Optimal. Leaf size=127 \[ \frac {\left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{5/2}}+\frac {3 e \sqrt {a+b x+c x^2} (2 c d-b e)}{4 c^2}+\frac {e (d+e x) \sqrt {a+b x+c x^2}}{2 c} \]
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Rubi [A] time = 0.12, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {742, 640, 621, 206} \begin {gather*} \frac {\left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{5/2}}+\frac {3 e \sqrt {a+b x+c x^2} (2 c d-b e)}{4 c^2}+\frac {e (d+e x) \sqrt {a+b x+c x^2}}{2 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 640
Rule 742
Rubi steps
\begin {align*} \int \frac {(d+e x)^2}{\sqrt {a+b x+c x^2}} \, dx &=\frac {e (d+e x) \sqrt {a+b x+c x^2}}{2 c}+\frac {\int \frac {\frac {1}{2} \left (4 c d^2-e (b d+2 a e)\right )+\frac {3}{2} e (2 c d-b e) x}{\sqrt {a+b x+c x^2}} \, dx}{2 c}\\ &=\frac {3 e (2 c d-b e) \sqrt {a+b x+c x^2}}{4 c^2}+\frac {e (d+e x) \sqrt {a+b x+c x^2}}{2 c}+\frac {\left (-\frac {3}{2} b e (2 c d-b e)+c \left (4 c d^2-e (b d+2 a e)\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{4 c^2}\\ &=\frac {3 e (2 c d-b e) \sqrt {a+b x+c x^2}}{4 c^2}+\frac {e (d+e x) \sqrt {a+b x+c x^2}}{2 c}+\frac {\left (-\frac {3}{2} b e (2 c d-b e)+c \left (4 c d^2-e (b d+2 a e)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{2 c^2}\\ &=\frac {3 e (2 c d-b e) \sqrt {a+b x+c x^2}}{4 c^2}+\frac {e (d+e x) \sqrt {a+b x+c x^2}}{2 c}+\frac {\left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 103, normalized size = 0.81 \begin {gather*} \frac {\left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )}{8 c^{5/2}}+\frac {e \sqrt {a+x (b+c x)} (-3 b e+8 c d+2 c e x)}{4 c^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.49, size = 114, normalized size = 0.90 \begin {gather*} \frac {\left (4 a c e^2-3 b^2 e^2+8 b c d e-8 c^2 d^2\right ) \log \left (-2 c^{5/2} \sqrt {a+b x+c x^2}+b c^2+2 c^3 x\right )}{8 c^{5/2}}+\frac {\sqrt {a+b x+c x^2} \left (-3 b e^2+8 c d e+2 c e^2 x\right )}{4 c^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 247, normalized size = 1.94 \begin {gather*} \left [-\frac {{\left (8 \, c^{2} d^{2} - 8 \, b c d e + {\left (3 \, b^{2} - 4 \, a c\right )} e^{2}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} + 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) - 4 \, {\left (2 \, c^{2} e^{2} x + 8 \, c^{2} d e - 3 \, b c e^{2}\right )} \sqrt {c x^{2} + b x + a}}{16 \, c^{3}}, -\frac {{\left (8 \, c^{2} d^{2} - 8 \, b c d e + {\left (3 \, b^{2} - 4 \, a c\right )} e^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) - 2 \, {\left (2 \, c^{2} e^{2} x + 8 \, c^{2} d e - 3 \, b c e^{2}\right )} \sqrt {c x^{2} + b x + a}}{8 \, c^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 105, normalized size = 0.83 \begin {gather*} \frac {1}{4} \, \sqrt {c x^{2} + b x + a} {\left (\frac {2 \, x e^{2}}{c} + \frac {8 \, c d e - 3 \, b e^{2}}{c^{2}}\right )} - \frac {{\left (8 \, c^{2} d^{2} - 8 \, b c d e + 3 \, b^{2} e^{2} - 4 \, a c e^{2}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{8 \, c^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 198, normalized size = 1.56 \begin {gather*} -\frac {a \,e^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}+\frac {3 b^{2} e^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {5}{2}}}-\frac {b d e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}+\frac {d^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{\sqrt {c}}+\frac {\sqrt {c \,x^{2}+b x +a}\, e^{2} x}{2 c}-\frac {3 \sqrt {c \,x^{2}+b x +a}\, b \,e^{2}}{4 c^{2}}+\frac {2 \sqrt {c \,x^{2}+b x +a}\, d e}{c} \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 [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^2}{\sqrt {c\,x^2+b\,x+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d + e x\right )^{2}}{\sqrt {a + b x + c x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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