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

Optimal. Leaf size=350 \[ -\frac{2 a \left (a^2-b^2 c\right )^3 \left (a+b \sqrt{c+d x}\right )^{p+1}}{b^8 d^4 (p+1)}+\frac{2 \left (a^2-b^2 c\right )^2 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{p+2}}{b^8 d^4 (p+2)}-\frac{6 a \left (7 a^2-3 b^2 c\right ) \left (a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{p+3}}{b^8 d^4 (p+3)}-\frac{10 a \left (7 a^2-3 b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{p+5}}{b^8 d^4 (p+5)}+\frac{6 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{p+6}}{b^8 d^4 (p+6)}+\frac{2 \left (35 a^4-30 a^2 b^2 c+3 b^4 c^2\right ) \left (a+b \sqrt{c+d x}\right )^{p+4}}{b^8 d^4 (p+4)}-\frac{14 a \left (a+b \sqrt{c+d x}\right )^{p+7}}{b^8 d^4 (p+7)}+\frac{2 \left (a+b \sqrt{c+d x}\right )^{p+8}}{b^8 d^4 (p+8)} \]

[Out]

(-2*a*(a^2 - b^2*c)^3*(a + b*Sqrt[c + d*x])^(1 + p))/(b^8*d^4*(1 + p)) + (2*(a^2
- b^2*c)^2*(7*a^2 - b^2*c)*(a + b*Sqrt[c + d*x])^(2 + p))/(b^8*d^4*(2 + p)) - (
6*a*(7*a^2 - 3*b^2*c)*(a^2 - b^2*c)*(a + b*Sqrt[c + d*x])^(3 + p))/(b^8*d^4*(3 +
p)) + (2*(35*a^4 - 30*a^2*b^2*c + 3*b^4*c^2)*(a + b*Sqrt[c + d*x])^(4 + p))/(b^
8*d^4*(4 + p)) - (10*a*(7*a^2 - 3*b^2*c)*(a + b*Sqrt[c + d*x])^(5 + p))/(b^8*d^4
*(5 + p)) + (6*(7*a^2 - b^2*c)*(a + b*Sqrt[c + d*x])^(6 + p))/(b^8*d^4*(6 + p))
- (14*a*(a + b*Sqrt[c + d*x])^(7 + p))/(b^8*d^4*(7 + p)) + (2*(a + b*Sqrt[c + d*
x])^(8 + p))/(b^8*d^4*(8 + p))

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Rubi [A]  time = 0.616953, antiderivative size = 350, 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 \[ -\frac{2 a \left (a^2-b^2 c\right )^3 \left (a+b \sqrt{c+d x}\right )^{p+1}}{b^8 d^4 (p+1)}+\frac{2 \left (a^2-b^2 c\right )^2 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{p+2}}{b^8 d^4 (p+2)}-\frac{6 a \left (7 a^2-3 b^2 c\right ) \left (a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{p+3}}{b^8 d^4 (p+3)}-\frac{10 a \left (7 a^2-3 b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{p+5}}{b^8 d^4 (p+5)}+\frac{6 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{p+6}}{b^8 d^4 (p+6)}+\frac{2 \left (35 a^4-30 a^2 b^2 c+3 b^4 c^2\right ) \left (a+b \sqrt{c+d x}\right )^{p+4}}{b^8 d^4 (p+4)}-\frac{14 a \left (a+b \sqrt{c+d x}\right )^{p+7}}{b^8 d^4 (p+7)}+\frac{2 \left (a+b \sqrt{c+d x}\right )^{p+8}}{b^8 d^4 (p+8)} \]

Antiderivative was successfully verified.

