Optimal. Leaf size=143 \[ \frac {\sqrt {a+b x+c x^2} \left (-16 a B c-2 c x (5 b B-6 A c)-18 A b c+15 b^2 B\right )}{24 c^3}-\frac {\left (8 a A c^2-12 a b B c-6 A b^2 c+5 b^3 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{7/2}}+\frac {B x^2 \sqrt {a+b x+c x^2}}{3 c} \]
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Rubi [A] time = 0.12, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {832, 779, 621, 206} \begin {gather*} \frac {\sqrt {a+b x+c x^2} \left (-16 a B c-2 c x (5 b B-6 A c)-18 A b c+15 b^2 B\right )}{24 c^3}-\frac {\left (8 a A c^2-12 a b B c-6 A b^2 c+5 b^3 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{7/2}}+\frac {B x^2 \sqrt {a+b x+c x^2}}{3 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 779
Rule 832
Rubi steps
\begin {align*} \int \frac {x^2 (A+B x)}{\sqrt {a+b x+c x^2}} \, dx &=\frac {B x^2 \sqrt {a+b x+c x^2}}{3 c}+\frac {\int \frac {x \left (-2 a B-\frac {1}{2} (5 b B-6 A c) x\right )}{\sqrt {a+b x+c x^2}} \, dx}{3 c}\\ &=\frac {B x^2 \sqrt {a+b x+c x^2}}{3 c}+\frac {\left (15 b^2 B-18 A b c-16 a B c-2 c (5 b B-6 A c) x\right ) \sqrt {a+b x+c x^2}}{24 c^3}-\frac {\left (5 b^3 B-6 A b^2 c-12 a b B c+8 a A c^2\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{16 c^3}\\ &=\frac {B x^2 \sqrt {a+b x+c x^2}}{3 c}+\frac {\left (15 b^2 B-18 A b c-16 a B c-2 c (5 b B-6 A c) x\right ) \sqrt {a+b x+c x^2}}{24 c^3}-\frac {\left (5 b^3 B-6 A b^2 c-12 a b B c+8 a A c^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 )}{8 c^3}\\ &=\frac {B x^2 \sqrt {a+b x+c x^2}}{3 c}+\frac {\left (15 b^2 B-18 A b c-16 a B c-2 c (5 b B-6 A c) x\right ) \sqrt {a+b x+c x^2}}{24 c^3}-\frac {\left (5 b^3 B-6 A b^2 c-12 a b B c+8 a A c^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 126, normalized size = 0.88 \begin {gather*} \frac {2 \sqrt {c} \sqrt {a+x (b+c x)} \left (4 c (c x (3 A+2 B x)-4 a B)-2 b c (9 A+5 B x)+15 b^2 B\right )-3 \left (8 a A c^2-12 a b B c-6 A b^2 c+5 b^3 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )}{48 c^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.52, size = 125, normalized size = 0.87 \begin {gather*} \frac {\sqrt {a+b x+c x^2} \left (-16 a B c-18 A b c+12 A c^2 x+15 b^2 B-10 b B c x+8 B c^2 x^2\right )}{24 c^3}+\frac {\left (8 a A c^2-12 a b B c-6 A b^2 c+5 b^3 B\right ) \log \left (-2 \sqrt {c} \sqrt {a+b x+c x^2}+b+2 c x\right )}{16 c^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 295, normalized size = 2.06 \begin {gather*} \left [\frac {3 \, {\left (5 \, B b^{3} + 8 \, A a c^{2} - 6 \, {\left (2 \, B a b + A b^{2}\right )} c\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 (8 \, B c^{3} x^{2} + 15 \, B b^{2} c - 2 \, {\left (8 \, B a + 9 \, A b\right )} c^{2} - 2 \, {\left (5 \, B b c^{2} - 6 \, A c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{96 \, c^{4}}, \frac {3 \, {\left (5 \, B b^{3} + 8 \, A a c^{2} - 6 \, {\left (2 \, B a b + A b^{2}\right )} c\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 (8 \, B c^{3} x^{2} + 15 \, B b^{2} c - 2 \, {\left (8 \, B a + 9 \, A b\right )} c^{2} - 2 \, {\left (5 \, B b c^{2} - 6 \, A c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{48 \, c^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 128, normalized size = 0.90 \begin {gather*} \frac {1}{24} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (\frac {4 \, B x}{c} - \frac {5 \, B b c - 6 \, A c^{2}}{c^{3}}\right )} x + \frac {15 \, B b^{2} - 16 \, B a c - 18 \, A b c}{c^{3}}\right )} + \frac {{\left (5 \, B b^{3} - 12 \, B a b c - 6 \, A b^{2} c + 8 \, A a c^{2}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{16 \, c^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 254, normalized size = 1.78 \begin {gather*} \frac {\sqrt {c \,x^{2}+b x +a}\, B \,x^{2}}{3 c}-\frac {A a \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 A \,b^{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 {3 B a b \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{4 c^{\frac {5}{2}}}-\frac {5 B \,b^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{16 c^{\frac {7}{2}}}+\frac {\sqrt {c \,x^{2}+b x +a}\, A x}{2 c}-\frac {5 \sqrt {c \,x^{2}+b x +a}\, B b x}{12 c^{2}}-\frac {3 \sqrt {c \,x^{2}+b x +a}\, A b}{4 c^{2}}-\frac {2 \sqrt {c \,x^{2}+b x +a}\, B a}{3 c^{2}}+\frac {5 \sqrt {c \,x^{2}+b x +a}\, B \,b^{2}}{8 c^{3}} \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 {x^2\,\left (A+B\,x\right )}{\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 {x^{2} \left (A + B x\right )}{\sqrt {a + b x + c x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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