3.630 \(\int \log (c (d+e (f+g x)^2)^q) \, dx\)

Optimal. Leaf size=63 \[ \frac{(f+g x) \log \left (c \left (d+e (f+g x)^2\right )^q\right )}{g}+\frac{2 \sqrt{d} q \tan ^{-1}\left (\frac{\sqrt{e} (f+g x)}{\sqrt{d}}\right )}{\sqrt{e} g}-2 q x \]

[Out]

-2*q*x + (2*Sqrt[d]*q*ArcTan[(Sqrt[e]*(f + g*x))/Sqrt[d]])/(Sqrt[e]*g) + ((f + g*x)*Log[c*(d + e*(f + g*x)^2)^
q])/g

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Rubi [A]  time = 0.047829, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2483, 2448, 321, 205} \[ \frac{(f+g x) \log \left (c \left (d+e (f+g x)^2\right )^q\right )}{g}+\frac{2 \sqrt{d} q \tan ^{-1}\left (\frac{\sqrt{e} (f+g x)}{\sqrt{d}}\right )}{\sqrt{e} g}-2 q x \]

Antiderivative was successfully verified.

[In]

Int[Log[c*(d + e*(f + g*x)^2)^q],x]

[Out]

-2*q*x + (2*Sqrt[d]*q*ArcTan[(Sqrt[e]*(f + g*x))/Sqrt[d]])/(Sqrt[e]*g) + ((f + g*x)*Log[c*(d + e*(f + g*x)^2)^
q])/g

Rule 2483

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*((f_.) + (g_.)*(x_))^(n_))^(p_.)]*(b_.))^(q_.), x_Symbol] :> Dist[1/g, Su
bst[Int[(a + b*Log[c*(d + e*x^n)^p])^q, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && IGtQ[q
, 0] && (EqQ[q, 1] || IntegerQ[n])

Rule 2448

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]

Rule 321

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

Rule 205

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

Rubi steps

\begin{align*} \int \log \left (c \left (d+e (f+g x)^2\right )^q\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \log \left (c \left (d+e x^2\right )^q\right ) \, dx,x,f+g x\right )}{g}\\ &=\frac{(f+g x) \log \left (c \left (d+e (f+g x)^2\right )^q\right )}{g}-\frac{(2 e q) \operatorname{Subst}\left (\int \frac{x^2}{d+e x^2} \, dx,x,f+g x\right )}{g}\\ &=-2 q x+\frac{(f+g x) \log \left (c \left (d+e (f+g x)^2\right )^q\right )}{g}+\frac{(2 d q) \operatorname{Subst}\left (\int \frac{1}{d+e x^2} \, dx,x,f+g x\right )}{g}\\ &=-2 q x+\frac{2 \sqrt{d} q \tan ^{-1}\left (\frac{\sqrt{e} (f+g x)}{\sqrt{d}}\right )}{\sqrt{e} g}+\frac{(f+g x) \log \left (c \left (d+e (f+g x)^2\right )^q\right )}{g}\\ \end{align*}

Mathematica [A]  time = 0.0351679, size = 63, normalized size = 1. \[ \frac{(f+g x) \log \left (c \left (d+e (f+g x)^2\right )^q\right )}{g}+\frac{2 \sqrt{d} q \tan ^{-1}\left (\frac{\sqrt{e} (f+g x)}{\sqrt{d}}\right )}{\sqrt{e} g}-2 q x \]

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(d + e*(f + g*x)^2)^q],x]

[Out]

-2*q*x + (2*Sqrt[d]*q*ArcTan[(Sqrt[e]*(f + g*x))/Sqrt[d]])/(Sqrt[e]*g) + ((f + g*x)*Log[c*(d + e*(f + g*x)^2)^
q])/g

