Optimal. Leaf size=158 \[ -\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^4 (-a B e-A b e+2 b B d)}{4 e^3 (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^3 (b d-a e) (B d-A e)}{3 e^3 (a+b x)}+\frac{b B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^5}{5 e^3 (a+b x)} \]
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Rubi [A] time = 0.118247, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061, Rules used = {770, 77} \[ -\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^4 (-a B e-A b e+2 b B d)}{4 e^3 (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^3 (b d-a e) (B d-A e)}{3 e^3 (a+b x)}+\frac{b B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^5}{5 e^3 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 770
Rule 77
Rubi steps
\begin{align*} \int (A+B x) (d+e x)^2 \sqrt{a^2+2 a b x+b^2 x^2} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (a b+b^2 x\right ) (A+B x) (d+e x)^2 \, 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) (d+e x)^2}{e^2}+\frac{b (-2 b B d+A b e+a B e) (d+e x)^3}{e^2}+\frac{b^2 B (d+e x)^4}{e^2}\right ) \, dx}{a b+b^2 x}\\ &=\frac{(b d-a e) (B d-A e) (d+e x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^3 (a+b x)}-\frac{(2 b B d-A b e-a B e) (d+e x)^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 e^3 (a+b x)}+\frac{b B (d+e x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^3 (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.0532615, size = 120, normalized size = 0.76 \[ \frac{x \sqrt{(a+b x)^2} \left (5 a \left (4 A \left (3 d^2+3 d e x+e^2 x^2\right )+B x \left (6 d^2+8 d e x+3 e^2 x^2\right )\right )+b x \left (5 A \left (6 d^2+8 d e x+3 e^2 x^2\right )+2 B x \left (10 d^2+15 d e x+6 e^2 x^2\right )\right )\right )}{60 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 128, normalized size = 0.8 \begin{align*}{\frac{x \left ( 12\,bB{e}^{2}{x}^{4}+15\,{x}^{3}Ab{e}^{2}+15\,{x}^{3}aB{e}^{2}+30\,{x}^{3}bBde+20\,{x}^{2}aA{e}^{2}+40\,{x}^{2}Abde+40\,{x}^{2}aBde+20\,{x}^{2}bB{d}^{2}+60\,xaAde+30\,xAb{d}^{2}+30\,xBa{d}^{2}+60\,aA{d}^{2} \right ) }{60\,bx+60\,a}\sqrt{ \left ( bx+a \right ) ^{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 = 1.53035, size = 215, normalized size = 1.36 \begin{align*} \frac{1}{5} \, B b e^{2} x^{5} + A a d^{2} x + \frac{1}{4} \,{\left (2 \, B b d e +{\left (B a + A b\right )} e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (B b d^{2} + A a e^{2} + 2 \,{\left (B a + A b\right )} d e\right )} x^{3} + \frac{1}{2} \,{\left (2 \, A a d e +{\left (B a + A b\right )} d^{2}\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.115422, size = 116, normalized size = 0.73 \begin{align*} A a d^{2} x + \frac{B b e^{2} x^{5}}{5} + x^{4} \left (\frac{A b e^{2}}{4} + \frac{B a e^{2}}{4} + \frac{B b d e}{2}\right ) + x^{3} \left (\frac{A a e^{2}}{3} + \frac{2 A b d e}{3} + \frac{2 B a d e}{3} + \frac{B b d^{2}}{3}\right ) + x^{2} \left (A a d e + \frac{A b d^{2}}{2} + \frac{B a d^{2}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12298, size = 250, normalized size = 1.58 \begin{align*} \frac{1}{5} \, B b x^{5} e^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{2} \, B b d x^{4} e \mathrm{sgn}\left (b x + a\right ) + \frac{1}{3} \, B b d^{2} x^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{4} \, B a x^{4} e^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{4} \, A b x^{4} e^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{2}{3} \, B a d x^{3} e \mathrm{sgn}\left (b x + a\right ) + \frac{2}{3} \, A b d x^{3} e \mathrm{sgn}\left (b x + a\right ) + \frac{1}{2} \, B a d^{2} x^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{2} \, A b d^{2} x^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{3} \, A a x^{3} e^{2} \mathrm{sgn}\left (b x + a\right ) + A a d x^{2} e \mathrm{sgn}\left (b x + a\right ) + A a d^{2} x \mathrm{sgn}\left (b x + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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