Optimal. Leaf size=198 \[ -\frac {-x \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )-3 b \left (a e^2+c d^2\right )+4 a c d e+2 b^2 d e}{\left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {2 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}-\frac {(d+e x) (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]
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Rubi [A] time = 0.22, antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {738, 638, 618, 206} \begin {gather*} -\frac {-x \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )-3 b \left (a e^2+c d^2\right )+4 a c d e+2 b^2 d e}{\left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {2 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}-\frac {(d+e x) (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 638
Rule 738
Rubi steps
\begin {align*} \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^3} \, dx &=-\frac {(d+e x) (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {\int \frac {2 \left (3 c d^2-e (2 b d-a e)\right )+2 e (2 c d-b e) x}{\left (a+b x+c x^2\right )^2} \, dx}{2 \left (b^2-4 a c\right )}\\ &=-\frac {(d+e x) (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {2 b^2 d e+4 a c d e-3 b \left (c d^2+a e^2\right )-\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) x}{\left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) \int \frac {1}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^2}\\ &=-\frac {(d+e x) (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {2 b^2 d e+4 a c d e-3 b \left (c d^2+a e^2\right )-\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) x}{\left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {\left (2 \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{\left (b^2-4 a c\right )^2}\\ &=-\frac {(d+e x) (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {2 b^2 d e+4 a c d e-3 b \left (c d^2+a e^2\right )-\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) x}{\left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {2 \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 203, normalized size = 1.03 \begin {gather*} \frac {1}{2} \left (\frac {(b+2 c x) \left (2 c e (a e-3 b d)+b^2 e^2+6 c^2 d^2\right )}{c \left (b^2-4 a c\right )^2 (a+x (b+c x))}+\frac {a b e^2-2 a c e (2 d+e x)+b^2 e^2 x+b c d (d-2 e x)+2 c^2 d^2 x}{c \left (4 a c-b^2\right ) (a+x (b+c x))^2}+\frac {4 \left (2 c e (a e-3 b d)+b^2 e^2+6 c^2 d^2\right ) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{5/2}}\right ) \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 {(d+e x)^2}{\left (a+b x+c x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.47, size = 1555, normalized size = 7.85
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 = 307, normalized size = 1.55 \begin {gather*} \frac {2 \, {\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2} + 2 \, a c e^{2}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {12 \, c^{3} d^{2} x^{3} - 12 \, b c^{2} d x^{3} e + 18 \, b c^{2} d^{2} x^{2} + 2 \, b^{2} c x^{3} e^{2} + 4 \, a c^{2} x^{3} e^{2} - 18 \, b^{2} c d x^{2} e + 4 \, b^{2} c d^{2} x + 20 \, a c^{2} d^{2} x + 3 \, b^{3} x^{2} e^{2} + 6 \, a b c x^{2} e^{2} - 4 \, b^{3} d x e - 20 \, a b c d x e - b^{3} d^{2} + 10 \, a b c d^{2} + 10 \, a b^{2} x e^{2} - 4 \, a^{2} c x e^{2} - 2 \, a b^{2} d e - 16 \, a^{2} c d e + 6 \, a^{2} b e^{2}}{2 \, {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} {\left (c x^{2} + b x + a\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 508, normalized size = 2.57 \begin {gather*} \frac {4 a c \,e^{2} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \sqrt {4 a c -b^{2}}}+\frac {2 b^{2} e^{2} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \sqrt {4 a c -b^{2}}}-\frac {12 b c d e \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \sqrt {4 a c -b^{2}}}+\frac {12 c^{2} d^{2} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \sqrt {4 a c -b^{2}}}+\frac {\frac {\left (2 a c \,e^{2}+b^{2} e^{2}-6 b c d e +6 c^{2} d^{2}\right ) c \,x^{3}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {3 \left (2 a c \,e^{2}+b^{2} e^{2}-6 b c d e +6 c^{2} d^{2}\right ) b \,x^{2}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {\left (2 a^{2} c \,e^{2}-5 a \,b^{2} e^{2}+10 