3.2262 \(\int \left (a+b \sqrt{x}\right )^p x^3 \, dx\)

Optimal. Leaf size=204 \[ -\frac{2 a^7 \left (a+b \sqrt{x}\right )^{p+1}}{b^8 (p+1)}+\frac{14 a^6 \left (a+b \sqrt{x}\right )^{p+2}}{b^8 (p+2)}-\frac{42 a^5 \left (a+b \sqrt{x}\right )^{p+3}}{b^8 (p+3)}+\frac{70 a^4 \left (a+b \sqrt{x}\right )^{p+4}}{b^8 (p+4)}-\frac{70 a^3 \left (a+b \sqrt{x}\right )^{p+5}}{b^8 (p+5)}+\frac{42 a^2 \left (a+b \sqrt{x}\right )^{p+6}}{b^8 (p+6)}-\frac{14 a \left (a+b \sqrt{x}\right )^{p+7}}{b^8 (p+7)}+\frac{2 \left (a+b \sqrt{x}\right )^{p+8}}{b^8 (p+8)} \]

[Out]

(-2*a^7*(a + b*Sqrt[x])^(1 + p))/(b^8*(1 + p)) + (14*a^6*(a + b*Sqrt[x])^(2 + p)
)/(b^8*(2 + p)) - (42*a^5*(a + b*Sqrt[x])^(3 + p))/(b^8*(3 + p)) + (70*a^4*(a +
b*Sqrt[x])^(4 + p))/(b^8*(4 + p)) - (70*a^3*(a + b*Sqrt[x])^(5 + p))/(b^8*(5 + p
)) + (42*a^2*(a + b*Sqrt[x])^(6 + p))/(b^8*(6 + p)) - (14*a*(a + b*Sqrt[x])^(7 +
p))/(b^8*(7 + p)) + (2*(a + b*Sqrt[x])^(8 + p))/(b^8*(8 + p))

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Rubi [A]  time = 0.273027, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{2 a^7 \left (a+b \sqrt{x}\right )^{p+1}}{b^8 (p+1)}+\frac{14 a^6 \left (a+b \sqrt{x}\right )^{p+2}}{b^8 (p+2)}-\frac{42 a^5 \left (a+b \sqrt{x}\right )^{p+3}}{b^8 (p+3)}+\frac{70 a^4 \left (a+b \sqrt{x}\right )^{p+4}}{b^8 (p+4)}-\frac{70 a^3 \left (a+b \sqrt{x}\right )^{p+5}}{b^8 (p+5)}+\frac{42 a^2 \left (a+b \sqrt{x}\right )^{p+6}}{b^8 (p+6)}-\frac{14 a \left (a+b \sqrt{x}\right )^{p+7}}{b^8 (p+7)}+\frac{2 \left (a+b \sqrt{x}\right )^{p+8}}{b^8 (p+8)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])^p*x^3,x]

[Out]

(-2*a^7*(a + b*Sqrt[x])^(1 + p))/(b^8*(1 + p)) + (14*a^6*(a + b*Sqrt[x])^(2 + p)
)/(b^8*(2 + p)) - (42*a^5*(a + b*Sqrt[x])^(3 + p))/(b^8*(3 + p)) + (70*a^4*(a +
b*Sqrt[x])^(4 + p))/(b^8*(4 + p)) - (70*a^3*(a + b*Sqrt[x])^(5 + p))/(b^8*(5 + p
)) + (42*a^2*(a + b*Sqrt[x])^(6 + p))/(b^8*(6 + p)) - (14*a*(a + b*Sqrt[x])^(7 +
p))/(b^8*(7 + p)) + (2*(a + b*Sqrt[x])^(8 + p))/(b^8*(8 + p))

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Rubi in Sympy [A]  time = 50.0283, size = 184, normalized size = 0.9 \[ - \frac{2 a^{7} \left (a + b \sqrt{x}\right )^{p + 1}}{b^{8} \left (p + 1\right )} + \frac{14 a^{6} \left (a + b \sqrt{x}\right )^{p + 2}}{b^{8} \left (p + 2\right )} - \frac{42 a^{5} \left (a + b \sqrt{x}\right )^{p + 3}}{b^{8} \left (p + 3\right )} + \frac{70 a^{4} \left (a + b \sqrt{x}\right )^{p + 4}}{b^{8} \left (p + 4\right )} - \frac{70 a^{3} \left (a + b \sqrt{x}\right )^{p + 5}}{b^{8} \left (p + 5\right )} + \frac{42 a^{2} \left (a + b \sqrt{x}\right )^{p + 6}}{b^{8} \left (p + 6\right )} - \frac{14 a \left (a + b \sqrt{x}\right )^{p + 7}}{b^{8} \left (p + 7\right )} + \frac{2 \left (a + b \sqrt{x}\right )^{p + 8}}{b^{8} \left (p + 8\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**7*(a + b*sqrt(x))**(p + 1)/(b**8*(p + 1)) + 14*a**6*(a + b*sqrt(x))**(p +
2)/(b**8*(p + 2)) - 42*a**5*(a + b*sqrt(x))**(p + 3)/(b**8*(p + 3)) + 70*a**4*(a
+ b*sqrt(x))**(p + 4)/(b**8*(p + 4)) - 70*a**3*(a + b*sqrt(x))**(p + 5)/(b**8*(
p + 5)) + 42*a**2*(a + b*sqrt(x))**(p + 6)/(b**8*(p + 6)) - 14*a*(a + b*sqrt(x))
**(p + 7)/(b**8*(p + 7)) + 2*(a + b*sqrt(x))**(p + 8)/(b**8*(p + 8))

