a c^2 d-3 b^3 e+11 a b c e\right ) x}{c^3 \left (b^2-4 a c\right )}-\frac {\left (2 b c d-3 b^2 e+8 a c e\right ) x^2}{2 c^2 \left (b^2-4 a c\right )}+\frac {x^3 \left (a (2 c d-b e)+\left (b c d-b^2 e+2 a c e\right ) x\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\left (2 b c d-3 b^2 e+2 a c e\right ) \log \left (a+b x+c x^2\right )}{2 c^4}-\frac {\left (2 b^4 c d-12 a b^2 c^2 d+12 a^2 c^3 d-3 b^5 e+20 a b^3 c e-30 a^2 b c^2 e\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{c^4 \left (b^2-4 a c\right )}\\ &=\frac {\left (2 b^2 c d-6 a c^2 d-3 b^3 e+11 a b c e\right ) x}{c^3 \left (b^2-4 a c\right )}-\frac {\left (2 b c d-3 b^2 e+8 a c e\right ) x^2}{2 c^2 \left (b^2-4 a c\right )}+\frac {x^3 \left (a (2 c d-b e)+\left (b c d-b^2 e+2 a c e\right ) x\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\left (2 b^4 c d-12 a b^2 c^2 d+12 a^2 c^3 d-3 b^5 e+20 a b^3 c e-30 a^2 b c^2 e\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^4 \left (b^2-4 a c\right )^{3/2}}-\frac {\left (2 b c d-3 b^2 e+2 a c e\right ) \log \left (a+b x+c x^2\right )}{2 c^4}\\ \end {align*}
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Mathematica [A] time = 0.36, size = 249, normalized size = 0.95 \begin {gather*} \frac {\frac {2 \left (30 a^2 b c^2 e-12 a^2 c^3 d-20 a b^3 c e+12 a b^2 c^2 d+3 b^5 e-2 b^4 c d\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 {2 \left (2 a^3 c^2 e+a^2 c \left (-4 b^2 e+b c (3 d+5 e x)-2 c^2 d x\right )+a b^2 \left (b^2 e-b c (d+5 e x)+4 c^2 d x\right )+b^4 x (b e-c d)\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}+\left (-2 a c e+3 b^2 e-2 b c d\right ) \log (a+x (b+c x))+2 c x (c d-2 b e)+c^2 e x^2}{2 c^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^4 (d+e x)}{\left (a+b x+c x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.48, size = 1696, normalized size = 6.47
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 297, normalized size = 1.13 \begin {gather*} \frac {{\left (2 \, b^{4} c d - 12 \, a b^{2} c^{2} d + 12 \, a^{2} c^{3} d - 3 \, b^{5} e + 20 \, a b^{3} c e - 30 \, a^{2} b c^{2} e\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} c^{4} - 4 \, a c^{5}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {{\left (2 \, b c d - 3 \, b^{2} e + 2 \, a c e\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c^{4}} + \frac {c^{2} x^{2} e + 2 \, c^{2} d x - 4 \, b c x e}{2 \, c^{4}} - \frac {a b^{3} c d - 3 \, a^{2} b c^{2} d - a b^{4} e + 4 \, a^{2} b^{2} c e - 2 \, a^{3} c^{2} e + {\left (b^{4} c d - 4 \, a b^{2} c^{2} d + 2 \, a^{2} c^{3} d - b^{5} e + 5 \, a b^{3} c e - 5 \, a^{2} b c^{2} e\right )} x}{{\left (c x^{2} + b x + a\right )} {\left (b^{2} - 4 \, a c\right )} c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 809, normalized size = 3.09 \begin {gather*} -\frac {5 a^{2} b e x}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) c^{2}}+\frac {30 a^{2} b e \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 {2 a^{2} d x}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) c}-\frac {12 a^{2} d \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 {5 a \,b^{3} e x}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) c^{3}}-\frac {20 a \,b^{3} e \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 {4 a \,b^{2} d x}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) c^{2}}+\frac {12 a \,b^{2} d \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^{5} e x}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) c^{4}}+\frac {3 b^{5} e \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^{4}}+\frac {b^{4} d x}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) c^{3}}-\frac {2 b^{4} d \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 {2 a^{3} e}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) c^{2}}+\frac {4 a^{2} b^{2} e}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) c^{3}}-\frac {3 a^{2} b d}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) c^{2}}-\frac {4 a^{2} e \ln \left (c \,x^{2}+b x +a \right )}{\left (4 a c -b^{2}\right ) c^{2}}-\frac {a \,b^{4} e}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) c^{4}}+\frac {a \,b^{3} d}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) c^{3}}+\frac {7 a \,b^{2} e \ln \left (c \,x^{2}+b x +a \right )}{\left (4 a c -b^{2}\right ) c^{3}}-\frac {4 a b d \ln \left (c \,x^{2}+b x +a \right )}{\left (4 a c -b^{2}\right ) c^{2}}-\frac {3 b^{4} e \ln \left (c \,x^{2}+b x +a \right )}{2 \left (4 a c -b^{2}\right ) c^{4}}+\frac {b^{3} d \ln \left (c \,x^{2}+b x +a \right )}{\left (4 a c -b^{2}\right ) c^{3}}+\frac {e \,x^{2}}{2 c^{2}}-\frac {2 b e x}{c^{3}}+\frac {d x}{c^{2}} \end {gather*}
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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mupad [B] time = 1.98, size = 427, normalized size = 1.63 \begin {gather*} x\,\left (\frac {d}{c^2}-\frac {2\,b\,e}{c^3}\right )-\frac {\frac {a\,\left (2\,e\,a^2\,c^2-4\,e\,a\,b^2\,c+3\,d\,a\,b\,c^2+e\,b^4-d\,b^3\,c\right )}{c\,\left (4\,a\,c-b^2\right )}+\frac {x\,\left (5\,e\,a^2\,b\,c^2-2\,d\,a^2\,c^3-5\,e\,a\,b^3\,c+4\,d\,a\,b^2\,c^2+e\,b^5-d\,b^4\,c\right )}{c\,\left (4\,a\,c-b^2\right )}}{c^4\,x^2+b\,c^3\,x+a\,c^3}+\frac {e\,x^2}{2\,c^2}-\frac {\ln \left (c\,x^2+b\,x+a\right )\,\left (128\,e\,a^4\,c^4-288\,e\,a^3\,b^2\,c^3+128\,d\,a^3\,b\,c^4+168\,e\,a^2\,b^4\,c^2-96\,d\,a^2\,b^3\,c^3-38\,e\,a\,b^6\,c+24\,d\,a\,b^5\,c^2+3\,e\,b^8-2\,d\,b^7\,c\right )}{2\,\left (64\,a^3\,c^7-48\,a^2\,b^2\,c^6+12\,a\,b^4\,c^5-b^6\,c^4\right )}+\frac {\mathrm {atan}\left (\frac {2\,c\,x}{\sqrt {4\,a\,c-b^2}}-\frac {b^3\,c^3-4\,a\,b\,c^4}{c^3\,{\left (4\,a\,c-b^2\right )}^{3/2}}\right )\,\left (30\,e\,a^2\,b\,c^2-12\,d\,a^2\,c^3-20\,e\,a\,b^3\,c+12\,d\,a\,b^2\,c^2+3\,e\,b^5-2\,d\,b^4\,c\right )}{c^4\,{\left (4\,a\,c-b^2\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 7.71, size = 1572, normalized size = 6.00
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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