Optimal. Leaf size=49 \[ \frac {2 (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{b^{3/2} \sqrt {c}}-\frac {2 A}{b \sqrt {x}} \]
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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, 78, 63, 205} \begin {gather*} \frac {2 (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{b^{3/2} \sqrt {c}}-\frac {2 A}{b \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 78
Rule 205
Rule 781
Rubi steps
\begin {align*} \int \frac {A+B x}{\sqrt {x} \left (b x+c x^2\right )} \, dx &=\int \frac {A+B x}{x^{3/2} (b+c x)} \, dx\\ &=-\frac {2 A}{b \sqrt {x}}+\frac {\left (2 \left (\frac {b B}{2}-\frac {A c}{2}\right )\right ) \int \frac {1}{\sqrt {x} (b+c x)} \, dx}{b}\\ &=-\frac {2 A}{b \sqrt {x}}+\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 )}{b}\\ &=-\frac {2 A}{b \sqrt {x}}+\frac {2 (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{b^{3/2} \sqrt {c}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 49, normalized size = 1.00 \begin {gather*} \frac {2 (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{b^{3/2} \sqrt {c}}-\frac {2 A}{b \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 49, normalized size = 1.00 \begin {gather*} \frac {2 (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{b^{3/2} \sqrt {c}}-\frac {2 A}{b \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 112, normalized size = 2.29 \begin {gather*} \left [-\frac {2 \, A b c \sqrt {x} - {\left (B b - A c\right )} \sqrt {-b c} x \log \left (\frac {c x - b + 2 \, \sqrt {-b c} \sqrt {x}}{c x + b}\right )}{b^{2} c x}, -\frac {2 \, {\left (A b c \sqrt {x} + {\left (B b - A c\right )} \sqrt {b c} x \arctan \left (\frac {\sqrt {b c}}{c \sqrt {x}}\right )\right )}}{b^{2} c x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 39, normalized size = 0.80 \begin {gather*} \frac {2 \, {\left (B b - A c\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c} b} - \frac {2 \, A}{b \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 53, normalized size = 1.08 \begin {gather*} -\frac {2 A c \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c}\, b}+\frac {2 B \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c}}-\frac {2 A}{b \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.19, size = 39, normalized size = 0.80 \begin {gather*} \frac {2 \, {\left (B b - A c\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c} b} - \frac {2 \, A}{b \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 50, normalized size = 1.02 \begin {gather*} \frac {2\,B\,\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {x}}{\sqrt {b}}\right )}{\sqrt {b}\,\sqrt {c}}-\frac {2\,A\,\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {x}}{\sqrt {b}}\right )}{b^{3/2}}-\frac {2\,A}{b\,\sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.40, size = 216, normalized size = 4.41 \begin {gather*} \begin {cases} \tilde {\infty } \left (- \frac {2 A}{3 x^{\frac {3}{2}}} - \frac {2 B}{\sqrt {x}}\right ) & \text {for}\: b = 0 \wedge c = 0 \\\frac {- \frac {2 A}{3 x^{\frac {3}{2}}} - \frac {2 B}{\sqrt {x}}}{c} & \text {for}\: b = 0 \\\frac {- \frac {2 A}{\sqrt {x}} + 2 B \sqrt {x}}{b} & \text {for}\: c = 0 \\- \frac {2 A}{b \sqrt {x}} + \frac {i A \log {\left (- i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{b^{\frac {3}{2}} \sqrt {\frac {1}{c}}} - \frac {i A \log {\left (i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{b^{\frac {3}{2}} \sqrt {\frac {1}{c}}} - \frac {i B \log {\left (- i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{\sqrt {b} c \sqrt {\frac {1}{c}}} + \frac {i B \log {\left (i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{\sqrt {b} c \sqrt {\frac {1}{c}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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