Optimal. Leaf size=90 \[ -\frac {2 b^{3/2} (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{c^{7/2}}+\frac {2 b \sqrt {x} (b B-A c)}{c^3}-\frac {2 x^{3/2} (b B-A c)}{3 c^2}+\frac {2 B x^{5/2}}{5 c} \]
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Rubi [A] time = 0.05, antiderivative size = 90, 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, 80, 50, 63, 205} \begin {gather*} -\frac {2 b^{3/2} (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{c^{7/2}}-\frac {2 x^{3/2} (b B-A c)}{3 c^2}+\frac {2 b \sqrt {x} (b B-A c)}{c^3}+\frac {2 B x^{5/2}}{5 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 205
Rule 781
Rubi steps
\begin {align*} \int \frac {x^{5/2} (A+B x)}{b x+c x^2} \, dx &=\int \frac {x^{3/2} (A+B x)}{b+c x} \, dx\\ &=\frac {2 B x^{5/2}}{5 c}+\frac {\left (2 \left (-\frac {5 b B}{2}+\frac {5 A c}{2}\right )\right ) \int \frac {x^{3/2}}{b+c x} \, dx}{5 c}\\ &=-\frac {2 (b B-A c) x^{3/2}}{3 c^2}+\frac {2 B x^{5/2}}{5 c}+\frac {(b (b B-A c)) \int \frac {\sqrt {x}}{b+c x} \, dx}{c^2}\\ &=\frac {2 b (b B-A c) \sqrt {x}}{c^3}-\frac {2 (b B-A c) x^{3/2}}{3 c^2}+\frac {2 B x^{5/2}}{5 c}-\frac {\left (b^2 (b B-A c)\right ) \int \frac {1}{\sqrt {x} (b+c x)} \, dx}{c^3}\\ &=\frac {2 b (b B-A c) \sqrt {x}}{c^3}-\frac {2 (b B-A c) x^{3/2}}{3 c^2}+\frac {2 B x^{5/2}}{5 c}-\frac {\left (2 b^2 (b B-A c)\right ) \operatorname {Subst}\left (\int \frac {1}{b+c x^2} \, dx,x,\sqrt {x}\right )}{c^3}\\ &=\frac {2 b (b B-A c) \sqrt {x}}{c^3}-\frac {2 (b B-A c) x^{3/2}}{3 c^2}+\frac {2 B x^{5/2}}{5 c}-\frac {2 b^{3/2} (b B-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 = 81, normalized size = 0.90 \begin {gather*} \frac {2 \sqrt {x} \left (-5 b c (3 A+B x)+c^2 x (5 A+3 B x)+15 b^2 B\right )}{15 c^3}-\frac {2 b^{3/2} (b B-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 = 0.10, size = 88, normalized size = 0.98 \begin {gather*} \frac {2 \sqrt {x} \left (-15 A b c+5 A c^2 x+15 b^2 B-5 b B c x+3 B c^2 x^2\right )}{15 c^3}-\frac {2 \left (b^{5/2} B-A b^{3/2} c\right ) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{c^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 180, normalized size = 2.00 \begin {gather*} \left [-\frac {15 \, {\left (B b^{2} - A b c\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 (3 \, B c^{2} x^{2} + 15 \, B b^{2} - 15 \, A b c - 5 \, {\left (B b c - A c^{2}\right )} x\right )} \sqrt {x}}{15 \, c^{3}}, -\frac {2 \, {\left (15 \, {\left (B b^{2} - A b c\right )} \sqrt {\frac {b}{c}} \arctan \left (\frac {c \sqrt {x} \sqrt {\frac {b}{c}}}{b}\right ) - {\left (3 \, B c^{2} x^{2} + 15 \, B b^{2} - 15 \, A b c - 5 \, {\left (B b c - A c^{2}\right )} x\right )} \sqrt {x}\right )}}{15 \, c^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 91, normalized size = 1.01 \begin {gather*} -\frac {2 \, {\left (B b^{3} - A b^{2} c\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c} c^{3}} + \frac {2 \, {\left (3 \, B c^{4} x^{\frac {5}{2}} - 5 \, B b c^{3} x^{\frac {3}{2}} + 5 \, A c^{4} x^{\frac {3}{2}} + 15 \, B b^{2} c^{2} \sqrt {x} - 15 \, A b c^{3} \sqrt {x}\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 = 102, normalized size = 1.13 \begin {gather*} \frac {2 B \,x^{\frac {5}{2}}}{5 c}+\frac {2 A \,b^{2} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c}\, c^{2}}-\frac {2 B \,b^{3} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c}\, c^{3}}+\frac {2 A \,x^{\frac {3}{2}}}{3 c}-\frac {2 B b \,x^{\frac {3}{2}}}{3 c^{2}}-\frac {2 A b \sqrt {x}}{c^{2}}+\frac {2 B \,b^{2} \sqrt {x}}{c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.23, size = 82, normalized size = 0.91 \begin {gather*} -\frac {2 \, {\left (B b^{3} - A b^{2} c\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c} c^{3}} + \frac {2 \, {\left (3 \, B c^{2} x^{\frac {5}{2}} - 5 \, {\left (B b c - A c^{2}\right )} x^{\frac {3}{2}} + 15 \, {\left (B b^{2} - A b c\right )} \sqrt {x}\right )}}{15 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.05, size = 101, normalized size = 1.12 \begin {gather*} x^{3/2}\,\left (\frac {2\,A}{3\,c}-\frac {2\,B\,b}{3\,c^2}\right )+\frac {2\,B\,x^{5/2}}{5\,c}-\frac {2\,b^{3/2}\,\mathrm {atan}\left (\frac {b^{3/2}\,\sqrt {c}\,\sqrt {x}\,\left (A\,c-B\,b\right )}{B\,b^3-A\,b^2\,c}\right )\,\left (A\,c-B\,b\right )}{c^{7/2}}-\frac {b\,\sqrt {x}\,\left (\frac {2\,A}{c}-\frac {2\,B\,b}{c^2}\right )}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 16.35, size = 245, normalized size = 2.72 \begin {gather*} \begin {cases} - \frac {i A b^{\frac {3}{2}} \log {\left (- i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{c^{3} \sqrt {\frac {1}{c}}} + \frac {i A b^{\frac {3}{2}} \log {\left (i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{c^{3} \sqrt {\frac {1}{c}}} - \frac {2 A b \sqrt {x}}{c^{2}} + \frac {2 A x^{\frac {3}{2}}}{3 c} + \frac {i B b^{\frac {5}{2}} \log {\left (- i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{c^{4} \sqrt {\frac {1}{c}}} - \frac {i B b^{\frac {5}{2}} \log {\left (i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{c^{4} \sqrt {\frac {1}{c}}} + \frac {2 B b^{2} \sqrt {x}}{c^{3}} - \frac {2 B b x^{\frac {3}{2}}}{3 c^{2}} + \frac {2 B x^{\frac {5}{2}}}{5 c} & \text {for}\: c \neq 0 \\\frac {\frac {2 A x^{\frac {5}{2}}}{5} + \frac {2 B x^{\frac {7}{2}}}{7}}{b} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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