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

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 = 0.0401228, 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, Rules used = {247, 190, 43} \[ \frac{2 \left (a+b \sqrt{c+d x}\right )^{p+2}}{b^2 d (p+2)}-\frac{2 a \left (a+b \sqrt{c+d x}\right )^{p+1}}{b^2 d (p+1)} \]

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))

Rule 247

Int[((a_.) + (b_.)*(v_)^(n_))^(p_), x_Symbol] :> Dist[1/Coefficient[v, x, 1], Subst[Int[(a + b*x^n)^p, x], x,
v], x] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && NeQ[v, x]

Rule 190

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(1/n - 1)*(a + b*x)^p, x], x, x^n], x] /
; FreeQ[{a, b, p}, x] && FractionQ[n] && IntegerQ[1/n]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \left (a+b \sqrt{c+d x}\right )^p \, dx &=\frac{\operatorname{Subst}\left (\int \left (a+b \sqrt{x}\right )^p \, dx,x,c+d x\right )}{d}\\ &=\frac{2 \operatorname{Subst}\left (\int x (a+b x)^p \, dx,x,\sqrt{c+d x}\right )}{d}\\ &=\frac{2 \operatorname{Subst}\left (\int \left (-\frac{a (a+b x)^p}{b}+\frac{(a+b x)^{1+p}}{b}\right ) \, dx,x,\sqrt{c+d x}\right )}{d}\\ &=-\frac{2 a \left (a+b \sqrt{c+d x}\right )^{1+p}}{b^2 d (1+p)}+\frac{2 \left (a+b \sqrt{c+d x}\right )^{2+p}}{b^2 d (2+p)}\\ \end{align*}

Mathematica [A]  time = 0.0358123, size = 53, normalized size = 0.85 \[ \frac{2 \left (a+b \sqrt{c+d x}\right )^{p+1} \left (b (p+1) \sqrt{c+d x}-a\right )}{b^2 d (p+1) (p+2)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[c + d*x])^p,x]

[Out]

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

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Maple [F]  time = 0.003, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\sqrt{dx+c} \right ) ^{p}\, dx \end{align*}

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 = 1.13776, size = 81, normalized size = 1.31 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*(d*x+c)^(1/2))^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 = 2.01997, size = 174, normalized size = 2.81 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*(d*x+c)^(1/2))^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. \begin{align*} \int \left (a + b \sqrt{c + d x}\right )^{p}\, dx \end{align*}

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 [B]  time = 1.4505, size = 817, normalized size = 13.18 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*(d*x+c)^(1/2))^p,x, algorithm="giac")

[Out]

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