Optimal. Leaf size=150 \[ \frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2}}{5 e^3 (a+b x)}-\frac{4 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)}{3 e^3 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^2}{e^3 (a+b x)} \]
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Rubi [A] time = 0.0694768, antiderivative size = 150, 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} \[ \frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2}}{5 e^3 (a+b x)}-\frac{4 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)}{3 e^3 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^2}{e^3 (a+b x)} \]
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}}{\sqrt{d+e x}} \, 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 )}{\sqrt{d+e x}} \, 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}{\sqrt{d+e x}} \, 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 \sqrt{d+e x}}-\frac{2 b (b d-a e) \sqrt{d+e x}}{e^2}+\frac{b^2 (d+e x)^{3/2}}{e^2}\right ) \, dx}{a b+b^2 x}\\ &=\frac{2 (b d-a e)^2 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}{e^3 (a+b x)}-\frac{4 b (b d-a e) (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^3 (a+b x)}+\frac{2 b^2 (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^3 (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.0503818, size = 78, normalized size = 0.52 \[ \frac{2 \sqrt{(a+b x)^2} \sqrt{d+e x} \left (15 a^2 e^2+10 a b e (e x-2 d)+b^2 \left (8 d^2-4 d e x+3 e^2 x^2\right )\right )}{15 e^3 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 79, normalized size = 0.5 \begin{align*}{\frac{6\,{x}^{2}{b}^{2}{e}^{2}+20\,xab{e}^{2}-8\,x{b}^{2}de+30\,{a}^{2}{e}^{2}-40\,abde+16\,{b}^{2}{d}^{2}}{15\, \left ( bx+a \right ){e}^{3}}\sqrt{ex+d}\sqrt{ \left ( bx+a \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.1848, size = 161, normalized size = 1.07 \begin{align*} \frac{2 \,{\left (b e^{2} x^{2} - 2 \, b d^{2} + 3 \, a d e -{\left (b d e - 3 \, a e^{2}\right )} x\right )} a}{3 \, \sqrt{e x + d} e^{2}} + \frac{2 \,{\left (3 \, b e^{3} x^{3} + 8 \, b d^{3} - 10 \, a d^{2} e -{\left (b d e^{2} - 5 \, a e^{3}\right )} x^{2} +{\left (4 \, b d^{2} e - 5 \, a d e^{2}\right )} x\right )} b}{15 \, \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 = 0.983919, size = 146, normalized size = 0.97 \begin{align*} \frac{2 \,{\left (3 \, b^{2} e^{2} x^{2} + 8 \, b^{2} d^{2} - 20 \, a b d e + 15 \, a^{2} e^{2} - 2 \,{\left (2 \, b^{2} d e - 5 \, a b e^{2}\right )} x\right )} \sqrt{e x + d}}{15 \, e^{3}} \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}}}{\sqrt{d + e x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14104, size = 139, normalized size = 0.93 \begin{align*} \frac{2}{15} \,{\left (10 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} a b e^{\left (-1\right )} \mathrm{sgn}\left (b x + a\right ) +{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{x e + d} d^{2}\right )} b^{2} e^{\left (-2\right )} \mathrm{sgn}\left (b x + a\right ) + 15 \, \sqrt{x e + d} a^{2} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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