Optimal. Leaf size=69 \[ \frac {2 \sqrt {b} (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{c^{5/2}}-\frac {2 \sqrt {x} (b B-A c)}{c^2}+\frac {2 B x^{3/2}}{3 c} \]
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Rubi [A] time = 0.04, antiderivative size = 69, 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, 80, 50, 63, 205} \begin {gather*} -\frac {2 \sqrt {x} (b B-A c)}{c^2}+\frac {2 \sqrt {b} (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{c^{5/2}}+\frac {2 B x^{3/2}}{3 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^{3/2} (A+B x)}{b x+c x^2} \, dx &=\int \frac {\sqrt {x} (A+B x)}{b+c x} \, dx\\ &=\frac {2 B x^{3/2}}{3 c}+\frac {\left (2 \left (-\frac {3 b B}{2}+\frac {3 A c}{2}\right )\right ) \int \frac {\sqrt {x}}{b+c x} \, dx}{3 c}\\ &=-\frac {2 (b B-A c) \sqrt {x}}{c^2}+\frac {2 B x^{3/2}}{3 c}+\frac {(b (b B-A c)) \int \frac {1}{\sqrt {x} (b+c x)} \, dx}{c^2}\\ &=-\frac {2 (b B-A c) \sqrt {x}}{c^2}+\frac {2 B x^{3/2}}{3 c}+\frac {(2 b (b B-A c)) \operatorname {Subst}\left (\int \frac {1}{b+c x^2} \, dx,x,\sqrt {x}\right )}{c^2}\\ &=-\frac {2 (b B-A c) \sqrt {x}}{c^2}+\frac {2 B x^{3/2}}{3 c}+\frac {2 \sqrt {b} (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{c^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 63, normalized size = 0.91 \begin {gather*} \frac {2 \sqrt {b} (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{c^{5/2}}+\frac {2 \sqrt {x} (3 A c-3 b B+B c x)}{3 c^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.07, size = 76, normalized size = 1.10 \begin {gather*} \frac {2 \left (b^{3/2} B-A \sqrt {b} c\right ) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{c^{5/2}}+\frac {2 \left (3 A c \sqrt {x}-3 b B \sqrt {x}+B c x^{3/2}\right )}{3 c^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 129, normalized size = 1.87 \begin {gather*} \left [-\frac {3 \, {\left (B b - A 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 (B c x - 3 \, B b + 3 \, A c\right )} \sqrt {x}}{3 \, c^{2}}, \frac {2 \, {\left (3 \, {\left (B b - A c\right )} \sqrt {\frac {b}{c}} \arctan \left (\frac {c \sqrt {x} \sqrt {\frac {b}{c}}}{b}\right ) + {\left (B c x - 3 \, B b + 3 \, A c\right )} \sqrt {x}\right )}}{3 \, c^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 64, normalized size = 0.93 \begin {gather*} \frac {2 \, {\left (B b^{2} - A b c\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c} c^{2}} + \frac {2 \, {\left (B c^{2} x^{\frac {3}{2}} - 3 \, B b c \sqrt {x} + 3 \, A c^{2} \sqrt {x}\right )}}{3 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 78, normalized size = 1.13 \begin {gather*} -\frac {2 A b \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c}\, c}+\frac {2 B \,b^{2} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c}\, c^{2}}+\frac {2 B \,x^{\frac {3}{2}}}{3 c}+\frac {2 A \sqrt {x}}{c}-\frac {2 B b \sqrt {x}}{c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.51, size = 58, normalized size = 0.84 \begin {gather*} \frac {2 \, {\left (B b^{2} - A b c\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c} c^{2}} + \frac {2 \, {\left (B c x^{\frac {3}{2}} - 3 \, {\left (B b - A c\right )} \sqrt {x}\right )}}{3 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 76, normalized size = 1.10 \begin {gather*} \sqrt {x}\,\left (\frac {2\,A}{c}-\frac {2\,B\,b}{c^2}\right )+\frac {2\,B\,x^{3/2}}{3\,c}+\frac {2\,\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {c}\,\sqrt {x}\,\left (A\,c-B\,b\right )}{B\,b^2-A\,b\,c}\right )\,\left (A\,c-B\,b\right )}{c^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.70, size = 212, normalized size = 3.07 \begin {gather*} \begin {cases} \frac {i A \sqrt {b} \log {\left (- i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{c^{2} \sqrt {\frac {1}{c}}} - \frac {i A \sqrt {b} \log {\left (i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{c^{2} \sqrt {\frac {1}{c}}} + \frac {2 A \sqrt {x}}{c} - \frac {i B b^{\frac {3}{2}} \log {\left (- i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{c^{3} \sqrt {\frac {1}{c}}} + \frac {i B b^{\frac {3}{2}} \log {\left (i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{c^{3} \sqrt {\frac {1}{c}}} - \frac {2 B b \sqrt {x}}{c^{2}} + \frac {2 B x^{\frac {3}{2}}}{3 c} & \text {for}\: c \neq 0 \\\frac {\frac {2 A x^{\frac {3}{2}}}{3} + \frac {2 B x^{\frac {5}{2}}}{5}}{b} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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