Optimal. Leaf size=109 \[ \frac {\sqrt {b} (5 b B-3 A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{c^{7/2}}-\frac {\sqrt {x} (5 b B-3 A c)}{c^3}+\frac {x^{3/2} (5 b B-3 A c)}{3 b c^2}-\frac {x^{5/2} (b B-A c)}{b c (b+c x)} \]
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Rubi [A] time = 0.06, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {781, 78, 50, 63, 205} \begin {gather*} \frac {x^{3/2} (5 b B-3 A c)}{3 b c^2}-\frac {\sqrt {x} (5 b B-3 A c)}{c^3}+\frac {\sqrt {b} (5 b B-3 A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{c^{7/2}}-\frac {x^{5/2} (b B-A c)}{b c (b+c x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 78
Rule 205
Rule 781
Rubi steps
\begin {align*} \int \frac {x^{7/2} (A+B x)}{\left (b x+c x^2\right )^2} \, dx &=\int \frac {x^{3/2} (A+B x)}{(b+c x)^2} \, dx\\ &=-\frac {(b B-A c) x^{5/2}}{b c (b+c x)}-\frac {\left (-\frac {5 b B}{2}+\frac {3 A c}{2}\right ) \int \frac {x^{3/2}}{b+c x} \, dx}{b c}\\ &=\frac {(5 b B-3 A c) x^{3/2}}{3 b c^2}-\frac {(b B-A c) x^{5/2}}{b c (b+c x)}-\frac {(5 b B-3 A c) \int \frac {\sqrt {x}}{b+c x} \, dx}{2 c^2}\\ &=-\frac {(5 b B-3 A c) \sqrt {x}}{c^3}+\frac {(5 b B-3 A c) x^{3/2}}{3 b c^2}-\frac {(b B-A c) x^{5/2}}{b c (b+c x)}+\frac {(b (5 b B-3 A c)) \int \frac {1}{\sqrt {x} (b+c x)} \, dx}{2 c^3}\\ &=-\frac {(5 b B-3 A c) \sqrt {x}}{c^3}+\frac {(5 b B-3 A c) x^{3/2}}{3 b c^2}-\frac {(b B-A c) x^{5/2}}{b c (b+c x)}+\frac {(b (5 b B-3 A c)) \operatorname {Subst}\left (\int \frac {1}{b+c x^2} \, dx,x,\sqrt {x}\right )}{c^3}\\ &=-\frac {(5 b B-3 A c) \sqrt {x}}{c^3}+\frac {(5 b B-3 A c) x^{3/2}}{3 b c^2}-\frac {(b B-A c) x^{5/2}}{b c (b+c x)}+\frac {\sqrt {b} (5 b B-3 A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{c^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 88, normalized size = 0.81 \begin {gather*} \frac {\sqrt {x} \left (b c (9 A-10 B x)+2 c^2 x (3 A+B x)-15 b^2 B\right )}{3 c^3 (b+c x)}+\frac {\sqrt {b} (5 b B-3 A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{c^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.13, size = 95, normalized size = 0.87 \begin {gather*} \frac {\left (5 b^{3/2} B-3 A \sqrt {b} c\right ) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{c^{7/2}}+\frac {\sqrt {x} \left (9 A b c+6 A c^2 x-15 b^2 B-10 b B c x+2 B c^2 x^2\right )}{3 c^3 (b+c x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 231, normalized size = 2.12 \begin {gather*} \left [-\frac {3 \, {\left (5 \, B b^{2} - 3 \, A b c + {\left (5 \, B b c - 3 \, A c^{2}\right )} x\right )} \sqrt {-\frac {b}{c}} \log \left (\frac {c x - 2 \, c \sqrt {x} \sqrt {-\frac {b}{c}} - b}{c x + b}\right ) - 2 \, {\left (2 \, B c^{2} x^{2} - 15 \, B b^{2} + 9 \, A b c - 2 \, {\left (5 \, B b c - 3 \, A c^{2}\right )} x\right )} \sqrt {x}}{6 \, {\left (c^{4} x + b c^{3}\right )}}, \frac {3 \, {\left (5 \, B b^{2} - 3 \, A b c + {\left (5 \, B b c - 3 \, A c^{2}\right )} x\right )} \sqrt {\frac {b}{c}} \arctan \left (\frac {c \sqrt {x} \sqrt {\frac {b}{c}}}{b}\right ) + {\left (2 \, B c^{2} x^{2} - 15 \, B b^{2} + 9 \, A b c - 2 \, {\left (5 \, B b c - 3 \, A c^{2}\right )} x\right )} \sqrt {x}}{3 \, {\left (c^{4} x + b c^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 95, normalized size = 0.87 \begin {gather*} \frac {{\left (5 \, B b^{2} - 3 \, A b c\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c} c^{3}} - \frac {B b^{2} \sqrt {x} - A b c \sqrt {x}}{{\left (c x + b\right )} c^{3}} + \frac {2 \, {\left (B c^{4} x^{\frac {3}{2}} - 6 \, B b c^{3} \sqrt {x} + 3 \, A c^{4} \sqrt {x}\right )}}{3 \, c^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 113, normalized size = 1.04 \begin {gather*} -\frac {3 A b \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c}\, c^{2}}+\frac {5 B \,b^{2} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c}\, c^{3}}+\frac {A b \sqrt {x}}{\left (c x +b \right ) c^{2}}-\frac {B \,b^{2} \sqrt {x}}{\left (c x +b \right ) c^{3}}+\frac {2 B \,x^{\frac {3}{2}}}{3 c^{2}}+\frac {2 A \sqrt {x}}{c^{2}}-\frac {4 B b \sqrt {x}}{c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.20, size = 88, normalized size = 0.81 \begin {gather*} -\frac {{\left (B b^{2} - A b c\right )} \sqrt {x}}{c^{4} x + b c^{3}} + \frac {{\left (5 \, B b^{2} - 3 \, A b c\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c} c^{3}} + \frac {2 \, {\left (B c x^{\frac {3}{2}} - 3 \, {\left (2 \, B b - A c\right )} \sqrt {x}\right )}}{3 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.09, size = 107, normalized size = 0.98 \begin {gather*} \sqrt {x}\,\left (\frac {2\,A}{c^2}-\frac {4\,B\,b}{c^3}\right )-\frac {\sqrt {x}\,\left (B\,b^2-A\,b\,c\right )}{x\,c^4+b\,c^3}+\frac {2\,B\,x^{3/2}}{3\,c^2}+\frac {\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {c}\,\sqrt {x}\,\left (3\,A\,c-5\,B\,b\right )}{5\,B\,b^2-3\,A\,b\,c}\right )\,\left (3\,A\,c-5\,B\,b\right )}{c^{7/2}} \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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