3.321 \(\int \frac{(f+g x^2) \log (c (d+e x^2)^p)}{x^4} \, dx\)

Optimal. Leaf size=108 \[ -\frac{f \log \left (c \left (d+e x^2\right )^p\right )}{3 x^3}-\frac{g \log \left (c \left (d+e x^2\right )^p\right )}{x}-\frac{2 e^{3/2} f p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{3 d^{3/2}}-\frac{2 e f p}{3 d x}+\frac{2 \sqrt{e} g p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d}} \]

[Out]

(-2*e*f*p)/(3*d*x) - (2*e^(3/2)*f*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(3*d^(3/2)) + (2*Sqrt[e]*g*p*ArcTan[(Sqrt[e]*
x)/Sqrt[d]])/Sqrt[d] - (f*Log[c*(d + e*x^2)^p])/(3*x^3) - (g*Log[c*(d + e*x^2)^p])/x

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Rubi [A]  time = 0.0998344, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {2476, 2455, 325, 205} \[ -\frac{f \log \left (c \left (d+e x^2\right )^p\right )}{3 x^3}-\frac{g \log \left (c \left (d+e x^2\right )^p\right )}{x}-\frac{2 e^{3/2} f p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{3 d^{3/2}}-\frac{2 e f p}{3 d x}+\frac{2 \sqrt{e} g p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d}} \]

Antiderivative was successfully verified.

[In]

Int[((f + g*x^2)*Log[c*(d + e*x^2)^p])/x^4,x]

[Out]

(-2*e*f*p)/(3*d*x) - (2*e^(3/2)*f*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(3*d^(3/2)) + (2*Sqrt[e]*g*p*ArcTan[(Sqrt[e]*
x)/Sqrt[d]])/Sqrt[d] - (f*Log[c*(d + e*x^2)^p])/(3*x^3) - (g*Log[c*(d + e*x^2)^p])/x

Rule 2476

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_) + (g_.)*(x_)^(s_))^(r_.),
x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x^n)^p])^q, x^m*(f + g*x^s)^r, x], x] /; FreeQ[{a, b, c,
d, e, f, g, m, n, p, q, r, s}, x] && IGtQ[q, 0] && IntegerQ[m] && IntegerQ[r] && IntegerQ[s]

Rule 2455

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

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && 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 \frac{\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^4} \, dx &=\int \left (\frac{f \log \left (c \left (d+e x^2\right )^p\right )}{x^4}+\frac{g \log \left (c \left (d+e x^2\right )^p\right )}{x^2}\right ) \, dx\\ &=f \int \frac{\log \left (c \left (d+e x^2\right )^p\right )}{x^4} \, dx+g \int \frac{\log \left (c \left (d+e x^2\right )^p\right )}{x^2} \, dx\\ &=-\frac{f \log \left (c \left (d+e x^2\right )^p\right )}{3 x^3}-\frac{g \log \left (c \left (d+e x^2\right )^p\right )}{x}+\frac{1}{3} (2 e f p) \int \frac{1}{x^2 \left (d+e x^2\right )} \, dx+(2 e g p) \int \frac{1}{d+e x^2} \, dx\\ &=-\frac{2 e f p}{3 d x}+\frac{2 \sqrt{e} g p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d}}-\frac{f \log \left (c \left (d+e x^2\right )^p\right )}{3 x^3}-\frac{g \log \left (c \left (d+e x^2\right )^p\right )}{x}-\frac{\left (2 e^2 f p\right ) \int \frac{1}{d+e x^2} \, dx}{3 d}\\ &=-\frac{2 e f p}{3 d x}-\frac{2 e^{3/2} f p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{3 d^{3/2}}+\frac{2 \sqrt{e} g p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d}}-\frac{f \log \left (c \left (d+e x^2\right )^p\right )}{3 x^3}-\frac{g \log \left (c \left (d+e x^2\right )^p\right )}{x}\\ \end{align*}

Mathematica [C]  time = 0.0394848, size = 96, normalized size = 0.89 \[ -\frac{f \log \left (c \left (d+e x^2\right )^p\right )}{3 x^3}-\frac{g \log \left (c \left (d+e x^2\right )^p\right )}{x}-\frac{2 e f p \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};-\frac{e x^2}{d}\right )}{3 d x}+\frac{2 \sqrt{e} g p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d}} \]

Antiderivative was successfully verified.

