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

Optimal. Leaf size=60 \[ a x+b x \log \left (c \left (d+e \sqrt {x}\right )^n\right )-\frac {b d^2 n \log \left (d+e \sqrt {x}\right )}{e^2}+\frac {b d n \sqrt {x}}{e}-\frac {b n x}{2} \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2448, 266, 43} \[ a x+b x \log \left (c \left (d+e \sqrt {x}\right )^n\right )-\frac {b d^2 n \log \left (d+e \sqrt {x}\right )}{e^2}+\frac {b d n \sqrt {x}}{e}-\frac {b n x}{2} \]

Antiderivative was successfully verified.

[In]

Int[a + b*Log[c*(d + e*Sqrt[x])^n],x]

[Out]

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

Rule 266

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

Rule 2448

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

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 60, normalized size = 1.00 \[ a x+b x \log \left (c \left (d+e \sqrt {x}\right )^n\right )-\frac {b d^2 n \log \left (d+e \sqrt {x}\right )}{e^2}+\frac {b d n \sqrt {x}}{e}-\frac {b n x}{2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.42, size = 65, normalized size = 1.08 \[ \frac {2 \, b e^{2} x \log \relax (c) + 2 \, b d e n \sqrt {x} - {\left (b e^{2} n - 2 \, a e^{2}\right )} x + 2 \, {\left (b e^{2} n x - b d^{2} n\right )} \log \left (e \sqrt {x} + d\right )}{2 \, e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.17, size = 107, normalized size = 1.78 \[ \frac {1}{2} \, {\left ({\left (2 \, {\left (\sqrt {x} e + d\right )}^{2} \log \left (\sqrt {x} e + d\right ) - 4 \, {\left (\sqrt {x} e + d\right )} d \log \left (\sqrt {x} e + d\right ) - {\left (\sqrt {x} e + d\right )}^{2} + 4 \, {\left (\sqrt {x} e + d\right )} d\right )} n e^{\left (-1\right )} + 2 \, {\left ({\left (\sqrt {x} e + d\right )}^{2} - 2 \, {\left (\sqrt {x} e + d\right )} d\right )} e^{\left (-1\right )} \log \relax (c)\right )} b e^{\left (-1\right )} + a x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.08, size = 53, normalized size = 0.88 \[ -\frac {b \,d^{2} n \ln \left (e \sqrt {x}+d \right )}{e^{2}}-\frac {b n x}{2}+b x \ln \left (c \left (e \sqrt {x}+d \right )^{n}\right )+\frac {b d n \sqrt {x}}{e}+a x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.47, size = 57, normalized size = 0.95 \[ -\frac {1}{2} \, {\left (e n {\left (\frac {2 \, d^{2} \log \left (e \sqrt {x} + d\right )}{e^{3}} + \frac {e x - 2 \, d \sqrt {x}}{e^{2}}\right )} - 2 \, x \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right )\right )} b + a x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(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))*b + a*x

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mupad [B]  time = 0.43, size = 52, normalized size = 0.87 \[ a\,x+b\,x\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )-\frac {b\,n\,\left (e^2\,x+2\,d^2\,\ln \left (d+e\,\sqrt {x}\right )-2\,d\,e\,\sqrt {x}\right )}{2\,e^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a + b*log(c*(d + e*x^(1/2))^n),x)

[Out]

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

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sympy [A]  time = 1.86, size = 66, normalized size = 1.10 \[ a x + b \left (- \frac {e n \left (\frac {2 d^{2} \left (\begin {cases} \frac {\sqrt {x}}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e \sqrt {x} \right )}}{e} & \text {otherwise} \end {cases}\right )}{e^{2}} - \frac {2 d \sqrt {x}}{e^{2}} + \frac {x}{e}\right )}{2} + x \log {\left (c \left (d + e \sqrt {x}\right )^{n} \right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x + b*(-e*n*(2*d**2*Piecewise((sqrt(x)/d, Eq(e, 0)), (log(d + e*sqrt(x))/e, True))/e**2 - 2*d*sqrt(x)/e**2 +
x/e)/2 + x*log(c*(d + e*sqrt(x))**n))

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