Optimal. Leaf size=162 \[ -\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (-a B e-A b e+2 b B d)}{3 e^3 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e) (B d-A e)}{e^3 (a+b x)}+\frac{2 b B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2}}{5 e^3 (a+b x)} \]
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Rubi [A] time = 0.0881532, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057, Rules used = {770, 77} \[ -\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (-a B e-A b e+2 b B d)}{3 e^3 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e) (B d-A e)}{e^3 (a+b x)}+\frac{2 b B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2}}{5 e^3 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 770
Rule 77
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{\left (a b+b^2 x\right ) (A+B x)}{\sqrt{d+e x}} \, dx}{a b+b^2 x}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (-\frac{b (b d-a e) (-B d+A e)}{e^2 \sqrt{d+e x}}+\frac{b (-2 b B d+A b e+a B e) \sqrt{d+e x}}{e^2}+\frac{b^2 B (d+e x)^{3/2}}{e^2}\right ) \, dx}{a b+b^2 x}\\ &=\frac{2 (b d-a e) (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 (2 b B d-A b e-a B 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 B (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 = 0.0703842, size = 86, normalized size = 0.53 \[ \frac{2 \sqrt{(a+b x)^2} \sqrt{d+e x} \left (5 a e (3 A e-2 B d+B e x)+5 A b e (e x-2 d)+b B \left (8 d^2-4 d e x+3 e^2 x^2\right )\right )}{15 e^3 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 89, normalized size = 0.6 \begin{align*}{\frac{6\,B{x}^{2}b{e}^{2}+10\,Axb{e}^{2}+10\,aB{e}^{2}x-8\,Bxbde+30\,aA{e}^{2}-20\,Abde-20\,aBde+16\,Bb{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 = 1.06632, size = 161, normalized size = 0.99 \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 = 1.27167, size = 165, normalized size = 1.02 \begin{align*} \frac{2 \,{\left (3 \, B b e^{2} x^{2} + 8 \, B b d^{2} + 15 \, A a e^{2} - 10 \,{\left (B a + A b\right )} d e -{\left (4 \, B b d e - 5 \,{\left (B a + A b\right )} 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 = 1.12098, size = 180, normalized size = 1.11 \begin{align*} \frac{2}{15} \,{\left (5 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} B a e^{\left (-1\right )} \mathrm{sgn}\left (b x + a\right ) + 5 \,{\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 b e^{\left (-2\right )} \mathrm{sgn}\left (b x + a\right ) + 15 \, \sqrt{x e + d} A a \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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