Optimal. Leaf size=211 \[ \frac {(2 b d-a e) \log \left (a+b x+c x^2\right )}{2 a^3}-\frac {\log (x) (2 b d-a e)}{a^3}-\frac {-a b e-6 a c d+2 b^2 d}{a^2 x \left (b^2-4 a c\right )}-\frac {\left (6 a^2 b c e+12 a^2 c^2 d-a b^3 e-12 a b^2 c d+2 b^4 d\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^3 \left (b^2-4 a c\right )^{3/2}}+\frac {c x (b d-2 a e)-a b e-2 a c d+b^2 d}{a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \]
________________________________________________________________________________________
Rubi [A] time = 0.36, antiderivative size = 211, 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 = {822, 800, 634, 618, 206, 628} \begin {gather*} -\frac {\left (6 a^2 b c e+12 a^2 c^2 d-12 a b^2 c d-a b^3 e+2 b^4 d\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^3 \left (b^2-4 a c\right )^{3/2}}-\frac {-a b e-6 a c d+2 b^2 d}{a^2 x \left (b^2-4 a c\right )}+\frac {(2 b d-a e) \log \left (a+b x+c x^2\right )}{2 a^3}-\frac {\log (x) (2 b d-a e)}{a^3}+\frac {c x (b d-2 a e)-a b e-2 a c d+b^2 d}{a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 618
Rule 628
Rule 634
Rule 800
Rule 822
Rubi steps
\begin {align*} \int \frac {d+e x}{x^2 \left (a+b x+c x^2\right )^2} \, dx &=\frac {b^2 d-2 a c d-a b e+c (b d-2 a e) x}{a \left (b^2-4 a c\right ) x \left (a+b x+c x^2\right )}-\frac {\int \frac {-2 b^2 d+6 a c d+a b e-2 c (b d-2 a e) x}{x^2 \left (a+b x+c x^2\right )} \, dx}{a \left (b^2-4 a c\right )}\\ &=\frac {b^2 d-2 a c d-a b e+c (b d-2 a e) x}{a \left (b^2-4 a c\right ) x \left (a+b x+c x^2\right )}-\frac {\int \left (\frac {-2 b^2 d+6 a c d+a b e}{a x^2}+\frac {\left (-b^2+4 a c\right ) (-2 b d+a e)}{a^2 x}+\frac {-2 b^4 d+10 a b^2 c d-6 a^2 c^2 d+a b^3 e-5 a^2 b c e-c \left (b^2-4 a c\right ) (2 b d-a e) x}{a^2 \left (a+b x+c x^2\right )}\right ) \, dx}{a \left (b^2-4 a c\right )}\\ &=-\frac {2 b^2 d-6 a c d-a b e}{a^2 \left (b^2-4 a c\right ) x}+\frac {b^2 d-2 a c d-a b e+c (b d-2 a e) x}{a \left (b^2-4 a c\right ) x \left (a+b x+c x^2\right )}-\frac {(2 b d-a e) \log (x)}{a^3}-\frac {\int \frac {-2 b^4 d+10 a b^2 c d-6 a^2 c^2 d+a b^3 e-5 a^2 b c e-c \left (b^2-4 a c\right ) (2 b d-a e) x}{a+b x+c x^2} \, dx}{a^3 \left (b^2-4 a c\right )}\\ &=-\frac {2 b^2 d-6 a c d-a b e}{a^2 \left (b^2-4 a c\right ) x}+\frac {b^2 d-2 a c d-a b e+c (b d-2 a e) x}{a \left (b^2-4 a c\right ) x \left (a+b x+c x^2\right )}-\frac {(2 b d-a e) \log (x)}{a^3}+\frac {(2 b d-a e) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 a^3}+\frac {\left (2 b^4 d-12 a b^2 c d+12 a^2 c^2 d-a b^3 e+6 a^2 b c e\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 a^3 \left (b^2-4 a c\right )}\\ &=-\frac {2 b^2 d-6 a c d-a b e}{a^2 \left (b^2-4 a c\right ) x}+\frac {b^2 d-2 a c d-a b e+c (b d-2 a e) x}{a \left (b^2-4 a c\right ) x \left (a+b x+c x^2\right )}-\frac {(2 b d-a e) \log (x)}{a^3}+\frac {(2 b d-a e) \log \left (a+b x+c x^2\right )}{2 a^3}-\frac {\left (2 b^4 d-12 a b^2 c d+12 a^2 c^2 d-a b^3 e+6 a^2 b c e\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{a^3 \left (b^2-4 a c\right )}\\ &=-\frac {2 b^2 d-6 a c d-a b e}{a^2 \left (b^2-4 a c\right ) x}+\frac {b^2 d-2 a c d-a b e+c (b d-2 a e) x}{a \left (b^2-4 a c\right ) x \left (a+b x+c x^2\right )}-\frac {\left (2 b^4 d-12 a b^2 c d+12 a^2 c^2 d-a b^3 e+6 a^2 b c e\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^3 \left (b^2-4 a c\right )^{3/2}}-\frac {(2 b d-a e) \log (x)}{a^3}+\frac {(2 b d-a e) \log \left (a+b x+c x^2\right )}{2 a^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.34, size = 192, normalized size = 0.91 \begin {gather*} \frac {-\frac {2 \left (6 a^2 b c e+12 a^2 c^2 d-a b^3 e-12 a b^2 c d+2 b^4 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 a \left (b^2 (c d x-a e)-a b c (3 d+e x)+2 a c (a e-c d x)+b^3 d\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}+(2 b d-a e) \log (a+x (b+c x))+2 \log (x) (a e-2 b d)-\frac {2 a d}{x}}{2 a^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {d+e x}{x^2 \left (a+b x+c x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 2.03, size = 1615, normalized size = 7.65
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.16, size = 245, normalized size = 1.16 \begin {gather*} \frac {{\left (2 \, b^{4} d - 12 \, a b^{2} c d + 12 \, a^{2} c^{2} d - a b^{3} e + 6 \, a^{2} b c e\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {2 \, b^{2} c d x^{2} - 6 \, a c^{2} d x^{2} - a b c x^{2} e + 2 \, b^{3} d x - 7 \, a b c d x - a b^{2} x e + 2 \, a^{2} c x e + a b^{2} d - 4 \, a^{2} c d}{{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} {\left (c x^{3} + b x^{2} + a x\right )}} + \frac {{\left (2 \, b d - a e\right )} \log \left (c x^{2} + b x + a\right )}{2 \, a^{3}} - \frac {{\left (2 \, b d - a e\right )} \log \left ({\left | x \right |}\right )}{a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.06, size = 582, normalized size = 2.76 \begin {gather*} -\frac {b c e x}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) a}-\frac {6 b c 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}} a}-\frac {2 c^{2} d x}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) a}-\frac {12 c^{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}} a}+\frac {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}} a^{2}}+\frac {b^{2} c d x}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) a^{2}}+\frac {12 b^{2} c 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}} a^{2}}-\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}} a^{3}}-\frac {b^{2} e}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) a}-\frac {3 b c d}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) a}-\frac {2 c e \ln \left (c \,x^{2}+b x +a \right )}{\left (4 a c -b^{2}\right ) a}+\frac {b^{3} d}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right ) a^{2}}+\frac {b^{2} e \ln \left (c \,x^{2}+b x +a \right )}{2 \left (4 a c -b^{2}\right ) a^{2}}+\frac {4 b c d \ln \left (c \,x^{2}+b x +a \right )}{\left (4 a c -b^{2}\right ) a^{2}}-\frac {b^{3} d \ln \left (c \,x^{2}+b x +a \right )}{\left (4 a c -b^{2}\right ) a^{3}}+\frac {2 c e}{\left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right )}+\frac {e \ln \relax (x )}{a^{2}}-\frac {2 b d \ln \relax (x )}{a^{3}}-\frac {d}{a^{2} x} \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 = 3.09, size = 1366, normalized size = 6.47 \begin {gather*} \ln \left (96\,a^5\,c^3\,e-2\,a^2\,b^6\,e+4\,a\,b^7\,d+4\,b^8\,d\,x+174\,a^3\,b^3\,c^2\,d-2\,a^2\,b^3\,e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+6\,a^3\,c^2\,d\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-84\,a^4\,b^2\,c^2\,e-2\,a\,b^7\,e\,x+4\,a\,b^4\,d\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-46\,a^2\,b^5\,c\,d-216\,a^4\,b\,c^3\,d+23\,a^3\,b^4\,c\,e+48\,a^4\,c^4\,d\,x+4\,b^5\,d\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+9\,a^3\,b\,c\,e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-2\,a\,b^4\,e\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+24\,a^2\,b^5\,c\,e\,x+120\,a^4\,b\,c^3\,e\,x-18\,a^2\,b^2\,c\,d\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+194\,a^2\,b^4\,c^2\,d\,x-276\,a^3\,b^2\,c^3\,d\,x-94\,a^3\,b^3\,c^2\,e\,x-12\,a^3\,c^2\,e\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-48\,a\,b^6\,c\,d\,x-24\,a\,b^3\,c\,d\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+30\,a^2\,b\,c^2\,d\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+12\,a^2\,b^2\,c\,e\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}\right )\,\left (\frac {b^4\,d\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+6\,a^2\,c^2\,d\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-\frac {a\,b^3\,e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}}{2}-6\,a\,b^2\,c\,d\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+3\,a^2\,b\,c\,e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}}{-64\,a^6\,c^3+48\,a^5\,b^2\,c^2-12\,a^4\,b^4\,c+a^3\,b^6}-\frac {e}{2\,a^2}+\frac {b\,d}{a^3}\right )-\ln \left (2\,a^2\,b^6\,e-96\,a^5\,c^3\,e-4\,a\,b^7\,d-4\,b^8\,d\,x-174\,a^3\,b^3\,c^2\,d-2\,a^2\,b^3\,e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+6\,a^3\,c^2\,d\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+84\,a^4\,b^2\,c^2\,e+2\,a\,b^7\,e\,x+4\,a\,b^4\,d\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+46\,a^2\,b^5\,c\,d+216\,a^4\,b\,c^3\,d-23\,a^3\,b^4\,c\,e-48\,a^4\,c^4\,d\,x+4\,b^5\,d\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+9\,a^3\,b\,c\,e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-2\,a\,b^4\,e\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-24\,a^2\,b^5\,c\,e\,x-120\,a^4\,b\,c^3\,e\,x-18\,a^2\,b^2\,c\,d\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-194\,a^2\,b^4\,c^2\,d\,x+276\,a^3\,b^2\,c^3\,d\,x+94\,a^3\,b^3\,c^2\,e\,x-12\,a^3\,c^2\,e\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+48\,a\,b^6\,c\,d\,x-24\,a\,b^3\,c\,d\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+30\,a^2\,b\,c^2\,d\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+12\,a^2\,b^2\,c\,e\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}\right )\,\left (\frac {b^4\,d\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+6\,a^2\,c^2\,d\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-\frac {a\,b^3\,e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}}{2}-6\,a\,b^2\,c\,d\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+3\,a^2\,b\,c\,e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}}{-64\,a^6\,c^3+48\,a^5\,b^2\,c^2-12\,a^4\,b^4\,c+a^3\,b^6}+\frac {e}{2\,a^2}-\frac {b\,d}{a^3}\right )-\frac {\frac {d}{a}-\frac {x\,\left (2\,c\,e\,a^2-e\,a\,b^2-7\,c\,d\,a\,b+2\,d\,b^3\right )}{a^2\,\left (4\,a\,c-b^2\right )}+\frac {c\,x^2\,\left (-2\,d\,b^2+a\,e\,b+6\,a\,c\,d\right )}{a^2\,\left (4\,a\,c-b^2\right )}}{c\,x^3+b\,x^2+a\,x}+\frac {\ln \relax (x)\,\left (a\,e-2\,b\,d\right )}{a^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] 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.
[In]
[Out]
________________________________________________________________________________________