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

[Out]

(-2*a*(a^2 - b^2*c)^3*(a + b*Sqrt[c + d*x])^(1 + p))/(b^8*d^4*(1 + p)) + (2*(a^2
- b^2*c)^2*(7*a^2 - b^2*c)*(a + b*Sqrt[c + d*x])^(2 + p))/(b^8*d^4*(2 + p)) - (
6*a*(7*a^2 - 3*b^2*c)*(a^2 - b^2*c)*(a + b*Sqrt[c + d*x])^(3 + p))/(b^8*d^4*(3 +
p)) + (2*(35*a^4 - 30*a^2*b^2*c + 3*b^4*c^2)*(a + b*Sqrt[c + d*x])^(4 + p))/(b^
8*d^4*(4 + p)) - (10*a*(7*a^2 - 3*b^2*c)*(a + b*Sqrt[c + d*x])^(5 + p))/(b^8*d^4
*(5 + p)) + (6*(7*a^2 - b^2*c)*(a + b*Sqrt[c + d*x])^(6 + p))/(b^8*d^4*(6 + p))
- (14*a*(a + b*Sqrt[c + d*x])^(7 + p))/(b^8*d^4*(7 + p)) + (2*(a + b*Sqrt[c + d*
x])^(8 + p))/(b^8*d^4*(8 + p))

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Rubi in Sympy [A]  time = 49.9932, size = 320, normalized size = 0.91 \[ - \frac{2 a \left (a + b \sqrt{c + d x}\right )^{p + 1} \left (a^{2} - b^{2} c\right )^{3}}{b^{8} d^{4} \left (p + 1\right )} - \frac{6 a \left (a + b \sqrt{c + d x}\right )^{p + 3} \left (a^{2} - b^{2} c\right ) \left (7 a^{2} - 3 b^{2} c\right )}{b^{8} d^{4} \left (p + 3\right )} - \frac{10 a \left (a + b \sqrt{c + d x}\right )^{p + 5} \left (7 a^{2} - 3 b^{2} c\right )}{b^{8} d^{4} \left (p + 5\right )} - \frac{14 a \left (a + b \sqrt{c + d x}\right )^{p + 7}}{b^{8} d^{4} \left (p + 7\right )} + \frac{2 \left (a + b \sqrt{c + d x}\right )^{p + 2} \left (a^{2} - b^{2} c\right )^{2} \left (7 a^{2} - b^{2} c\right )}{b^{8} d^{4} \left (p + 2\right )} + \frac{2 \left (a + b \sqrt{c + d x}\right )^{p + 4} \left (35 a^{4} - 30 a^{2} b^{2} c + 3 b^{4} c^{2}\right )}{b^{8} d^{4} \left (p + 4\right )} + \frac{6 \left (a + b \sqrt{c + d x}\right )^{p + 6} \left (7 a^{2} - b^{2} c\right )}{b^{8} d^{4} \left (p + 6\right )} + \frac{2 \left (a + b \sqrt{c + d x}\right )^{p + 8}}{b^{8} d^{4} \left (p + 8\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(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)**3/(b**8*d**4*(p + 1)) - 6*a
*(a + b*sqrt(c + d*x))**(p + 3)*(a**2 - b**2*c)*(7*a**2 - 3*b**2*c)/(b**8*d**4*(
p + 3)) - 10*a*(a + b*sqrt(c + d*x))**(p + 5)*(7*a**2 - 3*b**2*c)/(b**8*d**4*(p
+ 5)) - 14*a*(a + b*sqrt(c + d*x))**(p + 7)/(b**8*d**4*(p + 7)) + 2*(a + b*sqrt(
c + d*x))**(p + 2)*(a**2 - b**2*c)**2*(7*a**2 - b**2*c)/(b**8*d**4*(p + 2)) + 2*
(a + b*sqrt(c + d*x))**(p + 4)*(35*a**4 - 30*a**2*b**2*c + 3*b**4*c**2)/(b**8*d*
*4*(p + 4)) + 6*(a + b*sqrt(c + d*x))**(p + 6)*(7*a**2 - b**2*c)/(b**8*d**4*(p +
6)) + 2*(a + b*sqrt(c + d*x))**(p + 8)/(b**8*d**4*(p + 8))

