3.5.66 \(\int (a+b \log (c (d+e x^{2/3})^n)) \, dx\) [466]

Optimal. Leaf size=72 \[ \frac {2 b d n \sqrt [3]{x}}{e}+a x-\frac {2 b n x}{3}-\frac {2 b d^{3/2} n \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{e^{3/2}}+b x \log \left (c \left (d+e x^{2/3}\right )^n\right ) \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2498, 348, 308, 211} \begin {gather*} a x-\frac {2 b d^{3/2} n \text {ArcTan}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{e^{3/2}}+b x \log \left (c \left (d+e x^{2/3}\right )^n\right )+\frac {2 b d n \sqrt [3]{x}}{e}-\frac {2 b n x}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[a + b*Log[c*(d + e*x^(2/3))^n],x]

[Out]

(2*b*d*n*x^(1/3))/e + a*x - (2*b*n*x)/3 - (2*b*d^(3/2)*n*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]])/e^(3/2) + b*x*Log[
c*(d + e*x^(2/3))^n]

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 308

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 348

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

Rule 2498

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 x^{2/3}\right )^n\right )\right ) \, dx &=a x+b \int \log \left (c \left (d+e x^{2/3}\right )^n\right ) \, dx\\ &=a x+b x \log \left (c \left (d+e x^{2/3}\right )^n\right )-\frac {1}{3} (2 b e n) \int \frac {x^{2/3}}{d+e x^{2/3}} \, dx\\ &=a x+b x \log \left (c \left (d+e x^{2/3}\right )^n\right )-(2 b e n) \text {Subst}\left (\int \frac {x^4}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )\\ &=a x+b x \log \left (c \left (d+e x^{2/3}\right )^n\right )-(2 b e n) \text {Subst}\left (\int \left (-\frac {d}{e^2}+\frac {x^2}{e}+\frac {d^2}{e^2 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {2 b d n \sqrt [3]{x}}{e}+a x-\frac {2 b n x}{3}+b x \log \left (c \left (d+e x^{2/3}\right )^n\right )-\frac {\left (2 b d^2 n\right ) \text {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{e}\\ &=\frac {2 b d n \sqrt [3]{x}}{e}+a x-\frac {2 b n x}{3}-\frac {2 b d^{3/2} n \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{e^{3/2}}+b x \log \left (c \left (d+e x^{2/3}\right )^n\right )\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 72, normalized size = 1.00 \begin {gather*} \frac {2 b d n \sqrt [3]{x}}{e}+a x-\frac {2 b n x}{3}-\frac {2 b d^{3/2} n \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{e^{3/2}}+b x \log \left (c \left (d+e x^{2/3}\right )^n\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[a + b*Log[c*(d + e*x^(2/3))^n],x]

[Out]

(2*b*d*n*x^(1/3))/e + a*x - (2*b*n*x)/3 - (2*b*d^(3/2)*n*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]])/e^(3/2) + b*x*Log[
c*(d + e*x^(2/3))^n]

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Maple [A]
time = 0.03, size = 62, normalized size = 0.86

method	result	size
default	\(a x +b x \ln \left (c \left (d +e \,x^{\frac {2}{3}}\right )^{n}\right )-\frac {2 b n x}{3}+\frac {2 b d n \,x^{\frac {1}{3}}}{e}-\frac {2 b n \,d^{2} \arctan \left (\frac {x^{\frac {1}{3}} e}{\sqrt {e d}}\right )}{e \sqrt {e d}}\)	\(62\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a+b*ln(c*(d+e*x^(2/3))^n),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.51, size = 61, normalized size = 0.85 \begin {gather*} -\frac {1}{3} \, {\left (2 \, {\left (3 \, d^{\frac {3}{2}} \arctan \left (\frac {x^{\frac {1}{3}} e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {5}{2}\right )} + {\left (x e - 3 \, d x^{\frac {1}{3}}\right )} e^{\left (-2\right )}\right )} n e - 3 \, x \log \left ({\left (x^{\frac {2}{3}} e + d\right )}^{n} c\right )\right )} b + a x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*log(c*(d+e*x^(2/3))^n),x, algorithm="maxima")

[Out]

-1/3*(2*(3*d^(3/2)*arctan(x^(1/3)*e^(1/2)/sqrt(d))*e^(-5/2) + (x*e - 3*d*x^(1/3))*e^(-2))*n*e - 3*x*log((x^(2/
3)*e + d)^n*c))*b + a*x

