Optimal. Leaf size=113 \[ \frac {2 B \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{\sqrt {c} e}-\frac {(B d-A e) \tanh ^{-1}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{\sqrt {d} e \sqrt {c d-b e}} \]
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Rubi [A] time = 0.08, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {843, 620, 206, 724} \begin {gather*} \frac {2 B \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{\sqrt {c} e}-\frac {(B d-A e) \tanh ^{-1}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{\sqrt {d} e \sqrt {c d-b e}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 620
Rule 724
Rule 843
Rubi steps
\begin {align*} \int \frac {A+B x}{(d+e x) \sqrt {b x+c x^2}} \, dx &=\frac {B \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{e}+\frac {(-B d+A e) \int \frac {1}{(d+e x) \sqrt {b x+c x^2}} \, dx}{e}\\ &=\frac {(2 B) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{e}-\frac {(2 (-B d+A e)) \operatorname {Subst}\left (\int \frac {1}{4 c d^2-4 b d e-x^2} \, dx,x,\frac {-b d-(2 c d-b e) x}{\sqrt {b x+c x^2}}\right )}{e}\\ &=\frac {2 B \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{\sqrt {c} e}-\frac {(B d-A e) \tanh ^{-1}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{\sqrt {d} e \sqrt {c d-b e}}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 131, normalized size = 1.16 \begin {gather*} \frac {2 \sqrt {x} \left (\frac {\sqrt {b+c x} (A e-B d) \tanh ^{-1}\left (\frac {\sqrt {x} \sqrt {c d-b e}}{\sqrt {d} \sqrt {b+c x}}\right )}{\sqrt {d} \sqrt {c d-b e}}+\frac {\sqrt {b} B \sqrt {\frac {c x}{b}+1} \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{\sqrt {c}}\right )}{e \sqrt {x (b+c x)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.48, size = 124, normalized size = 1.10 \begin {gather*} -\frac {2 (B d-A e) \tanh ^{-1}\left (\frac {-e \sqrt {b x+c x^2}+\sqrt {c} d+\sqrt {c} e x}{\sqrt {d} \sqrt {c d-b e}}\right )}{\sqrt {d} e \sqrt {c d-b e}}-\frac {B \log \left (-2 \sqrt {c} e \sqrt {b x+c x^2}+b e+2 c e x\right )}{\sqrt {c} e} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 534, normalized size = 4.73 \begin {gather*} \left [\frac {{\left (B c d^{2} - B b d e\right )} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - {\left (B c d - A c e\right )} \sqrt {c d^{2} - b d e} \log \left (\frac {b d + {\left (2 \, c d - b e\right )} x + 2 \, \sqrt {c d^{2} - b d e} \sqrt {c x^{2} + b x}}{e x + d}\right )}{c^{2} d^{2} e - b c d e^{2}}, -\frac {2 \, {\left (B c d - A c e\right )} \sqrt {-c d^{2} + b d e} \arctan \left (-\frac {\sqrt {-c d^{2} + b d e} \sqrt {c x^{2} + b x}}{{\left (c d - b e\right )} x}\right ) - {\left (B c d^{2} - B b d e\right )} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{c^{2} d^{2} e - b c d e^{2}}, -\frac {2 \, {\left (B c d^{2} - B b d e\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + {\left (B c d - A c e\right )} \sqrt {c d^{2} - b d e} \log \left (\frac {b d + {\left (2 \, c d - b e\right )} x + 2 \, \sqrt {c d^{2} - b d e} \sqrt {c x^{2} + b x}}{e x + d}\right )}{c^{2} d^{2} e - b c d e^{2}}, -\frac {2 \, {\left ({\left (B c d - A c e\right )} \sqrt {-c d^{2} + b d e} \arctan \left (-\frac {\sqrt {-c d^{2} + b d e} \sqrt {c x^{2} + b x}}{{\left (c d - b e\right )} x}\right ) + {\left (B c d^{2} - B b d e\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right )\right )}}{c^{2} d^{2} e - b c d e^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 298, normalized size = 2.64 \begin {gather*} -\frac {A \ln \left (\frac {-\frac {2 \left (b e -c d \right ) d}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {\left (b e -c d \right ) d}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {\left (b e -c d \right ) d}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}}}{x +\frac {d}{e}}\right )}{\sqrt {-\frac {\left (b e -c d \right ) d}{e^{2}}}\, e}+\frac {B d \ln \left (\frac {-\frac {2 \left (b e -c d \right ) d}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {\left (b e -c d \right ) d}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {\left (b e -c d \right ) d}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}}}{x +\frac {d}{e}}\right )}{\sqrt {-\frac {\left (b e -c d \right ) d}{e^{2}}}\, e^{2}}+\frac {B \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{\sqrt {c}\, e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {A+B\,x}{\sqrt {c\,x^2+b\,x}\,\left (d+e\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A + B x}{\sqrt {x \left (b + c x\right )} \left (d + e x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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