3.186 \(\int (d+e x) \log (c (a+b x^2)^p) \, dx\)

Optimal. Leaf size=99 \[ \frac{(d+e x)^2 \log \left (c \left (a+b x^2\right )^p\right )}{2 e}-\frac{p \left (b d^2-a e^2\right ) \log \left (a+b x^2\right )}{2 b e}+\frac{2 \sqrt{a} d p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b}}-2 d p x-\frac{1}{2} e p x^2 \]

[Out]

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

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Rubi [A]  time = 0.078563, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {2463, 801, 635, 205, 260} \[ \frac{(d+e x)^2 \log \left (c \left (a+b x^2\right )^p\right )}{2 e}-\frac{p \left (b d^2-a e^2\right ) \log \left (a+b x^2\right )}{2 b e}+\frac{2 \sqrt{a} d p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b}}-2 d p x-\frac{1}{2} e p x^2 \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)*Log[c*(a + b*x^2)^p],x]

[Out]

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

Rule 2463

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.) + (g_.)*(x_))^(r_.), x_Symbol] :> Simp[((
f + g*x)^(r + 1)*(a + b*Log[c*(d + e*x^n)^p]))/(g*(r + 1)), x] - Dist[(b*e*n*p)/(g*(r + 1)), Int[(x^(n - 1)*(f
+ g*x)^(r + 1))/(d + e*x^n), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, r}, x] && (IGtQ[r, 0] || RationalQ[n
]) && NeQ[r, -1]

Rule 801

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
(d + e*x)^m*(f + g*x))/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Integer
Q[m]

Rule 635

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[-(a*c)]

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]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps

\begin{align*} \int (d+e x) \log \left (c \left (a+b x^2\right )^p\right ) \, dx &=\frac{(d+e x)^2 \log \left (c \left (a+b x^2\right )^p\right )}{2 e}-\frac{(b p) \int \frac{x (d+e x)^2}{a+b x^2} \, dx}{e}\\ &=\frac{(d+e x)^2 \log \left (c \left (a+b x^2\right )^p\right )}{2 e}-\frac{(b p) \int \left (\frac{2 d e}{b}+\frac{e^2 x}{b}-\frac{2 a d e-\left (b d^2-a e^2\right ) x}{b \left (a+b x^2\right )}\right ) \, dx}{e}\\ &=-2 d p x-\frac{1}{2} e p x^2+\frac{(d+e x)^2 \log \left (c \left (a+b x^2\right )^p\right )}{2 e}+\frac{p \int \frac{2 a d e-\left (b d^2-a e^2\right ) x}{a+b x^2} \, dx}{e}\\ &=-2 d p x-\frac{1}{2} e p x^2+\frac{(d+e x)^2 \log \left (c \left (a+b x^2\right )^p\right )}{2 e}+(2 a d p) \int \frac{1}{a+b x^2} \, dx+\frac{\left (\left (-b d^2+a e^2\right ) p\right ) \int \frac{x}{a+b x^2} \, dx}{e}\\ &=-2 d p x-\frac{1}{2} e p x^2+\frac{2 \sqrt{a} d p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b}}-\frac{\left (b d^2-a e^2\right ) p \log \left (a+b x^2\right )}{2 b e}+\frac{(d+e x)^2 \log \left (c \left (a+b x^2\right )^p\right )}{2 e}\\ \end{align*}

Mathematica [A]  time = 0.02559, size = 83, normalized size = 0.84 \[ d x \log \left (c \left (a+b x^2\right )^p\right )+\frac{1}{2} e \left (\frac{\left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{b}-p x^2\right )+\frac{2 \sqrt{a} d p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b}}-2 d p x \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.078, size = 93, normalized size = 0.9 \begin{align*} d\ln \left ( c \left ( b{x}^{2}+a \right ) ^{p} \right ) x-2\,dpx+2\,{\frac{dpa}{\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }+{\frac{e\ln \left ( c \left ( b{x}^{2}+a \right ) ^{p} \right ){x}^{2}}{2}}-{\frac{ep{x}^{2}}{2}}+{\frac{\ln \left ( c \left ( b{x}^{2}+a \right ) ^{p} \right ) ae}{2\,b}}-{\frac{ape}{2\,b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)*ln(c*(b*x^2+a)^p),x)

