Optimal. Leaf size=169 \[ \frac {3 (d+e x) \left (a e^2-b d e+c d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {12 \left (a e^2-b d e+c d^2\right )^2 \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}-\frac {(d+e x)^3 (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]
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Rubi [A] time = 0.12, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {722, 618, 206} \begin {gather*} \frac {3 (d+e x) \left (a e^2-b d e+c d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {12 \left (a e^2-b d e+c d^2\right )^2 \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)^3 (-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 722
Rubi steps
\begin {align*} \int \frac {(d+e x)^4}{\left (a+b x+c x^2\right )^3} \, dx &=-\frac {(d+e x)^3 (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 {\left (3 \left (c d^2-b d e+a e^2\right )\right ) \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^2} \, dx}{b^2-4 a c}\\ &=-\frac {(d+e x)^3 (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 {3 \left (c d^2-b d e+a e^2\right ) (d+e x) (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {\left (6 \left (c d^2-b d e+a e^2\right )^2\right ) \int \frac {1}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^2}\\ &=-\frac {(d+e x)^3 (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 {3 \left (c d^2-b d e+a e^2\right ) (d+e x) (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {\left (12 \left (c d^2-b d e+a e^2\right )^2\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)^3 (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 {3 \left (c d^2-b d e+a e^2\right ) (d+e x) (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {12 \left (c d^2-b d e+a e^2\right )^2 \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 [B] time = 0.55, size = 413, normalized size = 2.44 \begin {gather*} \frac {1}{2} \left (\frac {b c \left (-3 a^2 e^4+6 a c d e^2 (d+2 e x)+c^2 d^3 (d-4 e x)\right )+2 c^2 \left (a^2 e^3 (4 d+e x)-2 a c d^2 e (2 d+3 e x)+c^2 d^4 x\right )+b^3 e^3 (a e-4 c d x)+2 b^2 c e^2 \left (3 c d^2 x-2 a e (d+e x)\right )+b^4 e^4 x}{c^3 \left (4 a c-b^2\right ) (a+x (b+c x))^2}+\frac {2 b c^2 \left (11 a^2 e^4+6 a c d e^2 (d-2 e x)+3 c^2 d^3 (d-4 e x)\right )+4 c^3 \left (-a^2 e^3 (16 d+5 e x)+6 a c d^2 e^2 x+3 c^2 d^4 x\right )+2 b^3 c e^2 \left (3 c d^2-4 a e^2\right )+4 b^2 c^2 e \left (a e^2 (5 d+4 e x)-3 c d^2 (d-e x)\right )+b^5 e^4-2 b^4 c e^3 (2 d+e x)}{c^3 \left (b^2-4 a c\right )^2 (a+x (b+c x))}+\frac {24 \left (e (a e-b d)+c d^2\right )^2 \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)^4}{\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 = 0.45, size = 2631, normalized size = 15.57
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.20, size = 626, normalized size = 3.70 \begin {gather*} \frac {12 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\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^{5} d^{4} x^{3} - 24 \, b c^{4} d^{3} x^{3} e + 18 \, b c^{4} d^{4} x^{2} + 12 \, b^{2} c^{3} d^{2} x^{3} e^{2} + 24 \, a c^{4} d^{2} x^{3} e^{2} - 36 \, b^{2} c^{3} d^{3} x^{2} e + 4 \, b^{2} c^{3} d^{4} x + 20 \, a c^{4} d^{4} x - 24 \, a b c^{3} d x^{3} e^{3} + 18 \, b^{3} c^{2} d^{2} x^{2} e^{2} + 36 \, a b c^{3} d^{2} x^{2} e^{2} - 8 \, b^{3} c^{2} d^{3} x e - 40 \, a b c^{3} d^{3} x e - b^{3} c^{2} d^{4} + 10 \, a b c^{3} d^{4} - 2 \, b^{4} c x^{3} e^{4} + 16 \, a b^{2} c^{2} x^{3} e^{4} - 20 \, a^{2} c^{3} x^{3} e^{4} - 4 \, b^{4} c d x^{2} e^{3} - 4 \, a b^{2} c^{2} d x^{2} e^{3} - 64 \, a^{2} c^{3} d x^{2} e^{3} + 60 \, a