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

Optimal. Leaf size=242 \[ -\frac{2 a \left (a^2-b^2 c\right )^2 \left (a+b \sqrt{c+d x}\right )^{p+1}}{b^6 d^3 (p+1)}-\frac{4 a \left (5 a^2-3 b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{p+3}}{b^6 d^3 (p+3)}+\frac{4 \left (5 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{p+4}}{b^6 d^3 (p+4)}+\frac{2 \left (5 a^4-6 a^2 b^2 c+b^4 c^2\right ) \left (a+b \sqrt{c+d x}\right )^{p+2}}{b^6 d^3 (p+2)}-\frac{10 a \left (a+b \sqrt{c+d x}\right )^{p+5}}{b^6 d^3 (p+5)}+\frac{2 \left (a+b \sqrt{c+d x}\right )^{p+6}}{b^6 d^3 (p+6)} \]

[Out]

(-2*a*(a^2 - b^2*c)^2*(a + b*Sqrt[c + d*x])^(1 + p))/(b^6*d^3*(1 + p)) + (2*(5*a
^4 - 6*a^2*b^2*c + b^4*c^2)*(a + b*Sqrt[c + d*x])^(2 + p))/(b^6*d^3*(2 + p)) - (
4*a*(5*a^2 - 3*b^2*c)*(a + b*Sqrt[c + d*x])^(3 + p))/(b^6*d^3*(3 + p)) + (4*(5*a
^2 - b^2*c)*(a + b*Sqrt[c + d*x])^(4 + p))/(b^6*d^3*(4 + p)) - (10*a*(a + b*Sqrt
[c + d*x])^(5 + p))/(b^6*d^3*(5 + p)) + (2*(a + b*Sqrt[c + d*x])^(6 + p))/(b^6*d
^3*(6 + p))

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Rubi [A]  time = 0.410663, antiderivative size = 242, 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 \[ -\frac{2 a \left (a^2-b^2 c\right )^2 \left (a+b \sqrt{c+d x}\right )^{p+1}}{b^6 d^3 (p+1)}-\frac{4 a \left (5 a^2-3 b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{p+3}}{b^6 d^3 (p+3)}+\frac{4 \left (5 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{p+4}}{b^6 d^3 (p+4)}+\frac{2 \left (5 a^4-6 a^2 b^2 c+b^4 c^2\right ) \left (a+b \sqrt{c+d x}\right )^{p+2}}{b^6 d^3 (p+2)}-\frac{10 a \left (a+b \sqrt{c+d x}\right )^{p+5}}{b^6 d^3 (p+5)}+\frac{2 \left (a+b \sqrt{c+d x}\right )^{p+6}}{b^6 d^3 (p+6)} \]

Antiderivative was successfully verified.

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

[Out]

(-2*a*(a^2 - b^2*c)^2*(a + b*Sqrt[c + d*x])^(1 + p))/(b^6*d^3*(1 + p)) + (2*(5*a
^4 - 6*a^2*b^2*c + b^4*c^2)*(a + b*Sqrt[c + d*x])^(2 + p))/(b^6*d^3*(2 + p)) - (
4*a*(5*a^2 - 3*b^2*c)*(a + b*Sqrt[c + d*x])^(3 + p))/(b^6*d^3*(3 + p)) + (4*(5*a
^2 - b^2*c)*(a + b*Sqrt[c + d*x])^(4 + p))/(b^6*d^3*(4 + p)) - (10*a*(a + b*Sqrt
[c + d*x])^(5 + p))/(b^6*d^3*(5 + p)) + (2*(a + b*Sqrt[c + d*x])^(6 + p))/(b^6*d
^3*(6 + p))

