∫ ∘ _________ a + b√ ___

Optimal. Leaf size=56 \[ -\frac {4 a \left (a+b \sqrt {c+d x}\right )^{3/2}}{3 b^2 d}+\frac {4 \left (a+b \sqrt {c+d x}\right )^{5/2}}{5 b^2 d} \]

-4/3*a*(a+b*(d*x+c)^(1/2))^(3/2)/b^2/d+4/5*(a+b*(d*x+c)^(1/2))^(5/2)/b^2/d

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Rubi [A]
time = 0.02, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {253, 196, 45} \begin {gather*} \frac {4 \left (a+b \sqrt {c+d x}\right )^{5/2}}{5 b^2 d}-\frac {4 a \left (a+b \sqrt {c+d x}\right )^{3/2}}{3 b^2 d} \end {gather*}

(-4*a*(a + b*Sqrt[c + d*x])^(3/2))/(3*b^2*d) + (4*(a + b*Sqrt[c + d*x])^(5/2))/(5*b^2*d)

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])

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] /

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

\begin {align*} \int \sqrt {a+b \sqrt {c+d x}} \, dx &=\frac {\text {Subst}\left (\int \sqrt {a+b \sqrt {x}} \, dx,x,c+d x\right )}{d}\\ &=\frac {2 \text {Subst}\left (\int x \sqrt {a+b x} \, dx,x,\sqrt {c+d x}\right )}{d}\\ &=\frac {2 \text {Subst}\left (\int \left (-\frac {a \sqrt {a+b x}}{b}+\frac {(a+b x)^{3/2}}{b}\right ) \, dx,x,\sqrt {c+d x}\right )}{d}\\ &=-\frac {4 a \left (a+b \sqrt {c+d x}\right )^{3/2}}{3 b^2 d}+\frac {4 \left (a+b \sqrt {c+d x}\right )^{5/2}}{5 b^2 d}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 55, normalized size = 0.98 \begin {gather*} \frac {4 \sqrt {a+b \sqrt {c+d x}} \left (-2 a^2+a b \sqrt {c+d x}+3 b^2 (c+d x)\right )}{15 b^2 d} \end {gather*}

(4*Sqrt[a + b*Sqrt[c + d*x]]*(-2*a^2 + a*b*Sqrt[c + d*x] + 3*b^2*(c + d*x)))/(15*b^2*d)

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method	result	size

derivativedivides	\(\frac {\frac {4 \left (a +b \sqrt {d x +c}\right )^{\frac {5}{2}}}{5}-\frac {4 a \left (a +b \sqrt {d x +c}\right )^{\frac {3}{2}}}{3}}{b^{2} d}\)	\(41\)
default	\(\frac {\frac {4 \left (a +b \sqrt {d x +c}\right )^{\frac {5}{2}}}{5}-\frac {4 a \left (a +b \sqrt {d x +c}\right )^{\frac {3}{2}}}{3}}{b^{2} d}\)	\(41\)

4/d/b^2*(1/5*(a+b*(d*x+c)^(1/2))^(5/2)-1/3*a*(a+b*(d*x+c)^(1/2))^(3/2))

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Maxima [A]
time = 0.29, size = 43, normalized size = 0.77 \begin {gather*} \frac {4 \, {\left (\frac {3 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}}}{b^{2}} - \frac {5 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} a}{b^{2}}\right )}}{15 \, d} \end {gather*}

4/15*(3*(sqrt(d*x + c)*b + a)^(5/2)/b^2 - 5*(sqrt(d*x + c)*b + a)^(3/2)*a/b^2)/d

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Fricas [A]
time = 0.42, size = 50, normalized size = 0.89 \begin {gather*} \frac {4 \, {\left (3 \, b^{2} d x + 3 \, b^{2} c + \sqrt {d x + c} a b - 2 \, a^{2}\right )} \sqrt {\sqrt {d x + c} b + a}}{15 \, b^{2} d} \end {gather*}

4/15*(3*b^2*d*x + 3*b^2*c + sqrt(d*x + c)*a*b - 2*a^2)*sqrt(sqrt(d*x + c)*b + a)/(b^2*d)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {a + b \sqrt {c + d x}}\, dx \end {gather*}

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (44) = 88\).
time = 5.40, size = 99, normalized size = 1.77 \begin {gather*} \frac {4 \, {\left (\frac {5 \, {\left ({\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {\sqrt {d x + c} b + a} a\right )} a}{b} + \frac {3 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} - 10 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {\sqrt {d x + c} b + a} a^{2}}{b}\right )}}{15 \, b d} \end {gather*}

4/15*(5*((sqrt(d*x + c)*b + a)^(3/2) - 3*sqrt(sqrt(d*x + c)*b + a)*a)*a/b + (3*(sqrt(d*x + c)*b + a)^(5/2) - 1

0*(sqrt(d*x + c)*b + a)^(3/2)*a + 15*sqrt(sqrt(d*x + c)*b + a)*a^2)/b)/(b*d)

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Mupad [B]
time = 3.39, size = 44, normalized size = 0.79 \begin {gather*} \frac {4\,{\left (a+b\,\sqrt {c+d\,x}\right )}^{5/2}}{5\,b^2\,d}-\frac {4\,a\,{\left (a+b\,\sqrt {c+d\,x}\right )}^{3/2}}{3\,b^2\,d} \end {gather*}

(4*(a + b*(c + d*x)^(1/2))^(5/2))/(5*b^2*d) - (4*a*(a + b*(c + d*x)^(1/2))^(3/2))/(3*b^2*d)

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3.7.28 \(\int \sqrt {a+b \sqrt {c+d x}} \, dx\) [628]