3.184 \(\int (d+e x)^3 \log (c (a+b x^2)^p) \, dx\)

Optimal. Leaf size=178 \[ -\frac{p \left (a^2 e^4-6 a b d^2 e^2+b^2 d^4\right ) \log \left (a+b x^2\right )}{4 b^2 e}+\frac{2 \sqrt{a} d p \left (b d^2-a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{3/2}}+\frac{(d+e x)^4 \log \left (c \left (a+b x^2\right )^p\right )}{4 e}-\frac{e p x^2 \left (6 b d^2-a e^2\right )}{4 b}-\frac{2 d p x \left (b d^2-a e^2\right )}{b}-\frac{2}{3} d e^2 p x^3-\frac{1}{8} e^3 p x^4 \]

[Out]

(-2*d*(b*d^2 - a*e^2)*p*x)/b - (e*(6*b*d^2 - a*e^2)*p*x^2)/(4*b) - (2*d*e^2*p*x^3)/3 - (e^3*p*x^4)/8 + (2*Sqrt
[a]*d*(b*d^2 - a*e^2)*p*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(3/2) - ((b^2*d^4 - 6*a*b*d^2*e^2 + a^2*e^4)*p*Log[a +
b*x^2])/(4*b^2*e) + ((d + e*x)^4*Log[c*(a + b*x^2)^p])/(4*e)

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Rubi [A]  time = 0.163456, antiderivative size = 178, 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} \[ -\frac{p \left (a^2 e^4-6 a b d^2 e^2+b^2 d^4\right ) \log \left (a+b x^2\right )}{4 b^2 e}+\frac{2 \sqrt{a} d p \left (b d^2-a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{3/2}}+\frac{(d+e x)^4 \log \left (c \left (a+b x^2\right )^p\right )}{4 e}-\frac{e p x^2 \left (6 b d^2-a e^2\right )}{4 b}-\frac{2 d p x \left (b d^2-a e^2\right )}{b}-\frac{2}{3} d e^2 p x^3-\frac{1}{8} e^3 p x^4 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*d*(b*d^2 - a*e^2)*p*x)/b - (e*(6*b*d^2 - a*e^2)*p*x^2)/(4*b) - (2*d*e^2*p*x^3)/3 - (e^3*p*x^4)/8 + (2*Sqrt
[a]*d*(b*d^2 - a*e^2)*p*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(3/2) - ((b^2*d^4 - 6*a*b*d^2*e^2 + a^2*e^4)*p*Log[a +
b*x^2])/(4*b^2*e) + ((d + e*x)^4*Log[c*(a + b*x^2)^p])/(4*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)^3 \log \left (c \left (a+b x^2\right )^p\right ) \, dx &=\frac{(d+e x)^4 \log \left (c \left (a+b x^2\right )^p\right )}{4 e}-\frac{(b p) \int \frac{x (d+e x)^4}{a+b x^2} \, dx}{2 e}\\ &=\frac{(d+e x)^4 \log \left (c \left (a+b x^2\right )^p\right )}{4 e}-\frac{(b p) \int \left (\frac{4 d e \left (b d^2-a e^2\right )}{b^2}+\frac{e^2 \left (6 b d^2-a e^2\right ) x}{b^2}+\frac{4 d e^3 x^2}{b}+\frac{e^4 x^3}{b}-\frac{4 a d e \left (b d^2-a e^2\right )-\left (b^2 d^4-6 a b d^2 e^2+a^2 e^4\right ) x}{b^2 \left (a+b x^2\right )}\right ) \, dx}{2 e}\\ &=-\frac{2 d \left (b d^2-a e^2\right ) p x}{b}-\frac{e \left (6 b d^2-a e^2\right ) p x^2}{4 b}-\frac{2}{3} d e^2 p x^3-\frac{1}{8} e^3 p x^4+\frac{(d+e x)^4 \log \left (c \left (a+b x^2\right )^p\right )}{4 e}+\frac{p \int \frac{4 a d e \left (b d^2-a e^2\right )-\left (b^2 d^4-6 a b d^2 e^2+a^2 e^4\right ) x}{a+b x^2} \, dx}{2 b e}\\ &=-\frac{2 d \left (b d^2-a e^2\right ) p x}{b}-\frac{e \left (6 b d^2-a e^2\right ) p x^2}{4 b}-\frac{2}{3} d e^2 p x^3-\frac{1}{8} e^3 p x^4+\frac{(d+e x)^4 \log \left (c \left (a+b x^2\right )^p\right )}{4 e}+\frac{\left (2 a d \left (b d^2-a e^2\right ) p\right ) \int \frac{1}{a+b x^2} \, dx}{b}+\frac{\left (\left (-b^2 d^4+6 a b d^2 e^2-a^2 e^4\right ) p\right ) \int \frac{x}{a+b x^2} \, dx}{2 b e}\\ &=-\frac{2 d \left (b d^2-a e^2\right ) p x}{b}-\frac{e \left (6 b d^2-a e^2\right ) p x^2}{4 b}-\frac{2}{3} d e^2 p x^3-\frac{1}{8} e^3 p x^4+\frac{2 \sqrt{a} d \left (b d^2-a e^2\right ) p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{3/2}}-\frac{\left (b^2 d^4-6 a b d^2 e^2+a^2 e^4\right ) p \log \left (a+b x^2\right )}{4 b^2 e}+\frac{(d+e x)^4 \log \left (c \left (a+b x^2\right )^p\right )}{4 e}\\ \end{align*}

