Optimal. Leaf size=188 \[ \frac {e x \left (-c e (2 a e+3 b d)+b^2 e^2+6 c^2 d^2\right )}{c^2}+\frac {(2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{2 c^3}-\frac {e \sqrt {b^2-4 a c} \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^3}+\frac {e^2 x^2 (6 c d-b e)}{2 c}+\frac {2 e^3 x^3}{3} \]
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Rubi [A] time = 0.24, antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {800, 634, 618, 206, 628} \begin {gather*} \frac {(2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{2 c^3}+\frac {e x \left (-c e (2 a e+3 b d)+b^2 e^2+6 c^2 d^2\right )}{c^2}-\frac {e \sqrt {b^2-4 a c} \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^3}+\frac {e^2 x^2 (6 c d-b e)}{2 c}+\frac {2 e^3 x^3}{3} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 628
Rule 634
Rule 800
Rubi steps
\begin {align*} \int \frac {(b+2 c x) (d+e x)^3}{a+b x+c x^2} \, dx &=\int \left (\frac {e \left (6 c^2 d^2+b^2 e^2-c e (3 b d+2 a e)\right )}{c^2}+\frac {e^2 (6 c d-b e) x}{c}+2 e^3 x^2+\frac {-a b^2 e^3-2 a c e \left (3 c d^2-a e^2\right )+b c d \left (c d^2+3 a e^2\right )+(2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) x}{c^2 \left (a+b x+c x^2\right )}\right ) \, dx\\ &=\frac {e \left (6 c^2 d^2+b^2 e^2-c e (3 b d+2 a e)\right ) x}{c^2}+\frac {e^2 (6 c d-b e) x^2}{2 c}+\frac {2 e^3 x^3}{3}+\frac {\int \frac {-a b^2 e^3-2 a c e \left (3 c d^2-a e^2\right )+b c d \left (c d^2+3 a e^2\right )+(2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) x}{a+b x+c x^2} \, dx}{c^2}\\ &=\frac {e \left (6 c^2 d^2+b^2 e^2-c e (3 b d+2 a e)\right ) x}{c^2}+\frac {e^2 (6 c d-b e) x^2}{2 c}+\frac {2 e^3 x^3}{3}+\frac {\left (\left (b^2-4 a c\right ) e \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right )\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 c^3}+\frac {\left ((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 {b+2 c x}{a+b x+c x^2} \, dx}{2 c^3}\\ &=\frac {e \left (6 c^2 d^2+b^2 e^2-c e (3 b d+2 a e)\right ) x}{c^2}+\frac {e^2 (6 c d-b e) x^2}{2 c}+\frac {2 e^3 x^3}{3}+\frac {(2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \log \left (a+b x+c x^2\right )}{2 c^3}-\frac {\left (\left (b^2-4 a c\right ) e \left (3 c^2 d^2+b^2 e^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 )}{c^3}\\ &=\frac {e \left (6 c^2 d^2+b^2 e^2-c e (3 b d+2 a e)\right ) x}{c^2}+\frac {e^2 (6 c d-b e) x^2}{2 c}+\frac {2 e^3 x^3}{3}-\frac {\sqrt {b^2-4 a c} e \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^3}+\frac {(2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \log \left (a+b x+c x^2\right )}{2 c^3}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 177, normalized size = 0.94 \begin {gather*} \frac {c e x \left (-3 c e (4 a e+6 b d+b e x)+6 b^2 e^2+2 c^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )+3 (2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \log (a+x (b+c x))-6 e \sqrt {4 a c-b^2} \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right ) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{6 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)^3}{a+b x+c x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.44, size = 458, normalized size = 2.44 \begin {gather*} \left [\frac {4 \, c^{3} e^{3} x^{3} + 3 \, {\left (6 \, c^{3} d e^{2} - b c^{2} e^{3}\right )} x^{2} - 3 \, {\left (3 \, c^{2} d^{2} e - 3 \, b c d e^{2} + {\left (b^{2} - a c\right )} e^{3}\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 ) + 6 \, {\left (6 \, c^{3} d^{2} e - 3 \, b c^{2} d e^{2} + {\left (b^{2} c - 2 \, a c^{2}\right )} e^{3}\right )} x + 3 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e + 3 \, {\left (b^{2} c - 2 \, a c^{2}\right )} d e^{2} - {\left (b^{3} - 3 \, a b c\right )} e^{3}\right )} \log \left (c x^{2} + b x + a\right )}{6 \, c^{3}}, \frac {4 \, c^{3} e^{3} x^{3} + 3 \, {\left (6 \, c^{3} d e^{2} - b c^{2} e^{3}\right )} x^{2} - 6 \, {\left (3 \, c^{2} d^{2} e - 3 \, b c d e^{2} + {\left (b^{2} - a c\right )} e^{3}\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 ) + 6 \, {\left (6 \, c^{3} d^{2} e - 3 \, b c^{2} d e^{2} + {\left (b^{2} c - 2 \, a c^{2}\right )} e^{3}\right )} x + 3 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e + 3 \, {\left (b^{2} c - 2 \, a c^{2}\right )} d e^{2} - {\left (b^{3} - 3 \, a b c\right )} e^{3}\right )} \log \left (c x^{2} + b x + a\right )}{6 \, c^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 251, normalized size = 1.34 \begin {gather*} \frac {{\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e + 3 \, b^{2} c d e^{2} - 6 \, a c^{2} d e^{2} - b^{3} e^{3} + 3 \, a b c e^{3}\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c^{3}} + \frac {{\left (3 \, b^{2} c^{2} d^{2} e - 12 \, a c^{3} d^{2} e - 3 \, b^{3} c d e^{2} + 12 \, a b c^{2} d e^{2} + b^{4} e^{3} - 5 \, a b^{2} c e^{3} + 4 \, a^{2} c^{2} e^{3}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c} c^{3}} + \frac {4 \, c^{3} x^{3} e^{3} + 18 \, c^{3} d x^{2} e^{2} + 36 \, c^{3} d^{2} x e - 3 \, b c^{2} x^{2} e^{3} - 18 \, b c^{2} d x e^{2} + 6 \, b^{2} c x e^{3} - 12 \, a c^{2} x e^{3}}{6 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 492, normalized size = 2.62 \begin {gather*} \frac {2 e^{3} x^{3}}{3}+\frac {4 a^{2} e^{3} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c}-\frac {5 a \,b^{2} 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 {12 a b d \,e^{2} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c}-\frac {12 a \,d^{2} e \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}+\frac {b^{4} e^{3} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c^{3}}-\frac {3 b^{3} d \,e^{2} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c^{2}}+\frac {3 b^{2} d^{2} e \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c}-\frac {b \,e^{3} x^{2}}{2 c}+3 d \,e^{2} x^{2}+\frac {3 a b \,e^{3} \ln \left (c \,x^{2}+b x +a \right )}{2 c^{2}}-\frac {3 a