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Rubi in Sympy [A]  time = 33.7381, size = 221, normalized size = 0.91 \[ - \frac{2 a \left (a + b \sqrt{c + d x}\right )^{p + 1} \left (a^{2} - b^{2} c\right )^{2}}{b^{6} d^{3} \left (p + 1\right )} - \frac{4 a \left (a + b \sqrt{c + d x}\right )^{p + 3} \left (5 a^{2} - 3 b^{2} c\right )}{b^{6} d^{3} \left (p + 3\right )} - \frac{10 a \left (a + b \sqrt{c + d x}\right )^{p + 5}}{b^{6} d^{3} \left (p + 5\right )} + \frac{2 \left (a + b \sqrt{c + d x}\right )^{p + 2} \left (5 a^{4} - 6 a^{2} b^{2} c + b^{4} c^{2}\right )}{b^{6} d^{3} \left (p + 2\right )} + \frac{4 \left (a + b \sqrt{c + d x}\right )^{p + 4} \left (5 a^{2} - b^{2} c\right )}{b^{6} d^{3} \left (p + 4\right )} + \frac{2 \left (a + b \sqrt{c + d x}\right )^{p + 6}}{b^{6} d^{3} \left (p + 6\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a*(a + b*sqrt(c + d*x))**(p + 1)*(a**2 - b**2*c)**2/(b**6*d**3*(p + 1)) - 4*a
*(a + b*sqrt(c + d*x))**(p + 3)*(5*a**2 - 3*b**2*c)/(b**6*d**3*(p + 3)) - 10*a*(
a + b*sqrt(c + d*x))**(p + 5)/(b**6*d**3*(p + 5)) + 2*(a + b*sqrt(c + d*x))**(p
+ 2)*(5*a**4 - 6*a**2*b**2*c + b**4*c**2)/(b**6*d**3*(p + 2)) + 4*(a + b*sqrt(c
+ d*x))**(p + 4)*(5*a**2 - b**2*c)/(b**6*d**3*(p + 4)) + 2*(a + b*sqrt(c + d*x))
**(p + 6)/(b**6*d**3*(p + 6))

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Mathematica [A]  time = 0.57644, size = 285, normalized size = 1.18 \[ \frac{2 \left (a+b \sqrt{c+d x}\right )^{p+1} \left (-120 a^5+120 a^4 b (p+1) \sqrt{c+d x}+12 a^3 b^2 \left (-4 c \left (p^2+p-5\right )-5 d \left (p^2+3 p+2\right ) x\right )-4 a^2 b^3 (p+1) \sqrt{c+d x} \left (c \left (-2 p^2+8 p+60\right )-5 d \left (p^2+5 p+6\right ) x\right )-a b^4 \left (-8 c^2 \left (2 p^3+12 p^2+10 p-15\right )+4 c d \left (p^4+4 p^3-10 p^2-43 p-30\right ) x+5 d^2 \left (p^4+10 p^3+35 p^2+50 p+24\right ) x^2\right )+b^5 \left (p^3+9 p^2+23 p+15\right ) \sqrt{c+d x} \left (8 c^2-4 c d (p+2) x+d^2 \left (p^2+6 p+8\right ) x^2\right )\right )}{b^6 d^3 (p+1) (p+2) (p+3) (p+4) (p+5) (p+6)} \]

Antiderivative was successfully verified.

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

[Out]

(2*(a + b*Sqrt[c + d*x])^(1 + p)*(-120*a^5 + 120*a^4*b*(1 + p)*Sqrt[c + d*x] + 1
2*a^3*b^2*(-4*c*(-5 + p + p^2) - 5*d*(2 + 3*p + p^2)*x) - 4*a^2*b^3*(1 + p)*Sqrt
[c + d*x]*(c*(60 + 8*p - 2*p^2) - 5*d*(6 + 5*p + p^2)*x) + b^5*(15 + 23*p + 9*p^
2 + p^3)*Sqrt[c + d*x]*(8*c^2 - 4*c*d*(2 + p)*x + d^2*(8 + 6*p + p^2)*x^2) - a*b
^4*(-8*c^2*(-15 + 10*p + 12*p^2 + 2*p^3) + 4*c*d*(-30 - 43*p - 10*p^2 + 4*p^3 +
p^4)*x + 5*d^2*(24 + 50*p + 35*p^2 + 10*p^3 + p^4)*x^2)))/(b^6*d^3*(1 + p)*(2 +
p)*(3 + p)*(4 + p)*(5 + p)*(6 + p))

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Maple [F]  time = 0.006, size = 0, normalized size = 0. \[ \int{x}^{2} \left ( a+b\sqrt{dx+c} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int(x^2*(a+b*(d*x+c)^(1/2))^p,x)

