3.410 \(\int (a+b \log (c (d+e \sqrt{x})^n))^2 \, dx\)

Optimal. Leaf size=195 \[ -\frac{b n \left (d+e \sqrt{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{e^2}+\frac{\left (d+e \sqrt{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{e^2}-\frac{2 d \left (d+e \sqrt{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{e^2}+\frac{4 a b d n \sqrt{x}}{e}+\frac{4 b^2 d n \left (d+e \sqrt{x}\right ) \log \left (c \left (d+e \sqrt{x}\right )^n\right )}{e^2}+\frac{b^2 n^2 \left (d+e \sqrt{x}\right )^2}{2 e^2}-\frac{4 b^2 d n^2 \sqrt{x}}{e} \]

[Out]

(b^2*n^2*(d + e*Sqrt[x])^2)/(2*e^2) + (4*a*b*d*n*Sqrt[x])/e - (4*b^2*d*n^2*Sqrt[x])/e + (4*b^2*d*n*(d + e*Sqrt
[x])*Log[c*(d + e*Sqrt[x])^n])/e^2 - (b*n*(d + e*Sqrt[x])^2*(a + b*Log[c*(d + e*Sqrt[x])^n]))/e^2 - (2*d*(d +
e*Sqrt[x])*(a + b*Log[c*(d + e*Sqrt[x])^n])^2)/e^2 + ((d + e*Sqrt[x])^2*(a + b*Log[c*(d + e*Sqrt[x])^n])^2)/e^
2

________________________________________________________________________________________

Rubi [A]  time = 0.183552, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {2451, 2401, 2389, 2296, 2295, 2390, 2305, 2304} \[ -\frac{b n \left (d+e \sqrt{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{e^2}+\frac{\left (d+e \sqrt{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{e^2}-\frac{2 d \left (d+e \sqrt{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{e^2}+\frac{4 a b d n \sqrt{x}}{e}+\frac{4 b^2 d n \left (d+e \sqrt{x}\right ) \log \left (c \left (d+e \sqrt{x}\right )^n\right )}{e^2}+\frac{b^2 n^2 \left (d+e \sqrt{x}\right )^2}{2 e^2}-\frac{4 b^2 d n^2 \sqrt{x}}{e} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*(d + e*Sqrt[x])^n])^2,x]

[Out]

(b^2*n^2*(d + e*Sqrt[x])^2)/(2*e^2) + (4*a*b*d*n*Sqrt[x])/e - (4*b^2*d*n^2*Sqrt[x])/e + (4*b^2*d*n*(d + e*Sqrt
[x])*Log[c*(d + e*Sqrt[x])^n])/e^2 - (b*n*(d + e*Sqrt[x])^2*(a + b*Log[c*(d + e*Sqrt[x])^n]))/e^2 - (2*d*(d +
e*Sqrt[x])*(a + b*Log[c*(d + e*Sqrt[x])^n])^2)/e^2 + ((d + e*Sqrt[x])^2*(a + b*Log[c*(d + e*Sqrt[x])^n])^2)/e^
2

Rule 2451

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_), x_Symbol] :> With[{k = Denominator[n]}, Di
st[k, Subst[Int[x^(k - 1)*(a + b*Log[c*(d + e*x^(k*n))^p])^q, x], x, x^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p,
q}, x] && FractionQ[n]

Rule 2401

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Int[Exp
andIntegrand[(f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[
e*f - d*g, 0] && IGtQ[q, 0]

Rule 2389

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2296

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2390

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[((f*x)/d)^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 2305

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Lo
g[c*x^n])^p)/(d*(m + 1)), x] - Dist[(b*n*p)/(m + 1), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rubi steps

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

Mathematica [A]  time = 0.0820895, size = 150, normalized size = 0.77 \[ \frac{-2 a^2 \left (d^2-e^2 x\right )+2 b \left (d+e \sqrt{x}\right ) \left (-2 a d+2 a e \sqrt{x}+3 b d n-b e n \sqrt{x}\right ) \log \left (c \left (d+e \sqrt{x}\right )^n\right )-2 a b n \left (d-e \sqrt{x}\right )^2-2 b^2 \left (d^2-e^2 x\right ) \log ^2\left (c \left (d+e \sqrt{x}\right )^n\right )+b^2 e n^2 \sqrt{x} \left (e \sqrt{x}-6 d\right )}{2 e^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*(d + e*Sqrt[x])^n])^2,x]

[Out]

