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

Optimal. Leaf size=138 \[ \frac{\left (a^2-2 b^2 c\right ) (c+d x)^3}{3 d^3}-\frac{c \left (2 a^2-b^2 c\right ) (c+d x)^2}{2 d^3}+\frac{a^2 c^2 x}{d^2}+\frac{4 a b c^2 (c+d x)^{3/2}}{3 d^3}+\frac{4 a b (c+d x)^{7/2}}{7 d^3}-\frac{8 a b c (c+d x)^{5/2}}{5 d^3}+\frac{b^2 (c+d x)^4}{4 d^3} \]

[Out]

(a^2*c^2*x)/d^2 + (4*a*b*c^2*(c + d*x)^(3/2))/(3*d^3) - (c*(2*a^2 - b^2*c)*(c +
d*x)^2)/(2*d^3) - (8*a*b*c*(c + d*x)^(5/2))/(5*d^3) + ((a^2 - 2*b^2*c)*(c + d*x)
^3)/(3*d^3) + (4*a*b*(c + d*x)^(7/2))/(7*d^3) + (b^2*(c + d*x)^4)/(4*d^3)

_______________________________________________________________________________________

Rubi [A]  time = 0.358372, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{\left (a^2-2 b^2 c\right ) (c+d x)^3}{3 d^3}-\frac{c \left (2 a^2-b^2 c\right ) (c+d x)^2}{2 d^3}+\frac{a^2 c^2 x}{d^2}+\frac{4 a b c^2 (c+d x)^{3/2}}{3 d^3}+\frac{4 a b (c+d x)^{7/2}}{7 d^3}-\frac{8 a b c (c+d x)^{5/2}}{5 d^3}+\frac{b^2 (c+d x)^4}{4 d^3} \]

Antiderivative was successfully verified.

[In]  Int[x^2*(a + b*Sqrt[c + d*x])^2,x]

[Out]

(a^2*c^2*x)/d^2 + (4*a*b*c^2*(c + d*x)^(3/2))/(3*d^3) - (c*(2*a^2 - b^2*c)*(c +
d*x)^2)/(2*d^3) - (8*a*b*c*(c + d*x)^(5/2))/(5*d^3) + ((a^2 - 2*b^2*c)*(c + d*x)
^3)/(3*d^3) + (4*a*b*(c + d*x)^(7/2))/(7*d^3) + (b^2*(c + d*x)^4)/(4*d^3)

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{2 a^{2} c^{2} \int ^{\sqrt{c + d x}} x\, dx}{d^{3}} + \frac{4 a b c^{2} \left (c + d x\right )^{\frac{3}{2}}}{3 d^{3}} - \frac{8 a b c \left (c + d x\right )^{\frac{5}{2}}}{5 d^{3}} + \frac{4 a b \left (c + d x\right )^{\frac{7}{2}}}{7 d^{3}} + \frac{b^{2} \left (c + d x\right )^{4}}{4 d^{3}} - \frac{c \left (2 a^{2} - b^{2} c\right ) \left (c + d x\right )^{2}}{2 d^{3}} + \frac{\left (a^{2} - 2 b^{2} c\right ) \left (c + d x\right )^{3}}{3 d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(a+b*(d*x+c)**(1/2))**2,x)

[Out]

2*a**2*c**2*Integral(x, (x, sqrt(c + d*x)))/d**3 + 4*a*b*c**2*(c + d*x)**(3/2)/(
3*d**3) - 8*a*b*c*(c + d*x)**(5/2)/(5*d**3) + 4*a*b*(c + d*x)**(7/2)/(7*d**3) +
b**2*(c + d*x)**4/(4*d**3) - c*(2*a**2 - b**2*c)*(c + d*x)**2/(2*d**3) + (a**2 -
2*b**2*c)*(c + d*x)**3/(3*d**3)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0967523, size = 77, normalized size = 0.56 \[ \frac{a^2 x^3}{3}+\frac{4 a b \sqrt{c+d x} \left (8 c^3-4 c^2 d x+3 c d^2 x^2+15 d^3 x^3\right )}{105 d^3}+\frac{1}{12} b^2 x^3 (4 c+3 d x) \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*(a + b*Sqrt[c + d*x])^2,x]

[Out]

(a^2*x^3)/3 + (b^2*x^3*(4*c + 3*d*x))/12 + (4*a*b*Sqrt[c + d*x]*(8*c^3 - 4*c^2*d
*x + 3*c*d^2*x^2 + 15*d^3*x^3))/(105*d^3)

_______________________________________________________________________________________

Maple [A]  time = 0.003, size = 66, normalized size = 0.5 \[{b}^{2} \left ({\frac{d{x}^{4}}{4}}+{\frac{c{x}^{3}}{3}} \right ) +4\,{\frac{ab \left ( 1/7\, \left ( dx+c \right ) ^{7/2}-2/5\, \left ( dx+c \right ) ^{5/2}c+1/3\,{c}^{2} \left ( dx+c \right ) ^{3/2} \right ) }{{d}^{3}}}+{\frac{{a}^{2}{x}^{3}}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(a+b*(d*x+c)^(1/2))^2,x)

