Optimal. Leaf size=120 \[ -\frac{1}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{e (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}+\frac{e (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2} \]
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Rubi [A] time = 0.0839477, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {770, 21, 44} \[ -\frac{1}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{e (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}+\frac{e (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2} \]
Antiderivative was successfully verified.
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Rule 770
Rule 21
Rule 44
Rubi steps
\begin{align*} \int \frac{a+b x}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx &=\frac{\left (b^2 \left (a b+b^2 x\right )\right ) \int \frac{a+b x}{\left (a b+b^2 x\right )^3 (d+e x)} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{\left (a b+b^2 x\right ) \int \frac{1}{(a+b x)^2 (d+e x)} \, dx}{b \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{\left (a b+b^2 x\right ) \int \left (\frac{b}{(b d-a e) (a+b x)^2}-\frac{b e}{(b d-a e)^2 (a+b x)}+\frac{e^2}{(b d-a e)^2 (d+e x)}\right ) \, dx}{b \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{1}{(b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{e (a+b x) \log (a+b x)}{(b d-a e)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e (a+b x) \log (d+e x)}{(b d-a e)^2 \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0383337, size = 57, normalized size = 0.48 \[ \frac{e (a+b x) \log (d+e x)-e (a+b x) \log (a+b x)+a e-b d}{\sqrt{(a+b x)^2} (b d-a e)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 76, normalized size = 0.6 \begin{align*}{\frac{ \left ( \ln \left ( ex+d \right ) xbe-\ln \left ( bx+a \right ) xbe+\ln \left ( ex+d \right ) ae-\ln \left ( bx+a \right ) ae+ae-bd \right ) \left ( bx+a \right ) ^{2}}{ \left ( ae-bd \right ) ^{2}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \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 = 1.69395, size = 200, normalized size = 1.67 \begin{align*} -\frac{b d - a e +{\left (b e x + a e\right )} \log \left (b x + a\right ) -{\left (b e x + a e\right )} \log \left (e x + d\right )}{a b^{2} d^{2} - 2 \, a^{2} b d e + a^{3} e^{2} +{\left (b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}\right )} x} \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}{\left (d + e x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.18918, size = 274, normalized size = 2.28 \begin{align*} -\frac{a e \log \left ({\left | b + \frac{a}{x} \right |}\right )}{a b^{2} d^{2} \mathrm{sgn}\left (\frac{b}{x} + \frac{a}{x^{2}}\right ) - 2 \, a^{2} b d e \mathrm{sgn}\left (\frac{b}{x} + \frac{a}{x^{2}}\right ) + a^{3} e^{2} \mathrm{sgn}\left (\frac{b}{x} + \frac{a}{x^{2}}\right )} + \frac{d e \log \left ({\left | \frac{d}{x} + e \right |}\right )}{b^{2} d^{3} \mathrm{sgn}\left (\frac{b}{x} + \frac{a}{x^{2}}\right ) - 2 \, a b d^{2} e \mathrm{sgn}\left (\frac{b}{x} + \frac{a}{x^{2}}\right ) + a^{2} d e^{2} \mathrm{sgn}\left (\frac{b}{x} + \frac{a}{x^{2}}\right )} + \frac{b^{2} d - a b e}{{\left (b d - a e\right )}^{2} a{\left (b + \frac{a}{x}\right )} \mathrm{sgn}\left (\frac{b}{x} + \frac{a}{x^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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