a b c d e -10 a \,c^{2} d^{2}+2 b^{3} d e -2 b^{2} c \,d^{2}\right ) x}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {6 a^{2} b \,e^{2}-16 a^{2} c d e -2 a \,b^{2} d e +10 a b c \,d^{2}-b^{3} d^{2}}{32 a^{2} c^{2}-16 a \,b^{2} c +2 b^{4}}}{\left (c \,x^{2}+b x +a \right )^{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 = 1.05, size = 517, normalized size = 2.61 \begin {gather*} \frac {\frac {x\,\left (-2\,a^2\,c\,e^2+5\,a\,b^2\,e^2-10\,a\,b\,c\,d\,e+10\,a\,c^2\,d^2-2\,b^3\,d\,e+2\,b^2\,c\,d^2\right )}{16\,a^2\,c^2-8\,a\,b^2\,c+b^4}-\frac {-6\,a^2\,b\,e^2+16\,c\,a^2\,d\,e+2\,a\,b^2\,d\,e-10\,c\,a\,b\,d^2+b^3\,d^2}{2\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {3\,b\,x^2\,\left (b^2\,e^2-6\,b\,c\,d\,e+6\,c^2\,d^2+2\,a\,c\,e^2\right )}{2\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {c\,x^3\,\left (b^2\,e^2-6\,b\,c\,d\,e+6\,c^2\,d^2+2\,a\,c\,e^2\right )}{16\,a^2\,c^2-8\,a\,b^2\,c+b^4}}{x^2\,\left (b^2+2\,a\,c\right )+a^2+c^2\,x^4+2\,a\,b\,x+2\,b\,c\,x^3}+\frac {2\,\mathrm {atan}\left (\frac {\left (\frac {2\,c\,x\,\left (b^2\,e^2-6\,b\,c\,d\,e+6\,c^2\,d^2+2\,a\,c\,e^2\right )}{{\left (4\,a\,c-b^2\right )}^{5/2}}+\frac {\left (16\,a^2\,b\,c^2-8\,a\,b^3\,c+b^5\right )\,\left (b^2\,e^2-6\,b\,c\,d\,e+6\,c^2\,d^2+2\,a\,c\,e^2\right )}{{\left (4\,a\,c-b^2\right )}^{5/2}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}\right )\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}{b^2\,e^2-6\,b\,c\,d\,e+6\,c^2\,d^2+2\,a\,c\,e^2}\right )\,\left (b^2\,e^2-6\,b\,c\,d\,e+6\,c^2\,d^2+2\,a\,c\,e^2\right )}{{\left (4\,a\,c-b^2\right )}^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.71, size = 1052, normalized size = 5.31 \begin {gather*} - \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (2 a c e^{2} + b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right ) \log {\left (x + \frac {- 64 a^{3} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (2 a c e^{2} + b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right ) + 48 a^{2} b^{2} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (2 a c e^{2} + b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right ) - 12 a b^{4} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (2 a c e^{2} + b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right ) + 2 a b c e^{2} + b^{6} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (2 a c e^{2} + b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right ) + b^{3} e^{2} - 6 b^{2} c d e + 6 b c^{2} d^{2}}{4 a c^{2} e^{2} + 2 b^{2} c e^{2} - 12 b c^{2} d e + 12 c^{3} d^{2}} \right )} + \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (2 a c e^{2} + b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right ) \log {\left (x + \frac {64 a^{3} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (2 a c e^{2} + b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right ) - 48 a^{2} b^{2} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (2 a c e^{2} + b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right ) + 12 a b^{4} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (2 a c e^{2} + b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right ) + 2 a b c e^{2} - b^{6} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (2 a c e^{2} + b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right ) + b^{3} e^{2} - 6 b^{2} c d e + 6 b c^{2} d^{2}}{4 a c^{2} e^{2} + 2 b^{2} c e^{2} - 12 b c^{2} d e + 12 c^{3} d^{2}} \right )} + \frac {6 a^{2} b e^{2} - 16 a^{2} c d e - 2 a b^{2} d e + 10 a b c d^{2} - b^{3} d^{2} + x^{3} \left (4 a c^{2} e^{2} + 2 b^{2} c e^{2} - 12 b c^{2} d e + 12 c^{3} d^{2}\right ) + x^{2} \left (6 a b c e^{2} + 3 b^{3} e^{2} - 18 b^{2} c d e + 18 b c^{2} d^{2}\right ) + x \left (- 4 a^{2} c e^{2} + 10 a b^{2} e^{2} - 20 a b c d e + 20 a c^{2} d^{2} - 4 b^{3} d e + 4 b^{2} c d^{2}\right )}{32 a^{4} c^{2} - 16 a^{3} b^{2} c + 2 a^{2} b^{4} + x^{4} \left (32 a^{2} c^{4} - 16 a b^{2} c^{3} + 2 b^{4} c^{2}\right ) + x^{3} \left (64 a^{2} b c^{3} - 32 a b^{3} c^{2} + 4 b^{5} c\right ) + x^{2} \left (64 a^{3} c^{3} - 12 a b^{4} c + 2 b^{6}\right ) + x \left (64 a^{3} b c^{2} - 32 a^{2} b^{3} c + 4 a b^{5}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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