3.5.70 \(\int \frac {a+b \log (c (d+e x^{2/3})^n)}{x^4} \, dx\) [470]

Optimal. Leaf size=123 \[ -\frac {2 b e n}{21 d x^{7/3}}+\frac {2 b e^2 n}{15 d^2 x^{5/3}}-\frac {2 b e^3 n}{9 d^3 x}+\frac {2 b e^4 n}{3 d^4 \sqrt [3]{x}}+\frac {2 b e^{9/2} n \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{3 d^{9/2}}-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )}{3 x^3} \]

[Out]

-2/21*b*e*n/d/x^(7/3)+2/15*b*e^2*n/d^2/x^(5/3)-2/9*b*e^3*n/d^3/x+2/3*b*e^4*n/d^4/x^(1/3)+2/3*b*e^(9/2)*n*arcta
n(x^(1/3)*e^(1/2)/d^(1/2))/d^(9/2)+1/3*(-a-b*ln(c*(d+e*x^(2/3))^n))/x^3

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Rubi [A]
time = 0.05, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2505, 348, 331, 211} \begin {gather*} -\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )}{3 x^3}+\frac {2 b e^{9/2} n \text {ArcTan}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{3 d^{9/2}}+\frac {2 b e^4 n}{3 d^4 \sqrt [3]{x}}-\frac {2 b e^3 n}{9 d^3 x}+\frac {2 b e^2 n}{15 d^2 x^{5/3}}-\frac {2 b e n}{21 d x^{7/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*(d + e*x^(2/3))^n])/x^4,x]

[Out]

(-2*b*e*n)/(21*d*x^(7/3)) + (2*b*e^2*n)/(15*d^2*x^(5/3)) - (2*b*e^3*n)/(9*d^3*x) + (2*b*e^4*n)/(3*d^4*x^(1/3))
+ (2*b*e^(9/2)*n*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]])/(3*d^(9/2)) - (a + b*Log[c*(d + e*x^(2/3))^n])/(3*x^3)

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 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

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 2505

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

Rubi steps

\begin {align*} \int \frac {a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )}{x^4} \, dx &=-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )}{3 x^3}+\frac {1}{9} (2 b e n) \int \frac {1}{\left (d+e x^{2/3}\right ) x^{10/3}} \, dx\\ &=-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )}{3 x^3}+\frac {1}{3} (2 b e n) \text {Subst}\left (\int \frac {1}{x^8 \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {2 b e n}{21 d x^{7/3}}-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )}{3 x^3}-\frac {\left (2 b e^2 n\right ) \text {Subst}\left (\int \frac {1}{x^6 \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{3 d}\\ &=-\frac {2 b e n}{21 d x^{7/3}}+\frac {2 b e^2 n}{15 d^2 x^{5/3}}-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )}{3 x^3}+\frac {\left (2 b e^3 n\right ) \text {Subst}\left (\int \frac {1}{x^4 \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{3 d^2}\\ &=-\frac {2 b e n}{21 d x^{7/3}}+\frac {2 b e^2 n}{15 d^2 x^{5/3}}-\frac {2 b e^3 n}{9 d^3 x}-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )}{3 x^3}-\frac {\left (2 b e^4 n\right ) \text {Subst}\left (\int \frac {1}{x^2 \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{3 d^3}\\ &=-\frac {2 b e n}{21 d x^{7/3}}+\frac {2 b e^2 n}{15 d^2 x^{5/3}}-\frac {2 b e^3 n}{9 d^3 x}+\frac {2 b e^4 n}{3 d^4 \sqrt [3]{x}}-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )}{3 x^3}+\frac {\left (2 b e^5 n\right ) \text {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{3 d^4}\\ &=-\frac {2 b e n}{21 d x^{7/3}}+\frac {2 b e^2 n}{15 d^2 x^{5/3}}-\frac {2 b e^3 n}{9 d^3 x}+\frac {2 b e^4 n}{3 d^4 \sqrt [3]{x}}+\frac {2 b e^{9/2} n \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{3 d^{9/2}}-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )}{3 x^3}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.01, size = 65, normalized size = 0.53 \begin {gather*} -\frac {a}{3 x^3}-\frac {2 b e n \, _2F_1\left (-\frac {7}{2},1;-\frac {5}{2};-\frac {e x^{2/3}}{d}\right )}{21 d x^{7/3}}-\frac {b \log \left (c \left (d+e x^{2/3}\right )^n\right )}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*(d + e*x^(2/3))^n])/x^4,x]

[Out]

-1/3*a/x^3 - (2*b*e*n*Hypergeometric2F1[-7/2, 1, -5/2, -((e*x^(2/3))/d)])/(21*d*x^(7/3)) - (b*Log[c*(d + e*x^(
2/3))^n])/(3*x^3)

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {a +b \ln \left (c \left (d +e \,x^{\frac {2}{3}}\right )^{n}\right )}{x^{4}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*(d+e*x^(2/3))^n))/x^4,x)

[Out]

int((a+b*ln(c*(d+e*x^(2/3))^n))/x^4,x)