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Mathematica [A]  time = 0.34184, size = 265, normalized size = 1.3 \[ \frac{2 \left (a+b \sqrt{x}\right )^{p+1} \left (-5040 a^7+5040 a^6 b (p+1) \sqrt{x}-2520 a^5 b^2 \left (p^2+3 p+2\right ) x+840 a^4 b^3 \left (p^3+6 p^2+11 p+6\right ) x^{3/2}-210 a^3 b^4 \left (p^4+10 p^3+35 p^2+50 p+24\right ) x^2+42 a^2 b^5 \left (p^5+15 p^4+85 p^3+225 p^2+274 p+120\right ) x^{5/2}-7 a b^6 \left (p^6+21 p^5+175 p^4+735 p^3+1624 p^2+1764 p+720\right ) x^3+b^7 \left (p^7+28 p^6+322 p^5+1960 p^4+6769 p^3+13132 p^2+13068 p+5040\right ) x^{7/2}\right )}{b^8 (p+1) (p+2) (p+3) (p+4) (p+5) (p+6) (p+7) (p+8)} \]

Antiderivative was successfully verified.

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

[Out]

(2*(a + b*Sqrt[x])^(1 + p)*(-5040*a^7 + 5040*a^6*b*(1 + p)*Sqrt[x] - 2520*a^5*b^
2*(2 + 3*p + p^2)*x + 840*a^4*b^3*(6 + 11*p + 6*p^2 + p^3)*x^(3/2) - 210*a^3*b^4
*(24 + 50*p + 35*p^2 + 10*p^3 + p^4)*x^2 + 42*a^2*b^5*(120 + 274*p + 225*p^2 + 8
5*p^3 + 15*p^4 + p^5)*x^(5/2) - 7*a*b^6*(720 + 1764*p + 1624*p^2 + 735*p^3 + 175
*p^4 + 21*p^5 + p^6)*x^3 + b^7*(5040 + 13068*p + 13132*p^2 + 6769*p^3 + 1960*p^4
+ 322*p^5 + 28*p^6 + p^7)*x^(7/2)))/(b^8*(1 + p)*(2 + p)*(3 + p)*(4 + p)*(5 + p
)*(6 + p)*(7 + p)*(8 + p))

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Maple [F]  time = 0.022, size = 0, normalized size = 0. \[ \int{x}^{3} \left ( a+b\sqrt{x} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(a+b*x^(1/2))^p,x)

[Out]

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

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Maxima [A]  time = 1.48508, size = 386, normalized size = 1.89 \[ \frac{2 \,{\left ({\left (p^{7} + 28 \, p^{6} + 322 \, p^{5} + 1960 \, p^{4} + 6769 \, p^{3} + 13132 \, p^{2} + 13068 \, p + 5040\right )} b^{8} x^{4} +{\left (p^{7} + 21 \, p^{6} + 175 \, p^{5} + 735 \, p^{4} + 1624 \, p^{3} + 1764 \, p^{2} + 720 \, p\right )} a b^{7} x^{\frac{7}{2}} - 7 \,{\left (p^{6} + 15 \, p^{5} + 85 \, p^{4} + 225 \, p^{3} + 274 \, p^{2} + 120 \, p\right )} a^{2} b^{6} x^{3} + 42 \,{\left (p^{5} + 10 \, p^{4} + 35 \, p^{3} + 50 \, p^{2} + 24 \, p\right )} a^{3} b^{5} x^{\frac{5}{2}} - 210 \,{\left (p^{4} + 6 \, p^{3} + 11 \, p^{2} + 6 \, p\right )} a^{4} b^{4} x^{2} + 840 \,{\left (p^{3} + 3 \, p^{2} + 2 \, p\right )} a^{5} b^{3} x^{\frac{3}{2}} - 2520 \,{\left (p^{2} + p\right )} a^{6} b^{2} x + 5040 \, a^{7} b p \sqrt{x} - 5040 \, a^{8}\right )}{\left (b \sqrt{x} + a\right )}^{p}}{{\left (p^{8} + 36 \, p^{7} + 546 \, p^{6} + 4536 \, p^{5} + 22449 \, p^{4} + 67284 \, p^{3} + 118124 \, p^{2} + 109584 \, p + 40320\right )} b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*((p^7 + 28*p^6 + 322*p^5 + 1960*p^4 + 6769*p^3 + 13132*p^2 + 13068*p + 5040)*b
^8*x^4 + (p^7 + 21*p^6 + 175*p^5 + 735*p^4 + 1624*p^3 + 1764*p^2 + 720*p)*a*b^7*
x^(7/2) - 7*(p^6 + 15*p^5 + 85*p^4 + 225*p^3 + 274*p^2 + 120*p)*a^2*b^6*x^3 + 42
*(p^5 + 10*p^4 + 35*p^3 + 50*p^2 + 24*p)*a^3*b^5*x^(5/2) - 210*(p^4 + 6*p^3 + 11
*p^2 + 6*p)*a^4*b^4*x^2 + 840*(p^3 + 3*p^2 + 2*p)*a^5*b^3*x^(3/2) - 2520*(p^2 +
p)*a^6*b^2*x + 5040*a^7*b*p*sqrt(x) - 5040*a^8)*(b*sqrt(x) + 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)