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Mathematica [A]  time = 1.48302, size = 554, normalized size = 1.58 \[ -\frac{2 \left (a+b \sqrt{c+d x}\right )^{p+1} \left (5040 a^7-5040 a^6 b (p+1) \sqrt{c+d x}-360 a^5 b^2 \left (-6 c \left (p^2+p-7\right )-7 d \left (p^2+3 p+2\right ) x\right )+120 a^4 b^3 (p+1) \sqrt{c+d x} \left (c \left (-4 p^2+10 p+126\right )-7 d \left (p^2+5 p+6\right ) x\right )+6 a^3 b^4 \left (8 c^2 \left (p^4-14 p^3-139 p^2-124 p+315\right )+40 c d \left (p^4+4 p^3-16 p^2-61 p-42\right ) x+35 d^2 \left (p^4+10 p^3+35 p^2+50 p+24\right ) x^2\right )-6 a^2 b^5 (p+1) \sqrt{c+d x} \left (-24 c^2 \left (p^3+5 p^2-24 p-105\right )+4 c d \left (p^4-p^3-94 p^2-386 p-420\right ) x+7 d^2 \left (p^4+14 p^3+71 p^2+154 p+120\right ) x^2\right )+a b^6 \left (48 c^3 \left (3 p^4+38 p^3+138 p^2+103 p-105\right )-24 c^2 d \left (2 p^5+24 p^4+74 p^3-21 p^2-283 p-210\right ) x+6 c d^2 \left (p^6+11 p^5+10 p^4-265 p^3-1151 p^2-1726 p-840\right ) x^2+7 d^3 \left (p^6+21 p^5+175 p^4+735 p^3+1624 p^2+1764 p+720\right ) x^3\right )+b^7 \left (p^4+16 p^3+86 p^2+176 p+105\right ) \sqrt{c+d x} \left (48 c^3-24 c^2 d (p+2) x+6 c d^2 \left (p^2+6 p+8\right ) x^2-d^3 \left (p^3+12 p^2+44 p+48\right ) x^3\right )\right )}{b^8 d^4 (p+1) (p+2) (p+3) (p+4) (p+5) (p+6) (p+7) (p+8)} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(a + b*Sqrt[c + d*x])^(1 + p)*(5040*a^7 - 5040*a^6*b*(1 + p)*Sqrt[c + d*x] -
360*a^5*b^2*(-6*c*(-7 + p + p^2) - 7*d*(2 + 3*p + p^2)*x) + 120*a^4*b^3*(1 + p)
*Sqrt[c + d*x]*(c*(126 + 10*p - 4*p^2) - 7*d*(6 + 5*p + p^2)*x) + 6*a^3*b^4*(8*c
^2*(315 - 124*p - 139*p^2 - 14*p^3 + p^4) + 40*c*d*(-42 - 61*p - 16*p^2 + 4*p^3
+ p^4)*x + 35*d^2*(24 + 50*p + 35*p^2 + 10*p^3 + p^4)*x^2) - 6*a^2*b^5*(1 + p)*S
qrt[c + d*x]*(-24*c^2*(-105 - 24*p + 5*p^2 + p^3) + 4*c*d*(-420 - 386*p - 94*p^2
- p^3 + p^4)*x + 7*d^2*(120 + 154*p + 71*p^2 + 14*p^3 + p^4)*x^2) + b^7*(105 +
176*p + 86*p^2 + 16*p^3 + p^4)*Sqrt[c + d*x]*(48*c^3 - 24*c^2*d*(2 + p)*x + 6*c*
d^2*(8 + 6*p + p^2)*x^2 - d^3*(48 + 44*p + 12*p^2 + p^3)*x^3) + a*b^6*(48*c^3*(-
105 + 103*p + 138*p^2 + 38*p^3 + 3*p^4) - 24*c^2*d*(-210 - 283*p - 21*p^2 + 74*p
^3 + 24*p^4 + 2*p^5)*x + 6*c*d^2*(-840 - 1726*p - 1151*p^2 - 265*p^3 + 10*p^4 +
11*p^5 + p^6)*x^2 + 7*d^3*(720 + 1764*p + 1624*p^2 + 735*p^3 + 175*p^4 + 21*p^5
+ p^6)*x^3)))/(b^8*d^4*(1 + p)*(2 + p)*(3 + p)*(4 + p)*(5 + p)*(6 + p)*(7 + p)*(
8 + p))

