Optimal. Leaf size=89 \[ \frac{\left (a^2-b^2 c\right ) (c+d x)^2}{2 d^2}-\frac{a^2 c x}{d}+\frac{4 a b (c+d x)^{5/2}}{5 d^2}-\frac{4 a b c (c+d x)^{3/2}}{3 d^2}+\frac{b^2 (c+d x)^3}{3 d^2} \]
[Out]
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Rubi [A] time = 0.20363, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{\left (a^2-b^2 c\right ) (c+d x)^2}{2 d^2}-\frac{a^2 c x}{d}+\frac{4 a b (c+d x)^{5/2}}{5 d^2}-\frac{4 a b c (c+d x)^{3/2}}{3 d^2}+\frac{b^2 (c+d x)^3}{3 d^2} \]
Antiderivative was successfully verified.
[In] Int[x*(a + b*Sqrt[c + d*x])^2,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{2 a^{2} c \int ^{\sqrt{c + d x}} x\, dx}{d^{2}} - \frac{4 a b c \left (c + d x\right )^{\frac{3}{2}}}{3 d^{2}} + \frac{4 a b \left (c + d x\right )^{\frac{5}{2}}}{5 d^{2}} + \frac{b^{2} \left (c + d x\right )^{3}}{3 d^{2}} + \frac{\left (a^{2} - b^{2} c\right ) \left (c + d x\right )^{2}}{2 d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(a+b*(d*x+c)**(1/2))**2,x)
[Out]
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Mathematica [A] time = 0.0746338, size = 67, normalized size = 0.75 \[ -\frac{(c+d x) \left (15 a^2 (c-d x)+8 a b (2 c-3 d x) \sqrt{c+d x}+5 b^2 \left (c^2-c d x-2 d^2 x^2\right )\right )}{30 d^2} \]
Antiderivative was successfully verified.
[In] Integrate[x*(a + b*Sqrt[c + d*x])^2,x]
[Out]
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Maple [A] time = 0.004, size = 54, normalized size = 0.6 \[{b}^{2} \left ({\frac{d{x}^{3}}{3}}+{\frac{c{x}^{2}}{2}} \right ) +4\,{\frac{ab \left ( 1/5\, \left ( dx+c \right ) ^{5/2}-1/3\, \left ( dx+c \right ) ^{3/2}c \right ) }{{d}^{2}}}+{\frac{{a}^{2}{x}^{2}}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(a+b*(d*x+c)^(1/2))^2,x)
[Out]
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Maxima [A] time = 0.716909, size = 97, normalized size = 1.09 \[ \frac{10 \,{\left (d x + c\right )}^{3} b^{2} + 24 \,{\left (d x + c\right )}^{\frac{5}{2}} a b - 40 \,{\left (d x + c\right )}^{\frac{3}{2}} a b c - 30 \,{\left (d x + c\right )} a^{2} c - 15 \,{\left (b^{2} c - a^{2}\right )}{\left (d x + c\right )}^{2}}{30 \, d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((sqrt(d*x + c)*b + a)^2*x,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.294338, size = 90, normalized size = 1.01 \[ \frac{10 \, b^{2} d^{3} x^{3} + 15 \,{\left (b^{2} c + a^{2}\right )} d^{2} x^{2} + 8 \,{\left (3 \, a b d^{2} x^{2} + a b c d x - 2 \, a b c^{2}\right )} \sqrt{d x + c}}{30 \, d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((sqrt(d*x + c)*b + a)^2*x,x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.44111, size = 58, normalized size = 0.65 \[ \frac{a^{2} x^{2}}{2} + \frac{4 a b \left (- \frac{c \left (c + d x\right )^{\frac{3}{2}}}{3} + \frac{\left (c + d x\right )^{\frac{5}{2}}}{5}\right )}{d^{2}} + \frac{b^{2} c x^{2}}{2} + \frac{b^{2} d x^{3}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(a+b*(d*x+c)**(1/2))**2,x)
[Out]
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GIAC/XCAS [A] time = 0.27682, size = 115, normalized size = 1.29 \[ \frac{\frac{15 \,{\left ({\left (d x + c\right )}^{2} - 2 \,{\left (d x + c\right )} c\right )} a^{2}}{d} + \frac{8 \,{\left (3 \,{\left (d x + c\right )}^{\frac{5}{2}} - 5 \,{\left (d x + c\right )}^{\frac{3}{2}} c\right )} a b}{d} + \frac{5 \,{\left (2 \,{\left (d x + c\right )}^{3} - 3 \,{\left (d x + c\right )}^{2} c\right )} b^{2}}{d}}{30 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((sqrt(d*x + c)*b + a)^2*x,x, algorithm="giac")
[Out]