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Fricas [A]  time = 0.328416, size = 618, normalized size = 3.03 \[ -\frac{2 \,{\left (5040 \, a^{8} -{\left (b^{8} p^{7} + 28 \, b^{8} p^{6} + 322 \, b^{8} p^{5} + 1960 \, b^{8} p^{4} + 6769 \, b^{8} p^{3} + 13132 \, b^{8} p^{2} + 13068 \, b^{8} p + 5040 \, b^{8}\right )} x^{4} + 7 \,{\left (a^{2} b^{6} p^{6} + 15 \, a^{2} b^{6} p^{5} + 85 \, a^{2} b^{6} p^{4} + 225 \, a^{2} b^{6} p^{3} + 274 \, a^{2} b^{6} p^{2} + 120 \, a^{2} b^{6} p\right )} x^{3} + 210 \,{\left (a^{4} b^{4} p^{4} + 6 \, a^{4} b^{4} p^{3} + 11 \, a^{4} b^{4} p^{2} + 6 \, a^{4} b^{4} p\right )} x^{2} + 2520 \,{\left (a^{6} b^{2} p^{2} + a^{6} b^{2} p\right )} x -{\left (5040 \, a^{7} b p +{\left (a b^{7} p^{7} + 21 \, a b^{7} p^{6} + 175 \, a b^{7} p^{5} + 735 \, a b^{7} p^{4} + 1624 \, a b^{7} p^{3} + 1764 \, a b^{7} p^{2} + 720 \, a b^{7} p\right )} x^{3} + 42 \,{\left (a^{3} b^{5} p^{5} + 10 \, a^{3} b^{5} p^{4} + 35 \, a^{3} b^{5} p^{3} + 50 \, a^{3} b^{5} p^{2} + 24 \, a^{3} b^{5} p\right )} x^{2} + 840 \,{\left (a^{5} b^{3} p^{3} + 3 \, a^{5} b^{3} p^{2} + 2 \, a^{5} b^{3} p\right )} x\right )} \sqrt{x}\right )}{\left (b \sqrt{x} + a\right )}^{p}}{b^{8} p^{8} + 36 \, b^{8} p^{7} + 546 \, b^{8} p^{6} + 4536 \, b^{8} p^{5} + 22449 \, b^{8} p^{4} + 67284 \, b^{8} p^{3} + 118124 \, b^{8} p^{2} + 109584 \, b^{8} p + 40320 \, b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(5040*a^8 - (b^8*p^7 + 28*b^8*p^6 + 322*b^8*p^5 + 1960*b^8*p^4 + 6769*b^8*p^3
+ 13132*b^8*p^2 + 13068*b^8*p + 5040*b^8)*x^4 + 7*(a^2*b^6*p^6 + 15*a^2*b^6*p^5
+ 85*a^2*b^6*p^4 + 225*a^2*b^6*p^3 + 274*a^2*b^6*p^2 + 120*a^2*b^6*p)*x^3 + 210
*(a^4*b^4*p^4 + 6*a^4*b^4*p^3 + 11*a^4*b^4*p^2 + 6*a^4*b^4*p)*x^2 + 2520*(a^6*b^
2*p^2 + a^6*b^2*p)*x - (5040*a^7*b*p + (a*b^7*p^7 + 21*a*b^7*p^6 + 175*a*b^7*p^5
+ 735*a*b^7*p^4 + 1624*a*b^7*p^3 + 1764*a*b^7*p^2 + 720*a*b^7*p)*x^3 + 42*(a^3*
b^5*p^5 + 10*a^3*b^5*p^4 + 35*a^3*b^5*p^3 + 50*a^3*b^5*p^2 + 24*a^3*b^5*p)*x^2 +
840*(a^5*b^3*p^3 + 3*a^5*b^3*p^2 + 2*a^5*b^3*p)*x)*sqrt(x))*(b*sqrt(x) + a)^p/(
b^8*p^8 + 36*b^8*p^7 + 546*b^8*p^6 + 4536*b^8*p^5 + 22449*b^8*p^4 + 67284*b^8*p^
3 + 118124*b^8*p^2 + 109584*b^8*p + 40320*b^8)

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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*x**(1/2))**p,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.267367, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done