Optimal. Leaf size=198 \[ \frac {\left (b^2-4 a c\right )^2 \left (-4 a B c-12 A b c+7 b^2 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{1024 c^{9/2}}-\frac {\left (b^2-4 a c\right ) (b+2 c x) \sqrt {a+b x+c x^2} \left (-4 a B c-12 A b c+7 b^2 B\right )}{512 c^4}+\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 a B c-12 A b c+7 b^2 B\right )}{192 c^3}-\frac {\left (a+b x+c x^2\right )^{5/2} (-12 A c+7 b B-10 B c x)}{60 c^2} \]
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Rubi [A] time = 0.09, antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 5, 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) \left (a+b x+c x^2\right )^{3/2} \left (-4 a B c-12 A b c+7 b^2 B\right )}{192 c^3}-\frac {\left (b^2-4 a c\right ) (b+2 c x) \sqrt {a+b x+c x^2} \left (-4 a B c-12 A b c+7 b^2 B\right )}{512 c^4}+\frac {\left (b^2-4 a c\right )^2 \left (-4 a B c-12 A b c+7 b^2 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{1024 c^{9/2}}-\frac {\left (a+b x+c x^2\right )^{5/2} (-12 A c+7 b B-10 B c x)}{60 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) \left (a+b x+c x^2\right )^{3/2} \, dx &=-\frac {(7 b B-12 A c-10 B c x) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac {\left (7 b^2 B-12 A b c-4 a B c\right ) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{24 c^2}\\ &=\frac {\left (7 b^2 B-12 A b c-4 a B c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}-\frac {(7 b B-12 A c-10 B c x) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}-\frac {\left (\left (b^2-4 a c\right ) \left (7 b^2 B-12 A b c-4 a B c\right )\right ) \int \sqrt {a+b x+c x^2} \, dx}{128 c^3}\\ &=-\frac {\left (b^2-4 a c\right ) \left (7 b^2 B-12 A b c-4 a B c\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{512 c^4}+\frac {\left (7 b^2 B-12 A b c-4 a B c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}-\frac {(7 b B-12 A c-10 B c x) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac {\left (\left (b^2-4 a c\right )^2 \left (7 b^2 B-12 A b c-4 a B c\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{1024 c^4}\\ &=-\frac {\left (b^2-4 a c\right ) \left (7 b^2 B-12 A b c-4 a B c\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{512 c^4}+\frac {\left (7 b^2 B-12 A b c-4 a B c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}-\frac {(7 b B-12 A c-10 B c x) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac {\left (\left (b^2-4 a c\right )^2 \left (7 b^2 B-12 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 )}{512 c^4}\\ &=-\frac {\left (b^2-4 a c\right ) \left (7 b^2 B-12 A b c-4 a B c\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{512 c^4}+\frac {\left (7 b^2 B-12 A b c-4 a B c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}-\frac {(7 b B-12 A c-10 B c x) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac {\left (b^2-4 a c\right )^2 \left (7 b^2 B-12 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 )}{1024 c^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 156, normalized size = 0.79 \begin {gather*} \frac {\frac {5 \left (-4 a B c-12 A b c+7 b^2 B\right ) \left (2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)} \left (4 c \left (5 a+2 c x^2\right )-3 b^2+8 b c x\right )+3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )}{256 c^{5/2}}+(a+x (b+c x))^{5/2} (2 c (6 A+5 B x)-7 b B)}{60 c^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.26, size = 326, normalized size = 1.65 \begin {gather*} \frac {\sqrt {a+b x+c x^2} \left (1536 a^2 A c^3-1296 a^2 b B c^2+480 a^2 B c^3 x-1200 a A b^2 c^2+672 a A b c^3 x+3072 a A c^4 x^2+760 a b^3 B c-432 a b^2 B c^2 x+288 a b B c^3 x^2+2240 a B c^4 x^3+180 A b^4 c-120 A b^3 c^2 x+96 A b^2 c^3 x^2+2112 A b c^4 x^3+1536 A c^5 x^4-105 b^5 B+70 b^4 B c x-56 b^3 B c^2 x^2+48 b^2 B c^3 x^3+1664 b B c^4 x^4+1280 B c^5 x^5\right )}{7680 c^4}+\frac {\left (64 a^3 B c^3+192 a^2 A b c^3-144 a^2 b^2 B c^2-96 a A b^3 c^2+60 a b^4 B c+12 A b^5 c-7 b^6 B\right ) \log \left (-2 \sqrt {c} \sqrt {a+b x+c x^2}+b+2 c x\right )}{1024 c^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 669, normalized size = 3.38 \begin {gather*} \left [-\frac {15 \, {\left (7 \, B b^{6} - 64 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} c^{3} + 48 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} c^{2} - 12 \, {\left (5 \, B a b^{4} + A b^{5}\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 (1280 \, B c^{6} x^{5} - 105 \, B b^{5} c + 1536 \, A a^{2} c^{4} + 128 \, {\left (13 \, B b c^{5} + 12 \, A c^{6}\right )} x^{4} - 48 \, {\left (27 \, B a^{2} b + 25 \, A a b^{2}\right )} c^{3} + 16 \, {\left (3 \, B b^{2} c^{4} + 4 \, {\left (35 \, B a + 33 \, A b\right )} c^{5}\right )} x^{3} + 20 \, {\left (38 \, B a b^{3} + 9 \, A b^{4}\right )} c^{2} - 8 \, {\left (7 \, B b^{3} c^{3} - 384 \, A a c^{5} - 12 \, {\left (3 \, B a b + A b^{2}\right )} c^{4}\right )} x^{2} + 2 \, {\left (35 \, B b^{4} c^{2} + 48 \, {\left (5 \, B a^{2} + 7 \, A a b\right )} c^{4} - 12 \, {\left (18 \, B a b^{2} + 5 \, A b^{3}\right )} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{30720 \, c^{5}}, -\frac {15 \, {\left (7 \, B b^{6} - 64 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} c^{3} + 48 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} c^{2} - 12 \, {\left (5 \, B a b^{4} + A b^{5}\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 (1280 \, B c^{6} x^{5} - 105 \, B b^{5} c + 1536 \, A a^{2} c^{4} + 128 \, {\left (13 \, B b c^{5} + 12 \, A c^{6}\right )} x^{4} - 48 \, {\left (27 \, B a^{2} b + 25 \, A a b^{2}\right )} c^{3} + 16 \, {\left (3 \, B b^{2} c^{4} + 4 \, {\left (35 \, B a + 33 \, A b\right )} c^{5}\right )} x^{3} + 20 \, {\left (38 \, B a b^{3} + 9 \, A b^{4}\right )} c^{2} - 8 \, {\left (7 \, B b^{3} c^{3} - 384 \, A a c^{5} - 12 \, {\left (3 \, B a b + A b^{2}\right )} c^{4}\right )} x^{2} + 2 \, {\left (35 \, B b^{4} c^{2} + 48 \, {\left (5 \, B a^{2} + 7 \, A a b\right )} c^{4} - 12 \, {\left (18 \, B a b^{2} + 5 \, A b^{3}\right )} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{15360 \, c^{5}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 332, normalized size = 1.68 \begin {gather*} \frac {1}{7680} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, B c x + \frac {13 \, B b c^{5} + 12 \, A c^{6}}{c^{5}}\right )} x + \frac {3 \, B b^{2} c^{4} + 140 \, B a c^{5} + 132 \, A b c^{5}}{c^{5}}\right )} x - \frac {7 \, B b^{3} c^{3} - 36 \, B a b c^{4} - 12 \, A b^{2} c^{4} - 384 \, A a c^{5}}{c^{5}}\right )} x + \frac {35 \, B b^{4} c^{2} - 216 \, B a b^{2} c^{3} - 60 \, A b^{3} c^{3} + 240 \, B a^{2} c^{4} + 336 \, A a b c^{4}}{c^{5}}\right )} x - \frac {105 \, B b^{5} c - 760 \, B a b^{3} c^{2} - 180 \, A b^{4} c^{2} + 1296 \, B a^{2} b c^{3} + 1200 \, A a b^{2} c^{3} - 1536 \, A a^{2} c^{4}}{c^{5}}\right )} - \frac {{\left (7 \, B b^{6} - 60 \, B a b^{4} c - 12 \, A b^{5} c + 144 \, B a^{2} b^{2} c^{2} + 96 \, A a b^{3} c^{2} - 64 \, B a^{3} c^{3} - 192 \, A a^{2} b c^{3}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{1024 \, c^{\frac {9}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 644, normalized size = 3.25 \begin {gather*} -\frac {3 A \,a^{2} b \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{16 c^{\frac {3}{2}}}+\frac {3 A a \,b^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{32 c^{\frac {5}{2}}}-\frac {3 A \,b^{5} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{256 c^{\frac {7}{2}}}-\frac {B \,a^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{16 c^{\frac {3}{2}}}+\frac {9 B \,a^{2} b^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{64 c^{\frac {5}{2}}}-\frac {15 B a \,b^{4} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{256 c^{\frac {7}{2}}}+\frac {7 B \,b^{6} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{1024 c^{\frac {9}{2}}}-\frac {3 \sqrt {c \,x^{2}+b x +a}\, A a b x}{16 c}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, A \,b^{3} x}{64 c^{2}}-\frac {\sqrt {c \,x^{2}+b x +a}\, B \,a^{2} x}{16 c}+\frac {\sqrt {c \,x^{2}+b x +a}\, B a \,b^{2} x}{8 c^{2}}-\frac {7 \sqrt {c \,x^{2}+b x +a}\, B \,b^{4} x}{256 c^{3}}-\frac {3 \sqrt {c \,x^{2}+b x +a}\, A a \,b^{2}}{32 c^{2}}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, A \,b^{4}}{128 c^{3}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A b x}{8 c}-\frac {\sqrt {c \,x^{2}+b x +a}\, B \,a^{2} b}{32 c^{2}}+\frac {\sqrt {c \,x^{2}+b x +a}\, B a \,b^{3}}{16 c^{3}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B a x}{24 c}-\frac {7 \sqrt {c \,x^{2}+b x +a}\, B \,b^{5}}{512 c^{4}}+\frac {7 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B \,b^{2} x}{96 c^{2}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A \,b^{2}}{16 c^{2}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B a b}{48 c^{2}}+\frac {7 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B \,b^{3}}{192 c^{3}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} B x}{6 c}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} A}{5 c}-\frac {7 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} B b}{60 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 [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x\,\left (A+B\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2} \,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 x \left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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