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Maple [F]  time = 0.008, size = 0, normalized size = 0. \[ \int{x}^{3} \left ( a+b\sqrt{dx+c} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.723975, size = 983, normalized size = 2.81 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((sqrt(d*x + c)*b + a)^p*x^3,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^3/((p^2 + 3*p + 2)*b^2) - 3*((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^2/((p^4 + 10*p^3 + 35*p^2 + 50*
p + 24)*b^4) + 3*((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*c/((p^6 + 21*p^5 + 175*p^4 + 735*p^3 + 1624*p^2 + 1764*p
+ 720)*b^6) - ((p^7 + 28*p^6 + 322*p^5 + 1960*p^4 + 6769*p^3 + 13132*p^2 + 1306
8*p + 5040)*(d*x + c)^4*b^8 + (p^7 + 21*p^6 + 175*p^5 + 735*p^4 + 1624*p^3 + 176
4*p^2 + 720*p)*(d*x + c)^(7/2)*a*b^7 - 7*(p^6 + 15*p^5 + 85*p^4 + 225*p^3 + 274*
p^2 + 120*p)*(d*x + c)^3*a^2*b^6 + 42*(p^5 + 10*p^4 + 35*p^3 + 50*p^2 + 24*p)*(d
*x + c)^(5/2)*a^3*b^5 - 210*(p^4 + 6*p^3 + 11*p^2 + 6*p)*(d*x + c)^2*a^4*b^4 + 8
40*(p^3 + 3*p^2 + 2*p)*(d*x + c)^(3/2)*a^5*b^3 - 2520*(p^2 + p)*(d*x + c)*a^6*b^
2 + 5040*sqrt(d*x + c)*a^7*b*p - 5040*a^8)*(sqrt(d*x + c)*b + a)^p/((p^8 + 36*p^
7 + 546*p^6 + 4536*p^5 + 22449*p^4 + 67284*p^3 + 118124*p^2 + 109584*p + 40320)*
b^8))/d^4

