Optimal. Leaf size=75 \[ d^2 (b+2 c x) \sqrt {a+b x+c x^2}+\frac {\left (b^2-4 a c\right ) d^2 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {c}} \]
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Rubi [A]
time = 0.02, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {706, 635, 212}
\begin {gather*} \frac {d^2 \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {c}}+d^2 (b+2 c x) \sqrt {a+b x+c x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 635
Rule 706
Rubi steps
\begin {align*} \int \frac {(b d+2 c d x)^2}{\sqrt {a+b x+c x^2}} \, dx &=d^2 (b+2 c x) \sqrt {a+b x+c x^2}+\frac {1}{2} \left (\left (b^2-4 a c\right ) d^2\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx\\ &=d^2 (b+2 c x) \sqrt {a+b x+c x^2}+\left (\left (b^2-4 a c\right ) d^2\right ) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )\\ &=d^2 (b+2 c x) \sqrt {a+b x+c x^2}+\frac {\left (b^2-4 a c\right ) d^2 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {c}}\\ \end {align*}
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Mathematica [A]
time = 0.34, size = 69, normalized size = 0.92 \begin {gather*} d^2 \left ((b+2 c x) \sqrt {a+x (b+c x)}-\frac {\left (b^2-4 a c\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{2 \sqrt {c}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(198\) vs.
\(2(63)=126\).
time = 0.69, size = 199, normalized size = 2.65
method | result | size |
risch | \(d^{2} \left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}+\left (\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 \sqrt {c}}-2 a \sqrt {c}\, \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )\right ) d^{2}\) | \(93\) |
default | \(d^{2} \left (4 c^{2} \left (\frac {x \sqrt {c \,x^{2}+b x +a}}{2 c}-\frac {3 b \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}-\frac {a \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )+4 b c \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )+\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{\sqrt {c}}\right )\) | \(199\) |
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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Fricas [A]
time = 3.21, size = 195, normalized size = 2.60 \begin {gather*} \left [-\frac {{\left (b^{2} - 4 \, a c\right )} \sqrt {c} d^{2} \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} d^{2} x + b c d^{2}\right )} \sqrt {c x^{2} + b x + a}}{4 \, c}, -\frac {{\left (b^{2} - 4 \, a c\right )} \sqrt {-c} d^{2} \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} d^{2} x + b c d^{2}\right )} \sqrt {c x^{2} + b x + a}}{2 \, c}\right ] \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*} d^{2} \left (\int \frac {b^{2}}{\sqrt {a + b x + c x^{2}}}\, dx + \int \frac {4 c^{2} x^{2}}{\sqrt {a + b x + c x^{2}}}\, dx + \int \frac {4 b c x}{\sqrt {a + b x + c x^{2}}}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.25, size = 78, normalized size = 1.04 \begin {gather*} {\left (2 \, c d^{2} x + b d^{2}\right )} \sqrt {c x^{2} + b x + a} - \frac {{\left (b^{2} d^{2} - 4 \, a c d^{2}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{2 \, \sqrt {c}} \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 (b\,d+2\,c\,d\,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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