Optimal. Leaf size=66 \[ 4 \sqrt {c} d^2 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )-\frac {2 d^2 (b+2 c x)}{\sqrt {a+b x+c x^2}} \]
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Rubi [A] time = 0.03, antiderivative size = 66, 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 = {686, 621, 206} \[ 4 \sqrt {c} d^2 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )-\frac {2 d^2 (b+2 c x)}{\sqrt {a+b x+c x^2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 686
Rubi steps
\begin {align*} \int \frac {(b d+2 c d x)^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx &=-\frac {2 d^2 (b+2 c x)}{\sqrt {a+b x+c x^2}}+\left (4 c d^2\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx\\ &=-\frac {2 d^2 (b+2 c x)}{\sqrt {a+b x+c x^2}}+\left (8 c d^2\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 )\\ &=-\frac {2 d^2 (b+2 c x)}{\sqrt {a+b x+c x^2}}+4 \sqrt {c} d^2 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.19, size = 118, normalized size = 1.79 \[ d^2 \left (\frac {4 \sqrt {c} \sqrt {a+x (b+c x)} \sinh ^{-1}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {4 a-\frac {b^2}{c}}}\right )}{\sqrt {4 a-\frac {b^2}{c}} \sqrt {\frac {c (a+x (b+c x))}{4 a c-b^2}}}-\frac {2 (b+2 c x)}{\sqrt {a+x (b+c x)}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.37, size = 225, normalized size = 3.41 \[ \left [\frac {2 \, {\left ({\left (c d^{2} x^{2} + b d^{2} x + a d^{2}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) - {\left (2 \, c d^{2} x + b d^{2}\right )} \sqrt {c x^{2} + b x + a}\right )}}{c x^{2} + b x + a}, -\frac {2 \, {\left (2 \, {\left (c d^{2} x^{2} + b d^{2} x + a d^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + {\left (2 \, c d^{2} x + b d^{2}\right )} \sqrt {c x^{2} + b x + a}\right )}}{c x^{2} + b x + a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.29, size = 113, normalized size = 1.71 \[ -4 \, \sqrt {c} d^{2} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right ) - \frac {2 \, {\left (\frac {2 \, {\left (b^{2} c d^{2} - 4 \, a c^{2} d^{2}\right )} x}{b^{2} - 4 \, a c} + \frac {b^{3} d^{2} - 4 \, a b c d^{2}}{b^{2} - 4 \, a c}\right )}}{\sqrt {c x^{2} + b x + a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 72, normalized size = 1.09 \[ -\frac {4 c \,d^{2} x}{\sqrt {c \,x^{2}+b x +a}}+4 \sqrt {c}\, d^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )-\frac {2 b \,d^{2}}{\sqrt {c \,x^{2}+b x +a}} \]
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 = 0.79, size = 156, normalized size = 2.36 \[ 4\,\sqrt {c}\,d^2\,\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )+\frac {b^2\,d^2\,\left (\frac {b}{2}+c\,x\right )}{\left (a\,c-\frac {b^2}{4}\right )\,\sqrt {c\,x^2+b\,x+a}}+\frac {4\,c\,d^2\,\left (\frac {a\,b}{2}-x\,\left (a\,c-\frac {b^2}{2}\right )\right )}{\left (a\,c-\frac {b^2}{4}\right )\,\sqrt {c\,x^2+b\,x+a}}-\frac {4\,b\,c\,d^2\,\left (4\,a+2\,b\,x\right )}{\left (4\,a\,c-b^2\right )\,\sqrt {c\,x^2+b\,x+a}} \]
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 \[ d^{2} \left (\int \frac {b^{2}}{a \sqrt {a + b x + c x^{2}} + b x \sqrt {a + b x + c x^{2}} + c x^{2} \sqrt {a + b x + c x^{2}}}\, dx + \int \frac {4 c^{2} x^{2}}{a \sqrt {a + b x + c x^{2}} + b x \sqrt {a + b x + c x^{2}} + c x^{2} \sqrt {a + b x + c x^{2}}}\, dx + \int \frac {4 b c x}{a \sqrt {a + b x + c x^{2}} + b x \sqrt {a + b x + c x^{2}} + c x^{2} \sqrt {a + b x + c x^{2}}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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