3.190 \(\int \frac{\log (c (a+b x^2)^p)}{(d+e x)^3} \, dx\)

Optimal. Leaf size=174 \[ \frac{2 \sqrt{a} b^{3/2} d p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\left (a e^2+b d^2\right )^2}-\frac{\log \left (c \left (a+b x^2\right )^p\right )}{2 e (d+e x)^2}+\frac{b p \left (b d^2-a e^2\right ) \log \left (a+b x^2\right )}{2 e \left (a e^2+b d^2\right )^2}+\frac{b d p}{e (d+e x) \left (a e^2+b d^2\right )}-\frac{b p \left (b d^2-a e^2\right ) \log (d+e x)}{e \left (a e^2+b d^2\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.144985, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2463, 801, 635, 205, 260} \[ \frac{2 \sqrt{a} b^{3/2} d p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\left (a e^2+b d^2\right )^2}-\frac{\log \left (c \left (a+b x^2\right )^p\right )}{2 e (d+e x)^2}+\frac{b p \left (b d^2-a e^2\right ) \log \left (a+b x^2\right )}{2 e \left (a e^2+b d^2\right )^2}+\frac{b d p}{e (d+e x) \left (a e^2+b d^2\right )}-\frac{b p \left (b d^2-a e^2\right ) \log (d+e x)}{e \left (a e^2+b d^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[Log[c*(a + b*x^2)^p]/(d + e*x)^3,x]

[Out]

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

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 \frac{\log \left (c \left (a+b x^2\right )^p\right )}{(d+e x)^3} \, dx &=-\frac{\log \left (c \left (a+b x^2\right )^p\right )}{2 e (d+e x)^2}+\frac{(b p) \int \frac{x}{(d+e x)^2 \left (a+b x^2\right )} \, dx}{e}\\ &=-\frac{\log \left (c \left (a+b x^2\right )^p\right )}{2 e (d+e x)^2}+\frac{(b p) \int \left (-\frac{d e}{\left (b d^2+a e^2\right ) (d+e x)^2}+\frac{e \left (-b d^2+a e^2\right )}{\left (b d^2+a e^2\right )^2 (d+e x)}+\frac{b \left (2 a d e+\left (b d^2-a e^2\right ) x\right )}{\left (b d^2+a e^2\right )^2 \left (a+b x^2\right )}\right ) \, dx}{e}\\ &=\frac{b d p}{e \left (b d^2+a e^2\right ) (d+e x)}-\frac{b \left (b d^2-a e^2\right ) p \log (d+e x)}{e \left (b d^2+a e^2\right )^2}-\frac{\log \left (c \left (a+b x^2\right )^p\right )}{2 e (d+e x)^2}+\frac{\left (b^2 p\right ) \int \frac{2 a d e+\left (b d^2-a e^2\right ) x}{a+b x^2} \, dx}{e \left (b d^2+a e^2\right )^2}\\ &=\frac{b d p}{e \left (b d^2+a e^2\right ) (d+e x)}-\frac{b \left (b d^2-a e^2\right ) p \log (d+e x)}{e \left (b d^2+a e^2\right )^2}-\frac{\log \left (c \left (a+b x^2\right )^p\right )}{2 e (d+e x)^2}+\frac{\left (2 a b^2 d p\right ) \int \frac{1}{a+b x^2} \, dx}{\left (b d^2+a e^2\right )^2}+\frac{\left (b^2 \left (b d^2-a e^2\right ) p\right ) \int \frac{x}{a+b x^2} \, dx}{e \left (b d^2+a e^2\right )^2}\\ &=\frac{b d p}{e \left (b d^2+a e^2\right ) (d+e x)}+\frac{2 \sqrt{a} b^{3/2} d p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\left (b d^2+a e^2\right )^2}-\frac{b \left (b d^2-a e^2\right ) p \log (d+e x)}{e \left (b d^2+a e^2\right )^2}+\frac{b \left (b d^2-a e^2\right ) p \log \left (a+b x^2\right )}{2 e \left (b d^2+a e^2\right )^2}-\frac{\log \left (c \left (a+b x^2\right )^p\right )}{2 e (d+e x)^2}\\ \end{align*}

