3.4.21 \(\int \frac {(f+g x^2) \log (c (d+e x^2)^p)}{x^4} \, dx\) [321]

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

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 108, normalized size of antiderivative = 1.00, 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 = {2526, 2505, 331, 211} \begin {gather*} -\frac {2 e^{3/2} f p \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{3 d^{3/2}}+\frac {2 \sqrt {e} g p \text {ArcTan}\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 {2 e f p}{3 d x} \end {gather*}

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 211

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

Rule 331

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 2505

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 2526

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]

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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.03, size = 96, normalized size = 0.89 \begin {gather*} \frac {2 \sqrt {e} g p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d}}-\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 {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 {gather*}

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] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.18, size = 442, normalized size = 4.09

method	result	size
risch	\(-\frac {\left (3 g \,x^{2}+f \right ) \ln \left (\left (e \,x^{2}+d \right )^{p}\right )}{3 x^{3}}-\frac {3 i \pi \,d^{2} g \,x^{2} \mathrm {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2}-3 i \pi \,d^{2} g \,x^{2} \mathrm {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \right )-3 i \pi \,d^{2} g \,x^{2} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{3}+3 i \pi \,d^{2} g \,x^{2} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )+i \pi \,d^{2} f \,\mathrm {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2}-i \pi \,d^{2} f \,\mathrm {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \right )-i \pi \,d^{2} f \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{3}+i \pi \,d^{2} f \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )-6 \sqrt {-e d}\, p \ln \left (-e x -\sqrt {-e d}\right ) g d \,x^{3}+2 \sqrt {-e d}\, p \ln \left (-e x -\sqrt {-e d}\right ) e f \,x^{3}+6 \sqrt {-e d}\, p \ln \left (-e x +\sqrt {-e d}\right ) g d \,x^{3}-2 \sqrt {-e d}\, p \ln \left (-e x +\sqrt {-e d}\right ) e f \,x^{3}+6 \ln \left (c \right ) d^{2} g \,x^{2}+4 d e f p \,x^{2}+2 \ln \left (c \right ) d^{2} f}{6 d^{2} x^{3}}\)	\(442\)

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,method=_RETURNVERBOSE)

[Out]

-1/3*(3*g*x^2+f)/x^3*ln((e*x^2+d)^p)-1/6*(3*I*Pi*d^2*g*x^2*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2-3*I*Pi*
d^2*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^2*g*x^2*csgn(I*c*(e*x^2+d)^p)^3+3*I*Pi*
d^2*g*x^2*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)+I*Pi*d^2*f*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2-I*Pi*d^2*f*
csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)-I*Pi*d^2*f*csgn(I*c*(e*x^2+d)^p)^3+I*Pi*d^2*f*csgn(I*c*(e*
x^2+d)^p)^2*csgn(I*c)-6*(-e*d)^(1/2)*p*ln(-e*x-(-e*d)^(1/2))*g*d*x^3+2*(-e*d)^(1/2)*p*ln(-e*x-(-e*d)^(1/2))*e*
f*x^3+6*(-e*d)^(1/2)*p*ln(-e*x+(-e*d)^(1/2))*g*d*x^3-2*(-e*d)^(1/2)*p*ln(-e*x+(-e*d)^(1/2))*e*f*x^3+6*ln(c)*d^
2*g*x^2+4*d*e*f*p*x^2+2*ln(c)*d^2*f)/d^2/x^3

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Maxima [A]
time = 0.49, size = 66, normalized size = 0.61 \begin {gather*} \frac {2}{3} \, {\left (\frac {{\left (3 \, d g - f e\right )} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {1}{2}\right )}}{d^{\frac {3}{2}}} - \frac {f}{d x}\right )} p e - \frac {{\left (3 \, g x^{2} + f\right )} \log \left ({\left (x^{2} e + d\right )}^{p} c\right )}{3 \, x^{3}} \end {gather*}

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]

