3.231 \(\int \frac {\log (c (a+b x^2)^p)}{x^2 (d+e x)} \, dx\)

Optimal. Leaf size=306 \[ -\frac {e \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 d^2}+\frac {e \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{d^2}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{d x}-\frac {e p \text {Li}_2\left (\frac {b x^2}{a}+1\right )}{2 d^2}-\frac {e p \text {Li}_2\left (\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )}{d^2}-\frac {e p \text {Li}_2\left (\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )}{d^2}-\frac {e p \log (d+e x) \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {-a} e+\sqrt {b} d}\right )}{d^2}-\frac {e p \log (d+e x) \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right )}{d^2}+\frac {2 \sqrt {b} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d} \]

[Out]

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

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Rubi [A]  time = 0.35, antiderivative size = 306, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 11, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {2466, 2455, 205, 2454, 2394, 2315, 2462, 260, 2416, 2393, 2391} \[ -\frac {e p \text {PolyLog}\left (2,\frac {b x^2}{a}+1\right )}{2 d^2}-\frac {e p \text {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )}{d^2}-\frac {e p \text {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {-a} e+\sqrt {b} d}\right )}{d^2}-\frac {e \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 d^2}+\frac {e \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{d^2}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{d x}-\frac {e p \log (d+e x) \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {-a} e+\sqrt {b} d}\right )}{d^2}-\frac {e p \log (d+e x) \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right )}{d^2}+\frac {2 \sqrt {b} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[b]*p*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*d) - (e*p*Log[(e*(Sqrt[-a] - Sqrt[b]*x))/(Sqrt[b]*d + Sqrt[
-a]*e)]*Log[d + e*x])/d^2 - (e*p*Log[-((e*(Sqrt[-a] + Sqrt[b]*x))/(Sqrt[b]*d - Sqrt[-a]*e))]*Log[d + e*x])/d^2
- Log[c*(a + b*x^2)^p]/(d*x) - (e*Log[-((b*x^2)/a)]*Log[c*(a + b*x^2)^p])/(2*d^2) + (e*Log[d + e*x]*Log[c*(a
+ b*x^2)^p])/d^2 - (e*p*PolyLog[2, (Sqrt[b]*(d + e*x))/(Sqrt[b]*d - Sqrt[-a]*e)])/d^2 - (e*p*PolyLog[2, (Sqrt[
b]*(d + e*x))/(Sqrt[b]*d + Sqrt[-a]*e)])/d^2 - (e*p*PolyLog[2, 1 + (b*x^2)/a])/(2*d^2)

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]

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2394

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

Rule 2416

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

Rule 2454

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

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 2462

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

Rule 2466

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

Rubi steps

\begin {align*} \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^2 (d+e x)} \, dx &=\int \left (\frac {\log \left (c \left (a+b x^2\right )^p\right )}{d x^2}-\frac {e \log \left (c \left (a+b x^2\right )^p\right )}{d^2 x}+\frac {e^2 \log \left (c \left (a+b x^2\right )^p\right )}{d^2 (d+e x)}\right ) \, dx\\ &=\frac {\int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^2} \, dx}{d}-\frac {e \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x} \, dx}{d^2}+\frac {e^2 \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{d+e x} \, dx}{d^2}\\ &=-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{d x}+\frac {e \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{d^2}-\frac {e \operatorname {Subst}\left (\int \frac {\log \left (c (a+b x)^p\right )}{x} \, dx,x,x^2\right )}{2 d^2}+\frac {(2 b p) \int \frac {1}{a+b x^2} \, dx}{d}-\frac {(2 b e p) \int \frac {x \log (d+e x)}{a+b x^2} \, dx}{d^2}\\ &=\frac {2 \sqrt {b} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{d x}-\frac {e \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 d^2}+\frac {e \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{d^2}+\frac {(b e p) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {b x}{a}\right )}{a+b x} \, dx,x,x^2\right )}{2 d^2}-\frac {(2 b e p) \int \left (-\frac {\log (d+e x)}{2 \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\log (d+e x)}{2 \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )}\right ) \, dx}{d^2}\\ &=\frac {2 \sqrt {b} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{d x}-\frac {e \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 d^2}+\frac {e \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{d^2}-\frac {e p \text {Li}_2\left (1+\frac {b x^2}{a}\right )}{2 d^2}+\frac {\left (\sqrt {b} e p\right ) \int \frac {\log (d+e x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{d^2}-\frac {\left (\sqrt {b} e p\right ) \int \frac {\log (d+e x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{d^2}\\ &=\frac {2 \sqrt {b} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d}-\frac {e p \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)}{d^2}-\frac {e p \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right ) \log (d+e x)}{d^2}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{d x}-\frac {e \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 d^2}+\frac {e \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{d^2}-\frac {e p \text {Li}_2\left (1+\frac {b x^2}{a}\right )}{2 d^2}+\frac {\left (e^2 p\right ) \int \frac {\log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} d+\sqrt {-a} e}\right )}{d+e x} \, dx}{d^2}+\frac {\left (e^2 p\right ) \int \frac {\log \left (\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{-\sqrt {b} d+\sqrt {-a} e}\right )}{d+e x} \, dx}{d^2}\\ &=\frac {2 \sqrt {b} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d}-\frac {e p \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)}{d^2}-\frac {e p \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right ) \log (d+e x)}{d^2}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{d x}-\frac {e \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 d^2}+\frac {e \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{d^2}-\frac {e p \text {Li}_2\left (1+\frac {b x^2}{a}\right )}{2 d^2}+\frac {(e p) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {b} x}{-\sqrt {b} d+\sqrt {-a} e}\right )}{x} \, dx,x,d+e x\right )}{d^2}+\frac {(e p) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {b} x}{\sqrt {b} d+\sqrt {-a} e}\right )}{x} \, dx,x,d+e x\right )}{d^2}\\ &=\frac {2 \sqrt {b} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d}-\frac {e p \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)}{d^2}-\frac {e p \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right ) \log (d+e x)}{d^2}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{d x}-\frac {e \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 d^2}+\frac {e \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{d^2}-\frac {e p \text {Li}_2\left (\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )}{d^2}-\frac {e p \text {Li}_2\left (\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )}{d^2}-\frac {e p \text {Li}_2\left (1+\frac {b x^2}{a}\right )}{2 d^2}\\ \end {align*}