Mathematica [A]  time = 0.588592, size = 217, normalized size = 1.25 \[ \frac{\frac{b p (d+e x) \left ((d+e x) \left (\sqrt{-a} b d^2+2 a \sqrt{b} d e+(-a)^{3/2} e^2\right ) \log \left (\sqrt{-a}-\sqrt{b} x\right )+(d+e x) \left (\sqrt{-a} b d^2-2 a \sqrt{b} d e+(-a)^{3/2} e^2\right ) \log \left (\sqrt{-a}+\sqrt{b} x\right )+2 \sqrt{-a} \left (-(d+e x) \left (b d^2-a e^2\right ) \log (d+e x)+a d e^2+b d^3\right )\right )}{\sqrt{-a} \left (a e^2+b d^2\right )^2}-\log \left (c \left (a+b x^2\right )^p\right )}{2 e (d+e x)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(a + b*x^2)^p]/(d + e*x)^3,x]

[Out]

((b*p*(d + e*x)*((Sqrt[-a]*b*d^2 + 2*a*Sqrt[b]*d*e + (-a)^(3/2)*e^2)*(d + e*x)*Log[Sqrt[-a] - Sqrt[b]*x] + (Sq
rt[-a]*b*d^2 - 2*a*Sqrt[b]*d*e + (-a)^(3/2)*e^2)*(d + e*x)*Log[Sqrt[-a] + Sqrt[b]*x] + 2*Sqrt[-a]*(b*d^3 + a*d
*e^2 - (b*d^2 - a*e^2)*(d + e*x)*Log[d + e*x])))/(Sqrt[-a]*(b*d^2 + a*e^2)^2) - Log[c*(a + b*x^2)^p])/(2*e*(d
+ e*x)^2)