(-2*a*b*n*(d - e*Sqrt[x])^2 + b^2*e*n^2*(-6*d + e*Sqrt[x])*Sqrt[x] - 2*a^2*(d^2 - e^2*x) + 2*b*(d + e*Sqrt[x])
*(-2*a*d + 3*b*d*n + 2*a*e*Sqrt[x] - b*e*n*Sqrt[x])*Log[c*(d + e*Sqrt[x])^n] - 2*b^2*(d^2 - e^2*x)*Log[c*(d +
e*Sqrt[x])^n]^2)/(2*e^2)

________________________________________________________________________________________

Maple [F]  time = 0.092, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\ln \left ( c \left ( d+e\sqrt{x} \right ) ^{n} \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.05363, size = 242, normalized size = 1.24 \begin{align*} -{\left (e n{\left (\frac{2 \, d^{2} \log \left (e \sqrt{x} + d\right )}{e^{3}} + \frac{e x - 2 \, d \sqrt{x}}{e^{2}}\right )} - 2 \, x \log \left ({\left (e \sqrt{x} + d\right )}^{n} c\right )\right )} a b - \frac{1}{2} \,{\left (2 \, e n{\left (\frac{2 \, d^{2} \log \left (e \sqrt{x} + d\right )}{e^{3}} + \frac{e x - 2 \, d \sqrt{x}}{e^{2}}\right )} \log \left ({\left (e \sqrt{x} + d\right )}^{n} c\right ) - 2 \, x \log \left ({\left (e \sqrt{x} + d\right )}^{n} c\right )^{2} - \frac{{\left (2 \, d^{2} \log \left (e \sqrt{x} + d\right )^{2} + e^{2} x + 6 \, d^{2} \log \left (e \sqrt{x} + d\right ) - 6 \, d e \sqrt{x}\right )} n^{2}}{e^{2}}\right )} b^{2} + a^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(e*n*(2*d^2*log(e*sqrt(x) + d)/e^3 + (e*x - 2*d*sqrt(x))/e^2) - 2*x*log((e*sqrt(x) + d)^n*c))*a*b - 1/2*(2*e*
n*(2*d^2*log(e*sqrt(x) + d)/e^3 + (e*x - 2*d*sqrt(x))/e^2)*log((e*sqrt(x) + d)^n*c) - 2*x*log((e*sqrt(x) + d)^
n*c)^2 - (2*d^2*log(e*sqrt(x) + d)^2 + e^2*x + 6*d^2*log(e*sqrt(x) + d) - 6*d*e*sqrt(x))*n^2/e^2)*b^2 + a^2*x

________________________________________________________________________________________

Fricas [A]  time = 1.84546, size = 509, normalized size = 2.61 \begin{align*} \frac{2 \, b^{2} e^{2} x \log \left (c\right )^{2} + 2 \,{\left (b^{2} e^{2} n^{2} x - b^{2} d^{2} n^{2}\right )} \log \left (e \sqrt{x} + d\right )^{2} - 2 \,{\left (b^{2} e^{2} n - 2 \, a b e^{2}\right )} x \log \left (c\right ) +{\left (b^{2} e^{2} n^{2} - 2 \, a b e^{2} n + 2 \, a^{2} e^{2}\right )} x + 2 \,{\left (2 \, b^{2} d e n^{2} \sqrt{x} + 3 \, b^{2} d^{2} n^{2} - 2 \, a b d^{2} n -{\left (b^{2} e^{2} n^{2} - 2 \, a b e^{2} n\right )} x + 2 \,{\left (b^{2} e^{2} n x - b^{2} d^{2} n\right )} \log \left (c\right )\right )} \log \left (e \sqrt{x} + d\right ) - 2 \,{\left (3 \, b^{2} d e n^{2} - 2 \, b^{2} d e n \log \left (c\right ) - 2 \, a b d e n\right )} \sqrt{x}}{2 \, e^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*b^2*e^2*x*log(c)^2 + 2*(b^2*e^2*n^2*x - b^2*d^2*n^2)*log(e*sqrt(x) + d)^2 - 2*(b^2*e^2*n - 2*a*b*e^2)*x
*log(c) + (b^2*e^2*n^2 - 2*a*b*e^2*n + 2*a^2*e^2)*x + 2*(2*b^2*d*e*n^2*sqrt(x) + 3*b^2*d^2*n^2 - 2*a*b*d^2*n -
(b^2*e^2*n^2 - 2*a*b*e^2*n)*x + 2*(b^2*e^2*n*x - b^2*d^2*n)*log(c))*log(e*sqrt(x) + d) - 2*(3*b^2*d*e*n^2 - 2
*b^2*d*e*n*log(c) - 2*a*b*d*e*n)*sqrt(x))/e^2

