3.457 \(\int x^3 \left (a+b \sqrt{c+d x}\right )^2 \, dx\)

Optimal. Leaf size=185 \[ \frac{c^2 \left (3 a^2-b^2 c\right ) (c+d x)^2}{2 d^4}+\frac{\left (a^2-3 b^2 c\right ) (c+d x)^4}{4 d^4}-\frac{c \left (a^2-b^2 c\right ) (c+d x)^3}{d^4}-\frac{a^2 c^3 x}{d^3}-\frac{4 a b c^3 (c+d x)^{3/2}}{3 d^4}+\frac{12 a b c^2 (c+d x)^{5/2}}{5 d^4}+\frac{4 a b (c+d x)^{9/2}}{9 d^4}-\frac{12 a b c (c+d x)^{7/2}}{7 d^4}+\frac{b^2 (c+d x)^5}{5 d^4} \]

[Out]

-((a^2*c^3*x)/d^3) - (4*a*b*c^3*(c + d*x)^(3/2))/(3*d^4) + (c^2*(3*a^2 - b^2*c)*
(c + d*x)^2)/(2*d^4) + (12*a*b*c^2*(c + d*x)^(5/2))/(5*d^4) - (c*(a^2 - b^2*c)*(
c + d*x)^3)/d^4 - (12*a*b*c*(c + d*x)^(7/2))/(7*d^4) + ((a^2 - 3*b^2*c)*(c + d*x
)^4)/(4*d^4) + (4*a*b*(c + d*x)^(9/2))/(9*d^4) + (b^2*(c + d*x)^5)/(5*d^4)

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Rubi [A]  time = 0.511152, antiderivative size = 185, 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 \[ \frac{c^2 \left (3 a^2-b^2 c\right ) (c+d x)^2}{2 d^4}+\frac{\left (a^2-3 b^2 c\right ) (c+d x)^4}{4 d^4}-\frac{c \left (a^2-b^2 c\right ) (c+d x)^3}{d^4}-\frac{a^2 c^3 x}{d^3}-\frac{4 a b c^3 (c+d x)^{3/2}}{3 d^4}+\frac{12 a b c^2 (c+d x)^{5/2}}{5 d^4}+\frac{4 a b (c+d x)^{9/2}}{9 d^4}-\frac{12 a b c (c+d x)^{7/2}}{7 d^4}+\frac{b^2 (c+d x)^5}{5 d^4} \]

Antiderivative was successfully verified.

[In]  Int[x^3*(a + b*Sqrt[c + d*x])^2,x]

[Out]

-((a^2*c^3*x)/d^3) - (4*a*b*c^3*(c + d*x)^(3/2))/(3*d^4) + (c^2*(3*a^2 - b^2*c)*
(c + d*x)^2)/(2*d^4) + (12*a*b*c^2*(c + d*x)^(5/2))/(5*d^4) - (c*(a^2 - b^2*c)*(
c + d*x)^3)/d^4 - (12*a*b*c*(c + d*x)^(7/2))/(7*d^4) + ((a^2 - 3*b^2*c)*(c + d*x
)^4)/(4*d^4) + (4*a*b*(c + d*x)^(9/2))/(9*d^4) + (b^2*(c + d*x)^5)/(5*d^4)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{2 a^{2} c^{3} \int ^{\sqrt{c + d x}} x\, dx}{d^{4}} - \frac{4 a b c^{3} \left (c + d x\right )^{\frac{3}{2}}}{3 d^{4}} + \frac{12 a b c^{2} \left (c + d x\right )^{\frac{5}{2}}}{5 d^{4}} - \frac{12 a b c \left (c + d x\right )^{\frac{7}{2}}}{7 d^{4}} + \frac{4 a b \left (c + d x\right )^{\frac{9}{2}}}{9 d^{4}} + \frac{b^{2} \left (c + d x\right )^{5}}{5 d^{4}} + \frac{c^{2} \left (3 a^{2} - b^{2} c\right ) \left (c + d x\right )^{2}}{2 d^{4}} - \frac{c \left (a^{2} - b^{2} c\right ) \left (c + d x\right )^{3}}{d^{4}} + \frac{\left (a^{2} - 3 b^{2} c\right ) \left (c + d x\right )^{4}}{4 d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(a+b*(d*x+c)**(1/2))**2,x)