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Fricas [A]  time = 0.439962, size = 1912, normalized size = 5.46 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(5040*b^8*c^4 - 20160*a^2*b^6*c^3 + 30240*a^4*b^4*c^2 - 20160*a^6*b^2*c + 504
0*a^8 + 48*(b^8*c^4 + 6*a^2*b^6*c^3 + a^4*b^4*c^2)*p^4 - (b^8*d^4*p^7 + 28*b^8*d
^4*p^6 + 322*b^8*d^4*p^5 + 1960*b^8*d^4*p^4 + 6769*b^8*d^4*p^3 + 13132*b^8*d^4*p
^2 + 13068*b^8*d^4*p + 5040*b^8*d^4)*x^4 + 384*(2*b^8*c^4 + 7*a^2*b^6*c^3 - 3*a^
4*b^4*c^2)*p^3 - (b^8*c*d^3*p^7 + (22*b^8*c - 7*a^2*b^6)*d^3*p^6 + 5*(38*b^8*c -
21*a^2*b^6)*d^3*p^5 + 5*(164*b^8*c - 119*a^2*b^6)*d^3*p^4 + (1849*b^8*c - 1575*
a^2*b^6)*d^3*p^3 + 2*(1019*b^8*c - 959*a^2*b^6)*d^3*p^2 + 840*(b^8*c - a^2*b^6)*
d^3*p)*x^3 + 48*(86*b^8*c^4 + 81*a^2*b^6*c^3 - 124*a^4*b^4*c^2 + 45*a^6*b^2*c)*p
^2 + 6*(18*b^8*c^2*d^2*p^5 + (b^8*c^2 + a^2*b^6*c)*d^2*p^6 + (118*b^8*c^2 - 95*a
^2*b^6*c + 35*a^4*b^4)*d^2*p^4 + 6*(58*b^8*c^2 - 80*a^2*b^6*c + 35*a^4*b^4)*d^2*
p^3 + (457*b^8*c^2 - 806*a^2*b^6*c + 385*a^4*b^4)*d^2*p^2 + 210*(b^8*c^2 - 2*a^2
*b^6*c + a^4*b^4)*d^2*p)*x^2 + 192*(44*b^8*c^4 - 71*a^2*b^6*c^3 + 54*a^4*b^4*c^2
- 15*a^6*b^2*c)*p - 24*((b^8*c^3 + 3*a^2*b^6*c^2)*d*p^5 + 2*(8*b^8*c^3 + 9*a^2*
b^6*c^2 - 5*a^4*b^4*c)*d*p^4 + (86*b^8*c^3 - 57*a^2*b^6*c^2 + 15*a^4*b^4*c)*d*p^
3 + (176*b^8*c^3 - 387*a^2*b^6*c^2 + 340*a^4*b^4*c - 105*a^6*b^2)*d*p^2 + 105*(b
^8*c^3 - 3*a^2*b^6*c^2 + 3*a^4*b^4*c - a^6*b^2)*d*p)*x + (192*(a*b^7*c^3 + a^3*b
^5*c^2)*p^4 + 96*(27*a*b^7*c^3 + 2*a^3*b^5*c^2 - 5*a^5*b^3*c)*p^3 - (a*b^7*d^3*p
^7 + 21*a*b^7*d^3*p^6 + 175*a*b^7*d^3*p^5 + 735*a*b^7*d^3*p^4 + 1624*a*b^7*d^3*p
^3 + 1764*a*b^7*d^3*p^2 + 720*a*b^7*d^3*p)*x^3 + 192*(56*a*b^7*c^3 - 49*a^3*b^5*
c^2 + 15*a^5*b^3*c)*p^2 + 6*(2*a*b^7*c*d^2*p^6 + (33*a*b^7*c - 7*a^3*b^5)*d^2*p^
5 + 10*(20*a*b^7*c - 7*a^3*b^5)*d^2*p^4 + 5*(111*a*b^7*c - 49*a^3*b^5)*d^2*p^3 +
2*(349*a*b^7*c - 175*a^3*b^5)*d^2*p^2 + 24*(13*a*b^7*c - 7*a^3*b^5)*d^2*p)*x^2
+ 48*(279*a*b^7*c^3 - 511*a^3*b^5*c^2 + 385*a^5*b^3*c - 105*a^7*b)*p - 24*((3*a*
b^7*c^2 + a^3*b^5*c)*d*p^5 + 2*(21*a*b^7*c^2 - 5*a^3*b^5*c)*d*p^4 + (192*a*b^7*c
^2 - 135*a^3*b^5*c + 35*a^5*b^3)*d*p^3 + (327*a*b^7*c^2 - 320*a^3*b^5*c + 105*a^
5*b^3)*d*p^2 + 2*(87*a*b^7*c^2 - 98*a^3*b^5*c + 35*a^5*b^3)*d*p)*x)*sqrt(d*x + c
))*(sqrt(d*x + c)*b + a)^p/(b^8*d^4*p^8 + 36*b^8*d^4*p^7 + 546*b^8*d^4*p^6 + 453
6*b^8*d^4*p^5 + 22449*b^8*d^4*p^4 + 67284*b^8*d^4*p^3 + 118124*b^8*d^4*p^2 + 109
584*b^8*d^4*p + 40320*b^8*d^4)

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

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError