Optimal. Leaf size=186 \[ -\frac {e}{\left (c d^2-b d e+a e^2\right ) (d+e x)}-\frac {\left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right )^2}+\frac {e (2 c d-b e) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^2}-\frac {e (2 c d-b e) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^2} \]
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Rubi [A]
time = 0.19, antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {723, 814, 648,
632, 212, 642} \begin {gather*} -\frac {\left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (a e^2-b d e+c d^2\right )^2}-\frac {e (2 c d-b e) \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )^2}-\frac {e}{(d+e x) \left (a e^2-b d e+c d^2\right )}+\frac {e (2 c d-b e) \log (d+e x)}{\left (a e^2-b d e+c d^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 632
Rule 642
Rule 648
Rule 723
Rule 814
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )} \, dx &=-\frac {e}{\left (c d^2-b d e+a e^2\right ) (d+e x)}+\frac {\int \frac {c d-b e-c e x}{(d+e x) \left (a+b x+c x^2\right )} \, dx}{c d^2-b d e+a e^2}\\ &=-\frac {e}{\left (c d^2-b d e+a e^2\right ) (d+e x)}+\frac {\int \left (-\frac {e^2 (-2 c d+b e)}{\left (c d^2-b d e+a e^2\right ) (d+e x)}+\frac {c^2 d^2+b^2 e^2-c e (2 b d+a e)-c e (2 c d-b e) x}{\left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}\right ) \, dx}{c d^2-b d e+a e^2}\\ &=-\frac {e}{\left (c d^2-b d e+a e^2\right ) (d+e x)}+\frac {e (2 c d-b e) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^2}+\frac {\int \frac {c^2 d^2+b^2 e^2-c e (2 b d+a e)-c e (2 c d-b e) x}{a+b x+c x^2} \, dx}{\left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac {e}{\left (c d^2-b d e+a e^2\right ) (d+e x)}+\frac {e (2 c d-b e) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^2}-\frac {(e (2 c d-b e)) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 \left (c d^2-b d e+a e^2\right )^2}+\frac {\left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac {e}{\left (c d^2-b d e+a e^2\right ) (d+e x)}+\frac {e (2 c d-b e) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^2}-\frac {e (2 c d-b e) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^2}-\frac {\left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{\left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac {e}{\left (c d^2-b d e+a e^2\right ) (d+e x)}-\frac {\left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right )^2}+\frac {e (2 c d-b e) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^2}-\frac {e (2 c d-b e) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^2}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 151, normalized size = 0.81 \begin {gather*} \frac {-\frac {2 e \left (c d^2+e (-b d+a e)\right )}{d+e x}+\frac {2 \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}-2 e (-2 c d+b e) \log (d+e x)+e (-2 c d+b e) \log (a+x (b+c x))}{2 \left (c d^2+e (-b d+a e)\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.04, size = 197, normalized size = 1.06
method | result | size |
default | \(\frac {\frac {\left (b c \,e^{2}-2 c^{2} d e \right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (-a c \,e^{2}+b^{2} e^{2}-2 b c d e +c^{2} d^{2}-\frac {\left (b c \,e^{2}-2 c^{2} d e \right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{\left (e^{2} a -b d e +c \,d^{2}\right )^{2}}-\frac {e}{\left (e^{2} a -b d e +c \,d^{2}\right ) \left (e x +d \right )}-\frac {e \left (b e -2 c d \right ) \ln \left (e x +d \right )}{\left (e^{2} a -b d e +c \,d^{2}\right )^{2}}\) | \(197\) |
risch | \(\text {Expression too large to display}\) | \(12892\) |
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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 526 vs.
\(2 (188) = 376\).
time = 7.22, size = 1071, normalized size = 5.76 \begin {gather*} \left [-\frac {2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d^{2} e - 2 \, {\left (b^{3} - 4 \, a b c\right )} d e^{2} + {\left (2 \, c^{2} d^{3} + {\left (b^{2} - 2 \, a c\right )} x e^{3} - {\left (2 \, b c d x - {\left (b^{2} - 2 \, a c\right )} d\right )} e^{2} + 2 \, {\left (c^{2} d^{2} x - b c d^{2}\right )} e\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 ) + 2 \, {\left (a b^{2} - 4 \, a^{2} c\right )} e^{3} + {\left (2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d^{2} e - {\left (b^{3} - 4 \, a b c\right )} x e^{3} + {\left (2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d x - {\left (b^{3} - 4 \, a b c\right )} d\right )} e^{2}\right )} \log \left (c x^{2} + b x + a\right ) - 2 \, {\left (2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d^{2} e - {\left (b^{3} - 4 \, a b c\right )} x e^{3} + {\left (2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d x - {\left (b^{3} - 4 \, a b c\right )} d\right )} e^{2}\right )} \log \left (x e + d\right )}{2 \, {\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{5} + {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} x e^{5} - {\left (2 \, {\left (a b^{3} - 4 \, a^{2} b c\right )} d x - {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} d\right )} e^{4} + {\left ({\left (b^{4} - 2 \, a b^{2} c - 8 \, a^{2} c^{2}\right )} d^{2} x - 2 \, {\left (a b^{3} - 4 \, a^{2} b c\right )} d^{2}\right )} e^{3} - {\left (2 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} d^{3} x - {\left (b^{4} - 2 \, a b^{2} c - 8 \, a^{2} c^{2}\right )} d^{3}\right )} e^{2} + {\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{4} x - 2 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} d^{4}\right )} e\right )}}, -\frac {2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d^{2} e - 2 \, {\left (b^{3} - 4 \, a b c\right )} d e^{2} + 2 \, {\left (2 \, c^{2} d^{3} + {\left (b^{2} - 2 \, a c\right )} x e^{3} - {\left (2 \, b c d x - {\left (b^{2} - 2 \, a c\right )} d\right )} e^{2} + 2 \, {\left (c^{2} d^{2} x - b c d^{2}\right )} e\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 ) + 2 \, {\left (a b^{2} - 4 \, a^{2} c\right )} e^{3} + {\left (2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d^{2} e - {\left (b^{3} - 4 \, a b c\right )} x e^{3} + {\left (2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d x - {\left (b^{3} - 4 \, a b c\right )} d\right )} e^{2}\right )} \log \left (c x^{2} + b x + a\right ) - 2 \, {\left (2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d^{2} e - {\left (b^{3} - 4 \, a b c\right )} x e^{3} + {\left (2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d x - {\left (b^{3} - 4 \, a b c\right )} d\right )} e^{2}\right )} \log \left (x e + d\right )}{2 \, {\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{5} + {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} x e^{5} - {\left (2 \, {\left (a b^{3} - 4 \, a^{2} b c\right )} d x - {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} d\right )} e^{4} + {\left ({\left (b^{4} - 2 \, a b^{2} c - 8 \, a^{2} c^{2}\right )} d^{2} x - 2 \, {\left (a b^{3} - 4 \, a^{2} b c\right )} d^{2}\right )} e^{3} - {\left (2 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} d^{3} x - {\left (b^{4} - 2 \, a b^{2} c - 8 \, a^{2} c^{2}\right )} d^{3}\right )} e^{2} + {\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{4} x - 2 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} d^{4}\right )} e\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.46, size = 325, normalized size = 1.75 \begin {gather*} \frac {{\left (2 \, c^{2} d^{2} e^{2} - 2 \, b c d e^{3} + b^{2} e^{4} - 2 \, a c e^{4}\right )} \arctan \left (\frac {{\left (2 \, c d - \frac {2 \, c d^{2}}{x e + d} - b e + \frac {2 \, b d e}{x e + d} - \frac {2 \, a e^{2}}{x e + d}\right )} e^{\left (-1\right )}}{\sqrt {-b^{2} + 4 \, a c}}\right ) e^{\left (-2\right )}}{{\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 )} \sqrt {-b^{2} + 4 \, a c}} - \frac {{\left (2 \, c d e - b e^{2}\right )} \log \left (c - \frac {2 \, c d}{x e + d} + \frac {c d^{2}}{{\left (x e + d\right )}^{2}} + \frac {b e}{x e + d} - \frac {b d e}{{\left (x e + d\right )}^{2}} + \frac {a e^{2}}{{\left (x e + d\right )}^{2}}\right )}{2 \, {\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 )}} - \frac {e^{3}}{{\left (c d^{2} e^{2} - b d e^{3} + a e^{4}\right )} {\left (x e + d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 7.09, size = 1786, normalized size = 9.60 \begin {gather*} \frac {\ln \left (a\,e^4\,{\left (b^2-4\,a\,c\right )}^{5/2}+8\,a\,b^5\,e^4+8\,b^6\,e^4\,x-4\,c^3\,d^4\,{\left (b^2-4\,a\,c\right )}^{3/2}+4\,b^3\,c^3\,d^4+4\,b^3\,e^4\,x\,{\left (b^2-4\,a\,c\right )}^{3/2}-60\,a^2\,b^3\,c\,e^4+112\,a^3\,b\,c^2\,e^4+256\,a^2\,c^4\,d^3\,e-256\,a^3\,c^3\,d\,e^3+8\,b^4\,c^2\,d^3\,e-4\,b^5\,c\,d^2\,e^2-32\,a^3\,c^3\,e^4\,x+8\,b^2\,c^4\,d^4\,x+10\,b\,d\,e^3\,{\left (b^2-4\,a\,c\right )}^{5/2}+4\,b\,e^4\,x\,{\left (b^2-4\,a\,c\right )}^{5/2}-16\,a\,b\,c^4\,d^4-32\,a\,c^5\,d^4\,x+7\,a\,b^2\,e^4\,{\left (b^2-4\,a\,c\right )}^{3/2}-10\,b^3\,d\,e^3\,{\left (b^2-4\,a\,c\right )}^{3/2}-14\,c\,d^2\,e^2\,{\left (b^2-4\,a\,c\right )}^{5/2}-8\,c\,d\,e^3\,x\,{\left (b^2-4\,a\,c\right )}^{5/2}-24\,a\,b^4\,c\,d\,e^3-64\,a\,b^4\,c\,e^4\,x-32\,b^5\,c\,d\,e^3\,x-8\,b\,c^2\,d^3\,e\,{\left (b^2-4\,a\,c\right )}^{3/2}-32\,c^3\,d^3\,e\,x\,{\left (b^2-4\,a\,c\right )}^{3/2}-96\,a\,b^2\,c^3\,d^3\,e-16\,b^3\,c^3\,d^3\,e\,x+18\,b^2\,c\,d^2\,e^2\,{\left (b^2-4\,a\,c\right )}^{3/2}+56\,a\,b^3\,c^2\,d^2\,e^2-160\,a^2\,b\,c^3\,d^2\,e^2+160\,a^2\,b^2\,c^2\,d\,e^3+136\,a^2\,b^2\,c^2\,e^4\,x+448\,a^2\,c^4\,d^2\,e^2\,x+40\,b^4\,c^2\,d^2\,e^2\,x+48\,b\,c^2\,d^2\,e^2\,x\,{\left (b^2-4\,a\,c\right )}^{3/2}-272\,a\,b^2\,c^3\,d^2\,e^2\,x+64\,a\,b\,c^4\,d^3\,e\,x-24\,b^2\,c\,d\,e^3\,x\,{\left (b^2-4\,a\,c\right )}^{3/2}+240\,a\,b^3\,c^2\,d\,e^3\,x-448\,a^2\,b\,c^3\,d\,e^3\,x\right )\,\left (a\,\left (e^2\,\left (2\,b\,c+c\,\sqrt {b^2-4\,a\,c}\right )-4\,c^2\,d\,e\right )+e\,\left (b^2\,c\,d+b\,c\,d\,\sqrt {b^2-4\,a\,c}\right )-e^2\,\left (\frac {b^3}{2}+\frac {b^2\,\sqrt {b^2-4\,a\,c}}{2}\right )-c^2\,d^2\,\sqrt {b^2-4\,a\,c}\right )}{4\,a^3\,c\,e^4-a^2\,b^2\,e^4-8\,a^2\,b\,c\,d\,e^3+8\,a^2\,c^2\,d^2\,e^2+2\,a\,b^3\,d\,e^3+2\,a\,b^2\,c\,d^2\,e^2-8\,a\,b\,c^2\,d^3\,e+4\,a\,c^3\,d^4-b^4\,d^2\,e^2+2\,b^3\,c\,d^3\,e-b^2\,c^2\,d^4}+\frac {\ln \left (a\,e^4\,{\left (b^2-4\,a\,c\right )}^{5/2}-8\,a\,b^5\,e^4-8\,b^6\,e^4\,x-4\,c^3\,d^4\,{\left (b^2-4\,a\,c\right )}^{3/2}-4\,b^3\,c^3\,d^4+4\,b^3\,e^4\,x\,{\left (b^2-4\,a\,c\right )}^{3/2}+60\,a^2\,b^3\,c\,e^4-112\,a^3\,b\,c^2\,e^4-256\,a^2\,c^4\,d^3\,e+256\,a^3\,c^3\,d\,e^3-8\,b^4\,c^2\,d^3\,e+4\,b^5\,c\,d^2\,e^2+32\,a^3\,c^3\,e^4\,x-8\,b^2\,c^4\,d^4\,x+10\,b\,d\,e^3\,{\left (b^2-4\,a\,c\right )}^{5/2}+4\,b\,e^4\,x\,{\left (b^2-4\,a\,c\right )}^{5/2}+16\,a\,b\,c^4\,d^4+32\,a\,c^5\,d^4\,x+7\,a\,b^2\,e^4\,{\left (b^2-4\,a\,c\right )}^{3/2}-10\,b^3\,d\,e^3\,{\left (b^2-4\,a\,c\right )}^{3/2}-14\,c\,d^2\,e^2\,{\left (b^2-4\,a\,c\right )}^{5/2}-8\,c\,d\,e^3\,x\,{\left (b^2-4\,a\,c\right )}^{5/2}+24\,a\,b^4\,c\,d\,e^3+64\,a\,b^4\,c\,e^4\,x+32\,b^5\,c\,d\,e^3\,x-8\,b\,c^2\,d^3\,e\,{\left (b^2-4\,a\,c\right )}^{3/2}-32\,c^3\,d^3\,e\,x\,{\left (b^2-4\,a\,c\right )}^{3/2}+96\,a\,b^2\,c^3\,d^3\,e+16\,b^3\,c^3\,d^3\,e\,x+18\,b^2\,c\,d^2\,e^2\,{\left (b^2-4\,a\,c\right )}^{3/2}-56\,a\,b^3\,c^2\,d^2\,e^2+160\,a^2\,b\,c^3\,d^2\,e^2-160\,a^2\,b^2\,c^2\,d\,e^3-136\,a^2\,b^2\,c^2\,e^4\,x-448\,a^2\,c^4\,d^2\,e^2\,x-40\,b^4\,c^2\,d^2\,e^2\,x+48\,b\,c^2\,d^2\,e^2\,x\,{\left (b^2-4\,a\,c\right )}^{3/2}+272\,a\,b^2\,c^3\,d^2\,e^2\,x-64\,a\,b\,c^4\,d^3\,e\,x-24\,b^2\,c\,d\,e^3\,x\,{\left (b^2-4\,a\,c\right )}^{3/2}-240\,a\,b^3\,c^2\,d\,e^3\,x+448\,a^2\,b\,c^3\,d\,e^3\,x\right )\,\left (a\,\left (e^2\,\left (2\,b\,c-c\,\sqrt {b^2-4\,a\,c}\right )-4\,c^2\,d\,e\right )+e\,\left (b^2\,c\,d-b\,c\,d\,\sqrt {b^2-4\,a\,c}\right )-e^2\,\left (\frac {b^3}{2}-\frac {b^2\,\sqrt {b^2-4\,a\,c}}{2}\right )+c^2\,d^2\,\sqrt {b^2-4\,a\,c}\right )}{4\,a^3\,c\,e^4-a^2\,b^2\,e^4-8\,a^2\,b\,c\,d\,e^3+8\,a^2\,c^2\,d^2\,e^2+2\,a\,b^3\,d\,e^3+2\,a\,b^2\,c\,d^2\,e^2-8\,a\,b\,c^2\,d^3\,e+4\,a\,c^3\,d^4-b^4\,d^2\,e^2+2\,b^3\,c\,d^3\,e-b^2\,c^2\,d^4}-\frac {e}{\left (d+e\,x\right )\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}-\frac {\ln \left (d+e\,x\right )\,\left (b\,e^2-2\,c\,d\,e\right )}{a^2\,e^4-2\,a\,b\,d\,e^3+2\,a\,c\,d^2\,e^2+b^2\,d^2\,e^2-2\,b\,c\,d^3\,e+c^2\,d^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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