Optimal. Leaf size=95 \[ -\frac {2 A \left (b x+c x^2\right )^{5/2}}{5 b x^5}+2 B c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )-\frac {2 B c \sqrt {b x+c x^2}}{x}-\frac {2 B \left (b x+c x^2\right )^{3/2}}{3 x^3} \]
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Rubi [A] time = 0.10, antiderivative size = 95, 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, 662, 620, 206} \begin {gather*} -\frac {2 A \left (b x+c x^2\right )^{5/2}}{5 b x^5}+2 B c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )-\frac {2 B \left (b x+c x^2\right )^{3/2}}{3 x^3}-\frac {2 B c \sqrt {b x+c x^2}}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 620
Rule 662
Rule 792
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{x^5} \, dx &=-\frac {2 A \left (b x+c x^2\right )^{5/2}}{5 b x^5}+B \int \frac {\left (b x+c x^2\right )^{3/2}}{x^4} \, dx\\ &=-\frac {2 B \left (b x+c x^2\right )^{3/2}}{3 x^3}-\frac {2 A \left (b x+c x^2\right )^{5/2}}{5 b x^5}+(B c) \int \frac {\sqrt {b x+c x^2}}{x^2} \, dx\\ &=-\frac {2 B c \sqrt {b x+c x^2}}{x}-\frac {2 B \left (b x+c x^2\right )^{3/2}}{3 x^3}-\frac {2 A \left (b x+c x^2\right )^{5/2}}{5 b x^5}+\left (B c^2\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx\\ &=-\frac {2 B c \sqrt {b x+c x^2}}{x}-\frac {2 B \left (b x+c x^2\right )^{3/2}}{3 x^3}-\frac {2 A \left (b x+c x^2\right )^{5/2}}{5 b x^5}+\left (2 B c^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )\\ &=-\frac {2 B c \sqrt {b x+c x^2}}{x}-\frac {2 B \left (b x+c x^2\right )^{3/2}}{3 x^3}-\frac {2 A \left (b x+c x^2\right )^{5/2}}{5 b x^5}+2 B c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )\\ \end {align*}
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Mathematica [C] time = 0.08, size = 88, normalized size = 0.93 \begin {gather*} \frac {2 \sqrt {x (b+c x)} \left ((b+c x)^2 \sqrt {\frac {c x}{b}+1} (b B-A c)-b^3 B \, _2F_1\left (-\frac {5}{2},-\frac {5}{2};-\frac {3}{2};-\frac {c x}{b}\right )\right )}{5 b c x^3 \sqrt {\frac {c x}{b}+1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.39, size = 96, normalized size = 1.01 \begin {gather*} -\frac {2 \sqrt {b x+c x^2} \left (3 A b^2+6 A b c x+3 A c^2 x^2+5 b^2 B x+20 b B c x^2\right )}{15 b x^3}-B c^{3/2} \log \left (-2 \sqrt {c} \sqrt {b x+c x^2}+b+2 c x\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 188, normalized size = 1.98 \begin {gather*} \left [\frac {15 \, B b c^{\frac {3}{2}} x^{3} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 2 \, {\left (3 \, A b^{2} + {\left (20 \, B b c + 3 \, A c^{2}\right )} x^{2} + {\left (5 \, B b^{2} + 6 \, A b c\right )} x\right )} \sqrt {c x^{2} + b x}}{15 \, b x^{3}}, -\frac {2 \, {\left (15 \, B b \sqrt {-c} c x^{3} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + {\left (3 \, A b^{2} + {\left (20 \, B b c + 3 \, A c^{2}\right )} x^{2} + {\left (5 \, B b^{2} + 6 \, A b c\right )} x\right )} \sqrt {c x^{2} + b x}\right )}}{15 \, b x^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.25, size = 270, normalized size = 2.84 \begin {gather*} -B c^{\frac {3}{2}} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right ) + \frac {2 \, {\left (30 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} B b c^{\frac {3}{2}} + 15 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} A c^{\frac {5}{2}} + 15 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} B b^{2} c + 30 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} A b c^{2} + 5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B b^{3} \sqrt {c} + 30 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A b^{2} c^{\frac {3}{2}} + 15 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b^{3} c + 3 \, A b^{4} \sqrt {c}\right )}}{15 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} \sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 176, normalized size = 1.85 \begin {gather*} B \,c^{\frac {3}{2}} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )-\frac {4 \sqrt {c \,x^{2}+b x}\, B \,c^{3} x}{b^{2}}-\frac {2 \sqrt {c \,x^{2}+b x}\, B \,c^{2}}{b}-\frac {16 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} B \,c^{3}}{3 b^{3}}+\frac {16 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} B \,c^{2}}{3 b^{3} x^{2}}-\frac {4 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} B c}{3 b^{2} x^{3}}-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} B}{3 b \,x^{4}}-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} A}{5 b \,x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.95, size = 158, normalized size = 1.66 \begin {gather*} B c^{\frac {3}{2}} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - \frac {7 \, \sqrt {c x^{2} + b x} B c}{3 \, x} - \frac {2 \, \sqrt {c x^{2} + b x} A c^{2}}{5 \, b x} - \frac {\sqrt {c x^{2} + b x} B b}{3 \, x^{2}} + \frac {\sqrt {c x^{2} + b x} A c}{5 \, x^{2}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} B}{3 \, x^{3}} + \frac {3 \, \sqrt {c x^{2} + b x} A b}{5 \, x^{3}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} A}{x^{4}} \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^5} \,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^{5}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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