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

Optimal. Leaf size=62 \[ \frac{2 \left (a+b \sqrt{c+d x}\right )^{p+2}}{b^2 d (p+2)}-\frac{2 a \left (a+b \sqrt{c+d x}\right )^{p+1}}{b^2 d (p+1)} \]

[Out]

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

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Rubi [A]  time = 0.0832109, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{2 \left (a+b \sqrt{c+d x}\right )^{p+2}}{b^2 d (p+2)}-\frac{2 a \left (a+b \sqrt{c+d x}\right )^{p+1}}{b^2 d (p+1)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[c + d*x])^p,x]

[Out]

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

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Rubi in Sympy [A]  time = 6.21274, size = 51, normalized size = 0.82 \[ - \frac{2 a \left (a + b \sqrt{c + d x}\right )^{p + 1}}{b^{2} d \left (p + 1\right )} + \frac{2 \left (a + b \sqrt{c + d x}\right )^{p + 2}}{b^{2} d \left (p + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0529063, size = 64, normalized size = 1.03 \[ \frac{2 \left (a+b \sqrt{c+d x}\right )^p \left (-a^2+a b p \sqrt{c+d x}+b^2 (p+1) (c+d x)\right )}{b^2 d (p+1) (p+2)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.007, size = 0, normalized size = 0. \[ \int \left ( a+b\sqrt{dx+c} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*(d*x+c)^(1/2))^p,x)

[Out]

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

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Maxima [A]  time = 0.706817, size = 81, normalized size = 1.31 \[ \frac{2 \,{\left ({\left (d x + c\right )} b^{2}{\left (p + 1\right )} + \sqrt{d x + c} a b p - a^{2}\right )}{\left (\sqrt{d x + c} b + a\right )}^{p}}{{\left (p^{2} + 3 \, p + 2\right )} b^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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Fricas [A]  time = 0.298043, size = 109, normalized size = 1.76 \[ \frac{2 \,{\left (b^{2} c p + \sqrt{d x + c} a b p + b^{2} c - a^{2} +{\left (b^{2} d p + b^{2} d\right )} x\right )}{\left (\sqrt{d x + c} b + a\right )}^{p}}{b^{2} d p^{2} + 3 \, b^{2} d p + 2 \, b^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(b^2*c*p + sqrt(d*x + c)*a*b*p + b^2*c - a^2 + (b^2*d*p + b^2*d)*x)*(sqrt(d*x
+ c)*b + a)^p/(b^2*d*p^2 + 3*b^2*d*p + 2*b^2*d)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((a + b*sqrt(c + d*x))**p, x)

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GIAC/XCAS [A]  time = 0.356475, size = 836, normalized size = 13.48 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*((sqrt(d*x + c)*b + a)*a*b*p*e^(p*ln((sqrt(d*x + c)*b + a)*sign((sqrt(d*x + c)
*b + a)*b - a*b) - a*sign((sqrt(d*x + c)*b + a)*b - a*b) + a))*sign((sqrt(d*x +
c)*b + a)*b - a*b) - a^2*b*p*e^(p*ln((sqrt(d*x + c)*b + a)*sign((sqrt(d*x + c)*b
+ a)*b - a*b) - a*sign((sqrt(d*x + c)*b + a)*b - a*b) + a))*sign((sqrt(d*x + c)
*b + a)*b - a*b) + (sqrt(d*x + c)*b + a)^2*b*p*e^(p*ln((sqrt(d*x + c)*b + a)*sig
n((sqrt(d*x + c)*b + a)*b - a*b) - a*sign((sqrt(d*x + c)*b + a)*b - a*b) + a)) -
2*(sqrt(d*x + c)*b + a)*a*b*p*e^(p*ln((sqrt(d*x + c)*b + a)*sign((sqrt(d*x + c)
*b + a)*b - a*b) - a*sign((sqrt(d*x + c)*b + a)*b - a*b) + a)) + a^2*b*p*e^(p*ln
((sqrt(d*x + c)*b + a)*sign((sqrt(d*x + c)*b + a)*b - a*b) - a*sign((sqrt(d*x +
c)*b + a)*b - a*b) + a)) + (sqrt(d*x + c)*b + a)^2*b*e^(p*ln((sqrt(d*x + c)*b +
a)*sign((sqrt(d*x + c)*b + a)*b - a*b) - a*sign((sqrt(d*x + c)*b + a)*b - a*b) +
a)) - 2*(sqrt(d*x + c)*b + a)*a*b*e^(p*ln((sqrt(d*x + c)*b + a)*sign((sqrt(d*x
+ c)*b + a)*b - a*b) - a*sign((sqrt(d*x + c)*b + a)*b - a*b) + a)))/((p^2*sign((
sqrt(d*x + c)*b + a)*b - a*b) + 3*p*sign((sqrt(d*x + c)*b + a)*b - a*b) + 2*sign
((sqrt(d*x + c)*b + a)*b - a*b))*b^3*d)