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Maple [A]  time = 0.171, size = 98, normalized size = 1.6 \begin{align*} \ln \left ( c \left ( e{g}^{2}{x}^{2}+2\,efgx+e{f}^{2}+d \right ) ^{q} \right ) x-2\,qx+{\frac{qf\ln \left ( e{g}^{2}{x}^{2}+2\,efgx+e{f}^{2}+d \right ) }{g}}+2\,{\frac{qd}{g\sqrt{de}}\arctan \left ( 1/2\,{\frac{2\,e{g}^{2}x+2\,efg}{g\sqrt{de}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*(d+e*(g*x+f)^2)^q),x)

[Out]

ln(c*(e*g^2*x^2+2*e*f*g*x+e*f^2+d)^q)*x-2*q*x+q/g*f*ln(e*g^2*x^2+2*e*f*g*x+e*f^2+d)+2*q/g*d/(d*e)^(1/2)*arctan
(1/2*(2*e*g^2*x+2*e*f*g)/g/(d*e)^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(d+e*(g*x+f)^2)^q),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.6336, size = 458, normalized size = 7.27 \begin{align*} \left [-\frac{2 \, g q x - g x \log \left (c\right ) - q \sqrt{-\frac{d}{e}} \log \left (\frac{e g^{2} x^{2} + 2 \, e f g x + e f^{2} + 2 \,{\left (e g x + e f\right )} \sqrt{-\frac{d}{e}} - d}{e g^{2} x^{2} + 2 \, e f g x + e f^{2} + d}\right ) -{\left (g q x + f q\right )} \log \left (e g^{2} x^{2} + 2 \, e f g x + e f^{2} + d\right )}{g}, -\frac{2 \, g q x - g x \log \left (c\right ) - 2 \, q \sqrt{\frac{d}{e}} \arctan \left (\frac{{\left (e g x + e f\right )} \sqrt{\frac{d}{e}}}{d}\right ) -{\left (g q x + f q\right )} \log \left (e g^{2} x^{2} + 2 \, e f g x + e f^{2} + d\right )}{g}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(d+e*(g*x+f)^2)^q),x, algorithm="fricas")

[Out]

[-(2*g*q*x - g*x*log(c) - q*sqrt(-d/e)*log((e*g^2*x^2 + 2*e*f*g*x + e*f^2 + 2*(e*g*x + e*f)*sqrt(-d/e) - d)/(e
*g^2*x^2 + 2*e*f*g*x + e*f^2 + d)) - (g*q*x + f*q)*log(e*g^2*x^2 + 2*e*f*g*x + e*f^2 + d))/g, -(2*g*q*x - g*x*
log(c) - 2*q*sqrt(d/e)*arctan((e*g*x + e*f)*sqrt(d/e)/d) - (g*q*x + f*q)*log(e*g^2*x^2 + 2*e*f*g*x + e*f^2 + d
))/g]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(c*(d+e*(g*x+f)**2)**q),x)

[Out]

Timed out

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Giac [A]  time = 1.26182, size = 130, normalized size = 2.06 \begin{align*} q x \log \left (g^{2} x^{2} e + 2 \, f g x e + f^{2} e + d\right ) + \frac{2 \, \sqrt{d} q \arctan \left (\frac{{\left (g x e + f e\right )} e^{\left (-\frac{1}{2}\right )}}{\sqrt{d}}\right ) e^{\left (-\frac{1}{2}\right )}}{g} - 2 \, q x + \frac{f q \log \left (g^{2} x^{2} e + 2 \, f g x e + f^{2} e + d\right )}{g} + x \log \left (c\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(d+e*(g*x+f)^2)^q),x, algorithm="giac")

[Out]

q*x*log(g^2*x^2*e + 2*f*g*x*e + f^2*e + d) + 2*sqrt(d)*q*arctan((g*x*e + f*e)*e^(-1/2)/sqrt(d))*e^(-1/2)/g - 2
*q*x + f*q*log(g^2*x^2*e + 2*f*g*x*e + f^2*e + d)/g + x*log(c)