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \log{\left (c \left (d + e \sqrt{x}\right )^{n} \right )}\right )^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*log(c*(d + e*sqrt(x))**n))**2, x)

________________________________________________________________________________________

Giac [B]  time = 1.33501, size = 487, normalized size = 2.5 \begin{align*} \frac{1}{2} \,{\left ({\left (2 \,{\left (\sqrt{x} e + d\right )}^{2} \log \left (\sqrt{x} e + d\right )^{2} - 4 \,{\left (\sqrt{x} e + d\right )} d \log \left (\sqrt{x} e + d\right )^{2} - 2 \,{\left (\sqrt{x} e + d\right )}^{2} \log \left (\sqrt{x} e + d\right ) + 8 \,{\left (\sqrt{x} e + d\right )} d \log \left (\sqrt{x} e + d\right ) +{\left (\sqrt{x} e + d\right )}^{2} - 8 \,{\left (\sqrt{x} e + d\right )} d\right )} b^{2} n^{2} e^{\left (-1\right )} + 2 \,{\left (2 \,{\left (\sqrt{x} e + d\right )}^{2} \log \left (\sqrt{x} e + d\right ) - 4 \,{\left (\sqrt{x} e + d\right )} d \log \left (\sqrt{x} e + d\right ) -{\left (\sqrt{x} e + d\right )}^{2} + 4 \,{\left (\sqrt{x} e + d\right )} d\right )} b^{2} n e^{\left (-1\right )} \log \left (c\right ) + 2 \,{\left ({\left (\sqrt{x} e + d\right )}^{2} - 2 \,{\left (\sqrt{x} e + d\right )} d\right )} b^{2} e^{\left (-1\right )} \log \left (c\right )^{2} + 2 \,{\left (2 \,{\left (\sqrt{x} e + d\right )}^{2} \log \left (\sqrt{x} e + d\right ) - 4 \,{\left (\sqrt{x} e + d\right )} d \log \left (\sqrt{x} e + d\right ) -{\left (\sqrt{x} e + d\right )}^{2} + 4 \,{\left (\sqrt{x} e + d\right )} d\right )} a b n e^{\left (-1\right )} + 4 \,{\left ({\left (\sqrt{x} e + d\right )}^{2} - 2 \,{\left (\sqrt{x} e + d\right )} d\right )} a b e^{\left (-1\right )} \log \left (c\right ) + 2 \,{\left ({\left (\sqrt{x} e + d\right )}^{2} - 2 \,{\left (\sqrt{x} e + d\right )} d\right )} a^{2} e^{\left (-1\right )}\right )} e^{\left (-1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*((2*(sqrt(x)*e + d)^2*log(sqrt(x)*e + d)^2 - 4*(sqrt(x)*e + d)*d*log(sqrt(x)*e + d)^2 - 2*(sqrt(x)*e + d)^
2*log(sqrt(x)*e + d) + 8*(sqrt(x)*e + d)*d*log(sqrt(x)*e + d) + (sqrt(x)*e + d)^2 - 8*(sqrt(x)*e + d)*d)*b^2*n
^2*e^(-1) + 2*(2*(sqrt(x)*e + d)^2*log(sqrt(x)*e + d) - 4*(sqrt(x)*e + d)*d*log(sqrt(x)*e + d) - (sqrt(x)*e +
d)^2 + 4*(sqrt(x)*e + d)*d)*b^2*n*e^(-1)*log(c) + 2*((sqrt(x)*e + d)^2 - 2*(sqrt(x)*e + d)*d)*b^2*e^(-1)*log(c
)^2 + 2*(2*(sqrt(x)*e + d)^2*log(sqrt(x)*e + d) - 4*(sqrt(x)*e + d)*d*log(sqrt(x)*e + d) - (sqrt(x)*e + d)^2 +
4*(sqrt(x)*e + d)*d)*a*b*n*e^(-1) + 4*((sqrt(x)*e + d)^2 - 2*(sqrt(x)*e + d)*d)*a*b*e^(-1)*log(c) + 2*((sqrt(
x)*e + d)^2 - 2*(sqrt(x)*e + d)*d)*a^2*e^(-1))*e^(-1)