3.459 \(\int x \left (a+b \sqrt{c+d x}\right )^2 \, dx\)

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]

-((a^2*c*x)/d) - (4*a*b*c*(c + d*x)^(3/2))/(3*d^2) + ((a^2 - b^2*c)*(c + d*x)^2)
/(2*d^2) + (4*a*b*(c + d*x)^(5/2))/(5*d^2) + (b^2*(c + d*x)^3)/(3*d^2)

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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]

-((a^2*c*x)/d) - (4*a*b*c*(c + d*x)^(3/2))/(3*d^2) + ((a^2 - b^2*c)*(c + d*x)^2)
/(2*d^2) + (4*a*b*(c + d*x)^(5/2))/(5*d^2) + (b^2*(c + d*x)^3)/(3*d^2)

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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]

-2*a**2*c*Integral(x, (x, sqrt(c + d*x)))/d**2 - 4*a*b*c*(c + d*x)**(3/2)/(3*d**
2) + 4*a*b*(c + d*x)**(5/2)/(5*d**2) + b**2*(c + d*x)**3/(3*d**2) + (a**2 - b**2
*c)*(c + d*x)**2/(2*d**2)

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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]

-((c + d*x)*(15*a^2*(c - d*x) + 8*a*b*(2*c - 3*d*x)*Sqrt[c + d*x] + 5*b^2*(c^2 -
c*d*x - 2*d^2*x^2)))/(30*d^2)

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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]

b^2*(1/3*d*x^3+1/2*c*x^2)+4*a*b/d^2*(1/5*(d*x+c)^(5/2)-1/3*(d*x+c)^(3/2)*c)+1/2*
a^2*x^2

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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]

1/30*(10*(d*x + c)^3*b^2 + 24*(d*x + c)^(5/2)*a*b - 40*(d*x + c)^(3/2)*a*b*c - 3
0*(d*x + c)*a^2*c - 15*(b^2*c - a^2)*(d*x + c)^2)/d^2

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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]

1/30*(10*b^2*d^3*x^3 + 15*(b^2*c + a^2)*d^2*x^2 + 8*(3*a*b*d^2*x^2 + a*b*c*d*x -
2*a*b*c^2)*sqrt(d*x + c))/d^2

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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]

a**2*x**2/2 + 4*a*b*(-c*(c + d*x)**(3/2)/3 + (c + d*x)**(5/2)/5)/d**2 + b**2*c*x
**2/2 + b**2*d*x**3/3

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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]

1/30*(15*((d*x + c)^2 - 2*(d*x + c)*c)*a^2/d + 8*(3*(d*x + c)^(5/2) - 5*(d*x + c
)^(3/2)*c)*a*b/d + 5*(2*(d*x + c)^3 - 3*(d*x + c)^2*c)*b^2/d)/d