Optimal. Leaf size=135 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^4 (-2 a B e+A b e+b B d)}{5 b^3}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^3 (A b-a B) (b d-a e)}{4 b^3}+\frac{B e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{6 b^3} \]
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Rubi [A] time = 0.143975, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {770, 77} \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^4 (-2 a B e+A b e+b B d)}{5 b^3}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^3 (A b-a B) (b d-a e)}{4 b^3}+\frac{B e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{6 b^3} \]
Antiderivative was successfully verified.
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Rule 770
Rule 77
Rubi steps
\begin{align*} \int (A+B x) (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (a b+b^2 x\right )^3 (A+B x) (d+e x) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (\frac{(A b-a B) (b d-a e) \left (a b+b^2 x\right )^3}{b^2}+\frac{(b B d+A b e-2 a B e) \left (a b+b^2 x\right )^4}{b^3}+\frac{B e \left (a b+b^2 x\right )^5}{b^4}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac{(A b-a B) (b d-a e) (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{4 b^3}+\frac{(b B d+A b e-2 a B e) (a+b x)^4 \sqrt{a^2+2 a b x+b^2 x^2}}{5 b^3}+\frac{B e (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 b^3}\\ \end{align*}
Mathematica [A] time = 0.0702317, size = 144, normalized size = 1.07 \[ \frac{x \sqrt{(a+b x)^2} \left (15 a^2 b x (A (6 d+4 e x)+B x (4 d+3 e x))+10 a^3 (3 A (2 d+e x)+B x (3 d+2 e x))+3 a b^2 x^2 (5 A (4 d+3 e x)+3 B x (5 d+4 e x))+b^3 x^3 (3 A (5 d+4 e x)+2 B x (6 d+5 e x))\right )}{60 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 180, normalized size = 1.3 \begin{align*}{\frac{x \left ( 10\,{b}^{3}Be{x}^{5}+12\,{x}^{4}A{b}^{3}e+36\,{x}^{4}Be{b}^{2}a+12\,{x}^{4}B{b}^{3}d+45\,{x}^{3}Aa{b}^{2}e+15\,{x}^{3}Ad{b}^{3}+45\,{x}^{3}Be{a}^{2}b+45\,{x}^{3}Ba{b}^{2}d+60\,{x}^{2}A{a}^{2}be+60\,{x}^{2}Ad{b}^{2}a+20\,{x}^{2}Be{a}^{3}+60\,{x}^{2}B{a}^{2}bd+30\,xA{a}^{3}e+90\,xAd{a}^{2}b+30\,xB{a}^{3}d+60\,Ad{a}^{3} \right ) }{60\, \left ( bx+a \right ) ^{3}} \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 = 1.51386, size = 320, normalized size = 2.37 \begin{align*} \frac{1}{6} \, B b^{3} e x^{6} + A a^{3} d x + \frac{1}{5} \,{\left (B b^{3} d +{\left (3 \, B a b^{2} + A b^{3}\right )} e\right )} x^{5} + \frac{1}{4} \,{\left ({\left (3 \, B a b^{2} + A b^{3}\right )} d + 3 \,{\left (B a^{2} b + A a b^{2}\right )} e\right )} x^{4} + \frac{1}{3} \,{\left (3 \,{\left (B a^{2} b + A a b^{2}\right )} d +{\left (B a^{3} + 3 \, A a^{2} b\right )} e\right )} x^{3} + \frac{1}{2} \,{\left (A a^{3} e +{\left (B a^{3} + 3 \, A a^{2} b\right )} d\right )} x^{2} \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 \left (A + B x\right ) \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 = 1.1185, size = 360, normalized size = 2.67 \begin{align*} \frac{1}{6} \, B b^{3} x^{6} e \mathrm{sgn}\left (b x + a\right ) + \frac{1}{5} \, B b^{3} d x^{5} \mathrm{sgn}\left (b x + a\right ) + \frac{3}{5} \, B a b^{2} x^{5} e \mathrm{sgn}\left (b x + a\right ) + \frac{1}{5} \, A b^{3} x^{5} e \mathrm{sgn}\left (b x + a\right ) + \frac{3}{4} \, B a b^{2} d x^{4} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{4} \, A b^{3} d x^{4} \mathrm{sgn}\left (b x + a\right ) + \frac{3}{4} \, B a^{2} b x^{4} e \mathrm{sgn}\left (b x + a\right ) + \frac{3}{4} \, A a b^{2} x^{4} e \mathrm{sgn}\left (b x + a\right ) + B a^{2} b d x^{3} \mathrm{sgn}\left (b x + a\right ) + A a b^{2} d x^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{3} \, B a^{3} x^{3} e \mathrm{sgn}\left (b x + a\right ) + A a^{2} b x^{3} e \mathrm{sgn}\left (b x + a\right ) + \frac{1}{2} \, B a^{3} d x^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{3}{2} \, A a^{2} b d x^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{2} \, A a^{3} x^{2} e \mathrm{sgn}\left (b x + a\right ) + A a^{3} d x \mathrm{sgn}\left (b x + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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