Optimal. Leaf size=197 \[ \frac {3 \left (-4 a B c-4 A b c+5 b^2 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{7/2}}-\frac {2 x^2 \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {\sqrt {a+b x+c x^2} \left (-2 c x \left (-12 a B c-4 A b c+5 b^2 B\right )+32 a A c^2-52 a b B c-12 A b^2 c+15 b^3 B\right )}{4 c^3 \left (b^2-4 a c\right )} \]
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Rubi [A] time = 0.14, antiderivative size = 197, 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 = {818, 779, 621, 206} \begin {gather*} -\frac {\sqrt {a+b x+c x^2} \left (-2 c x \left (-12 a B c-4 A b c+5 b^2 B\right )+32 a A c^2-52 a b B c-12 A b^2 c+15 b^3 B\right )}{4 c^3 \left (b^2-4 a c\right )}+\frac {3 \left (-4 a B c-4 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 )}{8 c^{7/2}}-\frac {2 x^2 \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 779
Rule 818
Rubi steps
\begin {align*} \int \frac {x^3 (A+B x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx &=-\frac {2 x^2 \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {2 \int \frac {x \left (2 a (b B-2 A c)+\frac {1}{2} \left (5 b^2 B-4 A b c-12 a B c\right ) x\right )}{\sqrt {a+b x+c x^2}} \, dx}{c \left (b^2-4 a c\right )}\\ &=-\frac {2 x^2 \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {\left (15 b^3 B-12 A b^2 c-52 a b B c+32 a A c^2-2 c \left (5 b^2 B-4 A b c-12 a B c\right ) x\right ) \sqrt {a+b x+c x^2}}{4 c^3 \left (b^2-4 a c\right )}+\frac {\left (3 \left (5 b^2 B-4 A b c-4 a B c\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{8 c^3}\\ &=-\frac {2 x^2 \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {\left (15 b^3 B-12 A b^2 c-52 a b B c+32 a A c^2-2 c \left (5 b^2 B-4 A b c-12 a B c\right ) x\right ) \sqrt {a+b x+c x^2}}{4 c^3 \left (b^2-4 a c\right )}+\frac {\left (3 \left (5 b^2 B-4 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 )}{4 c^3}\\ &=-\frac {2 x^2 \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {\left (15 b^3 B-12 A b^2 c-52 a b B c+32 a A c^2-2 c \left (5 b^2 B-4 A b c-12 a B c\right ) x\right ) \sqrt {a+b x+c x^2}}{4 c^3 \left (b^2-4 a c\right )}+\frac {3 \left (5 b^2 B-4 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 )}{8 c^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 221, normalized size = 1.12 \begin {gather*} \frac {2 \sqrt {c} \left (4 a^2 c (8 A c-13 b B+6 B c x)+a \left (-2 b^2 c (6 A+31 B x)-20 b c^2 x (B x-2 A)+8 c^3 x^2 (2 A+B x)+15 b^3 B\right )+b^2 x \left (b (5 B c x-12 A c)-2 c^2 x (2 A+B x)+15 b^2 B\right )\right )-3 \left (b^2-4 a c\right ) \sqrt {a+x (b+c x)} \left (-4 a B c-4 A b c+5 b^2 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )}{8 c^{7/2} \left (4 a c-b^2\right ) \sqrt {a+x (b+c x)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.97, size = 235, normalized size = 1.19 \begin {gather*} \frac {32 a^2 A c^2-52 a^2 b B c+24 a^2 B c^2 x-12 a A b^2 c+40 a A b c^2 x+16 a A c^3 x^2+15 a b^3 B-62 a b^2 B c x-20 a b B c^2 x^2+8 a B c^3 x^3-12 A b^3 c x-4 A b^2 c^2 x^2+15 b^4 B x+5 b^3 B c x^2-2 b^2 B c^2 x^3}{4 c^3 \left (4 a c-b^2\right ) \sqrt {a+b x+c x^2}}-\frac {3 \left (-4 a B c-4 A b c+5 b^2 B\right ) \log \left (-2 c^{7/2} \sqrt {a+b x+c x^2}+b c^3+2 c^4 x\right )}{8 c^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.71, size = 793, normalized size = 4.03 \begin {gather*} \left [-\frac {3 \, {\left (5 \, B a b^{4} + 16 \, {\left (B a^{3} + A a^{2} b\right )} c^{2} + {\left (5 \, B b^{4} c + 16 \, {\left (B a^{2} + A a b\right )} c^{3} - 4 \, {\left (6 \, B a b^{2} + A b^{3}\right )} c^{2}\right )} x^{2} - 4 \, {\left (6 \, B a^{2} b^{2} + A a b^{3}\right )} c + {\left (5 \, B b^{5} + 16 \, {\left (B a^{2} b + A a b^{2}\right )} c^{2} - 4 \, {\left (6 \, B a b^{3} + A b^{4}\right )} c\right )} x\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 (15 \, B a b^{3} c + 32 \, A a^{2} c^{3} - 2 \, {\left (B b^{2} c^{3} - 4 \, B a c^{4}\right )} x^{3} - 4 \, {\left (13 \, B a^{2} b + 3 \, A a b^{2}\right )} c^{2} + {\left (5 \, B b^{3} c^{2} + 16 \, A a c^{4} - 4 \, {\left (5 \, B a b + A b^{2}\right )} c^{3}\right )} x^{2} + {\left (15 \, B b^{4} c + 8 \, {\left (3 \, B a^{2} + 5 \, A a b\right )} c^{3} - 2 \, {\left (31 \, B a b^{2} + 6 \, A b^{3}\right )} c^{2}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{16 \, {\left (a b^{2} c^{4} - 4 \, a^{2} c^{5} + {\left (b^{2} c^{5} - 4 \, a c^{6}\right )} x^{2} + {\left (b^{3} c^{4} - 4 \, a b c^{5}\right )} x\right )}}, -\frac {3 \, {\left (5 \, B a b^{4} + 16 \, {\left (B a^{3} + A a^{2} b\right )} c^{2} + {\left (5 \, B b^{4} c + 16 \, {\left (B a^{2} + A a b\right )} c^{3} - 4 \, {\left (6 \, B a b^{2} + A b^{3}\right )} c^{2}\right )} x^{2} - 4 \, {\left (6 \, B a^{2} b^{2} + A a b^{3}\right )} c + {\left (5 \, B b^{5} + 16 \, {\left (B a^{2} b + A a b^{2}\right )} c^{2} - 4 \, {\left (6 \, B a b^{3} + A b^{4}\right )} c\right )} x\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 (15 \, B a b^{3} c + 32 \, A a^{2} c^{3} - 2 \, {\left (B b^{2} c^{3} - 4 \, B a c^{4}\right )} x^{3} - 4 \, {\left (13 \, B a^{2} b + 3 \, A a b^{2}\right )} c^{2} + {\left (5 \, B b^{3} c^{2} + 16 \, A a c^{4} - 4 \, {\left (5 \, B a b + A b^{2}\right )} c^{3}\right )} x^{2} + {\left (15 \, B b^{4} c + 8 \, {\left (3 \, B a^{2} + 5 \, A a b\right )} c^{3} - 2 \, {\left (31 \, B a b^{2} + 6 \, A b^{3}\right )} c^{2}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{8 \, {\left (a b^{2} c^{4} - 4 \, a^{2} c^{5} + {\left (b^{2} c^{5} - 4 \, a c^{6}\right )} x^{2} + {\left (b^{3} c^{4} - 4 \, a b c^{5}\right )} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 268, normalized size = 1.36 \begin {gather*} \frac {{\left ({\left (\frac {2 \, {\left (B b^{2} c^{2} - 4 \, B a c^{3}\right )} x}{b^{2} c^{3} - 4 \, a c^{4}} - \frac {5 \, B b^{3} c - 20 \, B a b c^{2} - 4 \, A b^{2} c^{2} + 16 \, A a c^{3}}{b^{2} c^{3} - 4 \, a c^{4}}\right )} x - \frac {15 \, B b^{4} - 62 \, B a b^{2} c - 12 \, A b^{3} c + 24 \, B a^{2} c^{2} + 40 \, A a b c^{2}}{b^{2} c^{3} - 4 \, a c^{4}}\right )} x - \frac {15 \, B a b^{3} - 52 \, B a^{2} b c - 12 \, A a b^{2} c + 32 \, A a^{2} c^{2}}{b^{2} c^{3} - 4 \, a c^{4}}}{4 \, \sqrt {c x^{2} + b x + a}} - \frac {3 \, {\left (5 \, B b^{2} - 4 \, B a c - 4 \, A b c\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{8 \, 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 = 576, normalized size = 2.92 \begin {gather*} \frac {4 A a b x}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, c}-\frac {3 A \,b^{3} x}{2 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, c^{2}}-\frac {13 B a \,b^{2} x}{2 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, c^{2}}+\frac {15 B \,b^{4} x}{8 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, c^{3}}+\frac {B \,x^{3}}{2 \sqrt {c \,x^{2}+b x +a}\, c}+\frac {2 A a \,b^{2}}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, c^{2}}-\frac {3 A \,b^{4}}{4 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, c^{3}}+\frac {A \,x^{2}}{\sqrt {c \,x^{2}+b x +a}\, c}-\frac {13 B a \,b^{3}}{4 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, c^{3}}+\frac {15 B \,b^{5}}{16 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, c^{4}}-\frac {5 B b \,x^{2}}{4 \sqrt {c \,x^{2}+b x +a}\, c^{2}}+\frac {3 A b x}{2 \sqrt {c \,x^{2}+b x +a}\, c^{2}}+\frac {3 B a x}{2 \sqrt {c \,x^{2}+b x +a}\, c^{2}}-\frac {15 B \,b^{2} x}{8 \sqrt {c \,x^{2}+b x +a}\, c^{3}}-\frac {3 A b \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {5}{2}}}-\frac {3 B a \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {5}{2}}}+\frac {15 B \,b^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {7}{2}}}+\frac {2 A a}{\sqrt {c \,x^{2}+b x +a}\, c^{2}}-\frac {3 A \,b^{2}}{4 \sqrt {c \,x^{2}+b x +a}\, c^{3}}-\frac {13 B a b}{4 \sqrt {c \,x^{2}+b x +a}\, c^{3}}+\frac {15 B \,b^{3}}{16 \sqrt {c \,x^{2}+b x +a}\, c^{4}} \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^3\,\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 \frac {x^{3} \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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