Optimal. Leaf size=172 \[ \frac {2 e^2 \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{c^3}-\frac {4 e (2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^3 \sqrt {b^2-4 a c}}-\frac {(d+e x)^4}{a+b x+c x^2}+\frac {4 e^3 x (3 c d-b e)}{c^2}+\frac {2 e^4 x^2}{c} \]
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Rubi [A] time = 0.22, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {768, 701, 634, 618, 206, 628} \begin {gather*} \frac {2 e^2 \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{c^3}-\frac {4 e (2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^3 \sqrt {b^2-4 a c}}-\frac {(d+e x)^4}{a+b x+c x^2}+\frac {4 e^3 x (3 c d-b e)}{c^2}+\frac {2 e^4 x^2}{c} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 628
Rule 634
Rule 701
Rule 768
Rubi steps
\begin {align*} \int \frac {(b+2 c x) (d+e x)^4}{\left (a+b x+c x^2\right )^2} \, dx &=-\frac {(d+e x)^4}{a+b x+c x^2}+(4 e) \int \frac {(d+e x)^3}{a+b x+c x^2} \, dx\\ &=-\frac {(d+e x)^4}{a+b x+c x^2}+(4 e) \int \left (\frac {e^2 (3 c d-b e)}{c^2}+\frac {e^3 x}{c}+\frac {c^2 d^3-3 a c d e^2+a b e^3+e \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right ) x}{c^2 \left (a+b x+c x^2\right )}\right ) \, dx\\ &=\frac {4 e^3 (3 c d-b e) x}{c^2}+\frac {2 e^4 x^2}{c}-\frac {(d+e x)^4}{a+b x+c x^2}+\frac {(4 e) \int \frac {c^2 d^3-3 a c d e^2+a b e^3+e \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right ) x}{a+b x+c x^2} \, dx}{c^2}\\ &=\frac {4 e^3 (3 c d-b e) x}{c^2}+\frac {2 e^4 x^2}{c}-\frac {(d+e x)^4}{a+b x+c x^2}+\frac {\left (2 e^2 \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right )\right ) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{c^3}+\frac {\left (2 e (2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )\right ) \int \frac {1}{a+b x+c x^2} \, dx}{c^3}\\ &=\frac {4 e^3 (3 c d-b e) x}{c^2}+\frac {2 e^4 x^2}{c}-\frac {(d+e x)^4}{a+b x+c x^2}+\frac {2 e^2 \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right ) \log \left (a+b x+c x^2\right )}{c^3}-\frac {\left (4 e (2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{c^3}\\ &=\frac {4 e^3 (3 c d-b e) x}{c^2}+\frac {2 e^4 x^2}{c}-\frac {(d+e x)^4}{a+b x+c x^2}-\frac {4 e (2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^3 \sqrt {b^2-4 a c}}+\frac {2 e^2 \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right ) \log \left (a+b x+c x^2\right )}{c^3}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 241, normalized size = 1.40 \begin {gather*} \frac {\frac {-c e^3 \left (a^2 e+2 a b (2 d+e x)+4 b^2 d x\right )+b^2 e^4 (a+b x)+2 c^2 d e^2 (3 a d+2 a e x+3 b d x)-c^3 d^3 (d+4 e x)}{a+x (b+c x)}+2 e^2 \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right ) \log (a+x (b+c x))+\frac {4 e (b e-2 c d) \left (c e (3 a e+b d)-b^2 e^2-c^2 d^2\right ) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\sqrt {4 a c-b^2}}+c e^3 x (8 c d-3 b e)+c^2 e^4 x^2}{c^3} \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 {(b+2 c x) (d+e x)^4}{\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 = 0.47, size = 1701, normalized size = 9.89
