Optimal. Leaf size=173 \[ \frac {(2 c d-b e) \left (-2 c e (b d-3 a e)-b^2 e^2+2 c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^2 \left (b^2-4 a c\right )^{3/2}}+\frac {e^2 x (2 c d-b e)}{c \left (b^2-4 a c\right )}-\frac {(d+e x)^2 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {e^3 \log \left (a+b x+c x^2\right )}{2 c^2} \]
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Rubi [A] time = 0.28, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {738, 773, 634, 618, 206, 628} \[ \frac {(2 c d-b e) \left (-2 c e (b d-3 a e)-b^2 e^2+2 c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^2 \left (b^2-4 a c\right )^{3/2}}+\frac {e^2 x (2 c d-b e)}{c \left (b^2-4 a c\right )}-\frac {(d+e x)^2 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {e^3 \log \left (a+b x+c x^2\right )}{2 c^2} \]
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 628
Rule 634
Rule 738
Rule 773
Rubi steps
\begin {align*} \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^2} \, dx &=-\frac {(d+e x)^2 (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\int \frac {(d+e x) \left (2 c d^2-e (3 b d-4 a e)-e (2 c d-b e) x\right )}{a+b x+c x^2} \, dx}{-b^2+4 a c}\\ &=\frac {e^2 (2 c d-b e) x}{c \left (b^2-4 a c\right )}-\frac {(d+e x)^2 (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\int \frac {a e^2 (2 c d-b e)+c d \left (2 c d^2-e (3 b d-4 a e)\right )+\left (-c d e (2 c d-b e)+b e^2 (2 c d-b e)+c e \left (2 c d^2-e (3 b d-4 a e)\right )\right ) x}{a+b x+c x^2} \, dx}{c \left (b^2-4 a c\right )}\\ &=\frac {e^2 (2 c d-b e) x}{c \left (b^2-4 a c\right )}-\frac {(d+e x)^2 (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {e^3 \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 c^2}-\frac {\left ((2 c d-b e) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right )\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 c^2 \left (b^2-4 a c\right )}\\ &=\frac {e^2 (2 c d-b e) x}{c \left (b^2-4 a c\right )}-\frac {(d+e x)^2 (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {e^3 \log \left (a+b x+c x^2\right )}{2 c^2}+\frac {\left ((2 c d-b e) \left (2 c^2 d^2-b^2 e^2-2 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^2 \left (b^2-4 a c\right )}\\ &=\frac {e^2 (2 c d-b e) x}{c \left (b^2-4 a c\right )}-\frac {(d+e x)^2 (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {(2 c d-b e) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^2 \left (b^2-4 a c\right )^{3/2}}+\frac {e^3 \log \left (a+b x+c x^2\right )}{2 c^2}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 201, normalized size = 1.16 \[ \frac {\frac {2 \left (-2 c \left (a^2 e^3-3 a c d e (d+e x)+c^2 d^3 x\right )+b^2 e^2 (a e-3 c d x)-b c \left (3 a e^2 (d+e x)+c d^2 (d-3 e x)\right )+b^3 e^3 x\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}+\frac {2 (b e-2 c d) \left (2 c e (b d-3 a e)+b^2 e^2-2 c^2 d^2\right ) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}}+e^3 \log (a+x (b+c x))}{2 c^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.93, size = 1164, normalized size = 6.73 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 236, normalized size = 1.36 \[ -\frac {{\left (4 \, c^{3} d^{3} - 6 \, b c^{2} d^{2} e + 12 \, a c^{2} d e^{2} + b^{3} e^{3} - 6 \, a b c e^{3}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {e^{3} \log \left (c x^{2} + b x + a\right )}{2 \, c^{2}} - \frac {b c^{2} d^{3} - 6 \, a c^{2} d^{2} e + 3 \, a b c d e^{2} - a b^{2} e^{3} + 2 \, a^{2} c e^{3} + {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e + 3 \, b^{2} c d e^{2} - 6 \, a c^{2} d e^{2} - b^{3} e^{3} + 3 \, a b c e^{3}\right )} x}{{\left (c x^{2} + b x + a\right )} {\left (b^{2} - 4 \, a c\right )} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 416, normalized size = 2.40 \[ -\frac {6 a b \,e^{3} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}} c}+\frac {12 a d \,e^{2} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}}}+\frac {b^{3} e^{3} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}} c^{2}}-\frac {6 b \,d^{2} e \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}}}+\frac {4 c \,d^{3} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}}}+\frac {2 a \,e^{3} \ln \left (c \,x^{2}+b x +a \right )}{\left (4 a c -b^{2}\right ) c}-\frac {b^{2} e^{3} \ln \left (c \,x^{2}+b x +a \right )}{2 \left (4 a c -b^{2}\right ) c^{2}}+\frac {\frac {\left (3 a b c \,e^{3}-6 c^{2} a d \,e^{2}-b^{3} e^{3}+3 b^{2} c d \,e^{2}-3 b \,c^{2} d^{2} e +2 c^{3} d^{3}\right ) x}{\left (4 a c -b^{2}\right ) c^{2}}+\frac {2 a^{2} c \,e^{3}-a \,b^{2} e^{3}+3 a b c d \,e^{2}-6 a \,c^{2} d^{2} e +b \,c^{2} d^{3}}{\left (4 a c -b^{2}\right ) c^{2}}}{c \,x^{2}+b x +a} \]
