Optimal. Leaf size=49 \[ \frac {2 B \sqrt {x}}{c}-\frac {2 (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{\sqrt {b} c^{3/2}} \]
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Rubi [A] time = 0.03, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {781, 80, 63, 205} \begin {gather*} \frac {2 B \sqrt {x}}{c}-\frac {2 (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{\sqrt {b} c^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 80
Rule 205
Rule 781
Rubi steps
\begin {align*} \int \frac {\sqrt {x} (A+B x)}{b x+c x^2} \, dx &=\int \frac {A+B x}{\sqrt {x} (b+c x)} \, dx\\ &=\frac {2 B \sqrt {x}}{c}+\frac {\left (2 \left (-\frac {b B}{2}+\frac {A c}{2}\right )\right ) \int \frac {1}{\sqrt {x} (b+c x)} \, dx}{c}\\ &=\frac {2 B \sqrt {x}}{c}+\frac {\left (4 \left (-\frac {b B}{2}+\frac {A c}{2}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{b+c x^2} \, dx,x,\sqrt {x}\right )}{c}\\ &=\frac {2 B \sqrt {x}}{c}-\frac {2 (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{\sqrt {b} c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 49, normalized size = 1.00 \begin {gather*} \frac {2 B \sqrt {x}}{c}-\frac {2 (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{\sqrt {b} c^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.05, size = 49, normalized size = 1.00 \begin {gather*} \frac {2 B \sqrt {x}}{c}-\frac {2 (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{\sqrt {b} c^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 102, normalized size = 2.08 \begin {gather*} \left [\frac {2 \, B b c \sqrt {x} + {\left (B b - A c\right )} \sqrt {-b c} \log \left (\frac {c x - b - 2 \, \sqrt {-b c} \sqrt {x}}{c x + b}\right )}{b c^{2}}, \frac {2 \, {\left (B b c \sqrt {x} + {\left (B b - A c\right )} \sqrt {b c} \arctan \left (\frac {\sqrt {b c}}{c \sqrt {x}}\right )\right )}}{b c^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 39, normalized size = 0.80 \begin {gather*} \frac {2 \, B \sqrt {x}}{c} - \frac {2 \, {\left (B b - A c\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 53, normalized size = 1.08 \begin {gather*} \frac {2 A \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c}}-\frac {2 B b \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c}\, c}+\frac {2 B \sqrt {x}}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.22, size = 39, normalized size = 0.80 \begin {gather*} \frac {2 \, B \sqrt {x}}{c} - \frac {2 \, {\left (B b - A c\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 37, normalized size = 0.76 \begin {gather*} \frac {2\,B\,\sqrt {x}}{c}+\frac {2\,\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {x}}{\sqrt {b}}\right )\,\left (A\,c-B\,b\right )}{\sqrt {b}\,c^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.19, size = 218, normalized size = 4.45 \begin {gather*} \begin {cases} \tilde {\infty } \left (- \frac {2 A}{\sqrt {x}} + 2 B \sqrt {x}\right ) & \text {for}\: b = 0 \wedge c = 0 \\\frac {- \frac {2 A}{\sqrt {x}} + 2 B \sqrt {x}}{c} & \text {for}\: b = 0 \\\frac {2 A \sqrt {x} + \frac {2 B x^{\frac {3}{2}}}{3}}{b} & \text {for}\: c = 0 \\- \frac {i A \log {\left (- i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{\sqrt {b} c \sqrt {\frac {1}{c}}} + \frac {i A \log {\left (i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{\sqrt {b} c \sqrt {\frac {1}{c}}} + \frac {i B \sqrt {b} \log {\left (- i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{c^{2} \sqrt {\frac {1}{c}}} - \frac {i B \sqrt {b} \log {\left (i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{c^{2} \sqrt {\frac {1}{c}}} + \frac {2 B \sqrt {x}}{c} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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