[In]

Integrate[((f + g*x^2)*Log[c*(d + e*x^2)^p])/x^4,x]

[Out]

(2*Sqrt[e]*g*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[d] - (2*e*f*p*Hypergeometric2F1[-1/2, 1, 1/2, -((e*x^2)/d)])/
(3*d*x) - (f*Log[c*(d + e*x^2)^p])/(3*x^3) - (g*Log[c*(d + e*x^2)^p])/x

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Maple [C]  time = 0.566, size = 430, normalized size = 4. \begin{align*} -{\frac{ \left ( 3\,g{x}^{2}+f \right ) \ln \left ( \left ( e{x}^{2}+d \right ) ^{p} \right ) }{3\,{x}^{3}}}+{\frac{-3\,i\pi \,dg{x}^{2}{\it csgn} \left ( i \left ( e{x}^{2}+d \right ) ^{p} \right ) \left ({\it csgn} \left ( ic \left ( e{x}^{2}+d \right ) ^{p} \right ) \right ) ^{2}+3\,i\pi \,dg{x}^{2}{\it csgn} \left ( i \left ( e{x}^{2}+d \right ) ^{p} \right ){\it csgn} \left ( ic \left ( e{x}^{2}+d \right ) ^{p} \right ){\it csgn} \left ( ic \right ) +3\,i\pi \,dg{x}^{2} \left ({\it csgn} \left ( ic \left ( e{x}^{2}+d \right ) ^{p} \right ) \right ) ^{3}-3\,i\pi \,dg{x}^{2} \left ({\it csgn} \left ( ic \left ( e{x}^{2}+d \right ) ^{p} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -i\pi \,df{\it csgn} \left ( i \left ( e{x}^{2}+d \right ) ^{p} \right ) \left ({\it csgn} \left ( ic \left ( e{x}^{2}+d \right ) ^{p} \right ) \right ) ^{2}+i\pi \,df{\it csgn} \left ( i \left ( e{x}^{2}+d \right ) ^{p} \right ){\it csgn} \left ( ic \left ( e{x}^{2}+d \right ) ^{p} \right ){\it csgn} \left ( ic \right ) +i\pi \,df \left ({\it csgn} \left ( ic \left ( e{x}^{2}+d \right ) ^{p} \right ) \right ) ^{3}-i\pi \,df \left ({\it csgn} \left ( ic \left ( e{x}^{2}+d \right ) ^{p} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -6\,\ln \left ( c \right ) dg{x}^{2}+2\,\sum _{{\it \_R}={\it RootOf} \left ( 9\,{d}^{2}e{g}^{2}{p}^{2}-6\,d{e}^{2}fg{p}^{2}+{e}^{3}{f}^{2}{p}^{2}+{d}^{3}{{\it \_Z}}^{2} \right ) }{\it \_R}\,\ln \left ( \left ( 18\,{d}^{2}e{g}^{2}{p}^{2}-12\,d{e}^{2}fg{p}^{2}+2\,{e}^{3}{f}^{2}{p}^{2}+3\,{{\it \_R}}^{2}{d}^{3} \right ) x+ \left ( -3\,{d}^{3}gp+{d}^{2}efp \right ){\it \_R} \right ) d{x}^{3}-4\,efp{x}^{2}-2\,\ln \left ( c \right ) df}{6\,d{x}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x^2+f)*ln(c*(e*x^2+d)^p)/x^4,x)

[Out]

-1/3*(3*g*x^2+f)/x^3*ln((e*x^2+d)^p)+1/6*(-3*I*Pi*d*g*x^2*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2+3*I*Pi*d
*g*x^2*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)+3*I*Pi*d*g*x^2*csgn(I*c*(e*x^2+d)^p)^3-3*I*Pi*d*g*x
^2*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)-I*Pi*d*f*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2+I*Pi*d*f*csgn(I*(e*x
^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)+I*Pi*d*f*csgn(I*c*(e*x^2+d)^p)^3-I*Pi*d*f*csgn(I*c*(e*x^2+d)^p)^2*csg
n(I*c)-6*ln(c)*d*g*x^2+2*sum(_R*ln((18*d^2*e*g^2*p^2-12*d*e^2*f*g*p^2+2*e^3*f^2*p^2+3*_R^2*d^3)*x+(-3*d^3*g*p+
d^2*e*f*p)*_R),_R=RootOf(9*d^2*e*g^2*p^2-6*d*e^2*f*g*p^2+e^3*f^2*p^2+_Z^2*d^3))*d*x^3-4*e*f*p*x^2-2*ln(c)*d*f)
/d/x^3