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 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.63, size = 483, normalized size = 2.79 \[ \frac {\frac {2\,a^2\,c\,e^3-a\,b^2\,e^3+3\,a\,b\,c\,d\,e^2-6\,a\,c^2\,d^2\,e+b\,c^2\,d^3}{c^2\,\left (4\,a\,c-b^2\right )}-\frac {x\,\left (b^3\,e^3-3\,b^2\,c\,d\,e^2+3\,b\,c^2\,d^2\,e-3\,a\,b\,c\,e^3-2\,c^3\,d^3+6\,a\,c^2\,d\,e^2\right )}{c^2\,\left (4\,a\,c-b^2\right )}}{c\,x^2+b\,x+a}-\frac {\ln \left (c\,x^2+b\,x+a\right )\,\left (-64\,a^3\,c^3\,e^3+48\,a^2\,b^2\,c^2\,e^3-12\,a\,b^4\,c\,e^3+b^6\,e^3\right )}{2\,\left (64\,a^3\,c^5-48\,a^2\,b^2\,c^4+12\,a\,b^4\,c^3-b^6\,c^2\right )}+\frac {\mathrm {atan}\left (\frac {c^2\,\left (\frac {2\,x\,\left (b\,e-2\,c\,d\right )\,\left (b^2\,e^2+2\,b\,c\,d\,e-2\,c^2\,d^2-6\,a\,c\,e^2\right )}{c\,{\left (4\,a\,c-b^2\right )}^3}-\frac {\left (b\,e-2\,c\,d\right )\,\left (b^3\,c-4\,a\,b\,c^2\right )\,\left (b^2\,e^2+2\,b\,c\,d\,e-2\,c^2\,d^2-6\,a\,c\,e^2\right )}{c^3\,{\left (4\,a\,c-b^2\right )}^4}\right )\,{\left (4\,a\,c-b^2\right )}^{5/2}}{b^3\,e^3-6\,b\,c^2\,d^2\,e-6\,a\,b\,c\,e^3+4\,c^3\,d^3+12\,a\,c^2\,d\,e^2}\right )\,\left (b\,e-2\,c\,d\right )\,\left (b^2\,e^2+2\,b\,c\,d\,e-2\,c^2\,d^2-6\,a\,c\,e^2\right )}{c^2\,{\left (4\,a\,c-b^2\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 6.72, size = 1238, normalized size = 7.16 \[ \left (\frac {e^{3}}{2 c^{2}} - \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (b e - 2 c d\right ) \left (6 a c e^{2} - b^{2} e^{2} - 2 b c d e + 2 c^{2} d^{2}\right )}{2 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}\right ) \log {\left (x + \frac {- 16 a^{2} c^{3} \left (\frac {e^{3}}{2 c^{2}} - \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (b e - 2 c d\right ) \left (6 a c e^{2} - b^{2} e^{2} - 2 b c d e + 2 c^{2} d^{2}\right )}{2 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}\right ) + 8 a^{2} c e^{3} + 8 a b^{2} c^{2} \left (\frac {e^{3}}{2 c^{2}} - \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (b e - 2 c d\right ) \left (6 a c e^{2} - b^{2} e^{2} - 2 b c d e + 2 c^{2} d^{2}\right )}{2 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}\right ) - a b^{2} e^{3} - 6 a b c d e^{2} - b^{4} c \left (\frac {e^{3}}{2 c^{2}} - \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (b e - 2 c d\right ) \left (6 a c e^{2} - b^{2} e^{2} - 2 b c d e + 2 c^{2} d^{2}\right )}{2 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}\right ) + 3 b^{2} c d^{2} e - 2 b c^{2} d^{3}}{6 a b c e^{3} - 12 a c^{2} d e^{2} - b^{3} e^{3} + 6 b c^{2} d^{2} e - 4 c^{3} d^{3}} \right )} + \left (\frac {e^{3}}{2 c^{2}} + \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (b e - 2 c d\right ) \left (6 a c e^{2} - b^{2} e^{2} - 2 b c d e + 2 c^{2} d^{2}\right )}{2 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}\right ) \log {\left (x + \frac {- 16 a^{2} c^{3} \left (\frac {e^{3}}{2 c^{2}} + \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (b e - 2 c d\right ) \left (6 a c e^{2} - b^{2} e^{2} - 2 b c d e + 2 c^{2} d^{2}\right )}{2 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}\right ) + 8 a^{2} c e^{3} + 8 a b^{2} c^{2} \left (\frac {e^{3}}{2 c^{2}} + \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (b e - 2 c d\right ) \left (6 a c e^{2} - b^{2} e^{2} - 2 b c d e + 2 c^{2} d^{2}\right )}{2 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}\right ) - a b^{2} e^{3} - 6 a b c d e^{2} - b^{4} c \left (\frac {e^{3}}{2 c^{2}} + \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \left (b e - 2 c d\right ) \left (6 a c e^{2} - b^{2} e^{2} - 2 b c d e + 2 c^{2} d^{2}\right )}{2 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}\right ) + 3 b^{2} c d^{2} e - 2 b c^{2} d^{3}}{6 a b c e^{3} - 12 a c^{2} d e^{2} - b^{3} e^{3} + 6 b c^{2} d^{2} e - 4 c^{3} d^{3}} \right )} + \frac {2 a^{2} c e^{3} - a b^{2} e^{3} + 3 a b c d e^{2} - 6 a c^{2} d^{2} e + b c^{2} d^{3} + x \left (3 a b c e^{3} - 6 a c^{2} d e^{2} - b^{3} e^{3} + 3 b^{2} c d e^{2} - 3 b c^{2} d^{2} e + 2 c^{3} d^{3}\right )}{4 a^{2} c^{3} - a b^{2} c^{2} + x^{2} \left (4 a c^{4} - b^{2} c^{3}\right ) + x \left (4 a b c^{3} - b^{3} c^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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