Optimal. Leaf size=148 \[ \frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2}}{3 e^3 (a+b x)}-\frac{4 b \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)}{e^3 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^3 (a+b x) \sqrt{d+e x}} \]
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Rubi [A] time = 0.0659191, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086, Rules used = {770, 21, 43} \[ \frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2}}{3 e^3 (a+b x)}-\frac{4 b \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)}{e^3 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^3 (a+b x) \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Rule 770
Rule 21
Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{(d+e x)^{3/2}} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{(a+b x) \left (a b+b^2 x\right )}{(d+e x)^{3/2}} \, dx}{a b+b^2 x}\\ &=\frac{\left (b \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int \frac{(a+b x)^2}{(d+e x)^{3/2}} \, dx}{a b+b^2 x}\\ &=\frac{\left (b \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac{(-b d+a e)^2}{e^2 (d+e x)^{3/2}}-\frac{2 b (b d-a e)}{e^2 \sqrt{d+e x}}+\frac{b^2 \sqrt{d+e x}}{e^2}\right ) \, dx}{a b+b^2 x}\\ &=-\frac{2 (b d-a e)^2 \sqrt{a^2+2 a b x+b^2 x^2}}{e^3 (a+b x) \sqrt{d+e x}}-\frac{4 b (b d-a e) \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}{e^3 (a+b x)}+\frac{2 b^2 (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^3 (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.0418883, size = 78, normalized size = 0.53 \[ -\frac{2 \sqrt{(a+b x)^2} \left (3 a^2 e^2-6 a b e (2 d+e x)+b^2 \left (8 d^2+4 d e x-e^2 x^2\right )\right )}{3 e^3 (a+b x) \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 79, normalized size = 0.5 \begin{align*} -{\frac{-2\,{x}^{2}{b}^{2}{e}^{2}-12\,xab{e}^{2}+8\,x{b}^{2}de+6\,{a}^{2}{e}^{2}-24\,abde+16\,{b}^{2}{d}^{2}}{3\, \left ( bx+a \right ){e}^{3}}\sqrt{ \left ( bx+a \right ) ^{2}}{\frac{1}{\sqrt{ex+d}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15796, size = 101, normalized size = 0.68 \begin{align*} \frac{2 \,{\left (b e x + 2 \, b d - a e\right )} a}{\sqrt{e x + d} e^{2}} + \frac{2 \,{\left (b e^{2} x^{2} - 8 \, b d^{2} + 6 \, a d e -{\left (4 \, b d e - 3 \, a e^{2}\right )} x\right )} b}{3 \, \sqrt{e x + d} e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.975368, size = 157, normalized size = 1.06 \begin{align*} \frac{2 \,{\left (b^{2} e^{2} x^{2} - 8 \, b^{2} d^{2} + 12 \, a b d e - 3 \, a^{2} e^{2} - 2 \,{\left (2 \, b^{2} d e - 3 \, a b e^{2}\right )} x\right )} \sqrt{e x + d}}{3 \,{\left (e^{4} x + d e^{3}\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{\left (a + b x\right ) \sqrt{\left (a + b x\right )^{2}}}{\left (d + e x\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10708, size = 161, normalized size = 1.09 \begin{align*} \frac{2}{3} \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} b^{2} e^{6} \mathrm{sgn}\left (b x + a\right ) - 6 \, \sqrt{x e + d} b^{2} d e^{6} \mathrm{sgn}\left (b x + a\right ) + 6 \, \sqrt{x e + d} a b e^{7} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-9\right )} - \frac{2 \,{\left (b^{2} d^{2} \mathrm{sgn}\left (b x + a\right ) - 2 \, a b d e \mathrm{sgn}\left (b x + a\right ) + a^{2} e^{2} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-3\right )}}{\sqrt{x e + d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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