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Maple [C]  time = 0.504, size = 3183, normalized size = 18.3 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/e/(e*x+d)^2*ln((b*x^2+a)^p)-1/4*b^2*(-2*I*Pi*a*b*d^2*e^2*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)*csgn(I
*c)-2*b^2*ln((-8*a^2*b*d*e^3+8*a*b^2*d^3*e-3*(-a*b)^(1/2)*a^2*e^4+10*(-a*b)^(1/2)*a*b*d^2*e^2-3*(-a*b)^(1/2)*b
^2*d^4)*x-10*a^2*b*d^2*e^2-8*(-a*b)^(1/2)*a^2*d*e^3+8*(-a*b)^(1/2)*a*b*d^3*e+3*a^3*e^4+3*a*b^2*d^4)*d^2*e^2*p*
x^2-4*b^2*ln((-8*a^2*b*d*e^3+8*a*b^2*d^3*e-3*(-a*b)^(1/2)*a^2*e^4+10*(-a*b)^(1/2)*a*b*d^2*e^2-3*(-a*b)^(1/2)*b
^2*d^4)*x-10*a^2*b*d^2*e^2-8*(-a*b)^(1/2)*a^2*d*e^3+8*(-a*b)^(1/2)*a*b*d^3*e+3*a^3*e^4+3*a*b^2*d^4)*d^3*e*p*x+
2*b*ln((-8*a^2*b*d*e^3+8*a*b^2*d^3*e+3*(-a*b)^(1/2)*a^2*e^4-10*(-a*b)^(1/2)*a*b*d^2*e^2+3*(-a*b)^(1/2)*b^2*d^4
)*x-10*a^2*b*d^2*e^2+8*(-a*b)^(1/2)*a^2*d*e^3-8*(-a*b)^(1/2)*a*b*d^3*e+3*a^3*e^4+3*a*b^2*d^4)*a*e^4*p*x^2+2*b*
ln((-8*a^2*b*d*e^3+8*a*b^2*d^3*e+3*(-a*b)^(1/2)*a^2*e^4-10*(-a*b)^(1/2)*a*b*d^2*e^2+3*(-a*b)^(1/2)*b^2*d^4)*x-
10*a^2*b*d^2*e^2+8*(-a*b)^(1/2)*a^2*d*e^3-8*(-a*b)^(1/2)*a*b*d^3*e+3*a^3*e^4+3*a*b^2*d^4)*a*d^2*e^2*p-2*b^2*ln
((-8*a^2*b*d*e^3+8*a*b^2*d^3*e+3*(-a*b)^(1/2)*a^2*e^4-10*(-a*b)^(1/2)*a*b*d^2*e^2+3*(-a*b)^(1/2)*b^2*d^4)*x-10
*a^2*b*d^2*e^2+8*(-a*b)^(1/2)*a^2*d*e^3-8*(-a*b)^(1/2)*a*b*d^3*e+3*a^3*e^4+3*a*b^2*d^4)*d^2*e^2*p*x^2-4*b^2*ln
((-8*a^2*b*d*e^3+8*a*b^2*d^3*e+3*(-a*b)^(1/2)*a^2*e^4-10*(-a*b)^(1/2)*a*b*d^2*e^2+3*(-a*b)^(1/2)*b^2*d^4)*x-10
*a^2*b*d^2*e^2+8*(-a*b)^(1/2)*a^2*d*e^3-8*(-a*b)^(1/2)*a*b*d^3*e+3*a^3*e^4+3*a*b^2*d^4)*d^3*e*p*x+2*b*ln((-8*a
^2*b*d*e^3+8*a*b^2*d^3*e-3*(-a*b)^(1/2)*a^2*e^4+10*(-a*b)^(1/2)*a*b*d^2*e^2-3*(-a*b)^(1/2)*b^2*d^4)*x-10*a^2*b
*d^2*e^2-8*(-a*b)^(1/2)*a^2*d*e^3+8*(-a*b)^(1/2)*a*b*d^3*e+3*a^3*e^4+3*a*b^2*d^4)*a*e^4*p*x^2+2*b*ln((-8*a^2*b
*d*e^3+8*a*b^2*d^3*e-3*(-a*b)^(1/2)*a^2*e^4+10*(-a*b)^(1/2)*a*b*d^2*e^2-3*(-a*b)^(1/2)*b^2*d^4)*x-10*a^2*b*d^2
*e^2-8*(-a*b)^(1/2)*a^2*d*e^3+8*(-a*b)^(1/2)*a*b*d^3*e+3*a^3*e^4+3*a*b^2*d^4)*a*d^2*e^2*p-8*ln(e*x+d)*a*b*d*e^
3*p*x+4*b*ln((-8*a^2*b*d*e^3+8*a*b^2*d^3*e-3*(-a*b)^(1/2)*a^2*e^4+10*(-a*b)^(1/2)*a*b*d^2*e^2-3*(-a*b)^(1/2)*b
^2*d^4)*x-10*a^2*b*d^2*e^2-8*(-a*b)^(1/2)*a^2*d*e^3+8*(-a*b)^(1/2)*a*b*d^3*e+3*a^3*e^4+3*a*b^2*d^4)*a*d*e^3*p*
x+4*b*ln((-8*a^2*b*d*e^3+8*a*b^2*d^3*e+3*(-a*b)^(1/2)*a^2*e^4-10*(-a*b)^(1/2)*a*b*d^2*e^2+3*(-a*b)^(1/2)*b^2*d
^4)*x-10*a^2*b*d^2*e^2+8*(-a*b)^(1/2)*a^2*d*e^3-8*(-a*b)^(1/2)*a*b*d^3*e+3*a^3*e^4+3*a*b^2*d^4)*a*d*e^3*p*x-4*
b*ln((-8*a^2*b*d*e^3+8*a*b^2*d^3*e-3*(-a*b)^(1/2)*a^2*e^4+10*(-a*b)^(1/2)*a*b*d^2*e^2-3*(-a*b)^(1/2)*b^2*d^4)*