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 = 285, normalized size = 1.66 \begin {gather*} \frac {2 \, {\left (3 \, c^{2} d^{2} e^{2} - 3 \, b c d e^{3} + b^{2} e^{4} - a c e^{4}\right )} \log \left (c x^{2} + b x + a\right )}{c^{3}} + \frac {4 \, {\left (2 \, c^{3} d^{3} e - 3 \, b c^{2} d^{2} e^{2} + 3 \, b^{2} c d e^{3} - 6 \, a c^{2} d e^{3} - b^{3} e^{4} + 3 \, a b c e^{4}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c} c^{3}} + \frac {c^{3} x^{2} e^{4} + 8 \, c^{3} d x e^{3} - 3 \, b c^{2} x e^{4}}{c^{4}} - \frac {c^{3} d^{4} - 6 \, a c^{2} d^{2} e^{2} + 4 \, a b c d e^{3} - a b^{2} e^{4} + a^{2} c e^{4} + {\left (4 \, c^{3} d^{3} e - 6 \, b c^{2} d^{2} e^{2} + 4 \, b^{2} c d e^{3} - 4 \, a c^{2} d e^{3} - b^{3} e^{4} + 2 \, a b c e^{4}\right )} x}{{\left (c x^{2} + b x + a\right )} c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 618, normalized size = 3.59 \begin {gather*} -\frac {2 a b \,e^{4} x}{\left (c \,x^{2}+b x +a \right ) c^{2}}+\frac {12 a b \,e^{4} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c^{2}}+\frac {4 a d \,e^{3} x}{\left (c \,x^{2}+b x +a \right ) c}-\frac {24 a d \,e^{3} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c}+\frac {b^{3} e^{4} x}{\left (c \,x^{2}+b x +a \right ) c^{3}}-\frac {4 b^{3} e^{4} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c^{3}}-\frac {4 b^{2} d \,e^{3} x}{\left (c \,x^{2}+b x +a \right ) c^{2}}+\frac {12 b^{2} d \,e^{3} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c^{2}}+\frac {6 b \,d^{2} e^{2} x}{\left (c \,x^{2}+b x +a \right ) c}-\frac {12 b \,d^{2} e^{2} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c}+\frac {e^{4} x^{2}}{c}-\frac {4 d^{3} e x}{c \,x^{2}+b x +a}+\frac {8 d^{3} e \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}-\frac {a^{2} e^{4}}{\left (c \,x^{2}+b x +a \right ) c^{2}}+\frac {a \,b^{2} e^{4}}{\left (c \,x^{2}+b x +a \right ) c^{3}}-\frac {4 a b d \,e^{3}}{\left (c \,x^{2}+b x +a \right ) c^{2}}+\frac {6 a \,d^{2} e^{2}}{\left (c \,x^{2}+b x +a \right ) c}-\frac {2 a \,e^{4} \ln \left (c \,x^{2}+b x +a \right )}{c^{2}}+\frac {2 b^{2} e^{4} \ln \left (c \,x^{2}+b x +a \right )}{c^{3}}-\frac {6 b d \,e^{3} \ln \left (c \,x^{2}+b x +a \right )}{c^{2}}-\frac {3 b \,e^{4} x}{c^{2}}+\frac {6 d^{2} e^{2} \ln \left (c \,x^{2}+b x +a \right )}{c}+\frac {8 d \,e^{3} x}{c}-\frac {d^{4}}{c \,x^{2}+b x +a} \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 = 0.32, size = 358, normalized size = 2.08 \begin {gather*} x\,\left (\frac {b\,e^4+8\,c\,d\,e^3}{c^2}-\frac {4\,b\,e^4}{c^2}\right )-\frac {\frac {a^2\,c\,e^4-a\,b^2\,e^4+4\,a\,b\,c\,d\,e^3-6\,a\,c^2\,d^2\,e^2+c^3\,d^4}{c}-\frac {x\,\left (b^3\,e^4-4\,b^2\,c\,d\,e^3+6\,b\,c^2\,d^2\,e^2-2\,a\,b\,c\,e^4-4\,c^3\,d^3\,e+4\,a\,c^2\,d\,e^3\right )}{c}}{c^3\,x^2+b\,c^2\,x+a\,c^2}-\frac {\ln \left (c\,x^2+b\,x+a\right )\,\left (16\,a^2\,c^2\,e^4-20\,a\,b^2\,c\,e^4+48\,a\,b\,c^2\,d\,e^3-48\,a\,c^3\,d^2\,e^2+4\,b^4\,e^4-12\,b^3\,c\,d\,e^3+12\,b^2\,c^2\,d^2\,e^2\right )}{2\,\left (4\,a\,c^4-b^2\,c^3\right )}+\frac {e^4\,x^2}{c}-\frac {4\,e\,\mathrm {atan}\left (\frac {b+2\,c\,x}{\sqrt {4\,a\,c-b^2}}\right )\,\left (b\,e-2\,c\,d\right )\,\left (b^2\,e^2-b\,c\,d\,e+c^2\,d^2-3\,a\,c\,e^2\right )}{c^3\,\sqrt {4\,a\,c-b^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 25.12, size = 1071, normalized size = 6.23 \begin {gather*} x \left (- \frac {3 b e^{4}}{c^{2}} + \frac {8 d e^{3}}{c}\right ) + \left (- \frac {2 e^{2} \left (a c e^{2} - b^{2} e^{2} + 3 b c d e - 3 c^{2} d^{2}\right )}{c^{3}} - \frac {2 e \sqrt {- 4 a c + b^{2}} \left (b e - 2 c d\right ) \left (3 a c e^{2} - b^{2} e^{2} + b c d e - c^{2} d^{2}\right )}{c^{3} \left (4 a c - b^{2}\right )}\right ) \log {\left (x + \frac {8 a^{2} c e^{4} - 4 a b^{2} e^{4} + 12 a b c d e^{3} + 4 a c^{3} \left (- \frac {2 e^{2} \left (a c e^{2} - b^{2} e^{2} + 3 b c d e - 3 c^{2} d^{2}\right )}{c^{3}} - \frac {2 e \sqrt {- 4 a c + b^{2}} \left (b e - 2 c d\right ) \left (3 a c e^{2} - b^{2} e^{2} + b c d e - c^{2} d^{2}\right )}{c^{3} \left (4 a c - b^{2}\right )}\right ) - 24 a c^{2} d^{2} e^{2} - b^{2} c^{2} \left (- \frac {2 e^{2} \left (a c e^{2} - b^{2} e^{2} + 3 b c d e - 3 c^{2} d^{2}\right )}{c^{3}} - \frac {2 e \sqrt {- 4 a c + b^{2}} \left (b e - 2 c d\right ) \left (3 a c e^{2} - b^{2} e^{2} + b c d e - c^{2} d^{2}\right )}{c^{3} \left (4 a c - b^{2}\right )}\right ) + 4 b c^{2} d^{3} e}{12 a b c e^{4} - 24 a c^{2} d e^{3} - 4 b^{3} e^{4} + 12 b^{2} c d e^{3} - 12 b c^{2} d^{2} e^{2} + 8 c^{3} d^{3} e} \right )} + \left (- \frac {2 e^{2} \left (a c e^{2} - b^{2} e^{2} + 3 b c d e - 3 c^{2} d^{2}\right )}{c^{3}} + \frac {2 e \sqrt {- 4 a c + b^{2}} \left (b e - 2 c d\right ) \left (3 a c e^{2} - b^{2} e^{2} + b c d e - c^{2} d^{2}\right )}{c^{3} \left (4 a c - b^{2}\right )}\right ) \log {\left (x + \frac {8 a^{2} c e^{4} - 4 a b^{2} e^{4} + 12 a b c d e^{3} + 4 a c^{3} \left (- \frac {2 e^{2} \left (a c e^{2} - b^{2} e^{2} + 3 b c d e - 3 c^{2} d^{2}\right )}{c^{3}} + \frac {2 e \sqrt {- 4 a c + b^{2}} \left (b e - 2 c d\right ) \left (3 a c e^{2} - b^{2} e^{2} + b c d e - c^{2} d^{2}\right )}{c^{3} \left (4 a c - b^{2}\right )}\right ) - 24 a c^{2} d^{2} e^{2} - b^{2} c^{2} \left (- \frac {2 e^{2} \left (a c e^{2} - b^{2} e^{2} + 3 b c d e - 3 c^{2} d^{2}\right )}{c^{3}} + \frac {2 e \sqrt {- 4 a c + b^{2}} \left (b e - 2 c d\right ) \left (3 a c e^{2} - b^{2} e^{2} + b c d e - c^{2} d^{2}\right )}{c^{3} \left (4 a c - b^{2}\right )}\right ) + 4 b c^{2} d^{3} e}{12 a b c e^{4} - 24 a c^{2} d e^{3} - 4 b^{3} e^{4} + 12 b^{2} c d e^{3} - 12 b c^{2} d^{2} e^{2} + 8 c^{3} d^{3} e} \right )} + \frac {- a^{2} c e^{4} + a b^{2} e^{4} - 4 a b c d e^{3} + 6 a c^{2} d^{2} e^{2} - c^{3} d^{4} + x \left (- 2 a b c e^{4} + 4 a c^{2} d e^{3} + b^{3} e^{4} - 4 b^{2} c d e^{3} + 6 b c^{2} d^{2} e^{2} - 4 c^{3} d^{3} e\right )}{a c^{3} + b c^{3} x + c^{4} x^{2}} + \frac {e^{4} x^{2}}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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