Optimal. Leaf size=205 \[ \frac {\left (a+b x+c x^2\right )^{3/2} \left (-32 a B c-6 c x (7 b B-10 A c)-50 A b c+35 b^2 B\right )}{240 c^3}+\frac {\left (b^2-4 a c\right ) \left (8 a A c^2-12 a b B c-10 A b^2 c+7 b^3 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{9/2}}-\frac {(b+2 c x) \sqrt {a+b x+c x^2} \left (8 a A c^2-12 a b B c-10 A b^2 c+7 b^3 B\right )}{128 c^4}+\frac {B x^2 \left (a+b x+c x^2\right )^{3/2}}{5 c} \]
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Rubi [A] time = 0.20, antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {832, 779, 612, 621, 206} \begin {gather*} \frac {\left (a+b x+c x^2\right )^{3/2} \left (-32 a B c-6 c x (7 b B-10 A c)-50 A b c+35 b^2 B\right )}{240 c^3}-\frac {(b+2 c x) \sqrt {a+b x+c x^2} \left (8 a A c^2-12 a b B c-10 A b^2 c+7 b^3 B\right )}{128 c^4}+\frac {\left (b^2-4 a c\right ) \left (8 a A c^2-12 a b B c-10 A b^2 c+7 b^3 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{9/2}}+\frac {B x^2 \left (a+b x+c x^2\right )^{3/2}}{5 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 621
Rule 779
Rule 832
Rubi steps
\begin {align*} \int x^2 (A+B x) \sqrt {a+b x+c x^2} \, dx &=\frac {B x^2 \left (a+b x+c x^2\right )^{3/2}}{5 c}+\frac {\int x \left (-2 a B-\frac {1}{2} (7 b B-10 A c) x\right ) \sqrt {a+b x+c x^2} \, dx}{5 c}\\ &=\frac {B x^2 \left (a+b x+c x^2\right )^{3/2}}{5 c}+\frac {\left (35 b^2 B-50 A b c-32 a B c-6 c (7 b B-10 A c) x\right ) \left (a+b x+c x^2\right )^{3/2}}{240 c^3}-\frac {\left (7 b^3 B-10 A b^2 c-12 a b B c+8 a A c^2\right ) \int \sqrt {a+b x+c x^2} \, dx}{32 c^3}\\ &=-\frac {\left (7 b^3 B-10 A b^2 c-12 a b B c+8 a A c^2\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{128 c^4}+\frac {B x^2 \left (a+b x+c x^2\right )^{3/2}}{5 c}+\frac {\left (35 b^2 B-50 A b c-32 a B c-6 c (7 b B-10 A c) x\right ) \left (a+b x+c x^2\right )^{3/2}}{240 c^3}+\frac {\left (\left (b^2-4 a c\right ) \left (7 b^3 B-10 A b^2 c-12 a b B c+8 a A c^2\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{256 c^4}\\ &=-\frac {\left (7 b^3 B-10 A b^2 c-12 a b B c+8 a A c^2\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{128 c^4}+\frac {B x^2 \left (a+b x+c x^2\right )^{3/2}}{5 c}+\frac {\left (35 b^2 B-50 A b c-32 a B c-6 c (7 b B-10 A c) x\right ) \left (a+b x+c x^2\right )^{3/2}}{240 c^3}+\frac {\left (\left (b^2-4 a c\right ) \left (7 b^3 B-10 A b^2 c-12 a b B c+8 a A c^2\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 )}{128 c^4}\\ &=-\frac {\left (7 b^3 B-10 A b^2 c-12 a b B c+8 a A c^2\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{128 c^4}+\frac {B x^2 \left (a+b x+c x^2\right )^{3/2}}{5 c}+\frac {\left (35 b^2 B-50 A b c-32 a B c-6 c (7 b B-10 A c) x\right ) \left (a+b x+c x^2\right )^{3/2}}{240 c^3}+\frac {\left (b^2-4 a c\right ) \left (7 b^3 B-10 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 )}{256 c^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 179, normalized size = 0.87 \begin {gather*} \frac {\frac {(a+x (b+c x))^{3/2} \left (4 c (15 A c x-8 a B)-2 b c (25 A+21 B x)+35 b^2 B\right )}{48 c^2}+\frac {5 \left (8 a A c^2-12 a b B c-10 A b^2 c+7 b^3 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 )}{256 c^{7/2}}+B x^2 (a+x (b+c x))^{3/2}}{5 c} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.94, size = 244, normalized size = 1.19 \begin {gather*} \frac {\sqrt {a+b x+c x^2} \left (-256 a^2 B c^2-520 a A b c^2+240 a A c^3 x+460 a b^2 B c-232 a b B c^2 x+128 a B c^3 x^2+150 A b^3 c-100 A b^2 c^2 x+80 A b c^3 x^2+480 A c^4 x^3-105 b^4 B+70 b^3 B c x-56 b^2 B c^2 x^2+48 b B c^3 x^3+384 B c^4 x^4\right )}{1920 c^4}+\frac {\left (32 a^2 A c^3-48 a^2 b B c^2-48 a A b^2 c^2+40 a b^3 B c+10 A b^4 c-7 b^5 B\right ) \log \left (-2 \sqrt {c} \sqrt {a+b x+c x^2}+b+2 c x\right )}{256 c^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 517, normalized size = 2.52 \begin {gather*} \left [-\frac {15 \, {\left (7 \, B b^{5} - 32 \, A a^{2} c^{3} + 48 \, {\left (B a^{2} b + A a b^{2}\right )} c^{2} - 10 \, {\left (4 \, B a b^{3} + A b^{4}\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 (384 \, B c^{5} x^{4} - 105 \, B b^{4} c - 8 \, {\left (32 \, B a^{2} + 65 \, A a b\right )} c^{3} + 48 \, {\left (B b c^{4} + 10 \, A c^{5}\right )} x^{3} + 10 \, {\left (46 \, B a b^{2} + 15 \, A b^{3}\right )} c^{2} - 8 \, {\left (7 \, B b^{2} c^{3} - 2 \, {\left (8 \, B a + 5 \, A b\right )} c^{4}\right )} x^{2} + 2 \, {\left (35 \, B b^{3} c^{2} + 120 \, A a c^{4} - 2 \, {\left (58 \, B a b + 25 \, A b^{2}\right )} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{7680 \, c^{5}}, -\frac {15 \, {\left (7 \, B b^{5} - 32 \, A a^{2} c^{3} + 48 \, {\left (B a^{2} b + A a b^{2}\right )} c^{2} - 10 \, {\left (4 \, B a b^{3} + A b^{4}\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 (384 \, B c^{5} x^{4} - 105 \, B b^{4} c - 8 \, {\left (32 \, B a^{2} + 65 \, A a b\right )} c^{3} + 48 \, {\left (B b c^{4} + 10 \, A c^{5}\right )} x^{3} + 10 \, {\left (46 \, B a b^{2} + 15 \, A b^{3}\right )} c^{2} - 8 \, {\left (7 \, B b^{2} c^{3} - 2 \, {\left (8 \, B a + 5 \, A b\right )} c^{4}\right )} x^{2} + 2 \, {\left (35 \, B b^{3} c^{2} + 120 \, A a c^{4} - 2 \, {\left (58 \, B a b + 25 \, A b^{2}\right )} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{3840 \, c^{5}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 245, normalized size = 1.20 \begin {gather*} \frac {1}{1920} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (6 \, {\left (8 \, B x + \frac {B b c^{3} + 10 \, A c^{4}}{c^{4}}\right )} x - \frac {7 \, B b^{2} c^{2} - 16 \, B a c^{3} - 10 \, A b c^{3}}{c^{4}}\right )} x + \frac {35 \, B b^{3} c - 116 \, B a b c^{2} - 50 \, A b^{2} c^{2} + 120 \, A a c^{3}}{c^{4}}\right )} x - \frac {105 \, B b^{4} - 460 \, B a b^{2} c - 150 \, A b^{3} c + 256 \, B a^{2} c^{2} + 520 \, A a b c^{2}}{c^{4}}\right )} - \frac {{\left (7 \, B b^{5} - 40 \, B a b^{3} c - 10 \, A b^{4} c + 48 \, B a^{2} b c^{2} + 48 \, A a b^{2} c^{2} - 32 \, A a^{2} c^{3}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{256 \, c^{\frac {9}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 497, normalized size = 2.42 \begin {gather*} -\frac {A \,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 A 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 A \,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 {3 B \,a^{2} b \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 a \,b^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{32 c^{\frac {7}{2}}}+\frac {7 B \,b^{5} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{256 c^{\frac {9}{2}}}-\frac {\sqrt {c \,x^{2}+b x +a}\, A a x}{8 c}+\frac {5 \sqrt {c \,x^{2}+b x +a}\, A \,b^{2} x}{32 c^{2}}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, B a b x}{16 c^{2}}-\frac {7 \sqrt {c \,x^{2}+b x +a}\, B \,b^{3} x}{64 c^{3}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B \,x^{2}}{5 c}-\frac {\sqrt {c \,x^{2}+b x +a}\, A a b}{16 c^{2}}+\frac {5 \sqrt {c \,x^{2}+b x +a}\, A \,b^{3}}{64 c^{3}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A x}{4 c}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, B a \,b^{2}}{32 c^{3}}-\frac {7 \sqrt {c \,x^{2}+b x +a}\, B \,b^{4}}{128 c^{4}}-\frac {7 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B b x}{40 c^{2}}-\frac {5 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A b}{24 c^{2}}-\frac {2 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B a}{15 c^{2}}+\frac {7 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B \,b^{2}}{48 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 [B] time = 1.72, size = 463, normalized size = 2.26 \begin {gather*} \frac {B\,x^2\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{5\,c}-\frac {A\,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\,A\,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 {2\,B\,a\,\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 )}{5\,c}+\frac {A\,x\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{4\,c}+\frac {7\,B\,b\,\left (\frac {5\,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 {x\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{4\,c}+\frac {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}\right )}{10\,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^{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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