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Maxima [A]  time = 0.729036, size = 543, normalized size = 2.24 \[ \frac{2 \,{\left (\frac{{\left ({\left (d x + c\right )} b^{2}{\left (p + 1\right )} + \sqrt{d x + c} a b p - a^{2}\right )}{\left (\sqrt{d x + c} b + a\right )}^{p} c^{2}}{{\left (p^{2} + 3 \, p + 2\right )} b^{2}} - \frac{2 \,{\left ({\left (p^{3} + 6 \, p^{2} + 11 \, p + 6\right )}{\left (d x + c\right )}^{2} b^{4} +{\left (p^{3} + 3 \, p^{2} + 2 \, p\right )}{\left (d x + c\right )}^{\frac{3}{2}} a b^{3} - 3 \,{\left (p^{2} + p\right )}{\left (d x + c\right )} a^{2} b^{2} + 6 \, \sqrt{d x + c} a^{3} b p - 6 \, a^{4}\right )}{\left (\sqrt{d x + c} b + a\right )}^{p} c}{{\left (p^{4} + 10 \, p^{3} + 35 \, p^{2} + 50 \, p + 24\right )} b^{4}} + \frac{{\left ({\left (p^{5} + 15 \, p^{4} + 85 \, p^{3} + 225 \, p^{2} + 274 \, p + 120\right )}{\left (d x + c\right )}^{3} b^{6} +{\left (p^{5} + 10 \, p^{4} + 35 \, p^{3} + 50 \, p^{2} + 24 \, p\right )}{\left (d x + c\right )}^{\frac{5}{2}} a b^{5} - 5 \,{\left (p^{4} + 6 \, p^{3} + 11 \, p^{2} + 6 \, p\right )}{\left (d x + c\right )}^{2} a^{2} b^{4} + 20 \,{\left (p^{3} + 3 \, p^{2} + 2 \, p\right )}{\left (d x + c\right )}^{\frac{3}{2}} a^{3} b^{3} - 60 \,{\left (p^{2} + p\right )}{\left (d x + c\right )} a^{4} b^{2} + 120 \, \sqrt{d x + c} a^{5} b p - 120 \, a^{6}\right )}{\left (\sqrt{d x + c} b + a\right )}^{p}}{{\left (p^{6} + 21 \, p^{5} + 175 \, p^{4} + 735 \, p^{3} + 1624 \, p^{2} + 1764 \, p + 720\right )} b^{6}}\right )}}{d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(((d*x + c)*b^2*(p + 1) + sqrt(d*x + c)*a*b*p - a^2)*(sqrt(d*x + c)*b + a)^p*c
^2/((p^2 + 3*p + 2)*b^2) - 2*((p^3 + 6*p^2 + 11*p + 6)*(d*x + c)^2*b^4 + (p^3 +
3*p^2 + 2*p)*(d*x + c)^(3/2)*a*b^3 - 3*(p^2 + p)*(d*x + c)*a^2*b^2 + 6*sqrt(d*x
+ c)*a^3*b*p - 6*a^4)*(sqrt(d*x + c)*b + a)^p*c/((p^4 + 10*p^3 + 35*p^2 + 50*p +
24)*b^4) + ((p^5 + 15*p^4 + 85*p^3 + 225*p^2 + 274*p + 120)*(d*x + c)^3*b^6 + (
p^5 + 10*p^4 + 35*p^3 + 50*p^2 + 24*p)*(d*x + c)^(5/2)*a*b^5 - 5*(p^4 + 6*p^3 +
11*p^2 + 6*p)*(d*x + c)^2*a^2*b^4 + 20*(p^3 + 3*p^2 + 2*p)*(d*x + c)^(3/2)*a^3*b
^3 - 60*(p^2 + p)*(d*x + c)*a^4*b^2 + 120*sqrt(d*x + c)*a^5*b*p - 120*a^6)*(sqrt
(d*x + c)*b + a)^p/((p^6 + 21*p^5 + 175*p^4 + 735*p^3 + 1624*p^2 + 1764*p + 720)
*b^6))/d^3