d \,e^{2} \ln \left (c \,x^{2}+b x +a \right )}{c}-\frac {2 a \,e^{3} x}{c}-\frac {b^{3} e^{3} \ln \left (c \,x^{2}+b x +a \right )}{2 c^{3}}+\frac {3 b^{2} d \,e^{2} \ln \left (c \,x^{2}+b x +a \right )}{2 c^{2}}+\frac {b^{2} e^{3} x}{c^{2}}-\frac {3 b \,d^{2} e \ln \left (c \,x^{2}+b x +a \right )}{2 c}-\frac {3 b d \,e^{2} x}{c}+d^{3} \ln \left (c \,x^{2}+b x +a \right )+6 d^{2} e x \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 = 2.00, size = 433, normalized size = 2.30 \begin {gather*} x^2\,\left (\frac {b\,e^3+6\,c\,d\,e^2}{2\,c}-\frac {b\,e^3}{c}\right )-x\,\left (\frac {b\,\left (\frac {b\,e^3+6\,c\,d\,e^2}{c}-\frac {2\,b\,e^3}{c}\right )}{c}+\frac {2\,a\,e^3}{c}-\frac {3\,d\,e\,\left (b\,e+2\,c\,d\right )}{c}\right )+\frac {2\,e^3\,x^3}{3}-\frac {\ln \left (b\,\sqrt {b^2-4\,a\,c}-4\,a\,c+b^2+2\,c\,x\,\sqrt {b^2-4\,a\,c}\right )\,\left (b^3\,e^3-2\,c^3\,d^3+b^2\,e^3\,\sqrt {b^2-4\,a\,c}-3\,a\,b\,c\,e^3-a\,c\,e^3\,\sqrt {b^2-4\,a\,c}+6\,a\,c^2\,d\,e^2+3\,b\,c^2\,d^2\,e-3\,b^2\,c\,d\,e^2+3\,c^2\,d^2\,e\,\sqrt {b^2-4\,a\,c}-3\,b\,c\,d\,e^2\,\sqrt {b^2-4\,a\,c}\right )}{2\,c^3}-\frac {\ln \left (4\,a\,c+b\,\sqrt {b^2-4\,a\,c}-b^2+2\,c\,x\,\sqrt {b^2-4\,a\,c}\right )\,\left (b^3\,e^3-2\,c^3\,d^3-b^2\,e^3\,\sqrt {b^2-4\,a\,c}-3\,a\,b\,c\,e^3+a\,c\,e^3\,\sqrt {b^2-4\,a\,c}+6\,a\,c^2\,d\,e^2+3\,b\,c^2\,d^2\,e-3\,b^2\,c\,d\,e^2-3\,c^2\,d^2\,e\,\sqrt {b^2-4\,a\,c}+3\,b\,c\,d\,e^2\,\sqrt {b^2-4\,a\,c}\right )}{2\,c^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 4.81, size = 561, normalized size = 2.98 \begin {gather*} \frac {2 e^{3} x^{3}}{3} + x^{2} \left (- \frac {b e^{3}}{2 c} + 3 d e^{2}\right ) + x \left (- \frac {2 a e^{3}}{c} + \frac {b^{2} e^{3}}{c^{2}} - \frac {3 b d e^{2}}{c} + 6 d^{2} e\right ) + \left (- \frac {e \sqrt {- 4 a c + b^{2}} \left (a c e^{2} - b^{2} e^{2} + 3 b c d e - 3 c^{2} d^{2}\right )}{2 c^{3}} + \frac {\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 )}{2 c^{3}}\right ) \log {\left (x + \frac {- a b e^{3} + 3 a c d e^{2} - c^{2} d^{3} + c^{2} \left (- \frac {e \sqrt {- 4 a c + b^{2}} \left (a c e^{2} - b^{2} e^{2} + 3 b c d e - 3 c^{2} d^{2}\right )}{2 c^{3}} + \frac {\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 )}{2 c^{3}}\right )}{a c e^{3} - b^{2} e^{3} + 3 b c d e^{2} - 3 c^{2} d^{2} e} \right )} + \left (\frac {e \sqrt {- 4 a c + b^{2}} \left (a c e^{2} - b^{2} e^{2} + 3 b c d e - 3 c^{2} d^{2}\right )}{2 c^{3}} + \frac {\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 )}{2 c^{3}}\right ) \log {\left (x + \frac {- a b e^{3} + 3 a c d e^{2} - c^{2} d^{3} + c^{2} \left (\frac {e \sqrt {- 4 a c + b^{2}} \left (a c e^{2} - b^{2} e^{2} + 3 b c d e - 3 c^{2} d^{2}\right )}{2 c^{3}} + \frac {\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 )}{2 c^{3}}\right )}{a c e^{3} - b^{2} e^{3} + 3 b c d e^{2} - 3 c^{2} d^{2} e} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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