[Out]

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

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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((e*x+d)*log(c*(b*x^2+a)^p),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.21228, size = 455, normalized size = 4.6 \begin{align*} \left [-\frac{b e p x^{2} - 2 \, b d p \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} + 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) + 4 \, b d p x -{\left (b e p x^{2} + 2 \, b d p x + a e p\right )} \log \left (b x^{2} + a\right ) -{\left (b e x^{2} + 2 \, b d x\right )} \log \left (c\right )}{2 \, b}, -\frac{b e p x^{2} - 4 \, b d p \sqrt{\frac{a}{b}} \arctan \left (\frac{b x \sqrt{\frac{a}{b}}}{a}\right ) + 4 \, b d p x -{\left (b e p x^{2} + 2 \, b d p x + a e p\right )} \log \left (b x^{2} + a\right ) -{\left (b e x^{2} + 2 \, b d x\right )} \log \left (c\right )}{2 \, b}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*(b*e*p*x^2 - 2*b*d*p*sqrt(-a/b)*log((b*x^2 + 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)) + 4*b*d*p*x - (b*e*p*x^2
+ 2*b*d*p*x + a*e*p)*log(b*x^2 + a) - (b*e*x^2 + 2*b*d*x)*log(c))/b, -1/2*(b*e*p*x^2 - 4*b*d*p*sqrt(a/b)*arct
an(b*x*sqrt(a/b)/a) + 4*b*d*p*x - (b*e*p*x^2 + 2*b*d*p*x + a*e*p)*log(b*x^2 + a) - (b*e*x^2 + 2*b*d*x)*log(c))
/b]

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Sympy [A]  time = 19.996, size = 160, normalized size = 1.62 \begin{align*} \begin{cases} \frac{i \sqrt{a} d p \log{\left (a + b x^{2} \right )}}{b \sqrt{\frac{1}{b}}} - \frac{2 i \sqrt{a} d p \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + x \right )}}{b \sqrt{\frac{1}{b}}} + \frac{a e p \log{\left (a + b x^{2} \right )}}{2 b} + d p x \log{\left (a + b x^{2} \right )} - 2 d p x + d x \log{\left (c \right )} + \frac{e p x^{2} \log{\left (a + b x^{2} \right )}}{2} - \frac{e p x^{2}}{2} + \frac{e x^{2} \log{\left (c \right )}}{2} & \text{for}\: b \neq 0 \\\left (d x + \frac{e x^{2}}{2}\right ) \log{\left (a^{p} c \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*ln(c*(b*x**2+a)**p),x)

[Out]

Piecewise((I*sqrt(a)*d*p*log(a + b*x**2)/(b*sqrt(1/b)) - 2*I*sqrt(a)*d*p*log(-I*sqrt(a)*sqrt(1/b) + x)/(b*sqrt
(1/b)) + a*e*p*log(a + b*x**2)/(2*b) + d*p*x*log(a + b*x**2) - 2*d*p*x + d*x*log(c) + e*p*x**2*log(a + b*x**2)
/2 - e*p*x**2/2 + e*x**2*log(c)/2, Ne(b, 0)), ((d*x + e*x**2/2)*log(a**p*c), True))

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Giac [A]  time = 1.28123, size = 135, normalized size = 1.36 \begin{align*} \frac{2 \, a d p \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b}} + \frac{b p x^{2} e \log \left (b x^{2} + a\right ) - b p x^{2} e + 2 \, b d p x \log \left (b x^{2} + a\right ) + b x^{2} e \log \left (c\right ) - 4 \, b d p x + a p e \log \left (b x^{2} + a\right ) + 2 \, b d x \log \left (c\right )}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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