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

Optimal. Leaf size=145 \[ -\frac{2 a \left (a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{p+1}}{b^4 d^2 (p+1)}+\frac{2 \left (3 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{p+2}}{b^4 d^2 (p+2)}-\frac{6 a \left (a+b \sqrt{c+d x}\right )^{p+3}}{b^4 d^2 (p+3)}+\frac{2 \left (a+b \sqrt{c+d x}\right )^{p+4}}{b^4 d^2 (p+4)} \]

[Out]

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

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Rubi [A]  time = 0.241775, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ -\frac{2 a \left (a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{p+1}}{b^4 d^2 (p+1)}+\frac{2 \left (3 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{p+2}}{b^4 d^2 (p+2)}-\frac{6 a \left (a+b \sqrt{c+d x}\right )^{p+3}}{b^4 d^2 (p+3)}+\frac{2 \left (a+b \sqrt{c+d x}\right )^{p+4}}{b^4 d^2 (p+4)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 17.4348, size = 129, normalized size = 0.89 \[ - \frac{2 a \left (a + b \sqrt{c + d x}\right )^{p + 1} \left (a^{2} - b^{2} c\right )}{b^{4} d^{2} \left (p + 1\right )} - \frac{6 a \left (a + b \sqrt{c + d x}\right )^{p + 3}}{b^{4} d^{2} \left (p + 3\right )} + \frac{2 \left (a + b \sqrt{c + d x}\right )^{p + 2} \left (3 a^{2} - b^{2} c\right )}{b^{4} d^{2} \left (p + 2\right )} + \frac{2 \left (a + b \sqrt{c + d x}\right )^{p + 4}}{b^{4} d^{2} \left (p + 4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(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)/(b**4*d**2*(p + 1)) - 6*a*(a
+ b*sqrt(c + d*x))**(p + 3)/(b**4*d**2*(p + 3)) + 2*(a + b*sqrt(c + d*x))**(p +
2)*(3*a**2 - b**2*c)/(b**4*d**2*(p + 2)) + 2*(a + b*sqrt(c + d*x))**(p + 4)/(b*
*4*d**2*(p + 4))

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Mathematica [A]  time = 0.292388, size = 128, normalized size = 0.88 \[ -\frac{2 \left (a+b \sqrt{c+d x}\right )^{p+1} \left (6 a^3-6 a^2 b (p+1) \sqrt{c+d x}+a b^2 \left (2 c \left (p^2+p-3\right )+3 d \left (p^2+3 p+2\right ) x\right )-b^3 \left (p^2+4 p+3\right ) \sqrt{c+d x} (d (p+2) x-2 c)\right )}{b^4 d^2 (p+1) (p+2) (p+3) (p+4)} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(a+b*(d*x+c)^(1/2))^p,x)

[Out]

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

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Maxima [A]  time = 0.71969, size = 252, normalized size = 1.74 \[ -\frac{2 \,{\left (\frac{{\left ({\left (d x + c\right )} b^{2}{\left (p + 1\right )} + \sqrt{d x + c} a b p - a^{2}\right )}{\left (\sqrt{d x + c} b + a\right )}^{p} c}{{\left (p^{2} + 3 \, p + 2\right )} b^{2}} - \frac{{\left ({\left (p^{3} + 6 \, p^{2} + 11 \, p + 6\right )}{\left (d x + c\right )}^{2} b^{4} +{\left (p^{3} + 3 \, p^{2} + 2 \, p\right )}{\left (d x + c\right )}^{\frac{3}{2}} a b^{3} - 3 \,{\left (p^{2} + p\right )}{\left (d x + c\right )} a^{2} b^{2} + 6 \, \sqrt{d x + c} a^{3} b p - 6 \, a^{4}\right )}{\left (\sqrt{d x + c} b + a\right )}^{p}}{{\left (p^{4} + 10 \, p^{3} + 35 \, p^{2} + 50 \, p + 24\right )} b^{4}}\right )}}{d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((sqrt(d*x + c)*b + a)^p*x,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/((p^2 + 3*p + 2)*b^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/((p^4 + 10*p^3 + 35*p^2 + 50*p + 24)*
b^4))/d^2

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Fricas [A]  time = 0.318172, size = 397, normalized size = 2.74 \[ -\frac{2 \,{\left (6 \, b^{4} c^{2} - 12 \, a^{2} b^{2} c + 6 \, a^{4} + 2 \,{\left (b^{4} c^{2} + a^{2} b^{2} c\right )} p^{2} -{\left (b^{4} d^{2} p^{3} + 6 \, b^{4} d^{2} p^{2} + 11 \, b^{4} d^{2} p + 6 \, b^{4} d^{2}\right )} x^{2} + 4 \,{\left (2 \, b^{4} c^{2} - a^{2} b^{2} c\right )} p -{\left (b^{4} c d p^{3} +{\left (4 \, b^{4} c - 3 \, a^{2} b^{2}\right )} d p^{2} + 3 \,{\left (b^{4} c - a^{2} b^{2}\right )} d p\right )} x +{\left (4 \, a b^{3} c p^{2} + 2 \,{\left (5 \, a b^{3} c - 3 \, a^{3} b\right )} p -{\left (a b^{3} d p^{3} + 3 \, a b^{3} d p^{2} + 2 \, a b^{3} d p\right )} x\right )} \sqrt{d x + c}\right )}{\left (\sqrt{d x + c} b + a\right )}^{p}}{b^{4} d^{2} p^{4} + 10 \, b^{4} d^{2} p^{3} + 35 \, b^{4} d^{2} p^{2} + 50 \, b^{4} d^{2} p + 24 \, b^{4} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(6*b^4*c^2 - 12*a^2*b^2*c + 6*a^4 + 2*(b^4*c^2 + a^2*b^2*c)*p^2 - (b^4*d^2*p^
3 + 6*b^4*d^2*p^2 + 11*b^4*d^2*p + 6*b^4*d^2)*x^2 + 4*(2*b^4*c^2 - a^2*b^2*c)*p
- (b^4*c*d*p^3 + (4*b^4*c - 3*a^2*b^2)*d*p^2 + 3*(b^4*c - a^2*b^2)*d*p)*x + (4*a
*b^3*c*p^2 + 2*(5*a*b^3*c - 3*a^3*b)*p - (a*b^3*d*p^3 + 3*a*b^3*d*p^2 + 2*a*b^3*
d*p)*x)*sqrt(d*x + c))*(sqrt(d*x + c)*b + a)^p/(b^4*d^2*p^4 + 10*b^4*d^2*p^3 + 3
5*b^4*d^2*p^2 + 50*b^4*d^2*p + 24*b^4*d^2)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.858036, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done