3.3.93 \(\int \frac {a+b \log (c x^n)}{(d+e x^2)^{3/2}} \, dx\) [293]

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

[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 = 0.02, 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 = {2351, 223, 212} \begin {gather*} \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}} \end {gather*}

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 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]

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) \text {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 = 0.06, size = 70, normalized size = 1.21 \begin {gather*} \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 (e x+\sqrt {e} \sqrt {d+e x^2}\right )}{\sqrt {e}}}{d} \end {gather*}

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 56, normalized size = 0.97 \begin {gather*} -\frac {b n \operatorname {arsinh}\left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {1}{2}\right )}}{d} + \frac {b x \log \left (c x^{n}\right )}{\sqrt {x^{2} e + d} d} + \frac {a x}{\sqrt {x^{2} e + d} d} \end {gather*}

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

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Fricas [A]
time = 0.39, size = 95, normalized size = 1.64 \begin {gather*} \frac {{\left (b n x^{2} e + b d 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 ) + 2 \, {\left (b n x e \log \left (x\right ) + b x e \log \left (c\right ) + a x e\right )} \sqrt {x^{2} e + d}}{2 \, {\left (d x^{2} e^{2} + d^{2} 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)^(3/2),x, algorithm="fricas")

[Out]

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

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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 {3}{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)**(3/2),x)

[Out]

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

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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)^(3/2),x, algorithm="giac")

[Out]

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

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

[Out]

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

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