Mathematica [A]  time = 0.731074, size = 249, normalized size = 1.4 \[ \frac{-6 p \left (a^2 e^4+4 \sqrt{-a} b^{3/2} d^3 e-6 a b d^2 e^2+4 (-a)^{3/2} \sqrt{b} d e^3+b^2 d^4\right ) \log \left (\sqrt{-a}-\sqrt{b} x\right )-6 p \left (a^2 e^4-4 \sqrt{-a} b^{3/2} d^3 e-6 a b d^2 e^2+4 \sqrt{-a} a \sqrt{b} d e^3+b^2 d^4\right ) \log \left (\sqrt{-a}+\sqrt{b} x\right )+b \left (6 b (d+e x)^4 \log \left (c \left (a+b x^2\right )^p\right )+6 a e^3 p x (8 d+e x)-b e p x \left (36 d^2 e x+48 d^3+16 d e^2 x^2+3 e^3 x^3\right )\right )}{24 b^2 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*(b^2*d^4 + 4*Sqrt[-a]*b^(3/2)*d^3*e - 6*a*b*d^2*e^2 + 4*(-a)^(3/2)*Sqrt[b]*d*e^3 + a^2*e^4)*p*Log[Sqrt[-a]
- Sqrt[b]*x] - 6*(b^2*d^4 - 4*Sqrt[-a]*b^(3/2)*d^3*e - 6*a*b*d^2*e^2 + 4*Sqrt[-a]*a*Sqrt[b]*d*e^3 + a^2*e^4)*
p*Log[Sqrt[-a] + Sqrt[b]*x] + b*(6*a*e^3*p*x*(8*d + e*x) - b*e*p*x*(48*d^3 + 36*d^2*e*x + 16*d*e^2*x^2 + 3*e^3
*x^3) + 6*b*(d + e*x)^4*Log[c*(a + b*x^2)^p]))/(24*b^2*e)