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Maxima [A]
time = 0.51, size = 88, normalized size = 0.72 \begin {gather*} \frac {2}{315} \, b n {\left (\frac {105 \, \arctan \left (\frac {x^{\frac {1}{3}} e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\frac {7}{2}}}{d^{\frac {9}{2}}} + \frac {21 \, d^{2} x^{\frac {2}{3}} e - 35 \, d x^{\frac {4}{3}} e^{2} - 15 \, d^{3} + 105 \, x^{2} e^{3}}{d^{4} x^{\frac {7}{3}}}\right )} e - \frac {b \log \left ({\left (x^{\frac {2}{3}} e + d\right )}^{n} c\right )}{3 \, x^{3}} - \frac {a}{3 \, x^{3}} \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^4,x, algorithm="maxima")

[Out]

2/315*b*n*(105*arctan(x^(1/3)*e^(1/2)/sqrt(d))*e^(7/2)/d^(9/2) + (21*d^2*x^(2/3)*e - 35*d*x^(4/3)*e^2 - 15*d^3
+ 105*x^2*e^3)/(d^4*x^(7/3)))*e - 1/3*b*log((x^(2/3)*e + d)^n*c)/x^3 - 1/3*a/x^3

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Fricas [A]
time = 0.40, size = 303, normalized size = 2.46 \begin {gather*} \left [\frac {105 \, b n x^{3} \sqrt {-\frac {e}{d}} e^{4} \log \left (-\frac {2 \, d^{2} x \sqrt {-\frac {e}{d}} e + d^{3} - x^{2} e^{3} - 2 \, {\left (d x \sqrt {-\frac {e}{d}} e^{2} + d^{2} e\right )} x^{\frac {2}{3}} - 2 \, {\left (d^{3} \sqrt {-\frac {e}{d}} - d x e^{2}\right )} x^{\frac {1}{3}}}{d^{3} + x^{2} e^{3}}\right ) - 105 \, b d^{4} n \log \left (x^{\frac {2}{3}} e + d\right ) + 42 \, b d^{2} n x^{\frac {4}{3}} e^{2} - 70 \, b d n x^{2} e^{3} - 105 \, b d^{4} \log \left (c\right ) - 105 \, a d^{4} - 30 \, {\left (b d^{3} n e - 7 \, b n x^{2} e^{4}\right )} x^{\frac {2}{3}}}{315 \, d^{4} x^{3}}, -\frac {105 \, b d^{4} n \log \left (x^{\frac {2}{3}} e + d\right ) - \frac {210 \, b n x^{3} \arctan \left (\frac {x^{\frac {1}{3}} e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\frac {9}{2}}}{\sqrt {d}} - 42 \, b d^{2} n x^{\frac {4}{3}} e^{2} + 70 \, b d n x^{2} e^{3} + 105 \, b d^{4} \log \left (c\right ) + 105 \, a d^{4} + 30 \, {\left (b d^{3} n e - 7 \, b n x^{2} e^{4}\right )} x^{\frac {2}{3}}}{315 \, d^{4} x^{3}}\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^4,x, algorithm="fricas")

[Out]

[1/315*(105*b*n*x^3*sqrt(-e/d)*e^4*log(-(2*d^2*x*sqrt(-e/d)*e + d^3 - x^2*e^3 - 2*(d*x*sqrt(-e/d)*e^2 + d^2*e)
*x^(2/3) - 2*(d^3*sqrt(-e/d) - d*x*e^2)*x^(1/3))/(d^3 + x^2*e^3)) - 105*b*d^4*n*log(x^(2/3)*e + d) + 42*b*d^2*
n*x^(4/3)*e^2 - 70*b*d*n*x^2*e^3 - 105*b*d^4*log(c) - 105*a*d^4 - 30*(b*d^3*n*e - 7*b*n*x^2*e^4)*x^(2/3))/(d^4
*x^3), -1/315*(105*b*d^4*n*log(x^(2/3)*e + d) - 210*b*n*x^3*arctan(x^(1/3)*e^(1/2)/sqrt(d))*e^(9/2)/sqrt(d) -
42*b*d^2*n*x^(4/3)*e^2 + 70*b*d*n*x^2*e^3 + 105*b*d^4*log(c) + 105*a*d^4 + 30*(b*d^3*n*e - 7*b*n*x^2*e^4)*x^(2
/3))/(d^4*x^3)]

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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**4,x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3879 deep

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Giac [A]
time = 4.92, size = 94, normalized size = 0.76 \begin {gather*} \frac {1}{315} \, {\left (2 \, {\left (\frac {105 \, \arctan \left (\frac {x^{\frac {1}{3}} e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\frac {7}{2}}}{d^{\frac {9}{2}}} + \frac {21 \, d^{2} x^{\frac {2}{3}} e - 35 \, d x^{\frac {4}{3}} e^{2} - 15 \, d^{3} + 105 \, x^{2} e^{3}}{d^{4} x^{\frac {7}{3}}}\right )} e - \frac {105 \, \log \left (x^{\frac {2}{3}} e + d\right )}{x^{3}}\right )} b n - \frac {b \log \left (c\right )}{3 \, x^{3}} - \frac {a}{3 \, x^{3}} \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^4,x, algorithm="giac")

[Out]

1/315*(2*(105*arctan(x^(1/3)*e^(1/2)/sqrt(d))*e^(7/2)/d^(9/2) + (21*d^2*x^(2/3)*e - 35*d*x^(4/3)*e^2 - 15*d^3
+ 105*x^2*e^3)/(d^4*x^(7/3)))*e - 105*log(x^(2/3)*e + d)/x^3)*b*n - 1/3*b*log(c)/x^3 - 1/3*a/x^3

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {a+b\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )}{x^4} \,d x \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^4,x)

[Out]

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

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