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Mathematica [A]  time = 0.21, size = 268, normalized size = 0.88 \[ -\frac {-2 e \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )+\frac {2 d \log \left (c \left (a+b x^2\right )^p\right )}{x}+e \left (\log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )+p \text {Li}_2\left (\frac {b x^2}{a}+1\right )\right )+2 e p \left (\text {Li}_2\left (\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )+\text {Li}_2\left (\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )+\log (d+e x) \left (\log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {-a} e+\sqrt {b} d}\right )+\log \left (\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {-a} e-\sqrt {b} d}\right )\right )\right )-\frac {4 \sqrt {b} d p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a}}}{2 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*((-4*Sqrt[b]*d*p*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[a] + (2*d*Log[c*(a + b*x^2)^p])/x - 2*e*Log[d + e*x]*L
og[c*(a + b*x^2)^p] + 2*e*p*((Log[(e*(Sqrt[-a] - Sqrt[b]*x))/(Sqrt[b]*d + Sqrt[-a]*e)] + Log[(e*(Sqrt[-a] + Sq
rt[b]*x))/(-(Sqrt[b]*d) + Sqrt[-a]*e)])*Log[d + e*x] + PolyLog[2, (Sqrt[b]*(d + e*x))/(Sqrt[b]*d - Sqrt[-a]*e)
] + PolyLog[2, (Sqrt[b]*(d + e*x))/(Sqrt[b]*d + Sqrt[-a]*e)]) + e*(Log[-((b*x^2)/a)]*Log[c*(a + b*x^2)^p] + p*
PolyLog[2, 1 + (b*x^2)/a]))/d^2

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fricas [F]  time = 0.80, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )}{e x^{3} + d x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )}{{\left (e x + d\right )} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.28, size = 831, normalized size = 2.72 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*I*Pi*csgn(I*c*(b*x^2+a)^p)^3/d/x-1/d/x*ln(c)-ln((b*x^2+a)^p)/d/x+ln((b*x^2+a)^p)*e/d^2*ln(e*x+d)-ln((b*x^2
+a)^p)*e/d^2*ln(x)+2*b*p/d/(a*b)^(1/2)*arctan(1/(a*b)^(1/2)*b*x)+p*e/d^2*ln(x)*ln((-b*x+(-a*b)^(1/2))/(-a*b)^(
1/2))+p*e/d^2*ln(x)*ln((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))+1/2*I*Pi*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)*csg
n(I*c)*e/d^2*ln(x)+1/d^2*e*ln(c)*ln(e*x+d)-1/d^2*e*ln(c)*ln(x)-1/2*I*Pi*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c)/d/x-
1/2*I*Pi*csgn(I*c*(b*x^2+a)^p)^3*e/d^2*ln(e*x+d)-p*e/d^2*dilog((b*d-(e*x+d)*b+(-a*b)^(1/2)*e)/(b*d+(-a*b)^(1/2
)*e))-p*e/d^2*dilog((-b*d+(e*x+d)*b+(-a*b)^(1/2)*e)/(-b*d+(-a*b)^(1/2)*e))+p*e/d^2*dilog((-b*x+(-a*b)^(1/2))/(
-a*b)^(1/2))+p*e/d^2*dilog((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))-1/2*I*Pi*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)
^2/d/x+1/2*I*Pi*csgn(I*c*(b*x^2+a)^p)^3*e/d^2*ln(x)-1/2*I*Pi*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)*csgn(I*
c)*e/d^2*ln(e*x+d)+1/2*I*Pi*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^2*e/d^2*ln(e*x+d)+1/2*I*Pi*csgn(I*(b*x^2
+a)^p)*csgn(I*c*(b*x^2+a)^p)*csgn(I*c)/d/x-1/2*I*Pi*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^2*e/d^2*ln(x)-1/
2*I*Pi*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c)*e/d^2*ln(x)-p*e/d^2*ln(e*x+d)*ln((b*d-(e*x+d)*b+(-a*b)^(1/2)*e)/(b*d+
(-a*b)^(1/2)*e))-p*e/d^2*ln(e*x+d)*ln((-b*d+(e*x+d)*b+(-a*b)^(1/2)*e)/(-b*d+(-a*b)^(1/2)*e))+1/2*I*Pi*csgn(I*c
*(b*x^2+a)^p)^2*csgn(I*c)*e/d^2*ln(e*x+d)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )}{{\left (e x + d\right )} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}{x^2\,\left (d+e\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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