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Fricas [A]  time = 0.359207, size = 961, normalized size = 3.97 \[ \frac{2 \,{\left (120 \, b^{6} c^{3} - 360 \, a^{2} b^{4} c^{2} + 360 \, a^{4} b^{2} c - 120 \, a^{6} + 8 \,{\left (b^{6} c^{3} + 3 \, a^{2} b^{4} c^{2}\right )} p^{3} +{\left (b^{6} d^{3} p^{5} + 15 \, b^{6} d^{3} p^{4} + 85 \, b^{6} d^{3} p^{3} + 225 \, b^{6} d^{3} p^{2} + 274 \, b^{6} d^{3} p + 120 \, b^{6} d^{3}\right )} x^{3} + 24 \,{\left (3 \, b^{6} c^{3} + 3 \, a^{2} b^{4} c^{2} - 2 \, a^{4} b^{2} c\right )} p^{2} +{\left (b^{6} c d^{2} p^{5} +{\left (11 \, b^{6} c - 5 \, a^{2} b^{4}\right )} d^{2} p^{4} +{\left (41 \, b^{6} c - 30 \, a^{2} b^{4}\right )} d^{2} p^{3} +{\left (61 \, b^{6} c - 55 \, a^{2} b^{4}\right )} d^{2} p^{2} + 30 \,{\left (b^{6} c - a^{2} b^{4}\right )} d^{2} p\right )} x^{2} + 8 \,{\left (23 \, b^{6} c^{3} - 24 \, a^{2} b^{4} c^{2} + 9 \, a^{4} b^{2} c\right )} p - 4 \,{\left ({\left (b^{6} c^{2} + a^{2} b^{4} c\right )} d p^{4} + 3 \,{\left (3 \, b^{6} c^{2} - a^{2} b^{4} c\right )} d p^{3} +{\left (23 \, b^{6} c^{2} - 34 \, a^{2} b^{4} c + 15 \, a^{4} b^{2}\right )} d p^{2} + 15 \,{\left (b^{6} c^{2} - 2 \, a^{2} b^{4} c + a^{4} b^{2}\right )} d p\right )} x +{\left (8 \,{\left (3 \, a b^{5} c^{2} + a^{3} b^{3} c\right )} p^{3} + 24 \,{\left (7 \, a b^{5} c^{2} - 3 \, a^{3} b^{3} c\right )} p^{2} +{\left (a b^{5} d^{2} p^{5} + 10 \, a b^{5} d^{2} p^{4} + 35 \, a b^{5} d^{2} p^{3} + 50 \, a b^{5} d^{2} p^{2} + 24 \, a b^{5} d^{2} p\right )} x^{2} + 8 \,{\left (33 \, a b^{5} c^{2} - 40 \, a^{3} b^{3} c + 15 \, a^{5} b\right )} p - 4 \,{\left (2 \, a b^{5} c d p^{4} + 5 \,{\left (3 \, a b^{5} c - a^{3} b^{3}\right )} d p^{3} +{\left (31 \, a b^{5} c - 15 \, a^{3} b^{3}\right )} d p^{2} + 2 \,{\left (9 \, a b^{5} c - 5 \, a^{3} b^{3}\right )} d p\right )} x\right )} \sqrt{d x + c}\right )}{\left (\sqrt{d x + c} b + a\right )}^{p}}{b^{6} d^{3} p^{6} + 21 \, b^{6} d^{3} p^{5} + 175 \, b^{6} d^{3} p^{4} + 735 \, b^{6} d^{3} p^{3} + 1624 \, b^{6} d^{3} p^{2} + 1764 \, b^{6} d^{3} p + 720 \, b^{6} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(120*b^6*c^3 - 360*a^2*b^4*c^2 + 360*a^4*b^2*c - 120*a^6 + 8*(b^6*c^3 + 3*a^2*
b^4*c^2)*p^3 + (b^6*d^3*p^5 + 15*b^6*d^3*p^4 + 85*b^6*d^3*p^3 + 225*b^6*d^3*p^2
+ 274*b^6*d^3*p + 120*b^6*d^3)*x^3 + 24*(3*b^6*c^3 + 3*a^2*b^4*c^2 - 2*a^4*b^2*c
)*p^2 + (b^6*c*d^2*p^5 + (11*b^6*c - 5*a^2*b^4)*d^2*p^4 + (41*b^6*c - 30*a^2*b^4
)*d^2*p^3 + (61*b^6*c - 55*a^2*b^4)*d^2*p^2 + 30*(b^6*c - a^2*b^4)*d^2*p)*x^2 +
8*(23*b^6*c^3 - 24*a^2*b^4*c^2 + 9*a^4*b^2*c)*p - 4*((b^6*c^2 + a^2*b^4*c)*d*p^4
+ 3*(3*b^6*c^2 - a^2*b^4*c)*d*p^3 + (23*b^6*c^2 - 34*a^2*b^4*c + 15*a^4*b^2)*d*
p^2 + 15*(b^6*c^2 - 2*a^2*b^4*c + a^4*b^2)*d*p)*x + (8*(3*a*b^5*c^2 + a^3*b^3*c)
*p^3 + 24*(7*a*b^5*c^2 - 3*a^3*b^3*c)*p^2 + (a*b^5*d^2*p^5 + 10*a*b^5*d^2*p^4 +
35*a*b^5*d^2*p^3 + 50*a*b^5*d^2*p^2 + 24*a*b^5*d^2*p)*x^2 + 8*(33*a*b^5*c^2 - 40
*a^3*b^3*c + 15*a^5*b)*p - 4*(2*a*b^5*c*d*p^4 + 5*(3*a*b^5*c - a^3*b^3)*d*p^3 +
(31*a*b^5*c - 15*a^3*b^3)*d*p^2 + 2*(9*a*b^5*c - 5*a^3*b^3)*d*p)*x)*sqrt(d*x + c
))*(sqrt(d*x + c)*b + a)^p/(b^6*d^3*p^6 + 21*b^6*d^3*p^5 + 175*b^6*d^3*p^4 + 735
*b^6*d^3*p^3 + 1624*b^6*d^3*p^2 + 1764*b^6*d^3*p + 720*b^6*d^3)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 2.27939, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done