Optimal. Leaf size=118 \[ \frac {\left (b x+c x^2\right )^{3/2} (4 A c+b B)}{2 b x}+\frac {3}{4} \sqrt {b x+c x^2} (4 A c+b B)+\frac {3 b (4 A c+b B) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{4 \sqrt {c}}-\frac {2 A \left (b x+c x^2\right )^{5/2}}{b x^3} \]
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Rubi [A] time = 0.12, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {792, 664, 620, 206} \begin {gather*} \frac {\left (b x+c x^2\right )^{3/2} (4 A c+b B)}{2 b x}+\frac {3}{4} \sqrt {b x+c x^2} (4 A c+b B)+\frac {3 b (4 A c+b B) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{4 \sqrt {c}}-\frac {2 A \left (b x+c x^2\right )^{5/2}}{b x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 620
Rule 664
Rule 792
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{x^3} \, dx &=-\frac {2 A \left (b x+c x^2\right )^{5/2}}{b x^3}+\frac {\left (2 \left (-3 (-b B+A c)+\frac {5}{2} (-b B+2 A c)\right )\right ) \int \frac {\left (b x+c x^2\right )^{3/2}}{x^2} \, dx}{b}\\ &=\frac {(b B+4 A c) \left (b x+c x^2\right )^{3/2}}{2 b x}-\frac {2 A \left (b x+c x^2\right )^{5/2}}{b x^3}+\frac {1}{4} (3 (b B+4 A c)) \int \frac {\sqrt {b x+c x^2}}{x} \, dx\\ &=\frac {3}{4} (b B+4 A c) \sqrt {b x+c x^2}+\frac {(b B+4 A c) \left (b x+c x^2\right )^{3/2}}{2 b x}-\frac {2 A \left (b x+c x^2\right )^{5/2}}{b x^3}+\frac {1}{8} (3 b (b B+4 A c)) \int \frac {1}{\sqrt {b x+c x^2}} \, dx\\ &=\frac {3}{4} (b B+4 A c) \sqrt {b x+c x^2}+\frac {(b B+4 A c) \left (b x+c x^2\right )^{3/2}}{2 b x}-\frac {2 A \left (b x+c x^2\right )^{5/2}}{b x^3}+\frac {1}{4} (3 b (b B+4 A c)) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )\\ &=\frac {3}{4} (b B+4 A c) \sqrt {b x+c x^2}+\frac {(b B+4 A c) \left (b x+c x^2\right )^{3/2}}{2 b x}-\frac {2 A \left (b x+c x^2\right )^{5/2}}{b x^3}+\frac {3 b (b B+4 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{4 \sqrt {c}}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 94, normalized size = 0.80 \begin {gather*} \frac {\sqrt {x (b+c x)} \left (\frac {3 \sqrt {b} \sqrt {x} (4 A c+b B) \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{\sqrt {c} \sqrt {\frac {c x}{b}+1}}+A (4 c x-8 b)+B x (5 b+2 c x)\right )}{4 x} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.50, size = 90, normalized size = 0.76 \begin {gather*} \frac {\sqrt {b x+c x^2} \left (-8 A b+4 A c x+5 b B x+2 B c x^2\right )}{4 x}-\frac {3 \left (4 A b c+b^2 B\right ) \log \left (-2 \sqrt {c} \sqrt {b x+c x^2}+b+2 c x\right )}{8 \sqrt {c}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 186, normalized size = 1.58 \begin {gather*} \left [\frac {3 \, {\left (B b^{2} + 4 \, A b c\right )} \sqrt {c} x \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) + 2 \, {\left (2 \, B c^{2} x^{2} - 8 \, A b c + {\left (5 \, B b c + 4 \, A c^{2}\right )} x\right )} \sqrt {c x^{2} + b x}}{8 \, c x}, -\frac {3 \, {\left (B b^{2} + 4 \, A b c\right )} \sqrt {-c} x \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) - {\left (2 \, B c^{2} x^{2} - 8 \, A b c + {\left (5 \, B b c + 4 \, A c^{2}\right )} x\right )} \sqrt {c x^{2} + b x}}{4 \, c x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 109, normalized size = 0.92 \begin {gather*} \frac {2 \, A b^{2}}{\sqrt {c} x - \sqrt {c x^{2} + b x}} + \frac {1}{4} \, {\left (2 \, B c x + \frac {5 \, B b c + 4 \, A c^{2}}{c}\right )} \sqrt {c x^{2} + b x} - \frac {3 \, {\left (B b^{2} + 4 \, A b c\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{8 \, \sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 232, normalized size = 1.97 \begin {gather*} \frac {3 A b \sqrt {c}\, \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{2}+\frac {3 B \,b^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 \sqrt {c}}-\frac {6 \sqrt {c \,x^{2}+b x}\, A \,c^{2} x}{b}-\frac {3 \sqrt {c \,x^{2}+b x}\, B c x}{2}-3 \sqrt {c \,x^{2}+b x}\, A c -\frac {3 \sqrt {c \,x^{2}+b x}\, B b}{4}-\frac {8 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} A \,c^{2}}{b^{2}}-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} B c}{b}+\frac {8 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} A c}{b^{2} x^{2}}+\frac {2 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} B}{b \,x^{2}}-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} A}{b \,x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.87, size = 129, normalized size = 1.09 \begin {gather*} \frac {3 \, B b^{2} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{8 \, \sqrt {c}} + \frac {3}{2} \, A b \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) + \frac {3}{4} \, \sqrt {c x^{2} + b x} B b + \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} B}{2 \, x} - \frac {3 \, \sqrt {c x^{2} + b x} A b}{x} + \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} A}{x^{2}} \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 {{\left (c\,x^2+b\,x\right )}^{3/2}\,\left (A+B\,x\right )}{x^3} \,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 (x \left (b + c x\right )\right )^{\frac {3}{2}} \left (A + B x\right )}{x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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