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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((g*x^2+f)*log(c*(e*x^2+d)^p)/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.40017, size = 441, normalized size = 4.08 \begin{align*} \left [-\frac{{\left (e f - 3 \, d g\right )} p x^{3} \sqrt{-\frac{e}{d}} \log \left (\frac{e x^{2} + 2 \, d x \sqrt{-\frac{e}{d}} - d}{e x^{2} + d}\right ) + 2 \, e f p x^{2} +{\left (3 \, d g p x^{2} + d f p\right )} \log \left (e x^{2} + d\right ) +{\left (3 \, d g x^{2} + d f\right )} \log \left (c\right )}{3 \, d x^{3}}, -\frac{2 \,{\left (e f - 3 \, d g\right )} p x^{3} \sqrt{\frac{e}{d}} \arctan \left (x \sqrt{\frac{e}{d}}\right ) + 2 \, e f p x^{2} +{\left (3 \, d g p x^{2} + d f p\right )} \log \left (e x^{2} + d\right ) +{\left (3 \, d g x^{2} + d f\right )} \log \left (c\right )}{3 \, d x^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x^2+f)*log(c*(e*x^2+d)^p)/x^4,x, algorithm="fricas")

[Out]

[-1/3*((e*f - 3*d*g)*p*x^3*sqrt(-e/d)*log((e*x^2 + 2*d*x*sqrt(-e/d) - d)/(e*x^2 + d)) + 2*e*f*p*x^2 + (3*d*g*p
*x^2 + d*f*p)*log(e*x^2 + d) + (3*d*g*x^2 + d*f)*log(c))/(d*x^3), -1/3*(2*(e*f - 3*d*g)*p*x^3*sqrt(e/d)*arctan
(x*sqrt(e/d)) + 2*e*f*p*x^2 + (3*d*g*p*x^2 + d*f*p)*log(e*x^2 + d) + (3*d*g*x^2 + d*f)*log(c))/(d*x^3)]

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Sympy [A]  time = 131.902, size = 1469, normalized size = 13.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x**2+f)*ln(c*(e*x**2+d)**p)/x**4,x)

[Out]