x-10*a^2*b*d^2*e^2-8*(-a*b)^(1/2)*a^2*d*e^3+8*(-a*b)^(1/2)*a*b*d^3*e+3*a^3*e^4+3*a*b^2*d^4)*(-a*b)^(1/2)*d*e^3
*p*x^2+2*I*Pi*a*b*d^2*e^2*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^2+2*I*Pi*a*b*d^2*e^2*csgn(I*c*(b*x^2+a)^p)
^2*csgn(I*c)+4*ln(e*x+d)*b^2*d^4*p-4*a*d^2*b*p*e^2+4*b*ln((-8*a^2*b*d*e^3+8*a*b^2*d^3*e+3*(-a*b)^(1/2)*a^2*e^4
-10*(-a*b)^(1/2)*a*b*d^2*e^2+3*(-a*b)^(1/2)*b^2*d^4)*x-10*a^2*b*d^2*e^2+8*(-a*b)^(1/2)*a^2*d*e^3-8*(-a*b)^(1/2
)*a*b*d^3*e+3*a^3*e^4+3*a*b^2*d^4)*(-a*b)^(1/2)*d*e^3*p*x^2-8*b*ln((-8*a^2*b*d*e^3+8*a*b^2*d^3*e-3*(-a*b)^(1/2
)*a^2*e^4+10*(-a*b)^(1/2)*a*b*d^2*e^2-3*(-a*b)^(1/2)*b^2*d^4)*x-10*a^2*b*d^2*e^2-8*(-a*b)^(1/2)*a^2*d*e^3+8*(-
a*b)^(1/2)*a*b*d^3*e+3*a^3*e^4+3*a*b^2*d^4)*(-a*b)^(1/2)*d^2*e^2*p*x+8*b*ln((-8*a^2*b*d*e^3+8*a*b^2*d^3*e+3*(-
a*b)^(1/2)*a^2*e^4-10*(-a*b)^(1/2)*a*b*d^2*e^2+3*(-a*b)^(1/2)*b^2*d^4)*x-10*a^2*b*d^2*e^2+8*(-a*b)^(1/2)*a^2*d
*e^3-8*(-a*b)^(1/2)*a*b*d^3*e+3*a^3*e^4+3*a*b^2*d^4)*(-a*b)^(1/2)*d^2*e^2*p*x-4*b^2*d^4*p-2*b^2*ln((-8*a^2*b*d
*e^3+8*a*b^2*d^3*e-3*(-a*b)^(1/2)*a^2*e^4+10*(-a*b)^(1/2)*a*b*d^2*e^2-3*(-a*b)^(1/2)*b^2*d^4)*x-10*a^2*b*d^2*e
^2-8*(-a*b)^(1/2)*a^2*d*e^3+8*(-a*b)^(1/2)*a*b*d^3*e+3*a^3*e^4+3*a*b^2*d^4)*d^4*p-2*b^2*ln((-8*a^2*b*d*e^3+8*a
*b^2*d^3*e+3*(-a*b)^(1/2)*a^2*e^4-10*(-a*b)^(1/2)*a*b*d^2*e^2+3*(-a*b)^(1/2)*b^2*d^4)*x-10*a^2*b*d^2*e^2+8*(-a
*b)^(1/2)*a^2*d*e^3-8*(-a*b)^(1/2)*a*b*d^3*e+3*a^3*e^4+3*a*b^2*d^4)*d^4*p-I*Pi*a^2*e^4*csgn(I*(b*x^2+a)^p)*csg
n(I*c*(b*x^2+a)^p)*csgn(I*c)-I*Pi*a^2*e^4*csgn(I*c*(b*x^2+a)^p)^3+I*Pi*b^2*d^4*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*
c)-4*a*d*p*e^3*x*b+I*Pi*a^2*e^4*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^2+I*Pi*a^2*e^4*csgn(I*c*(b*x^2+a)^p)
^2*csgn(I*c)+I*Pi*b^2*d^4*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^2+4*ln(c)*a*b*d^2*e^2+2*ln(c)*b^2*d^4+2*ln
(c)*a^2*e^4-4*d^3*p*x*b^2*e-2*I*Pi*a*b*d^2*e^2*csgn(I*c*(b*x^2+a)^p)^3-I*Pi*b^2*d^4*csgn(I*(b*x^2+a)^p)*csgn(I
*c*(b*x^2+a)^p)*csgn(I*c)-I*Pi*b^2*d^4*csgn(I*c*(b*x^2+a)^p)^3-4*b*ln((-8*a^2*b*d*e^3+8*a*b^2*d^3*e-3*(-a*b)^(
1/2)*a^2*e^4+10*(-a*b)^(1/2)*a*b*d^2*e^2-3*(-a*b)^(1/2)*b^2*d^4)*x-10*a^2*b*d^2*e^2-8*(-a*b)^(1/2)*a^2*d*e^3+8
*(-a*b)^(1/2)*a*b*d^3*e+3*a^3*e^4+3*a*b^2*d^4)*(-a*b)^(1/2)*d^3*e*p+4*b*ln((-8*a^2*b*d*e^3+8*a*b^2*d^3*e+3*(-a
*b)^(1/2)*a^2*e^4-10*(-a*b)^(1/2)*a*b*d^2*e^2+3*(-a*b)^(1/2)*b^2*d^4)*x-10*a^2*b*d^2*e^2+8*(-a*b)^(1/2)*a^2*d*
e^3-8*(-a*b)^(1/2)*a*b*d^3*e+3*a^3*e^4+3*a*b^2*d^4)*(-a*b)^(1/2)*d^3*e*p-4*ln(e*x+d)*a*b*e^4*p*x^2+4*ln(e*x+d)
*b^2*d^2*e^2*p*x^2+8*ln(e*x+d)*b^2*d^3*e*p*x-4*ln(e*x+d)*a*b*d^2*e^2*p)/(e*x+d)^2/(b*d-e*(-a*b)^(1/2))^2/(e*(-
a*b)^(1/2)+b*d)^2/e