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Maple [C]  time = 0.796, size = 1330, normalized size = 7.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*d^3*p*x-1/8*I*e^3*Pi*x^4*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)*csgn(I*c)+1/2*I*e^2*Pi*d*x^3*csgn(I*(b*x
^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^2+1/2*I*e^2*Pi*d*x^3*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c)+3/4*I*e*Pi*d^2*x^2*csgn(
I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^2+3/4*I*e*Pi*d^2*x^2*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c)-1/2*I*Pi*d^3*csgn(
I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)*csgn(I*c)*x+ln(c)*d^3*x+1/4*e^3*ln(c)*x^4-1/4/e*p*ln(-a^2*d*e^3+a*b*d^3*e
-(-a^3*b*d^2*e^6+2*a^2*b^2*d^4*e^4-a*b^3*d^6*e^2)^(1/2)*x)*d^4-1/4/e*p*ln(-a^2*d*e^3+a*b*d^3*e+(-a^3*b*d^2*e^6
+2*a^2*b^2*d^4*e^4-a*b^3*d^6*e^2)^(1/2)*x)*d^4+3/2*e*ln(c)*d^2*x^2+e^2*ln(c)*d*x^3-1/8*e^3*p*x^4-2/3*d*e^2*p*x
^3-3/2*d^2*e*p*x^2+1/4*(e*x+d)^4/e*ln((b*x^2+a)^p)+2/b*a*d*p*e^2*x-1/2*I*e^2*Pi*d*x^3*csgn(I*(b*x^2+a)^p)*csgn
(I*c*(b*x^2+a)^p)*csgn(I*c)-3/4*I*e*Pi*d^2*x^2*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)*csgn(I*c)-1/4/b^2*e^3
*p*ln(-a^2*d*e^3+a*b*d^3*e-(-a^3*b*d^2*e^6+2*a^2*b^2*d^4*e^4-a*b^3*d^6*e^2)^(1/2)*x)*a^2-1/4/b^2*e^3*p*ln(-a^2
*d*e^3+a*b*d^3*e+(-a^3*b*d^2*e^6+2*a^2*b^2*d^4*e^4-a*b^3*d^6*e^2)^(1/2)*x)*a^2+1/b^2/e*p*ln(-a^2*d*e^3+a*b*d^3
*e-(-a^3*b*d^2*e^6+2*a^2*b^2*d^4*e^4-a*b^3*d^6*e^2)^(1/2)*x)*(-a^3*b*d^2*e^6+2*a^2*b^2*d^4*e^4-a*b^3*d^6*e^2)^
(1/2)-1/b^2/e*p*ln(-a^2*d*e^3+a*b*d^3*e+(-a^3*b*d^2*e^6+2*a^2*b^2*d^4*e^4-a*b^3*d^6*e^2)^(1/2)*x)*(-a^3*b*d^2*
e^6+2*a^2*b^2*d^4*e^4-a*b^3*d^6*e^2)^(1/2)-1/2*I*Pi*d^3*csgn(I*c*(b*x^2+a)^p)^3*x-1/8*I*e^3*Pi*x^4*csgn(I*c*(b
*x^2+a)^p)^3+1/4/b*a*e^3*p*x^2+1/8*I*e^3*Pi*x^4*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^2+1/8*I*e^3*Pi*x^4*c
sgn(I*c*(b*x^2+a)^p)^2*csgn(I*c)-1/2*I*e^2*Pi*d*x^3*csgn(I*c*(b*x^2+a)^p)^3-3/4*I*e*Pi*d^2*x^2*csgn(I*c*(b*x^2
+a)^p)^3+1/2*I*Pi*d^3*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^2*x+1/2*I*Pi*d^3*csgn(I*c*(b*x^2+a)^p)^2*csgn(
I*c)*x+3/2/b*e*p*ln(-a^2*d*e^3+a*b*d^3*e-(-a^3*b*d^2*e^6+2*a^2*b^2*d^4*e^4-a*b^3*d^6*e^2)^(1/2)*x)*a*d^2+3/2/b
*e*p*ln(-a^2*d*e^3+a*b*d^3*e+(-a^3*b*d^2*e^6+2*a^2*b^2*d^4*e^4-a*b^3*d^6*e^2)^(1/2)*x)*a*d^2