Piecewise(((-f/(3*x**3) - g/x)*log(0**p*c), Eq(d, 0) & Eq(e, 0)), (-f*p*log(e)/(3*x**3) - 2*f*p*log(x)/(3*x**3
) - 2*f*p/(9*x**3) - f*log(c)/(3*x**3) - g*p*log(e)/x - 2*g*p*log(x)/x - 2*g*p/x - g*log(c)/x, Eq(d, 0)), ((-f
/(3*x**3) - g/x)*log(c*d**p), Eq(e, 0)), (-I*d**(7/2)*f*p*sqrt(1/e)*log(d + e*x**2)/(3*I*d**(7/2)*x**3*sqrt(1/
e) + 3*I*d**(5/2)*e*x**5*sqrt(1/e)) - I*d**(7/2)*f*sqrt(1/e)*log(c)/(3*I*d**(7/2)*x**3*sqrt(1/e) + 3*I*d**(5/2
)*e*x**5*sqrt(1/e)) - 3*I*d**(7/2)*g*p*x**2*sqrt(1/e)*log(d + e*x**2)/(3*I*d**(7/2)*x**3*sqrt(1/e) + 3*I*d**(5
/2)*e*x**5*sqrt(1/e)) - 3*I*d**(7/2)*g*x**2*sqrt(1/e)*log(c)/(3*I*d**(7/2)*x**3*sqrt(1/e) + 3*I*d**(5/2)*e*x**
5*sqrt(1/e)) - I*d**(5/2)*f*p*x**2*sqrt(1/e)*log(d + e*x**2)/(3*I*d**(7/2)*x**3*sqrt(1/e)/e + 3*I*d**(5/2)*x**
5*sqrt(1/e)) - 2*I*d**(5/2)*f*p*x**2*sqrt(1/e)/(3*I*d**(7/2)*x**3*sqrt(1/e)/e + 3*I*d**(5/2)*x**5*sqrt(1/e)) -
I*d**(5/2)*f*x**2*sqrt(1/e)*log(c)/(3*I*d**(7/2)*x**3*sqrt(1/e)/e + 3*I*d**(5/2)*x**5*sqrt(1/e)) - 3*I*d**(5/
2)*g*p*x**4*sqrt(1/e)*log(d + e*x**2)/(3*I*d**(7/2)*x**3*sqrt(1/e)/e + 3*I*d**(5/2)*x**5*sqrt(1/e)) - 3*I*d**(
5/2)*g*x**4*sqrt(1/e)*log(c)/(3*I*d**(7/2)*x**3*sqrt(1/e)/e + 3*I*d**(5/2)*x**5*sqrt(1/e)) - 2*I*d**(3/2)*e*f*
p*x**4*sqrt(1/e)/(3*I*d**(7/2)*x**3*sqrt(1/e)/e + 3*I*d**(5/2)*x**5*sqrt(1/e)) - 3*d**3*g*p*x**3*log(d + e*x**
2)/(3*I*d**(7/2)*x**3*sqrt(1/e) + 3*I*d**(5/2)*e*x**5*sqrt(1/e)) + 6*d**3*g*p*x**3*log(-I*sqrt(d)*sqrt(1/e) +
x)/(3*I*d**(7/2)*x**3*sqrt(1/e) + 3*I*d**(5/2)*e*x**5*sqrt(1/e)) - 3*d**3*g*x**3*log(c)/(3*I*d**(7/2)*x**3*sqr
t(1/e) + 3*I*d**(5/2)*e*x**5*sqrt(1/e)) + d**2*f*p*x**3*log(d + e*x**2)/(3*I*d**(7/2)*x**3*sqrt(1/e)/e + 3*I*d
**(5/2)*x**5*sqrt(1/e)) - 2*d**2*f*p*x**3*log(-I*sqrt(d)*sqrt(1/e) + x)/(3*I*d**(7/2)*x**3*sqrt(1/e)/e + 3*I*d
**(5/2)*x**5*sqrt(1/e)) + d**2*f*x**3*log(c)/(3*I*d**(7/2)*x**3*sqrt(1/e)/e + 3*I*d**(5/2)*x**5*sqrt(1/e)) - 3
*d**2*g*p*x**5*log(d + e*x**2)/(3*I*d**(7/2)*x**3*sqrt(1/e)/e + 3*I*d**(5/2)*x**5*sqrt(1/e)) + 6*d**2*g*p*x**5
*log(-I*sqrt(d)*sqrt(1/e) + x)/(3*I*d**(7/2)*x**3*sqrt(1/e)/e + 3*I*d**(5/2)*x**5*sqrt(1/e)) - 3*d**2*g*x**5*l
og(c)/(3*I*d**(7/2)*x**3*sqrt(1/e)/e + 3*I*d**(5/2)*x**5*sqrt(1/e)) + d*e*f*p*x**5*log(d + e*x**2)/(3*I*d**(7/
2)*x**3*sqrt(1/e)/e + 3*I*d**(5/2)*x**5*sqrt(1/e)) - 2*d*e*f*p*x**5*log(-I*sqrt(d)*sqrt(1/e) + x)/(3*I*d**(7/2
)*x**3*sqrt(1/e)/e + 3*I*d**(5/2)*x**5*sqrt(1/e)) + d*e*f*x**5*log(c)/(3*I*d**(7/2)*x**3*sqrt(1/e)/e + 3*I*d**
(5/2)*x**5*sqrt(1/e)), True))

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Giac [A]  time = 1.24153, size = 124, normalized size = 1.15 \begin{align*} \frac{2 \,{\left (3 \, d g p e - f p e^{2}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{1}{2}\right )}}{3 \, d^{\frac{3}{2}}} - \frac{3 \, d g p x^{2} \log \left (x^{2} e + d\right ) + 2 \, f p x^{2} e + 3 \, d g x^{2} \log \left (c\right ) + d f p \log \left (x^{2} e + d\right ) + d f \log \left (c\right )}{3 \, d x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x^2+f)*log(c*(e*x^2+d)^p)/x^4,x, algorithm="giac")

[Out]

2/3*(3*d*g*p*e - f*p*e^2)*arctan(x*e^(1/2)/sqrt(d))*e^(-1/2)/d^(3/2) - 1/3*(3*d*g*p*x^2*log(x^2*e + d) + 2*f*p
*x^2*e + 3*d*g*x^2*log(c) + d*f*p*log(x^2*e + d) + d*f*log(c))/(d*x^3)