[Out]

-2*a**2*c**3*Integral(x, (x, sqrt(c + d*x)))/d**4 - 4*a*b*c**3*(c + d*x)**(3/2)/
(3*d**4) + 12*a*b*c**2*(c + d*x)**(5/2)/(5*d**4) - 12*a*b*c*(c + d*x)**(7/2)/(7*
d**4) + 4*a*b*(c + d*x)**(9/2)/(9*d**4) + b**2*(c + d*x)**5/(5*d**4) + c**2*(3*a
**2 - b**2*c)*(c + d*x)**2/(2*d**4) - c*(a**2 - b**2*c)*(c + d*x)**3/d**4 + (a**
2 - 3*b**2*c)*(c + d*x)**4/(4*d**4)

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Mathematica [A]  time = 0.114188, size = 88, normalized size = 0.48 \[ \frac{a^2 x^4}{4}+\frac{4 a b \sqrt{c+d x} \left (-16 c^4+8 c^3 d x-6 c^2 d^2 x^2+5 c d^3 x^3+35 d^4 x^4\right )}{315 d^4}+\frac{1}{20} b^2 x^4 (5 c+4 d x) \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*(a + b*Sqrt[c + d*x])^2,x]

[Out]

(a^2*x^4)/4 + (b^2*x^4*(5*c + 4*d*x))/20 + (4*a*b*Sqrt[c + d*x]*(-16*c^4 + 8*c^3
*d*x - 6*c^2*d^2*x^2 + 5*c*d^3*x^3 + 35*d^4*x^4))/(315*d^4)

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Maple [A]  time = 0.004, size = 78, normalized size = 0.4 \[{b}^{2} \left ({\frac{{x}^{5}d}{5}}+{\frac{c{x}^{4}}{4}} \right ) +4\,{\frac{ab \left ( 1/9\, \left ( dx+c \right ) ^{9/2}-3/7\, \left ( dx+c \right ) ^{7/2}c+3/5\, \left ( dx+c \right ) ^{5/2}{c}^{2}-1/3\,{c}^{3} \left ( dx+c \right ) ^{3/2} \right ) }{{d}^{4}}}+{\frac{{a}^{2}{x}^{4}}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(a+b*(d*x+c)^(1/2))^2,x)

[Out]

b^2*(1/5*x^5*d+1/4*c*x^4)+4*a*b/d^4*(1/9*(d*x+c)^(9/2)-3/7*(d*x+c)^(7/2)*c+3/5*(
d*x+c)^(5/2)*c^2-1/3*c^3*(d*x+c)^(3/2))+1/4*a^2*x^4

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Maxima [A]  time = 0.69538, size = 204, normalized size = 1.1 \[ \frac{252 \,{\left (d x + c\right )}^{5} b^{2} + 560 \,{\left (d x + c\right )}^{\frac{9}{2}} a b - 2160 \,{\left (d x + c\right )}^{\frac{7}{2}} a b c + 3024 \,{\left (d x + c\right )}^{\frac{5}{2}} a b c^{2} - 1680 \,{\left (d x + c\right )}^{\frac{3}{2}} a b c^{3} - 1260 \,{\left (d x + c\right )} a^{2} c^{3} - 315 \,{\left (3 \, b^{2} c - a^{2}\right )}{\left (d x + c\right )}^{4} + 1260 \,{\left (b^{2} c^{2} - a^{2} c\right )}{\left (d x + c\right )}^{3} - 630 \,{\left (b^{2} c^{3} - 3 \, a^{2} c^{2}\right )}{\left (d x + c\right )}^{2}}{1260 \, d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((sqrt(d*x + c)*b + a)^2*x^3,x, algorithm="maxima")

[Out]

