Optimal. Leaf size=100 \[ \frac {(3 A c+b B) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{4 b^{5/2} c^{3/2}}+\frac {\sqrt {x} (3 A c+b B)}{4 b^2 c (b+c x)}-\frac {\sqrt {x} (b B-A c)}{2 b c (b+c x)^2} \]
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Rubi [A] time = 0.05, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {781, 78, 51, 63, 205} \[ \frac {(3 A c+b B) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{4 b^{5/2} c^{3/2}}+\frac {\sqrt {x} (3 A c+b B)}{4 b^2 c (b+c x)}-\frac {\sqrt {x} (b B-A c)}{2 b c (b+c x)^2} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 78
Rule 205
Rule 781
Rubi steps
\begin {align*} \int \frac {x^{5/2} (A+B x)}{\left (b x+c x^2\right )^3} \, dx &=\int \frac {A+B x}{\sqrt {x} (b+c x)^3} \, dx\\ &=-\frac {(b B-A c) \sqrt {x}}{2 b c (b+c x)^2}+\frac {(b B+3 A c) \int \frac {1}{\sqrt {x} (b+c x)^2} \, dx}{4 b c}\\ &=-\frac {(b B-A c) \sqrt {x}}{2 b c (b+c x)^2}+\frac {(b B+3 A c) \sqrt {x}}{4 b^2 c (b+c x)}+\frac {(b B+3 A c) \int \frac {1}{\sqrt {x} (b+c x)} \, dx}{8 b^2 c}\\ &=-\frac {(b B-A c) \sqrt {x}}{2 b c (b+c x)^2}+\frac {(b B+3 A c) \sqrt {x}}{4 b^2 c (b+c x)}+\frac {(b B+3 A c) \operatorname {Subst}\left (\int \frac {1}{b+c x^2} \, dx,x,\sqrt {x}\right )}{4 b^2 c}\\ &=-\frac {(b B-A c) \sqrt {x}}{2 b c (b+c x)^2}+\frac {(b B+3 A c) \sqrt {x}}{4 b^2 c (b+c x)}+\frac {(b B+3 A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{4 b^{5/2} c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 91, normalized size = 0.91 \[ \frac {\sqrt {x} \left (\frac {b^2 (A c-b B)}{(b+c x)^2}-\frac {1}{2} (-3 A c-b B) \left (\frac {b}{b+c x}+\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{\sqrt {c} \sqrt {x}}\right )\right )}{2 b^3 c} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 291, normalized size = 2.91 \[ \left [-\frac {{\left (B b^{3} + 3 \, A b^{2} c + {\left (B b c^{2} + 3 \, A c^{3}\right )} x^{2} + 2 \, {\left (B b^{2} c + 3 \, A b c^{2}\right )} x\right )} \sqrt {-b c} \log \left (\frac {c x - b - 2 \, \sqrt {-b c} \sqrt {x}}{c x + b}\right ) + 2 \, {\left (B b^{3} c - 5 \, A b^{2} c^{2} - {\left (B b^{2} c^{2} + 3 \, A b c^{3}\right )} x\right )} \sqrt {x}}{8 \, {\left (b^{3} c^{4} x^{2} + 2 \, b^{4} c^{3} x + b^{5} c^{2}\right )}}, -\frac {{\left (B b^{3} + 3 \, A b^{2} c + {\left (B b c^{2} + 3 \, A c^{3}\right )} x^{2} + 2 \, {\left (B b^{2} c + 3 \, A b c^{2}\right )} x\right )} \sqrt {b c} \arctan \left (\frac {\sqrt {b c}}{c \sqrt {x}}\right ) + {\left (B b^{3} c - 5 \, A b^{2} c^{2} - {\left (B b^{2} c^{2} + 3 \, A b c^{3}\right )} x\right )} \sqrt {x}}{4 \, {\left (b^{3} c^{4} x^{2} + 2 \, b^{4} c^{3} x + b^{5} c^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 82, normalized size = 0.82 \[ \frac {{\left (B b + 3 \, A c\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{4 \, \sqrt {b c} b^{2} c} + \frac {B b c x^{\frac {3}{2}} + 3 \, A c^{2} x^{\frac {3}{2}} - B b^{2} \sqrt {x} + 5 \, A b c \sqrt {x}}{4 \, {\left (c x + b\right )}^{2} b^{2} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 95, normalized size = 0.95 \[ \frac {3 A \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{4 \sqrt {b c}\, b^{2}}+\frac {B \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{4 \sqrt {b c}\, b c}+\frac {\frac {\left (3 A c +b B \right ) x^{\frac {3}{2}}}{4 b^{2}}+\frac {\left (5 A c -b B \right ) \sqrt {x}}{4 b c}}{\left (c x +b \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.23, size = 94, normalized size = 0.94 \[ \frac {{\left (B b c + 3 \, A c^{2}\right )} x^{\frac {3}{2}} - {\left (B b^{2} - 5 \, A b c\right )} \sqrt {x}}{4 \, {\left (b^{2} c^{3} x^{2} + 2 \, b^{3} c^{2} x + b^{4} c\right )}} + \frac {{\left (B b + 3 \, A c\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{4 \, \sqrt {b c} b^{2} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.09, size = 84, normalized size = 0.84 \[ \frac {\frac {x^{3/2}\,\left (3\,A\,c+B\,b\right )}{4\,b^2}+\frac {\sqrt {x}\,\left (5\,A\,c-B\,b\right )}{4\,b\,c}}{b^2+2\,b\,c\,x+c^2\,x^2}+\frac {\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {x}}{\sqrt {b}}\right )\,\left (3\,A\,c+B\,b\right )}{4\,b^{5/2}\,c^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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