3.4.6 \(\int \frac {a+b \log (c x^n)}{(d+e x^2)^{5/2}} \, dx\) [306]

Optimal. Leaf size=113 \[ -\frac {b n x}{3 d^2 \sqrt {d+e x^2}}-\frac {2 b n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{3 d^2 \sqrt {e}}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^2 \sqrt {d+e x^2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.05, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {2360, 2351, 223, 212, 197} \begin {gather*} \frac {2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac {b n x}{3 d^2 \sqrt {d+e x^2}}-\frac {2 b n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{3 d^2 \sqrt {e}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*x^n])/(d + e*x^2)^(5/2),x]

[Out]

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

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 2351

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

Rule 2360

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Simp[(-x)*(d + e*x^2)^(q +
1)*((a + b*Log[c*x^n])/(2*d*(q + 1))), x] + (Dist[(2*q + 3)/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*(a + b*Log[
c*x^n]), x], x] + Dist[b*(n/(2*d*(q + 1))), Int[(d + e*x^2)^(q + 1), x], x]) /; FreeQ[{a, b, c, d, e, n}, x] &
& LtQ[q, -1]

Rubi steps

\begin {align*} \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^{5/2}} \, dx &=\frac {x \left (a+b \log \left (c x^n\right )\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^{3/2}} \, dx}{3 d}-\frac {(b n) \int \frac {1}{\left (d+e x^2\right )^{3/2}} \, dx}{3 d}\\ &=-\frac {b n x}{3 d^2 \sqrt {d+e x^2}}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^2 \sqrt {d+e x^2}}-\frac {(2 b n) \int \frac {1}{\sqrt {d+e x^2}} \, dx}{3 d^2}\\ &=-\frac {b n x}{3 d^2 \sqrt {d+e x^2}}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^2 \sqrt {d+e x^2}}-\frac {(2 b n) \text {Subst}\left (\int \frac {1}{1-e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{3 d^2}\\ &=-\frac {b n x}{3 d^2 \sqrt {d+e x^2}}-\frac {2 b n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{3 d^2 \sqrt {e}}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^2 \sqrt {d+e x^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.08, size = 116, normalized size = 1.03 \begin {gather*} \frac {\sqrt {e} x \left (-b n \left (d+e x^2\right )+a \left (3 d+2 e x^2\right )\right )+b \sqrt {e} x \left (3 d+2 e x^2\right ) \log \left (c x^n\right )-2 b n \left (d+e x^2\right )^{3/2} \log \left (e x+\sqrt {e} \sqrt {d+e x^2}\right )}{3 d^2 \sqrt {e} \left (d+e x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*x^n])/(d + e*x^2)^(5/2),x]

[Out]

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

________________________________________________________________________________________

Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {a +b \ln \left (c \,x^{n}\right )}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x^n))/(e*x^2+d)^(5/2),x)

[Out]

int((a+b*ln(c*x^n))/(e*x^2+d)^(5/2),x)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/(e*x^2+d)^(5/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.38, size = 167, normalized size = 1.48 \begin {gather*} \frac {{\left (b n x^{4} e^{2} + 2 \, b d n x^{2} e + b d^{2} n\right )} e^{\frac {1}{2}} \log \left (-2 \, x^{2} e + 2 \, \sqrt {x^{2} e + d} x e^{\frac {1}{2}} - d\right ) - {\left ({\left (b n - 2 \, a\right )} x^{3} e^{2} + {\left (b d n - 3 \, a d\right )} x e - {\left (2 \, b x^{3} e^{2} + 3 \, b d x e\right )} \log \left (c\right ) - {\left (2 \, b n x^{3} e^{2} + 3 \, b d n x e\right )} \log \left (x\right )\right )} \sqrt {x^{2} e + d}}{3 \, {\left (d^{2} x^{4} e^{3} + 2 \, d^{3} x^{2} e^{2} + d^{4} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/(e*x^2+d)^(5/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + b \log {\left (c x^{n} \right )}}{\left (d + e x^{2}\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*x**n))/(e*x**2+d)**(5/2),x)

[Out]

Integral((a + b*log(c*x**n))/(d + e*x**2)**(5/2), x)

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/(e*x^2+d)^(5/2),x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)/(x^2*e + d)^(5/2), x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {a+b\,\ln \left (c\,x^n\right )}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*log(c*x^n))/(d + e*x^2)^(5/2),x)

[Out]

int((a + b*log(c*x^n))/(d + e*x^2)^(5/2), x)

________________________________________________________________________________________