3.621 \(\int (a+b \sqrt{c+d x})^2 \, dx\)

Optimal. Leaf size=41 \[ a^2 x+\frac{4 a b (c+d x)^{3/2}}{3 d}+\frac{b^2 (c+d x)^2}{2 d} \]

[Out]

a^2*x + (4*a*b*(c + d*x)^(3/2))/(3*d) + (b^2*(c + d*x)^2)/(2*d)

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Rubi [A]  time = 0.0303472, antiderivative size = 41, 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} \[ a^2 x+\frac{4 a b (c+d x)^{3/2}}{3 d}+\frac{b^2 (c+d x)^2}{2 d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[c + d*x])^2,x]

[Out]

a^2*x + (4*a*b*(c + d*x)^(3/2))/(3*d) + (b^2*(c + d*x)^2)/(2*d)

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 )^2 \, dx &=\frac{\operatorname{Subst}\left (\int \left (a+b \sqrt{x}\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac{2 \operatorname{Subst}\left (\int x (a+b x)^2 \, dx,x,\sqrt{c+d x}\right )}{d}\\ &=\frac{2 \operatorname{Subst}\left (\int \left (a^2 x+2 a b x^2+b^2 x^3\right ) \, dx,x,\sqrt{c+d x}\right )}{d}\\ &=a^2 x+\frac{4 a b (c+d x)^{3/2}}{3 d}+\frac{b^2 (c+d x)^2}{2 d}\\ \end{align*}

Mathematica [A]  time = 0.0270579, size = 40, normalized size = 0.98 \[ \frac{6 a^2 d x+8 a b (c+d x)^{3/2}+3 b^2 (c+d x)^2}{6 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[c + d*x])^2,x]

[Out]

(6*a^2*d*x + 8*a*b*(c + d*x)^(3/2) + 3*b^2*(c + d*x)^2)/(6*d)

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Maple [A]  time = 0.002, size = 35, normalized size = 0.9 \begin{align*}{b}^{2} \left ({\frac{d{x}^{2}}{2}}+cx \right ) +{\frac{4\,ab}{3\,d} \left ( dx+c \right ) ^{{\frac{3}{2}}}}+x{a}^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^2*(1/2*d*x^2+c*x)+4/3*a*b*(d*x+c)^(3/2)/d+x*a^2

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Maxima [A]  time = 1.1046, size = 47, normalized size = 1.15 \begin{align*} \frac{1}{2} \,{\left (d x^{2} + 2 \, c x\right )} b^{2} + a^{2} x + \frac{4 \,{\left (d x + c\right )}^{\frac{3}{2}} a b}{3 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(d*x^2 + 2*c*x)*b^2 + a^2*x + 4/3*(d*x + c)^(3/2)*a*b/d

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Fricas [A]  time = 1.93548, size = 109, normalized size = 2.66 \begin{align*} \frac{3 \, b^{2} d^{2} x^{2} + 6 \,{\left (b^{2} c + a^{2}\right )} d x + 8 \,{\left (a b d x + a b c\right )} \sqrt{d x + c}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(3*b^2*d^2*x^2 + 6*(b^2*c + a^2)*d*x + 8*(a*b*d*x + a*b*c)*sqrt(d*x + c))/d

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Sympy [A]  time = 0.190655, size = 68, normalized size = 1.66 \begin{align*} \begin{cases} a^{2} x + \frac{4 a b c \sqrt{c + d x}}{3 d} + \frac{4 a b x \sqrt{c + d x}}{3} + b^{2} c x + \frac{b^{2} d x^{2}}{2} & \text{for}\: d \neq 0 \\x \left (a + b \sqrt{c}\right )^{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**2*x + 4*a*b*c*sqrt(c + d*x)/(3*d) + 4*a*b*x*sqrt(c + d*x)/3 + b**2*c*x + b**2*d*x**2/2, Ne(d, 0)
), (x*(a + b*sqrt(c))**2, True))

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Giac [A]  time = 1.23808, size = 53, normalized size = 1.29 \begin{align*} \frac{3 \,{\left (d x + c\right )}^{2} b^{2} + 8 \,{\left (d x + c\right )}^{\frac{3}{2}} a b + 6 \,{\left (d x + c\right )} a^{2}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(3*(d*x + c)^2*b^2 + 8*(d*x + c)^(3/2)*a*b + 6*(d*x + c)*a^2)/d