Optimal. Leaf size=123 \[ -\frac {d^2 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{32 c^{3/2}}-\frac {d^2 \left (b^2-4 a c\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{16 c}+\frac {d^2 (b+2 c x)^3 \sqrt {a+b x+c x^2}}{8 c} \]
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Rubi [A] time = 0.06, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {685, 692, 621, 206} \[ -\frac {d^2 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{32 c^{3/2}}-\frac {d^2 \left (b^2-4 a c\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{16 c}+\frac {d^2 (b+2 c x)^3 \sqrt {a+b x+c x^2}}{8 c} \]
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 685
Rule 692
Rubi steps
\begin {align*} \int (b d+2 c d x)^2 \sqrt {a+b x+c x^2} \, dx &=\frac {d^2 (b+2 c x)^3 \sqrt {a+b x+c x^2}}{8 c}-\frac {\left (b^2-4 a c\right ) \int \frac {(b d+2 c d x)^2}{\sqrt {a+b x+c x^2}} \, dx}{16 c}\\ &=-\frac {\left (b^2-4 a c\right ) d^2 (b+2 c x) \sqrt {a+b x+c x^2}}{16 c}+\frac {d^2 (b+2 c x)^3 \sqrt {a+b x+c x^2}}{8 c}-\frac {\left (\left (b^2-4 a c\right )^2 d^2\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{32 c}\\ &=-\frac {\left (b^2-4 a c\right ) d^2 (b+2 c x) \sqrt {a+b x+c x^2}}{16 c}+\frac {d^2 (b+2 c x)^3 \sqrt {a+b x+c x^2}}{8 c}-\frac {\left (\left (b^2-4 a c\right )^2 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 )}{16 c}\\ &=-\frac {\left (b^2-4 a c\right ) d^2 (b+2 c x) \sqrt {a+b x+c x^2}}{16 c}+\frac {d^2 (b+2 c x)^3 \sqrt {a+b x+c x^2}}{8 c}-\frac {\left (b^2-4 a c\right )^2 d^2 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{32 c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.36, size = 140, normalized size = 1.14 \[ \frac {d^2 \sqrt {a+x (b+c x)} \left (2 (b+2 c x) \left (4 c \left (a+2 c x^2\right )+b^2+8 b c x\right )-\frac {c^{3/2} \sqrt {4 a-\frac {b^2}{c}} (a+x (b+c x)) \sinh ^{-1}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {4 a-\frac {b^2}{c}}}\right )}{\left (\frac {c (a+x (b+c x))}{4 a c-b^2}\right )^{3/2}}\right )}{32 c} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.04, size = 309, normalized size = 2.51 \[ \left [\frac {{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {c} d^{2} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} + 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (16 \, c^{4} d^{2} x^{3} + 24 \, b c^{3} d^{2} x^{2} + 2 \, {\left (5 \, b^{2} c^{2} + 4 \, a c^{3}\right )} d^{2} x + {\left (b^{3} c + 4 \, a b c^{2}\right )} d^{2}\right )} \sqrt {c x^{2} + b x + a}}{64 \, c^{2}}, \frac {{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {-c} d^{2} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + 2 \, {\left (16 \, c^{4} d^{2} x^{3} + 24 \, b c^{3} d^{2} x^{2} + 2 \, {\left (5 \, b^{2} c^{2} + 4 \, a c^{3}\right )} d^{2} x + {\left (b^{3} c + 4 \, a b c^{2}\right )} d^{2}\right )} \sqrt {c x^{2} + b x + a}}{32 \, c^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 155, normalized size = 1.26 \[ \frac {1}{16} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, c^{2} d^{2} x + 3 \, b c d^{2}\right )} x + \frac {5 \, b^{2} c^{3} d^{2} + 4 \, a c^{4} d^{2}}{c^{3}}\right )} x + \frac {b^{3} c^{2} d^{2} + 4 \, a b c^{3} d^{2}}{c^{3}}\right )} + \frac {{\left (b^{4} d^{2} - 8 \, a b^{2} c d^{2} + 16 \, a^{2} c^{2} d^{2}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{32 \, c^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 230, normalized size = 1.87 \[ -\frac {a^{2} \sqrt {c}\, d^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2}+\frac {a \,b^{2} d^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{4 \sqrt {c}}-\frac {b^{4} d^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{32 c^{\frac {3}{2}}}-\frac {\sqrt {c \,x^{2}+b x +a}\, a c \,d^{2} x}{2}+\frac {\sqrt {c \,x^{2}+b x +a}\, b^{2} d^{2} x}{8}-\frac {\sqrt {c \,x^{2}+b x +a}\, a b \,d^{2}}{4}+\frac {\sqrt {c \,x^{2}+b x +a}\, b^{3} d^{2}}{16 c}+\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} c \,d^{2} x +\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b \,d^{2}}{2} \]
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 = 1.07, size = 335, normalized size = 2.72 \[ b^2\,d^2\,\left (\frac {x}{2}+\frac {b}{4\,c}\right )\,\sqrt {c\,x^2+b\,x+a}-a\,c\,d^2\,\left (\left (\frac {x}{2}+\frac {b}{4\,c}\right )\,\sqrt {c\,x^2+b\,x+a}+\frac {\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{2\,c^{3/2}}\right )-\frac {5\,b\,c\,d^2\,\left (\frac {\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x+a}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}+\frac {\left (-3\,b^2+2\,c\,x\,b+8\,c\,\left (c\,x^2+a\right )\right )\,\sqrt {c\,x^2+b\,x+a}}{24\,c^2}\right )}{2}+c\,d^2\,x\,{\left (c\,x^2+b\,x+a\right )}^{3/2}+\frac {b\,d^2\,\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x+a}\right )\,\left (b^3-4\,a\,b\,c\right )}{4\,c^{3/2}}+\frac {b\,d^2\,\left (-3\,b^2+2\,c\,x\,b+8\,c\,\left (c\,x^2+a\right )\right )\,\sqrt {c\,x^2+b\,x+a}}{6\,c}+\frac {b^2\,d^2\,\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{2\,c^{3/2}} \]
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 \[ d^{2} \left (\int b^{2} \sqrt {a + b x + c x^{2}}\, dx + \int 4 c^{2} x^{2} \sqrt {a + b x + c x^{2}}\, dx + \int 4 b c x \sqrt {a + b x + c x^{2}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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