Optimal. Leaf size=128 \[ \frac{\sqrt{b x+c x^2} (B d-A e)}{d (d+e x) (c d-b e)}-\frac{(A b e-2 A c d+b B d) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{2 d^{3/2} (c d-b e)^{3/2}} \]
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Rubi [A] time = 0.0848131, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {806, 724, 206} \[ \frac{\sqrt{b x+c x^2} (B d-A e)}{d (d+e x) (c d-b e)}-\frac{(A b e-2 A c d+b B d) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{2 d^{3/2} (c d-b e)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 806
Rule 724
Rule 206
Rubi steps
\begin{align*} \int \frac{A+B x}{(d+e x)^2 \sqrt{b x+c x^2}} \, dx &=\frac{(B d-A e) \sqrt{b x+c x^2}}{d (c d-b e) (d+e x)}-\frac{(b B d-2 A c d+A b e) \int \frac{1}{(d+e x) \sqrt{b x+c x^2}} \, dx}{2 d (c d-b e)}\\ &=\frac{(B d-A e) \sqrt{b x+c x^2}}{d (c d-b e) (d+e x)}+\frac{(b B d-2 A c d+A b 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 )}{d (c d-b e)}\\ &=\frac{(B d-A e) \sqrt{b x+c x^2}}{d (c d-b e) (d+e x)}-\frac{(b B d-2 A c d+A b 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 )}{2 d^{3/2} (c d-b e)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.195715, size = 133, normalized size = 1.04 \[ \frac{\sqrt{x} \left (\frac{\sqrt{d} \sqrt{x} (b+c x) (B d-A e)}{(d+e x) (c d-b e)}+\frac{\sqrt{b+c x} (A b e-2 A c d+b B d) \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{(b e-c d)^{3/2}}\right )}{d^{3/2} \sqrt{x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.012, size = 849, normalized size = 6.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58358, size = 842, normalized size = 6.58 \begin{align*} \left [-\frac{{\left (A b d e +{\left (B b - 2 \, A c\right )} d^{2} +{\left (A b e^{2} +{\left (B b - 2 \, A c\right )} d e\right )} x\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 ) - 2 \,{\left (B c d^{3} + A b d e^{2} -{\left (B b + A c\right )} d^{2} e\right )} \sqrt{c x^{2} + b x}}{2 \,{\left (c^{2} d^{5} - 2 \, b c d^{4} e + b^{2} d^{3} e^{2} +{\left (c^{2} d^{4} e - 2 \, b c d^{3} e^{2} + b^{2} d^{2} e^{3}\right )} x\right )}}, -\frac{{\left (A b d e +{\left (B b - 2 \, A c\right )} d^{2} +{\left (A b e^{2} +{\left (B b - 2 \, A c\right )} d e\right )} x\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^{3} + A b d e^{2} -{\left (B b + A c\right )} d^{2} e\right )} \sqrt{c x^{2} + b x}}{c^{2} d^{5} - 2 \, b c d^{4} e + b^{2} d^{3} e^{2} +{\left (c^{2} d^{4} e - 2 \, b c d^{3} e^{2} + b^{2} d^{2} e^{3}\right )} x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{A + B x}{\sqrt{x \left (b + c x\right )} \left (d + e x\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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