[Out]

b^2*(1/4*d*x^4+1/3*c*x^3)+4*a*b/d^3*(1/7*(d*x+c)^(7/2)-2/5*(d*x+c)^(5/2)*c+1/3*c
^2*(d*x+c)^(3/2))+1/3*a^2*x^3

_______________________________________________________________________________________

Maxima [A]  time = 0.70696, size = 151, normalized size = 1.09 \[ \frac{105 \,{\left (d x + c\right )}^{4} b^{2} + 240 \,{\left (d x + c\right )}^{\frac{7}{2}} a b - 672 \,{\left (d x + c\right )}^{\frac{5}{2}} a b c + 560 \,{\left (d x + c\right )}^{\frac{3}{2}} a b c^{2} + 420 \,{\left (d x + c\right )} a^{2} c^{2} - 140 \,{\left (2 \, b^{2} c - a^{2}\right )}{\left (d x + c\right )}^{3} + 210 \,{\left (b^{2} c^{2} - 2 \, a^{2} c\right )}{\left (d x + c\right )}^{2}}{420 \, d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((sqrt(d*x + c)*b + a)^2*x^2,x, algorithm="maxima")

[Out]

1/420*(105*(d*x + c)^4*b^2 + 240*(d*x + c)^(7/2)*a*b - 672*(d*x + c)^(5/2)*a*b*c
+ 560*(d*x + c)^(3/2)*a*b*c^2 + 420*(d*x + c)*a^2*c^2 - 140*(2*b^2*c - a^2)*(d*
x + c)^3 + 210*(b^2*c^2 - 2*a^2*c)*(d*x + c)^2)/d^3

_______________________________________________________________________________________

Fricas [A]  time = 0.288538, size = 109, normalized size = 0.79 \[ \frac{105 \, b^{2} d^{4} x^{4} + 140 \,{\left (b^{2} c + a^{2}\right )} d^{3} x^{3} + 16 \,{\left (15 \, a b d^{3} x^{3} + 3 \, a b c d^{2} x^{2} - 4 \, a b c^{2} d x + 8 \, a b c^{3}\right )} \sqrt{d x + c}}{420 \, d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((sqrt(d*x + c)*b + a)^2*x^2,x, algorithm="fricas")

[Out]

1/420*(105*b^2*d^4*x^4 + 140*(b^2*c + a^2)*d^3*x^3 + 16*(15*a*b*d^3*x^3 + 3*a*b*
c*d^2*x^2 - 4*a*b*c^2*d*x + 8*a*b*c^3)*sqrt(d*x + c))/d^3

_______________________________________________________________________________________

Sympy [A]  time = 2.43874, size = 73, normalized size = 0.53 \[ \frac{a^{2} x^{3}}{3} + \frac{4 a b \left (\frac{c^{2} \left (c + d x\right )^{\frac{3}{2}}}{3} - \frac{2 c \left (c + d x\right )^{\frac{5}{2}}}{5} + \frac{\left (c + d x\right )^{\frac{7}{2}}}{7}\right )}{d^{3}} + \frac{b^{2} c x^{3}}{3} + \frac{b^{2} d x^{4}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(a+b*(d*x+c)**(1/2))**2,x)

[Out]

a**2*x**3/3 + 4*a*b*(c**2*(c + d*x)**(3/2)/3 - 2*c*(c + d*x)**(5/2)/5 + (c + d*x
)**(7/2)/7)/d**3 + b**2*c*x**3/3 + b**2*d*x**4/4

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.27438, size = 163, normalized size = 1.18 \[ \frac{140 \,{\left (d x^{3} + \frac{c^{3}}{d^{2}}\right )} a^{2} + \frac{35 \,{\left (3 \,{\left (d x + c\right )}^{4} d^{6} - 8 \,{\left (d x + c\right )}^{3} c d^{6} + 6 \,{\left (d x + c\right )}^{2} c^{2} d^{6}\right )} b^{2}}{d^{8}} + \frac{16 \,{\left (15 \,{\left (d x + c\right )}^{\frac{7}{2}} d^{12} - 42 \,{\left (d x + c\right )}^{\frac{5}{2}} c d^{12} + 35 \,{\left (d x + c\right )}^{\frac{3}{2}} c^{2} d^{12}\right )} a b}{d^{14}}}{420 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((sqrt(d*x + c)*b + a)^2*x^2,x, algorithm="giac")

[Out]

1/420*(140*(d*x^3 + c^3/d^2)*a^2 + 35*(3*(d*x + c)^4*d^6 - 8*(d*x + c)^3*c*d^6 +
6*(d*x + c)^2*c^2*d^6)*b^2/d^8 + 16*(15*(d*x + c)^(7/2)*d^12 - 42*(d*x + c)^(5/
2)*c*d^12 + 35*(d*x + c)^(3/2)*c^2*d^12)*a*b/d^14)/d