Optimal. Leaf size=218 \[ a^{3/2} (-A) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )+\frac {\left (\left (b^2-4 a c\right ) \left (-12 a B c-8 A b c+3 b^2 B\right )+64 a A b c^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{128 c^{5/2}}-\frac {\sqrt {a+b x+c x^2} \left (2 c x \left (-12 a B c-8 A b c+3 b^2 B\right )-64 a A c^2-12 a b B c-8 A b^2 c+3 b^3 B\right )}{64 c^2}+\frac {\left (a+b x+c x^2\right )^{3/2} (8 A c+3 b B+6 B c x)}{24 c} \]
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Rubi [A] time = 0.25, antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {814, 843, 621, 206, 724} \begin {gather*} a^{3/2} (-A) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )-\frac {\sqrt {a+b x+c x^2} \left (2 c x \left (-12 a B c-8 A b c+3 b^2 B\right )-64 a A c^2-12 a b B c-8 A b^2 c+3 b^3 B\right )}{64 c^2}+\frac {\left (\left (b^2-4 a c\right ) \left (-12 a B c-8 A b c+3 b^2 B\right )+64 a A 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^{5/2}}+\frac {\left (a+b x+c x^2\right )^{3/2} (8 A c+3 b B+6 B c x)}{24 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 724
Rule 814
Rule 843
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x} \, dx &=\frac {(3 b B+8 A c+6 B c x) \left (a+b x+c x^2\right )^{3/2}}{24 c}-\frac {\int \frac {\left (-8 a A c-\frac {1}{2} \left (8 A b c-3 B \left (b^2-4 a c\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{x} \, dx}{8 c}\\ &=-\frac {\left (3 b^3 B-8 A b^2 c-12 a b B c-64 a A c^2+2 c \left (3 b^2 B-8 A b c-12 a B c\right ) x\right ) \sqrt {a+b x+c x^2}}{64 c^2}+\frac {(3 b B+8 A c+6 B c x) \left (a+b x+c x^2\right )^{3/2}}{24 c}+\frac {\int \frac {32 a^2 A c^2+\frac {1}{4} \left (64 a A b c^2-\left (b^2-4 a c\right ) \left (8 A b c-3 B \left (b^2-4 a c\right )\right )\right ) x}{x \sqrt {a+b x+c x^2}} \, dx}{32 c^2}\\ &=-\frac {\left (3 b^3 B-8 A b^2 c-12 a b B c-64 a A c^2+2 c \left (3 b^2 B-8 A b c-12 a B c\right ) x\right ) \sqrt {a+b x+c x^2}}{64 c^2}+\frac {(3 b B+8 A c+6 B c x) \left (a+b x+c x^2\right )^{3/2}}{24 c}+\left (a^2 A\right ) \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx+\frac {1}{128} \left (64 a A b+\frac {\left (b^2-4 a c\right ) \left (3 b^2 B-8 A b c-12 a B c\right )}{c^2}\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx\\ &=-\frac {\left (3 b^3 B-8 A b^2 c-12 a b B c-64 a A c^2+2 c \left (3 b^2 B-8 A b c-12 a B c\right ) x\right ) \sqrt {a+b x+c x^2}}{64 c^2}+\frac {(3 b B+8 A c+6 B c x) \left (a+b x+c x^2\right )^{3/2}}{24 c}-\left (2 a^2 A\right ) \operatorname {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b x}{\sqrt {a+b x+c x^2}}\right )+\frac {1}{64} \left (64 a A b+\frac {\left (b^2-4 a c\right ) \left (3 b^2 B-8 A b c-12 a B c\right )}{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 )\\ &=-\frac {\left (3 b^3 B-8 A b^2 c-12 a b B c-64 a A c^2+2 c \left (3 b^2 B-8 A b c-12 a B c\right ) x\right ) \sqrt {a+b x+c x^2}}{64 c^2}+\frac {(3 b B+8 A c+6 B c x) \left (a+b x+c x^2\right )^{3/2}}{24 c}-a^{3/2} A \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )+\frac {\left (64 a A b+\frac {\left (b^2-4 a c\right ) \left (3 b^2 B-8 A b c-12 a B c\right )}{c^2}\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{128 \sqrt {c}}\\ \end {align*}
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Mathematica [A] time = 0.34, size = 206, normalized size = 0.94 \begin {gather*} a^{3/2} (-A) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+x (b+c x)}}\right )+\frac {\left (48 a^2 B c^2+96 a A b c^2-24 a b^2 B c-8 A b^3 c+3 b^4 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )}{128 c^{5/2}}+\frac {\sqrt {a+x (b+c x)} \left (4 b c (15 a B+2 c x (14 A+9 B x))+8 c^2 \left (32 a A+15 a B x+8 A c x^2+6 B c x^3\right )+6 b^2 c (4 A+B x)-9 b^3 B\right )}{192 c^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.95, size = 226, normalized size = 1.04 \begin {gather*} 2 a^{3/2} A \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}-\frac {\sqrt {a+b x+c x^2}}{\sqrt {a}}\right )+\frac {\left (-48 a^2 B c^2-96 a A b c^2+24 a b^2 B c+8 A b^3 c-3 b^4 B\right ) \log \left (-2 c^{5/2} \sqrt {a+b x+c x^2}+b c^2+2 c^3 x\right )}{128 c^{5/2}}+\frac {\sqrt {a+b x+c x^2} \left (256 a A c^2+60 a b B c+120 a B c^2 x+24 A b^2 c+112 A b c^2 x+64 A c^3 x^2-9 b^3 B+6 b^2 B c x+72 b B c^2 x^2+48 B c^3 x^3\right )}{192 c^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 5.17, size = 1023, normalized size = 4.69
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 390, normalized size = 1.79 \begin {gather*} -A \,a^{\frac {3}{2}} \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )+\frac {3 A a b \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{4 \sqrt {c}}-\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 {3}{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 \sqrt {c}}-\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 {3}{2}}}+\frac {3 B \,b^{4} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{128 c^{\frac {5}{2}}}+\frac {\sqrt {c \,x^{2}+b x +a}\, A b x}{4}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, B a x}{8}-\frac {3 \sqrt {c \,x^{2}+b x +a}\, B \,b^{2} x}{32 c}+\sqrt {c \,x^{2}+b x +a}\, A a +\frac {\sqrt {c \,x^{2}+b x +a}\, A \,b^{2}}{8 c}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, B a b}{16 c}-\frac {3 \sqrt {c \,x^{2}+b x +a}\, B \,b^{3}}{64 c^{2}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B x}{4}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A}{3}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B b}{8 c} \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 = -0.00 \begin {gather*} \int \frac {\left (A+B\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{x} \,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 {\left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac {3}{2}}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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