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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*(b*x^2+a)^p)/(e*x+d)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 3.02908, size = 1524, normalized size = 8.76 \begin{align*} \left [\frac{2 \,{\left (b^{2} d^{3} e + a b d e^{3}\right )} p x + 2 \,{\left (b d e^{3} p x^{2} + 2 \, b d^{2} e^{2} p x + b d^{3} e p\right )} \sqrt{-a b} \log \left (\frac{b x^{2} + 2 \, \sqrt{-a b} x - a}{b x^{2} + a}\right ) + 2 \,{\left (b^{2} d^{4} + a b d^{2} e^{2}\right )} p +{\left ({\left (b^{2} d^{2} e^{2} - a b e^{4}\right )} p x^{2} + 2 \,{\left (b^{2} d^{3} e - a b d e^{3}\right )} p x -{\left (3 \, a b d^{2} e^{2} + a^{2} e^{4}\right )} p\right )} \log \left (b x^{2} + a\right ) - 2 \,{\left ({\left (b^{2} d^{2} e^{2} - a b e^{4}\right )} p x^{2} + 2 \,{\left (b^{2} d^{3} e - a b d e^{3}\right )} p x +{\left (b^{2} d^{4} - a b d^{2} e^{2}\right )} p\right )} \log \left (e x + d\right ) -{\left (b^{2} d^{4} + 2 \, a b d^{2} e^{2} + a^{2} e^{4}\right )} \log \left (c\right )}{2 \,{\left (b^{2} d^{6} e + 2 \, a b d^{4} e^{3} + a^{2} d^{2} e^{5} +{\left (b^{2} d^{4} e^{3} + 2 \, a b d^{2} e^{5} + a^{2} e^{7}\right )} x^{2} + 2 \,{\left (b^{2} d^{5} e^{2} + 2 \, a b d^{3} e^{4} + a^{2} d e^{6}\right )} x\right )}}, \frac{2 \,{\left (b^{2} d^{3} e + a b d e^{3}\right )} p x + 4 \,{\left (b d e^{3} p x^{2} + 2 \, b d^{2} e^{2} p x + b d^{3} e p\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) + 2 \,{\left (b^{2} d^{4} + a b d^{2} e^{2}\right )} p +{\left ({\left (b^{2} d^{2} e^{2} - a b e^{4}\right )} p x^{2} + 2 \,{\left (b^{2} d^{3} e - a b d e^{3}\right )} p x -{\left (3 \, a b d^{2} e^{2} + a^{2} e^{4}\right )} p\right )} \log \left (b x^{2} + a\right ) - 2 \,{\left ({\left (b^{2} d^{2} e^{2} - a b e^{4}\right )} p x^{2} + 2 \,{\left (b^{2} d^{3} e - a b d e^{3}\right )} p x +{\left (b^{2} d^{4} - a b d^{2} e^{2}\right )} p\right )} \log \left (e x + d\right ) -{\left (b^{2} d^{4} + 2 \, a b d^{2} e^{2} + a^{2} e^{4}\right )} \log \left (c\right )}{2 \,{\left (b^{2} d^{6} e + 2 \, a b d^{4} e^{3} + a^{2} d^{2} e^{5} +{\left (b^{2} d^{4} e^{3} + 2 \, a b d^{2} e^{5} + a^{2} e^{7}\right )} x^{2} + 2 \,{\left (b^{2} d^{5} e^{2} + 2 \, a b d^{3} e^{4} + a^{2} d e^{6}\right )} x\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(b*x^2+a)^p)/(e*x+d)^3,x, algorithm="fricas")

