Optimal. Leaf size=144 \[ -\frac {\left (b^2-4 a c\right ) \left (-4 a B c-8 A b c+5 b^2 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{128 c^{7/2}}+\frac {(b+2 c x) \sqrt {a+b x+c x^2} \left (-4 a B c-8 A b c+5 b^2 B\right )}{64 c^3}-\frac {\left (a+b x+c x^2\right )^{3/2} (-8 A c+5 b B-6 B c x)}{24 c^2} \]
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Rubi [A] time = 0.06, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {779, 612, 621, 206} \begin {gather*} \frac {(b+2 c x) \sqrt {a+b x+c x^2} \left (-4 a B c-8 A b c+5 b^2 B\right )}{64 c^3}-\frac {\left (b^2-4 a c\right ) \left (-4 a B c-8 A b c+5 b^2 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{128 c^{7/2}}-\frac {\left (a+b x+c x^2\right )^{3/2} (-8 A c+5 b B-6 B c x)}{24 c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 621
Rule 779
Rubi steps
\begin {align*} \int x (A+B x) \sqrt {a+b x+c x^2} \, dx &=-\frac {(5 b B-8 A c-6 B c x) \left (a+b x+c x^2\right )^{3/2}}{24 c^2}+\frac {\left (5 b^2 B-8 A b c-4 a B c\right ) \int \sqrt {a+b x+c x^2} \, dx}{16 c^2}\\ &=\frac {\left (5 b^2 B-8 A b c-4 a B c\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{64 c^3}-\frac {(5 b B-8 A c-6 B c x) \left (a+b x+c x^2\right )^{3/2}}{24 c^2}-\frac {\left (\left (b^2-4 a c\right ) \left (5 b^2 B-8 A b c-4 a B c\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{128 c^3}\\ &=\frac {\left (5 b^2 B-8 A b c-4 a B c\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{64 c^3}-\frac {(5 b B-8 A c-6 B c x) \left (a+b x+c x^2\right )^{3/2}}{24 c^2}-\frac {\left (\left (b^2-4 a c\right ) \left (5 b^2 B-8 A b c-4 a B c\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 )}{64 c^3}\\ &=\frac {\left (5 b^2 B-8 A b c-4 a B c\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{64 c^3}-\frac {(5 b B-8 A c-6 B c x) \left (a+b x+c x^2\right )^{3/2}}{24 c^2}-\frac {\left (b^2-4 a c\right ) \left (5 b^2 B-8 A b c-4 a B c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{128 c^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 127, normalized size = 0.88 \begin {gather*} \frac {(a+x (b+c x))^{3/2} (8 A c-5 b B+6 B c x)-\frac {3 \left (-4 a B c-8 A b c+5 b^2 B\right ) \left (\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )-2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)}\right )}{16 c^{3/2}}}{24 c^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.63, size = 177, normalized size = 1.23 \begin {gather*} \frac {\left (16 a^2 B c^2+32 a A b c^2-24 a b^2 B c-8 A b^3 c+5 b^4 B\right ) \log \left (-2 \sqrt {c} \sqrt {a+b x+c x^2}+b+2 c x\right )}{128 c^{7/2}}+\frac {\sqrt {a+b x+c x^2} \left (64 a A c^2-52 a b B c+24 a B c^2 x-24 A b^2 c+16 A b c^2 x+64 A c^3 x^2+15 b^3 B-10 b^2 B c x+8 b B c^2 x^2+48 B c^3 x^3\right )}{192 c^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 393, normalized size = 2.73 \begin {gather*} \left [\frac {3 \, {\left (5 \, B b^{4} + 16 \, {\left (B a^{2} + 2 \, A a b\right )} c^{2} - 8 \, {\left (3 \, B a b^{2} + A b^{3}\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 (48 \, B c^{4} x^{3} + 15 \, B b^{3} c + 64 \, A a c^{3} - 4 \, {\left (13 \, B a b + 6 \, A b^{2}\right )} c^{2} + 8 \, {\left (B b c^{3} + 8 \, A c^{4}\right )} x^{2} - 2 \, {\left (5 \, B b^{2} c^{2} - 4 \, {\left (3 \, B a + 2 \, A b\right )} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{768 \, c^{4}}, \frac {3 \, {\left (5 \, B b^{4} + 16 \, {\left (B a^{2} + 2 \, A a b\right )} c^{2} - 8 \, {\left (3 \, B a b^{2} + A b^{3}\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 (48 \, B c^{4} x^{3} + 15 \, B b^{3} c + 64 \, A a c^{3} - 4 \, {\left (13 \, B a b + 6 \, A b^{2}\right )} c^{2} + 8 \, {\left (B b c^{3} + 8 \, A c^{4}\right )} x^{2} - 2 \, {\left (5 \, B b^{2} c^{2} - 4 \, {\left (3 \, B a + 2 \, A b\right )} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{384 \, 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 = 178, normalized size = 1.24 \begin {gather*} \frac {1}{192} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (6 \, B x + \frac {B b c^{2} + 8 \, A c^{3}}{c^{3}}\right )} x - \frac {5 \, B b^{2} c - 12 \, B a c^{2} - 8 \, A b c^{2}}{c^{3}}\right )} x + \frac {15 \, B b^{3} - 52 \, B a b c - 24 \, A b^{2} c + 64 \, A a c^{2}}{c^{3}}\right )} + \frac {{\left (5 \, B b^{4} - 24 \, B a b^{2} c - 8 \, A b^{3} c + 16 \, B a^{2} c^{2} + 32 \, A a b c^{2}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{128 \, c^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 352, normalized size = 2.44 \begin {gather*} -\frac {A a b \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{4 c^{\frac {3}{2}}}+\frac {A \,b^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{16 c^{\frac {5}{2}}}-\frac {B \,a^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}+\frac {3 B a \,b^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{16 c^{\frac {5}{2}}}-\frac {5 B \,b^{4} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{128 c^{\frac {7}{2}}}-\frac {\sqrt {c \,x^{2}+b x +a}\, A b x}{4 c}-\frac {\sqrt {c \,x^{2}+b x +a}\, B a x}{8 c}+\frac {5 \sqrt {c \,x^{2}+b x +a}\, B \,b^{2} x}{32 c^{2}}-\frac {\sqrt {c \,x^{2}+b x +a}\, A \,b^{2}}{8 c^{2}}-\frac {\sqrt {c \,x^{2}+b x +a}\, B a b}{16 c^{2}}+\frac {5 \sqrt {c \,x^{2}+b x +a}\, B \,b^{3}}{64 c^{3}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B x}{4 c}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A}{3 c}-\frac {5 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B b}{24 c^{2}} \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 [B] time = 1.51, size = 256, normalized size = 1.78 \begin {gather*} \frac {A\,\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 {B\,a\,\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 )}{4\,c}-\frac {5\,B\,b\,\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 )}{8\,c}+\frac {A\,\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}+\frac {B\,x\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{4\,c} \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 x \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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