Optimal. Leaf size=105 \[ -2 A b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )+\frac {2 A b \sqrt {b x+c x^2}}{\sqrt {x}}+\frac {2 A \left (b x+c x^2\right )^{3/2}}{3 x^{3/2}}+\frac {2 B \left (b x+c x^2\right )^{5/2}}{5 c x^{5/2}} \]
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Rubi [A] time = 0.09, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {794, 664, 660, 207} \begin {gather*} -2 A b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )+\frac {2 A \left (b x+c x^2\right )^{3/2}}{3 x^{3/2}}+\frac {2 A b \sqrt {b x+c x^2}}{\sqrt {x}}+\frac {2 B \left (b x+c x^2\right )^{5/2}}{5 c x^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 207
Rule 660
Rule 664
Rule 794
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{x^{5/2}} \, dx &=\frac {2 B \left (b x+c x^2\right )^{5/2}}{5 c x^{5/2}}+A \int \frac {\left (b x+c x^2\right )^{3/2}}{x^{5/2}} \, dx\\ &=\frac {2 A \left (b x+c x^2\right )^{3/2}}{3 x^{3/2}}+\frac {2 B \left (b x+c x^2\right )^{5/2}}{5 c x^{5/2}}+(A b) \int \frac {\sqrt {b x+c x^2}}{x^{3/2}} \, dx\\ &=\frac {2 A b \sqrt {b x+c x^2}}{\sqrt {x}}+\frac {2 A \left (b x+c x^2\right )^{3/2}}{3 x^{3/2}}+\frac {2 B \left (b x+c x^2\right )^{5/2}}{5 c x^{5/2}}+\left (A b^2\right ) \int \frac {1}{\sqrt {x} \sqrt {b x+c x^2}} \, dx\\ &=\frac {2 A b \sqrt {b x+c x^2}}{\sqrt {x}}+\frac {2 A \left (b x+c x^2\right )^{3/2}}{3 x^{3/2}}+\frac {2 B \left (b x+c x^2\right )^{5/2}}{5 c x^{5/2}}+\left (2 A b^2\right ) \operatorname {Subst}\left (\int \frac {1}{-b+x^2} \, dx,x,\frac {\sqrt {b x+c x^2}}{\sqrt {x}}\right )\\ &=\frac {2 A b \sqrt {b x+c x^2}}{\sqrt {x}}+\frac {2 A \left (b x+c x^2\right )^{3/2}}{3 x^{3/2}}+\frac {2 B \left (b x+c x^2\right )^{5/2}}{5 c x^{5/2}}-2 A b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )\\ \end {align*}
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Mathematica [A] time = 0.08, size = 100, normalized size = 0.95 \begin {gather*} \frac {2 \sqrt {x} \sqrt {b+c x} \left (\sqrt {b+c x} \left (b (20 A c+6 B c x)+c^2 x (5 A+3 B x)+3 b^2 B\right )-15 A b^{3/2} c \tanh ^{-1}\left (\frac {\sqrt {b+c x}}{\sqrt {b}}\right )\right )}{15 c \sqrt {x (b+c x)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.68, size = 93, normalized size = 0.89 \begin {gather*} \frac {2 \sqrt {b x+c x^2} \left (20 A b c+5 A c^2 x+3 b^2 B+6 b B c x+3 B c^2 x^2\right )}{15 c \sqrt {x}}-2 A b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {b x+c x^2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 195, normalized size = 1.86 \begin {gather*} \left [\frac {15 \, A b^{\frac {3}{2}} c x \log \left (-\frac {c x^{2} + 2 \, b x - 2 \, \sqrt {c x^{2} + b x} \sqrt {b} \sqrt {x}}{x^{2}}\right ) + 2 \, {\left (3 \, B c^{2} x^{2} + 3 \, B b^{2} + 20 \, A b c + {\left (6 \, B b c + 5 \, A c^{2}\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{15 \, c x}, \frac {2 \, {\left (15 \, A \sqrt {-b} b c x \arctan \left (\frac {\sqrt {-b} \sqrt {x}}{\sqrt {c x^{2} + b x}}\right ) + {\left (3 \, B c^{2} x^{2} + 3 \, B b^{2} + 20 \, A b c + {\left (6 \, B b c + 5 \, A c^{2}\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {x}\right )}}{15 \, c x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 123, normalized size = 1.17 \begin {gather*} \frac {2 \, A b^{2} \arctan \left (\frac {\sqrt {c x + b}}{\sqrt {-b}}\right )}{\sqrt {-b}} - \frac {2 \, {\left (15 \, A b^{2} c \arctan \left (\frac {\sqrt {b}}{\sqrt {-b}}\right ) + 3 \, B \sqrt {-b} b^{\frac {5}{2}} + 20 \, A \sqrt {-b} b^{\frac {3}{2}} c\right )}}{15 \, \sqrt {-b} c} + \frac {2 \, {\left (3 \, {\left (c x + b\right )}^{\frac {5}{2}} B c^{4} + 5 \, {\left (c x + b\right )}^{\frac {3}{2}} A c^{5} + 15 \, \sqrt {c x + b} A b c^{5}\right )}}{15 \, c^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 113, normalized size = 1.08 \begin {gather*} -\frac {2 \sqrt {\left (c x +b \right ) x}\, \left (-3 \sqrt {c x +b}\, B \,c^{2} x^{2}+15 A \,b^{\frac {3}{2}} c \arctanh \left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right )-5 \sqrt {c x +b}\, A \,c^{2} x -6 \sqrt {c x +b}\, B b c x -20 \sqrt {c x +b}\, A b c -3 \sqrt {c x +b}\, B \,b^{2}\right )}{15 \sqrt {c x +b}\, c \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} A b \int \frac {\sqrt {c x + b}}{x}\,{d x} + \frac {2 \, {\left (5 \, {\left (B b c + A c^{2}\right )} x^{2} + {\left (3 \, B c^{2} x^{2} + B b c x - 2 \, B b^{2}\right )} x + 5 \, {\left (B b^{2} + A b c\right )} x\right )} \sqrt {c x + b}}{15 \, c x} \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^{5/2}} \,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 {\left (x \left (b + c x\right )\right )^{\frac {3}{2}} \left (A + B x\right )}{x^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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