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]

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

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Rubi [A]  time = 0.18, antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, 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 2295

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

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 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 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 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 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]

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*}

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Mathematica [A]  time = 0.08, 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)

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fricas [A]  time = 0.44, size = 225, normalized size = 1.15 \[ \frac {2 \, b^{2} e^{2} x \log \relax (c)^{2} + 2 \, {\left (b^{2} e^{2} n^{2} x - b^{2} d^{2} n^{2}\right )} \log \left (e \sqrt {x} + d\right )^{2} - 2 \, {\left (b^{2} e^{2} n - 2 \, a b e^{2}\right )} x \log \relax (c) + {\left (b^{2} e^{2} n^{2} - 2 \, a b e^{2} n + 2 \, a^{2} e^{2}\right )} x + 2 \, {\left (2 \, b^{2} d e n^{2} \sqrt {x} + 3 \, b^{2} d^{2} n^{2} - 2 \, a b d^{2} n - {\left (b^{2} e^{2} n^{2} - 2 \, a b e^{2} n\right )} x + 2 \, {\left (b^{2} e^{2} n x - b^{2} d^{2} n\right )} \log \relax (c)\right )} \log \left (e \sqrt {x} + d\right ) - 2 \, {\left (3 \, b^{2} d e n^{2} - 2 \, b^{2} d e n \log \relax (c) - 2 \, a b d e n\right )} \sqrt {x}}{2 \, e^{2}} \]

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

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giac [B]  time = 0.19, size = 361, normalized size = 1.85 \[ \frac {1}{2} \, {\left ({\left (2 \, {\left (\sqrt {x} e + d\right )}^{2} \log \left (\sqrt {x} e + d\right )^{2} - 4 \, {\left (\sqrt {x} e + d\right )} d \log \left (\sqrt {x} e + d\right )^{2} - 2 \, {\left (\sqrt {x} e + d\right )}^{2} \log \left (\sqrt {x} e + d\right ) + 8 \, {\left (\sqrt {x} e + d\right )} d \log \left (\sqrt {x} e + d\right ) + {\left (\sqrt {x} e + d\right )}^{2} - 8 \, {\left (\sqrt {x} e + d\right )} d\right )} b^{2} n^{2} e^{\left (-1\right )} + 2 \, {\left (2 \, {\left (\sqrt {x} e + d\right )}^{2} \log \left (\sqrt {x} e + d\right ) - 4 \, {\left (\sqrt {x} e + d\right )} d \log \left (\sqrt {x} e + d\right ) - {\left (\sqrt {x} e + d\right )}^{2} + 4 \, {\left (\sqrt {x} e + d\right )} d\right )} b^{2} n e^{\left (-1\right )} \log \relax (c) + 2 \, {\left ({\left (\sqrt {x} e + d\right )}^{2} - 2 \, {\left (\sqrt {x} e + d\right )} d\right )} b^{2} e^{\left (-1\right )} \log \relax (c)^{2} + 2 \, {\left (2 \, {\left (\sqrt {x} e + d\right )}^{2} \log \left (\sqrt {x} e + d\right ) - 4 \, {\left (\sqrt {x} e + d\right )} d \log \left (\sqrt {x} e + d\right ) - {\left (\sqrt {x} e + d\right )}^{2} + 4 \, {\left (\sqrt {x} e + d\right )} d\right )} a b n e^{\left (-1\right )} + 4 \, {\left ({\left (\sqrt {x} e + d\right )}^{2} - 2 \, {\left (\sqrt {x} e + d\right )} d\right )} a b e^{\left (-1\right )} \log \relax (c) + 2 \, {\left ({\left (\sqrt {x} e + d\right )}^{2} - 2 \, {\left (\sqrt {x} e + d\right )} d\right )} a^{2} e^{\left (-1\right )}\right )} e^{\left (-1\right )} \]

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)

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maple [F]  time = 0.08, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \left (e \sqrt {x}+d \right )^{n}\right )+a \right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.54, size = 179, normalized size = 0.92 \[ -{\left (e n {\left (\frac {2 \, d^{2} \log \left (e \sqrt {x} + d\right )}{e^{3}} + \frac {e x - 2 \, d \sqrt {x}}{e^{2}}\right )} - 2 \, x \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right )\right )} a b - \frac {1}{2} \, {\left (2 \, e n {\left (\frac {2 \, d^{2} \log \left (e \sqrt {x} + d\right )}{e^{3}} + \frac {e x - 2 \, d \sqrt {x}}{e^{2}}\right )} \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) - 2 \, x \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right )^{2} - \frac {{\left (2 \, d^{2} \log \left (e \sqrt {x} + d\right )^{2} + e^{2} x + 6 \, d^{2} \log \left (e \sqrt {x} + d\right ) - 6 \, d e \sqrt {x}\right )} n^{2}}{e^{2}}\right )} b^{2} + a^{2} x \]

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

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mupad [B]  time = 0.47, size = 186, normalized size = 0.95 \[ x\,\left (a^2-a\,b\,n+\frac {b^2\,n^2}{2}\right )-\sqrt {x}\,\left (\frac {d\,\left (2\,a^2-2\,a\,b\,n+b^2\,n^2\right )}{e}-\frac {2\,d\,\left (a^2-b^2\,n^2\right )}{e}\right )+{\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}^2\,\left (b^2\,x-\frac {b^2\,d^2}{e^2}\right )-\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )\,\left (\sqrt {x}\,\left (\frac {2\,b\,d\,\left (2\,a-b\,n\right )}{e}-\frac {4\,a\,b\,d}{e}\right )-b\,x\,\left (2\,a-b\,n\right )\right )+\frac {\ln \left (d+e\,\sqrt {x}\right )\,\left (3\,b^2\,d^2\,n^2-2\,a\,b\,d^2\,n\right )}{e^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \log {\left (c \left (d + e \sqrt {x}\right )^{n} \right )}\right )^{2}\, dx \]

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)

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