[Out]

[1/2*(2*(b^2*d^3*e + a*b*d*e^3)*p*x + 2*(b*d*e^3*p*x^2 + 2*b*d^2*e^2*p*x + b*d^3*e*p)*sqrt(-a*b)*log((b*x^2 +
2*sqrt(-a*b)*x - a)/(b*x^2 + a)) + 2*(b^2*d^4 + a*b*d^2*e^2)*p + ((b^2*d^2*e^2 - a*b*e^4)*p*x^2 + 2*(b^2*d^3*e
- a*b*d*e^3)*p*x - (3*a*b*d^2*e^2 + a^2*e^4)*p)*log(b*x^2 + a) - 2*((b^2*d^2*e^2 - a*b*e^4)*p*x^2 + 2*(b^2*d^
3*e - a*b*d*e^3)*p*x + (b^2*d^4 - a*b*d^2*e^2)*p)*log(e*x + d) - (b^2*d^4 + 2*a*b*d^2*e^2 + a^2*e^4)*log(c))/(
b^2*d^6*e + 2*a*b*d^4*e^3 + a^2*d^2*e^5 + (b^2*d^4*e^3 + 2*a*b*d^2*e^5 + a^2*e^7)*x^2 + 2*(b^2*d^5*e^2 + 2*a*b
*d^3*e^4 + a^2*d*e^6)*x), 1/2*(2*(b^2*d^3*e + a*b*d*e^3)*p*x + 4*(b*d*e^3*p*x^2 + 2*b*d^2*e^2*p*x + b*d^3*e*p)
*sqrt(a*b)*arctan(sqrt(a*b)*x/a) + 2*(b^2*d^4 + a*b*d^2*e^2)*p + ((b^2*d^2*e^2 - a*b*e^4)*p*x^2 + 2*(b^2*d^3*e
- a*b*d*e^3)*p*x - (3*a*b*d^2*e^2 + a^2*e^4)*p)*log(b*x^2 + a) - 2*((b^2*d^2*e^2 - a*b*e^4)*p*x^2 + 2*(b^2*d^
3*e - a*b*d*e^3)*p*x + (b^2*d^4 - a*b*d^2*e^2)*p)*log(e*x + d) - (b^2*d^4 + 2*a*b*d^2*e^2 + a^2*e^4)*log(c))/(
b^2*d^6*e + 2*a*b*d^4*e^3 + a^2*d^2*e^5 + (b^2*d^4*e^3 + 2*a*b*d^2*e^5 + a^2*e^7)*x^2 + 2*(b^2*d^5*e^2 + 2*a*b
*d^3*e^4 + a^2*d*e^6)*x)]