1/1260*(252*(d*x + c)^5*b^2 + 560*(d*x + c)^(9/2)*a*b - 2160*(d*x + c)^(7/2)*a*b
*c + 3024*(d*x + c)^(5/2)*a*b*c^2 - 1680*(d*x + c)^(3/2)*a*b*c^3 - 1260*(d*x + c
)*a^2*c^3 - 315*(3*b^2*c - a^2)*(d*x + c)^4 + 1260*(b^2*c^2 - a^2*c)*(d*x + c)^3
- 630*(b^2*c^3 - 3*a^2*c^2)*(d*x + c)^2)/d^4

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Fricas [A]  time = 0.36167, size = 127, normalized size = 0.69 \[ \frac{252 \, b^{2} d^{5} x^{5} + 315 \,{\left (b^{2} c + a^{2}\right )} d^{4} x^{4} + 16 \,{\left (35 \, a b d^{4} x^{4} + 5 \, a b c d^{3} x^{3} - 6 \, a b c^{2} d^{2} x^{2} + 8 \, a b c^{3} d x - 16 \, a b c^{4}\right )} \sqrt{d x + c}}{1260 \, d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((sqrt(d*x + c)*b + a)^2*x^3,x, algorithm="fricas")

[Out]

1/1260*(252*b^2*d^5*x^5 + 315*(b^2*c + a^2)*d^4*x^4 + 16*(35*a*b*d^4*x^4 + 5*a*b
*c*d^3*x^3 - 6*a*b*c^2*d^2*x^2 + 8*a*b*c^3*d*x - 16*a*b*c^4)*sqrt(d*x + c))/d^4

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Sympy [A]  time = 2.58479, size = 88, normalized size = 0.48 \[ \frac{a^{2} x^{4}}{4} + \frac{4 a b \left (- \frac{c^{3} \left (c + d x\right )^{\frac{3}{2}}}{3} + \frac{3 c^{2} \left (c + d x\right )^{\frac{5}{2}}}{5} - \frac{3 c \left (c + d x\right )^{\frac{7}{2}}}{7} + \frac{\left (c + d x\right )^{\frac{9}{2}}}{9}\right )}{d^{4}} + \frac{b^{2} c x^{4}}{4} + \frac{b^{2} d x^{5}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(a+b*(d*x+c)**(1/2))**2,x)

[Out]

a**2*x**4/4 + 4*a*b*(-c**3*(c + d*x)**(3/2)/3 + 3*c**2*(c + d*x)**(5/2)/5 - 3*c*
(c + d*x)**(7/2)/7 + (c + d*x)**(9/2)/9)/d**4 + b**2*c*x**4/4 + b**2*d*x**5/5

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GIAC/XCAS [A]  time = 0.276203, size = 205, normalized size = 1.11 \[ \frac{315 \,{\left (d x^{4} - \frac{c^{4}}{d^{3}}\right )} a^{2} + \frac{63 \,{\left (4 \,{\left (d x + c\right )}^{5} d^{12} - 15 \,{\left (d x + c\right )}^{4} c d^{12} + 20 \,{\left (d x + c\right )}^{3} c^{2} d^{12} - 10 \,{\left (d x + c\right )}^{2} c^{3} d^{12}\right )} b^{2}}{d^{15}} + \frac{16 \,{\left (35 \,{\left (d x + c\right )}^{\frac{9}{2}} d^{24} - 135 \,{\left (d x + c\right )}^{\frac{7}{2}} c d^{24} + 189 \,{\left (d x + c\right )}^{\frac{5}{2}} c^{2} d^{24} - 105 \,{\left (d x + c\right )}^{\frac{3}{2}} c^{3} d^{24}\right )} a b}{d^{27}}}{1260 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((sqrt(d*x + c)*b + a)^2*x^3,x, algorithm="giac")

[Out]

1/1260*(315*(d*x^4 - c^4/d^3)*a^2 + 63*(4*(d*x + c)^5*d^12 - 15*(d*x + c)^4*c*d^
12 + 20*(d*x + c)^3*c^2*d^12 - 10*(d*x + c)^2*c^3*d^12)*b^2/d^15 + 16*(35*(d*x +
c)^(9/2)*d^24 - 135*(d*x + c)^(7/2)*c*d^24 + 189*(d*x + c)^(5/2)*c^2*d^24 - 105
*(d*x + c)^(3/2)*c^3*d^24)*a*b/d^27)/d