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

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Fricas [A]
time = 0.39, size = 207, normalized size = 1.92 \begin {gather*} \left [-\frac {2 \, f p x^{2} e + {\left (3 \, d g p x^{3} - f p x^{3} e\right )} \sqrt {-\frac {e}{d}} \log \left (\frac {x^{2} e - 2 \, d x \sqrt {-\frac {e}{d}} - d}{x^{2} e + d}\right ) + {\left (3 \, d g p x^{2} + d f p\right )} \log \left (x^{2} e + d\right ) + {\left (3 \, d g x^{2} + d f\right )} \log \left (c\right )}{3 \, d x^{3}}, -\frac {2 \, f p x^{2} e - \frac {2 \, {\left (3 \, d g p x^{3} - f p x^{3} e\right )} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\frac {1}{2}}}{\sqrt {d}} + {\left (3 \, d g p x^{2} + d f p\right )} \log \left (x^{2} e + d\right ) + {\left (3 \, d g x^{2} + d f\right )} \log \left (c\right )}{3 \, d x^{3}}\right ] \end {gather*}

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 879 vs. \(2 (104) = 208\).
time = 40.83, size = 879, normalized size = 8.14 \begin {gather*} \begin {cases} \left (- \frac {f}{3 x^{3}} - \frac {g}{x}\right ) \log {\left (0^{p} c \right )} & \text {for}\: d = 0 \wedge e = 0 \\\left (- \frac {f}{3 x^{3}} - \frac {g}{x}\right ) \log {\left (c d^{p} \right )} & \text {for}\: e = 0 \\- \frac {2 f p}{9 x^{3}} - \frac {f \log {\left (c \left (e x^{2}\right )^{p} \right )}}{3 x^{3}} - \frac {2 g p}{x} - \frac {g \log {\left (c \left (e x^{2}\right )^{p} \right )}}{x} & \text {for}\: d = 0 \\- \frac {d^{2} f \sqrt {- \frac {d}{e}} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{3 d^{2} x^{3} \sqrt {- \frac {d}{e}} + 3 d e x^{5} \sqrt {- \frac {d}{e}}} + \frac {6 d^{2} g p x^{3} \log {\left (x - \sqrt {- \frac {d}{e}} \right )}}{3 d^{2} x^{3} \sqrt {- \frac {d}{e}} + 3 d e x^{5} \sqrt {- \frac {d}{e}}} - \frac {3 d^{2} g x^{3} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{3 d^{2} x^{3} \sqrt {- \frac {d}{e}} + 3 d e x^{5} \sqrt {- \frac {d}{e}}} - \frac {3 d^{2} g x^{2} \sqrt {- \frac {d}{e}} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{3 d^{2} x^{3} \sqrt {- \frac {d}{e}} + 3 d e x^{5} \sqrt {- \frac {d}{e}}} - \frac {2 d f p x^{3} \log {\left (x - \sqrt {- \frac {d}{e}} \right )}}{\frac {3 d^{2} x^{3} \sqrt {- \frac {d}{e}}}{e} + 3 d x^{5} \sqrt {- \frac {d}{e}}} - \frac {2 d f p x^{2} \sqrt {- \frac {d}{e}}}{\frac {3 d^{2} x^{3} \sqrt {- \frac {d}{e}}}{e} + 3 d x^{5} \sqrt {- \frac {d}{e}}} + \frac {d f x^{3} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{\frac {3 d^{2} x^{3} \sqrt {- \frac {d}{e}}}{e} + 3 d x^{5} \sqrt {- \frac {d}{e}}} - \frac {d f x^{2} \sqrt {- \frac {d}{e}} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{\frac {3 d^{2} x^{3} \sqrt {- \frac {d}{e}}}{e} + 3 d x^{5} \sqrt {- \frac {d}{e}}} + \frac {6 d g p x^{5} \log {\left (x - \sqrt {- \frac {d}{e}} \right )}}{\frac {3 d^{2} x^{3} \sqrt {- \frac {d}{e}}}{e} + 3 d x^{5} \sqrt {- \frac {d}{e}}} - \frac {3 d g x^{5} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{\frac {3 d^{2} x^{3} \sqrt {- \frac {d}{e}}}{e} + 3 d x^{5} \sqrt {- \frac {d}{e}}} - \frac {3 d g x^{4} \sqrt {- \frac {d}{e}} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{\frac {3 d^{2} x^{3} \sqrt {- \frac {d}{e}}}{e} + 3 d x^{5} \sqrt {- \frac {d}{e}}} - \frac {2 e f p x^{5} \log {\left (x - \sqrt {- \frac {d}{e}} \right )}}{\frac {3 d^{2} x^{3} \sqrt {- \frac {d}{e}}}{e} + 3 d x^{5} \sqrt {- \frac {d}{e}}} - \frac {2 e f p x^{4} \sqrt {- \frac {d}{e}}}{\frac {3 d^{2} x^{3} \sqrt {- \frac {d}{e}}}{e} + 3 d x^{5} \sqrt {- \frac {d}{e}}} + \frac {e f x^{5} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{\frac {3 d^{2} x^{3} \sqrt {- \frac {d}{e}}}{e} + 3 d x^{5} \sqrt {- \frac {d}{e}}} & \text {otherwise} \end {cases} \end {gather*}