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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*(b*x**2+a)**p)/(e*x+d)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.33172, size = 567, normalized size = 3.26 \begin{align*} \frac{2 \, a b^{2} d p \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{{\left (b^{2} d^{4} + 2 \, a b d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt{a b}} + \frac{{\left (b^{2} d^{2} p - a b p e^{2}\right )} \log \left (b x^{2} + a\right )}{2 \,{\left (b^{2} d^{4} e + 2 \, a b d^{2} e^{3} + a^{2} e^{5}\right )}} - \frac{2 \, b^{2} d^{2} p x^{2} e^{2} \log \left (x e + d\right ) + 4 \, b^{2} d^{3} p x e \log \left (x e + d\right ) - 2 \, b^{2} d^{3} p x e + b^{2} d^{4} p \log \left (b x^{2} + a\right ) + 2 \, b^{2} d^{4} p \log \left (x e + d\right ) - 2 \, b^{2} d^{4} p + 2 \, a b d^{2} p e^{2} \log \left (b x^{2} + a\right ) - 2 \, a b p x^{2} e^{4} \log \left (x e + d\right ) - 4 \, a b d p x e^{3} \log \left (x e + d\right ) - 2 \, a b d^{2} p e^{2} \log \left (x e + d\right ) + b^{2} d^{4} \log \left (c\right ) - 2 \, a b d p x e^{3} - 2 \, a b d^{2} p e^{2} + 2 \, a b d^{2} e^{2} \log \left (c\right ) + a^{2} p e^{4} \log \left (b x^{2} + a\right ) + a^{2} e^{4} \log \left (c\right )}{2 \,{\left (b^{2} d^{4} x^{2} e^{3} + 2 \, b^{2} d^{5} x e^{2} + b^{2} d^{6} e + 2 \, a b d^{2} x^{2} e^{5} + 4 \, a b d^{3} x e^{4} + 2 \, a b d^{4} e^{3} + a^{2} x^{2} e^{7} + 2 \, a^{2} d x e^{6} + a^{2} d^{2} e^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(b*x^2+a)^p)/(e*x+d)^3,x, algorithm="giac")

[Out]

2*a*b^2*d*p*arctan(b*x/sqrt(a*b))/((b^2*d^4 + 2*a*b*d^2*e^2 + a^2*e^4)*sqrt(a*b)) + 1/2*(b^2*d^2*p - a*b*p*e^2
)*log(b*x^2 + a)/(b^2*d^4*e + 2*a*b*d^2*e^3 + a^2*e^5) - 1/2*(2*b^2*d^2*p*x^2*e^2*log(x*e + d) + 4*b^2*d^3*p*x
*e*log(x*e + d) - 2*b^2*d^3*p*x*e + b^2*d^4*p*log(b*x^2 + a) + 2*b^2*d^4*p*log(x*e + d) - 2*b^2*d^4*p + 2*a*b*
d^2*p*e^2*log(b*x^2 + a) - 2*a*b*p*x^2*e^4*log(x*e + d) - 4*a*b*d*p*x*e^3*log(x*e + d) - 2*a*b*d^2*p*e^2*log(x
*e + d) + b^2*d^4*log(c) - 2*a*b*d*p*x*e^3 - 2*a*b*d^2*p*e^2 + 2*a*b*d^2*e^2*log(c) + a^2*p*e^4*log(b*x^2 + a)
+ a^2*e^4*log(c))/(b^2*d^4*x^2*e^3 + 2*b^2*d^5*x*e^2 + b^2*d^6*e + 2*a*b*d^2*x^2*e^5 + 4*a*b*d^3*x*e^4 + 2*a*
b*d^4*e^3 + a^2*x^2*e^7 + 2*a^2*d*x*e^6 + a^2*d^2*e^5)