Optimal. Leaf size=206 \[ \frac {\left (48 a^2 B c^2+96 a A b c^2-120 a b^2 B c-40 A b^3 c+35 b^4 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{128 c^{9/2}}-\frac {\sqrt {a+b x+c x^2} \left (-2 c x \left (-36 a B c-40 A b c+35 b^2 B\right )+128 a A c^2-220 a b B c-120 A b^2 c+105 b^3 B\right )}{192 c^4}-\frac {x^2 \sqrt {a+b x+c x^2} (7 b B-8 A c)}{24 c^2}+\frac {B x^3 \sqrt {a+b x+c x^2}}{4 c} \]
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Rubi [A] time = 0.24, antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {832, 779, 621, 206} \[ \frac {\left (48 a^2 B c^2+96 a A b c^2-120 a b^2 B c-40 A b^3 c+35 b^4 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{128 c^{9/2}}-\frac {\sqrt {a+b x+c x^2} \left (-2 c x \left (-36 a B c-40 A b c+35 b^2 B\right )+128 a A c^2-220 a b B c-120 A b^2 c+105 b^3 B\right )}{192 c^4}-\frac {x^2 \sqrt {a+b x+c x^2} (7 b B-8 A c)}{24 c^2}+\frac {B x^3 \sqrt {a+b x+c x^2}}{4 c} \]
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 779
Rule 832
Rubi steps
\begin {align*} \int \frac {x^3 (A+B x)}{\sqrt {a+b x+c x^2}} \, dx &=\frac {B x^3 \sqrt {a+b x+c x^2}}{4 c}+\frac {\int \frac {x^2 \left (-3 a B-\frac {1}{2} (7 b B-8 A c) x\right )}{\sqrt {a+b x+c x^2}} \, dx}{4 c}\\ &=-\frac {(7 b B-8 A c) x^2 \sqrt {a+b x+c x^2}}{24 c^2}+\frac {B x^3 \sqrt {a+b x+c x^2}}{4 c}+\frac {\int \frac {x \left (a (7 b B-8 A c)+\frac {1}{4} \left (35 b^2 B-40 A b c-36 a B c\right ) x\right )}{\sqrt {a+b x+c x^2}} \, dx}{12 c^2}\\ &=-\frac {(7 b B-8 A c) x^2 \sqrt {a+b x+c x^2}}{24 c^2}+\frac {B x^3 \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (105 b^3 B-120 A b^2 c-220 a b B c+128 a A c^2-2 c \left (35 b^2 B-40 A b c-36 a B c\right ) x\right ) \sqrt {a+b x+c x^2}}{192 c^4}+\frac {\left (35 b^4 B-40 A b^3 c-120 a b^2 B c+96 a A b c^2+48 a^2 B c^2\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{128 c^4}\\ &=-\frac {(7 b B-8 A c) x^2 \sqrt {a+b x+c x^2}}{24 c^2}+\frac {B x^3 \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (105 b^3 B-120 A b^2 c-220 a b B c+128 a A c^2-2 c \left (35 b^2 B-40 A b c-36 a B c\right ) x\right ) \sqrt {a+b x+c x^2}}{192 c^4}+\frac {\left (35 b^4 B-40 A b^3 c-120 a b^2 B c+96 a A b c^2+48 a^2 B 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 )}{64 c^4}\\ &=-\frac {(7 b B-8 A c) x^2 \sqrt {a+b x+c x^2}}{24 c^2}+\frac {B x^3 \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (105 b^3 B-120 A b^2 c-220 a b B c+128 a A c^2-2 c \left (35 b^2 B-40 A b c-36 a B c\right ) x\right ) \sqrt {a+b x+c x^2}}{192 c^4}+\frac {\left (35 b^4 B-40 A b^3 c-120 a b^2 B c+96 a A b c^2+48 a^2 B c^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{128 c^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 169, normalized size = 0.82 \[ \frac {\left (48 a^2 B c^2+96 a A b c^2-120 a b^2 B c-40 A b^3 c+35 b^4 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )}{128 c^{9/2}}+\frac {\sqrt {a+x (b+c x)} \left (4 b c (55 a B-2 c x (10 A+7 B x))+8 c^2 \left (-16 a A-9 a B x+8 A c x^2+6 B c x^3\right )+10 b^2 c (12 A+7 B x)-105 b^3 B\right )}{192 c^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 395, normalized