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/(3*x**3) - g/x)*log(c*d**p), Eq(e, 0)),
(-2*f*p/(9*x**3) - f*log(c*(e*x**2)**p)/(3*x**3) - 2*g*p/x - g*log(c*(e*x**2)**p)/x, Eq(d, 0)), (-d**2*f*sqrt
(-d/e)*log(c*(d + e*x**2)**p)/(3*d**2*x**3*sqrt(-d/e) + 3*d*e*x**5*sqrt(-d/e)) + 6*d**2*g*p*x**3*log(x - sqrt(
-d/e))/(3*d**2*x**3*sqrt(-d/e) + 3*d*e*x**5*sqrt(-d/e)) - 3*d**2*g*x**3*log(c*(d + e*x**2)**p)/(3*d**2*x**3*sq
rt(-d/e) + 3*d*e*x**5*sqrt(-d/e)) - 3*d**2*g*x**2*sqrt(-d/e)*log(c*(d + e*x**2)**p)/(3*d**2*x**3*sqrt(-d/e) +
3*d*e*x**5*sqrt(-d/e)) - 2*d*f*p*x**3*log(x - sqrt(-d/e))/(3*d**2*x**3*sqrt(-d/e)/e + 3*d*x**5*sqrt(-d/e)) - 2
*d*f*p*x**2*sqrt(-d/e)/(3*d**2*x**3*sqrt(-d/e)/e + 3*d*x**5*sqrt(-d/e)) + d*f*x**3*log(c*(d + e*x**2)**p)/(3*d
**2*x**3*sqrt(-d/e)/e + 3*d*x**5*sqrt(-d/e)) - d*f*x**2*sqrt(-d/e)*log(c*(d + e*x**2)**p)/(3*d**2*x**3*sqrt(-d
/e)/e + 3*d*x**5*sqrt(-d/e)) + 6*d*g*p*x**5*log(x - sqrt(-d/e))/(3*d**2*x**3*sqrt(-d/e)/e + 3*d*x**5*sqrt(-d/e
)) - 3*d*g*x**5*log(c*(d + e*x**2)**p)/(3*d**2*x**3*sqrt(-d/e)/e + 3*d*x**5*sqrt(-d/e)) - 3*d*g*x**4*sqrt(-d/e
)*log(c*(d + e*x**2)**p)/(3*d**2*x**3*sqrt(-d/e)/e + 3*d*x**5*sqrt(-d/e)) - 2*e*f*p*x**5*log(x - sqrt(-d/e))/(
3*d**2*x**3*sqrt(-d/e)/e + 3*d*x**5*sqrt(-d/e)) - 2*e*f*p*x**4*sqrt(-d/e)/(3*d**2*x**3*sqrt(-d/e)/e + 3*d*x**5
*sqrt(-d/e)) + e*f*x**5*log(c*(d + e*x**2)**p)/(3*d**2*x**3*sqrt(-d/e)/e + 3*d*x**5*sqrt(-d/e)), True))

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Giac [A]
time = 5.40, size = 92, normalized size = 0.85 \begin {gather*} \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 {gather*}

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)

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Mupad [B]
time = 0.37, size = 65, normalized size = 0.60 \begin {gather*} \frac {2\,\sqrt {e}\,p\,\mathrm {atan}\left (\frac {\sqrt {e}\,x}{\sqrt {d}}\right )\,\left (3\,d\,g-e\,f\right )}{3\,d^{3/2}}-\frac {2\,e\,f\,p}{3\,d\,x}-\frac {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\,\left (g\,x^2+\frac {f}{3}\right )}{x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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