3.293 \(\int \frac {a+b \log (c x^n)}{(d+e x^2)^{3/2}} \, dx\)

Optimal. Leaf size=58 \[ \frac {x \left (a+b \log \left (c x^n\right )\right )}{d \sqrt {d+e x^2}}-\frac {b n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{d \sqrt {e}} \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2314, 217, 206} \[ \frac {x \left (a+b \log \left (c x^n\right )\right )}{d \sqrt {d+e x^2}}-\frac {b n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{d \sqrt {e}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 206

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

Rule 217

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 2314

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]

Rubi steps

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

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Mathematica [A]  time = 0.10, size = 70, normalized size = 1.21 \[ \frac {\frac {a x}{\sqrt {d+e x^2}}+\frac {b x \log \left (c x^n\right )}{\sqrt {d+e x^2}}-\frac {b n \log \left (\sqrt {e} \sqrt {d+e x^2}+e x\right )}{\sqrt {e}}}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.44, size = 172, normalized size = 2.97 \[ \left [\frac {{\left (b e n x^{2} + b d n\right )} \sqrt {e} \log \left (-2 \, e x^{2} + 2 \, \sqrt {e x^{2} + d} \sqrt {e} x - d\right ) + 2 \, {\left (b e n x \log \relax (x) + b e x \log \relax (c) + a e x\right )} \sqrt {e x^{2} + d}}{2 \, {\left (d e^{2} x^{2} + d^{2} e\right )}}, \frac {{\left (b e n x^{2} + b d n\right )} \sqrt {-e} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) + {\left (b e n x \log \relax (x) + b e x \log \relax (c) + a e x\right )} \sqrt {e x^{2} + d}}{d e^{2} x^{2} + d^{2} e}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.33, size = 0, normalized size = 0.00 \[ \int \frac {b \ln \left (c \,x^{n}\right )+a}{\left (e \,x^{2}+d \right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.65, size = 56, normalized size = 0.97 \[ -\frac {b n \operatorname {arsinh}\left (\frac {e x}{\sqrt {d e}}\right )}{d \sqrt {e}} + \frac {b x \log \left (c x^{n}\right )}{\sqrt {e x^{2} + d} d} + \frac {a x}{\sqrt {e x^{2} + d} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {a+b\,\ln \left (c\,x^n\right )}{{\left (e\,x^2+d\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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