size = 1.92 \[ \left [\frac {3 \, {\left (35 \, B b^{4} + 48 \, {\left (B a^{2} + 2 \, A a b\right )} c^{2} - 40 \, {\left (3 \, B a b^{2} + A b^{3}\right )} c\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (48 \, B c^{4} x^{3} - 105 \, B b^{3} c - 128 \, A a c^{3} + 20 \, {\left (11 \, B a b + 6 \, A b^{2}\right )} c^{2} - 8 \, {\left (7 \, B b c^{3} - 8 \, A c^{4}\right )} x^{2} + 2 \, {\left (35 \, B b^{2} c^{2} - 4 \, {\left (9 \, B a + 10 \, A b\right )} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{768 \, c^{5}}, -\frac {3 \, {\left (35 \, B b^{4} + 48 \, {\left (B a^{2} + 2 \, A a b\right )} c^{2} - 40 \, {\left (3 \, B a b^{2} + A b^{3}\right )} c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) - 2 \, {\left (48 \, B c^{4} x^{3} - 105 \, B b^{3} c - 128 \, A a c^{3} + 20 \, {\left (11 \, B a b + 6 \, A b^{2}\right )} c^{2} - 8 \, {\left (7 \, B b c^{3} - 8 \, A c^{4}\right )} x^{2} + 2 \, {\left (35 \, B b^{2} c^{2} - 4 \, {\left (9 \, B a + 10 \, A b\right )} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{384 \, c^{5}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 183, normalized size = 0.89 \[ \frac {1}{192} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (\frac {6 \, B x}{c} - \frac {7 \, B b c^{2} - 8 \, A c^{3}}{c^{4}}\right )} x + \frac {35 \, B b^{2} c - 36 \, B a c^{2} - 40 \, A b c^{2}}{c^{4}}\right )} x - \frac {105 \, B b^{3} - 220 \, B a b c - 120 \, A b^{2} c + 128 \, A a c^{2}}{c^{4}}\right )} - \frac {{\left (35 \, B b^{4} - 120 \, B a b^{2} c - 40 \, A b^{3} c + 48 \, B a^{2} c^{2} + 96 \, A a b c^{2}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{128 \, c^{\frac {9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 379, normalized size = 1.84 \[ \frac {\sqrt {c \,x^{2}+b x +a}\, B \,x^{3}}{4 c}+\frac {\sqrt {c \,x^{2}+b x +a}\, A \,x^{2}}{3 c}-\frac {7 \sqrt {c \,x^{2}+b x +a}\, B b \,x^{2}}{24 c^{2}}+\frac {3 A a b \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{4 c^{\frac {5}{2}}}-\frac {5 A \,b^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{16 c^{\frac {7}{2}}}+\frac {3 B \,a^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {5}{2}}}-\frac {15 B a \,b^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{16 c^{\frac {7}{2}}}+\frac {35 B \,b^{4} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{128 c^{\frac {9}{2}}}-\frac {5 \sqrt {c \,x^{2}+b x +a}\, A b x}{12 c^{2}}-\frac {3 \sqrt {c \,x^{2}+b x +a}\, B a x}{8 c^{2}}+\frac {35 \sqrt {c \,x^{2}+b x +a}\, B \,b^{2} x}{96 c^{3}}-\frac {2 \sqrt {c \,x^{2}+b x +a}\, A a}{3 c^{2}}+\frac {5 \sqrt {c \,x^{2}+b x +a}\, A \,b^{2}}{8 c^{3}}+\frac {55 \sqrt {c \,x^{2}+b x +a}\, B a b}{48 c^{3}}-\frac {35 \sqrt {c \,x^{2}+b x +a}\, B \,b^{3}}{64 c^{4}} \]
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 [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^3\,\left (A+B\,x\right )}{\sqrt {c\,x^2+b\,x+a}} \,d x \]
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 \[ \int \frac {x^{3} \left (A + B x\right )}{\sqrt {a + b x + c x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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