b^{2} c^{2} d^{2} x e^{2} - 24 \, a^{2} c^{3} d^{2} x e^{2} - 4 \, a b^{2} c^{2} d^{3} e - 32 \, a^{2} c^{3} d^{3} e - b^{5} x^{2} e^{4} + 8 \, a b^{3} c x^{2} e^{4} + 2 \, a^{2} b c^{2} x^{2} e^{4} - 8 \, a b^{3} c d x e^{3} - 40 \, a^{2} b c^{2} d x e^{3} + 36 \, a^{2} b c^{2} d^{2} e^{2} - 2 \, a b^{4} x e^{4} + 20 \, a^{2} b^{2} c x e^{4} - 12 \, a^{3} c^{2} x e^{4} - 4 \, a^{2} b^{2} c d e^{3} - 32 \, a^{3} c^{2} d e^{3} - a^{2} b^{3} e^{4} + 10 \, a^{3} b c e^{4}}{2 \, {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\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 = 932, normalized size = 5.51 \begin {gather*} \frac {12 a^{2} e^{4} \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 {24 a b d \,e^{3} \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 {24 a c \,d^{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^{2} d^{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 {24 b c \,d^{3} 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^{4} \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 (10 c^{2} a^{2} e^{4}-8 a \,b^{2} c \,e^{4}+12 a b \,c^{2} d \,e^{3}-12 a \,c^{3} d^{2} e^{2}+b^{4} e^{4}-6 b^{2} c^{2} d^{2} e^{2}+12 b \,c^{3} d^{3} e -6 c^{4} d^{4}\right ) x^{3}}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) c}+\frac {\left (2 a^{2} b \,c^{2} e^{4}-64 a^{2} c^{3} d \,e^{3}+8 a \,b^{3} c \,e^{4}-4 a \,b^{2} c^{2} d \,e^{3}+36 a b \,c^{3} d^{2} e^{2}-b^{5} e^{4}-4 b^{4} c d \,e^{3}+18 b^{3} c^{2} d^{2} e^{2}-36 b^{2} c^{3} d^{3} e +18 b \,c^{4} d^{4}\right ) x^{2}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) c^{2}}-\frac {\left (6 a^{3} c^{2} e^{4}-10 a^{2} b^{2} c \,e^{4}+20 a^{2} b \,c^{2} d \,e^{3}+12 a^{2} c^{3} d^{2} e^{2}+a \,b^{4} e^{4}+4 a \,b^{3} c d \,e^{3}-30 a \,b^{2} c^{2} d^{2} e^{2}+20 a b \,c^{3} d^{3} e -10 a \,c^{4} d^{4}+4 b^{3} c^{2} d^{3} e -2 b^{2} c^{3} d^{4}\right ) x}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) c^{2}}+\frac {10 a^{3} b c \,e^{4}-32 a^{3} c^{2} d \,e^{3}-a^{2} b^{3} e^{4}-4 a^{2} b^{2} c d \,e^{3}+36 a^{2} b \,c^{2} d^{2} e^{2}-32 a^{2} c^{3} d^{3} e -4 a \,b^{2} c^{2} d^{3} e +10 a b \,c^{3} d^{4}-b^{3} c^{2} d^{4}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) c^{2}}}{\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 = 1.32, size = 798, normalized size = 4.72 \begin {gather*} \frac {12\,\mathrm {atan}\left (\frac {\left (\frac {6\,\left (16\,a^2\,b\,c^2-8\,a\,b^3\,c+b^5\right )\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^2}{{\left (4\,a\,c-b^2\right )}^{5/2}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {12\,c\,x\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^2}{{\left (4\,a\,c-b^2\right )}^{5/2}}\right )\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}{6\,a^2\,e^4-12\,a\,b\,d\,e^3+12\,a\,c\,d^2\,e^2+6\,b^2\,d^2\,e^2-12\,b\,c\,d^3\,e+6\,c^2\,d^4}\right )\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^2}{{\left (4\,a\,c-b^2\right )}^{5/2}}-\frac {\frac {-10\,a^3\,b\,c\,e^4+32\,a^3\,c^2\,d\,e^3+a^2\,b^3\,e^4+4\,a^2\,b^2\,c\,d\,e^3-36\,a^2\,b\,c^2\,d^2\,e^2+32\,a^2\,c^3\,d^3\,e+4\,a\,b^2\,c^2\,d^3\,e-10\,a\,b\,c^3\,d^4+b^3\,c^2\,d^4}{2\,c^2\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {x^2\,\left (-2\,a^2\,b\,c^2\,e^4+64\,a^2\,c^3\,d\,e^3-8\,a\,b^3\,c\,e^4+4\,a\,b^2\,c^2\,d\,e^3-36\,a\,b\,c^3\,d^2\,e^2+b^5\,e^4+4\,b^4\,c\,d\,e^3-18\,b^3\,c^2\,d^2\,e^2+36\,b^2\,c^3\,d^3\,e-18\,b\,c^4\,d^4\right )}{2\,c^2\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {x^3\,\left (10\,a^2\,c^2\,e^4-8\,a\,b^2\,c\,e^4+12\,a\,b\,c^2\,d\,e^3-12\,a\,c^3\,d^2\,e^2+b^4\,e^4-6\,b^2\,c^2\,d^2\,e^2+12\,b\,c^3\,d^3\,e-6\,c^4\,d^4\right )}{c\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {x\,\left (6\,a^3\,c^2\,e^4-10\,a^2\,b^2\,c\,e^4+20\,a^2\,b\,c^2\,d\,e^3+12\,a^2\,c^3\,d^2\,e^2+a\,b^4\,e^4+4\,a\,b^3\,c\,d\,e^3-30\,a\,b^2\,c^2\,d^2\,e^2+20\,a\,b\,c^3\,d^3\,e-10\,a\,c^4\,d^4+4\,b^3\,c^2\,d^3\,e-2\,b^2\,c^3\,d^4\right )}{c^2\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}}{x^2\,\left (b^2+2\,a\,c\right )+a^2+c^2\,x^4+2\,a\,b\,x+2\,b\,c\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 32.63, size = 1355, normalized size = 8.02 \begin {gather*} - 6 \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (a e^{2} - b d e + c d^{2}\right )^{2} \log {\left (x + \frac {- 384 a^{3} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (a e^{2} - b d e + c d^{2}\right )^{2} + 288 a^{2} b^{2} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (a e^{2} - b d e + c d^{2}\right )^{2} + 6 a^{2} b e^{4} - 72 a b^{4} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (a e^{2} - b d e + c d^{2}\right )^{2} - 12 a b^{2} d e^{3} + 12 a b c d^{2} e^{2} + 6 b^{6} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (a e^{2} - b d e + c d^{2}\right )^{2} + 6 b^{3} d^{2} e^{2} - 12 b^{2} c d^{3} e + 6 b c^{2} d^{4}}{12 a^{2} c e^{4} - 24 a b c d e^{3} + 24 a c^{2} d^{2} e^{2} + 12 b^{2} c d^{2} e^{2} - 24 b c^{2} d^{3} e + 12 c^{3} d^{4}} \right )} + 6 \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (a e^{2} - b d e + c d^{2}\right )^{2} \log {\left (x + \frac {384 a^{3} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (a e^{2} - b d e + c d^{2}\right )^{2} - 288 a^{2} b^{2} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (a e^{2} - b d e + c d^{2}\right )^{2} + 6 a^{2} b e^{4} + 72 a b^{4} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (a e^{2} - b d e + c d^{2}\right )^{2} - 12 a b^{2} d e^{3} + 12 a b c d^{2} e^{2} - 6 b^{6} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (a e^{2} - b d e + c d^{2}\right )^{2} + 6 b^{3} d^{2} e^{2} - 12 b^{2} c d^{3} e + 6 b c^{2} d^{4}}{12 a^{2} c e^{4} - 24 a b c d e^{3} + 24 a c^{2} d^{2} e^{2} + 12 b^{2} c d^{2} e^{2} - 24 b c^{2} d^{3} e + 12 c^{3} d^{4}} \right )} + \frac {10 a^{3} b c e^{4} - 32 a^{3} c^{2} d e^{3} - a^{2} b^{3} e^{4} - 4 a^{2} b^{2} c d e^{3} + 36 a^{2} b c^{2} d^{2} e^{2} - 32 a^{2} c^{3} d^{3} e - 4 a b^{2} c^{2} d^{3} e + 10 a b c^{3} d^{4} - b^{3} c^{2} d^{4} + x^{3} \left (- 20 a^{2} c^{3} e^{4} + 16 a b^{2} c^{2} e^{4} - 24 a b c^{3} d e^{3} + 24 a c^{4} d^{2} e^{2} - 2 b^{4} c e^{4} + 12 b^{2} c^{3} d^{2} e^{2} - 24 b c^{4} d^{3} e + 12 c^{5} d^{4}\right ) + x^{2} \left (2 a^{2} b c^{2} e^{4} - 64 a^{2} c^{3} d e^{3} + 8 a b^{3} c e^{4} - 4 a b^{2} c^{2} d e^{3} + 36 a b c^{3} d^{2} e^{2} - b^{5} e^{4} - 4 b^{4} c d e^{3} + 18 b^{3} c^{2} d^{2} e^{2} - 36 b^{2} c^{3} d^{3} e + 18 b c^{4} d^{4}\right ) + x \left (- 12 a^{3} c^{2} e^{4} + 20 a^{2} b^{2} c e^{4} - 40 a^{2} b c^{2} d e^{3} - 24 a^{2} c^{3} d^{2} e^{2} - 2 a b^{4} e^{4} - 8 a b^{3} c d e^{3} + 60 a b^{2} c^{2} d^{2} e^{2} - 40 a b c^{3} d^{3} e + 20 a c^{4} d^{4} - 8 b^{3} c^{2} d^{3} e + 4 b^{2} c^{3} d^{4}\right )}{32 a^{4} c^{4} - 16 a^{3} b^{2} c^{3} + 2 a^{2} b^{4} c^{2} + x^{4} \left (32 a^{2} c^{6} - 16 a b^{2} c^{5} + 2 b^{4} c^{4}\right ) + x^{3} \left (64 a^{2} b c^{5} - 32 a b^{3} c^{4} + 4 b^{5} c^{3}\right ) + x^{2} \left (64 a^{3} c^{5} - 12 a b^{4} c^{3} + 2 b^{6} c^{2}\right ) + x \left (64 a^{3} b c^{4} - 32 a^{2} b^{3} c^{3} + 4 a b^{5} c^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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