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Fricas [A]
time = 0.38, size = 218, normalized size = 3.03 \begin {gather*} \left [\frac {1}{3} \, {\left (3 \, b n x e \log \left (x^{\frac {2}{3}} e + d\right ) + 3 \, \sqrt {-d e^{\left (-1\right )}} b d n \log \left (\frac {2 \, \sqrt {-d e^{\left (-1\right )}} d x e^{2} - d^{3} + x^{2} e^{3} + 2 \, {\left (d^{2} e - \sqrt {-d e^{\left (-1\right )}} x e^{3}\right )} x^{\frac {2}{3}} - 2 \, {\left (\sqrt {-d e^{\left (-1\right )}} d^{2} e + d x e^{2}\right )} x^{\frac {1}{3}}}{d^{3} + x^{2} e^{3}}\right ) + 3 \, b x e \log \left (c\right ) + 6 \, b d n x^{\frac {1}{3}} - {\left (2 \, b n - 3 \, a\right )} x e\right )} e^{\left (-1\right )}, -\frac {1}{3} \, {\left (6 \, b d^{\frac {3}{2}} n \arctan \left (\frac {x^{\frac {1}{3}} e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {1}{2}\right )} - 3 \, b n x e \log \left (x^{\frac {2}{3}} e + d\right ) - 3 \, b x e \log \left (c\right ) - 6 \, b d n x^{\frac {1}{3}} + {\left (2 \, b n - 3 \, a\right )} x e\right )} e^{\left (-1\right )}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*log(c*(d+e*x^(2/3))^n),x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 2.40, size = 124, normalized size = 1.72 \begin {gather*} a x + b \left (- \frac {2 e n \left (\begin {cases} \tilde {\infty } x & \text {for}\: d = 0 \wedge e = 0 \\\frac {3 x^{\frac {5}{3}}}{5 d} & \text {for}\: e = 0 \\\frac {x}{e} & \text {for}\: d = 0 \\\frac {3 d^{2} \log {\left (\sqrt [3]{x} - \sqrt {- \frac {d}{e}} \right )}}{2 e^{3} \sqrt {- \frac {d}{e}}} - \frac {3 d^{2} \log {\left (\sqrt [3]{x} + \sqrt {- \frac {d}{e}} \right )}}{2 e^{3} \sqrt {- \frac {d}{e}}} - \frac {3 d \sqrt [3]{x}}{e^{2}} + \frac {x}{e} & \text {otherwise} \end {cases}\right )}{3} + x \log {\left (c \left (d + e x^{\frac {2}{3}}\right )^{n} \right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*ln(c*(d+e*x**(2/3))**n),x)

[Out]

a*x + b*(-2*e*n*Piecewise((zoo*x, Eq(d, 0) & Eq(e, 0)), (3*x**(5/3)/(5*d), Eq(e, 0)), (x/e, Eq(d, 0)), (3*d**2
*log(x**(1/3) - sqrt(-d/e))/(2*e**3*sqrt(-d/e)) - 3*d**2*log(x**(1/3) + sqrt(-d/e))/(2*e**3*sqrt(-d/e)) - 3*d*
x**(1/3)/e**2 + x/e, True))/3 + x*log(c*(d + e*x**(2/3))**n))

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Giac [A]
time = 4.11, size = 68, normalized size = 0.94 \begin {gather*} -\frac {1}{3} \, {\left ({\left (2 \, {\left (3 \, d^{\frac {3}{2}} \arctan \left (\frac {x^{\frac {1}{3}} e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {5}{2}\right )} - {\left (3 \, d x^{\frac {1}{3}} e - x e^{2}\right )} e^{\left (-3\right )}\right )} e - 3 \, x \log \left (x^{\frac {2}{3}} e + d\right )\right )} n - 3 \, x \log \left (c\right )\right )} b + a x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*log(c*(d+e*x^(2/3))^n),x, algorithm="giac")

[Out]

-1/3*((2*(3*d^(3/2)*arctan(x^(1/3)*e^(1/2)/sqrt(d))*e^(-5/2) - (3*d*x^(1/3)*e - x*e^2)*e^(-3))*e - 3*x*log(x^(
2/3)*e + d))*n - 3*x*log(c))*b + a*x

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Mupad [B]
time = 0.39, size = 56, normalized size = 0.78 \begin {gather*} a\,x+b\,x\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )-\frac {2\,b\,n\,x}{3}+\frac {2\,b\,d\,n\,x^{1/3}}{e}-\frac {2\,b\,d^{3/2}\,n\,\mathrm {atan}\left (\frac {\sqrt {e}\,x^{1/3}}{\sqrt {d}}\right )}{e^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a + b*log(c*(d + e*x^(2/3))^n),x)

[Out]

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

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