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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)^3*log(c*(b*x^2+a)^p),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.09646, size = 1044, normalized size = 5.87 \begin{align*} \left [-\frac{3 \, b^{2} e^{3} p x^{4} + 16 \, b^{2} d e^{2} p x^{3} + 6 \,{\left (6 \, b^{2} d^{2} e - a b e^{3}\right )} p x^{2} - 24 \,{\left (b^{2} d^{3} - a b d e^{2}\right )} p \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} + 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) + 48 \,{\left (b^{2} d^{3} - a b d e^{2}\right )} p x - 6 \,{\left (b^{2} e^{3} p x^{4} + 4 \, b^{2} d e^{2} p x^{3} + 6 \, b^{2} d^{2} e p x^{2} + 4 \, b^{2} d^{3} p x +{\left (6 \, a b d^{2} e - a^{2} e^{3}\right )} p\right )} \log \left (b x^{2} + a\right ) - 6 \,{\left (b^{2} e^{3} x^{4} + 4 \, b^{2} d e^{2} x^{3} + 6 \, b^{2} d^{2} e x^{2} + 4 \, b^{2} d^{3} x\right )} \log \left (c\right )}{24 \, b^{2}}, -\frac{3 \, b^{2} e^{3} p x^{4} + 16 \, b^{2} d e^{2} p x^{3} + 6 \,{\left (6 \, b^{2} d^{2} e - a b e^{3}\right )} p x^{2} - 48 \,{\left (b^{2} d^{3} - a b d e^{2}\right )} p \sqrt{\frac{a}{b}} \arctan \left (\frac{b x \sqrt{\frac{a}{b}}}{a}\right ) + 48 \,{\left (b^{2} d^{3} - a b d e^{2}\right )} p x - 6 \,{\left (b^{2} e^{3} p x^{4} + 4 \, b^{2} d e^{2} p x^{3} + 6 \, b^{2} d^{2} e p x^{2} + 4 \, b^{2} d^{3} p x +{\left (6 \, a b d^{2} e - a^{2} e^{3}\right )} p\right )} \log \left (b x^{2} + a\right ) - 6 \,{\left (b^{2} e^{3} x^{4} + 4 \, b^{2} d e^{2} x^{3} + 6 \, b^{2} d^{2} e x^{2} + 4 \, b^{2} d^{3} x\right )} \log \left (c\right )}{24 \, b^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/24*(3*b^2*e^3*p*x^4 + 16*b^2*d*e^2*p*x^3 + 6*(6*b^2*d^2*e - a*b*e^3)*p*x^2 - 24*(b^2*d^3 - a*b*d*e^2)*p*sq
rt(-a/b)*log((b*x^2 + 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)) + 48*(b^2*d^3 - a*b*d*e^2)*p*x - 6*(b^2*e^3*p*x^4 + 4
*b^2*d*e^2*p*x^3 + 6*b^2*d^2*e*p*x^2 + 4*b^2*d^3*p*x + (6*a*b*d^2*e - a^2*e^3)*p)*log(b*x^2 + a) - 6*(b^2*e^3*
x^4 + 4*b^2*d*e^2*x^3 + 6*b^2*d^2*e*x^2 + 4*b^2*d^3*x)*log(c))/b^2, -1/24*(3*b^2*e^3*p*x^4 + 16*b^2*d*e^2*p*x^
3 + 6*(6*b^2*d^2*e - a*b*e^3)*p*x^2 - 48*(b^2*d^3 - a*b*d*e^2)*p*sqrt(a/b)*arctan(b*x*sqrt(a/b)/a) + 48*(b^2*d
^3 - a*b*d*e^2)*p*x - 6*(b^2*e^3*p*x^4 + 4*b^2*d*e^2*p*x^3 + 6*b^2*d^2*e*p*x^2 + 4*b^2*d^3*p*x + (6*a*b*d^2*e
- a^2*e^3)*p)*log(b*x^2 + a) - 6*(b^2*e^3*x^4 + 4*b^2*d*e^2*x^3 + 6*b^2*d^2*e*x^2 + 4*b^2*d^3*x)*log(c))/b^2]

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Sympy [A]  time = 82.9278, size = 422, normalized size = 2.37 \begin{align*} \begin{cases} - \frac{i a^{\frac{3}{2}} d e^{2} p \log{\left (a + b x^{2} \right )}}{b^{2} \sqrt{\frac{1}{b}}} + \frac{2 i a^{\frac{3}{2}} d e^{2} p \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + x \right )}}{b^{2} \sqrt{\frac{1}{b}}} + \frac{i \sqrt{a} d^{3} p \log{\left (a + b x^{2} \right )}}{b \sqrt{\frac{1}{b}}} - \frac{2 i \sqrt{a} d^{3} p \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + x \right )}}{b \sqrt{\frac{1}{b}}} - \frac{a^{2} e^{3} p \log{\left (a + b x^{2} \right )}}{4 b^{2}} + \frac{3 a d^{2} e p \log{\left (a + b x^{2} \right )}}{2 b} + \frac{2 a d e^{2} p x}{b} + \frac{a e^{3} p x^{2}}{4 b} + d^{3} p x \log{\left (a + b x^{2} \right )} - 2 d^{3} p x + d^{3} x \log{\left (c \right )} + \frac{3 d^{2} e p x^{2} \log{\left (a + b x^{2} \right )}}{2} - \frac{3 d^{2} e p x^{2}}{2} + \frac{3 d^{2} e x^{2} \log{\left (c \right )}}{2} + d e^{2} p x^{3} \log{\left (a + b x^{2} \right )} - \frac{2 d e^{2} p x^{3}}{3} + d e^{2} x^{3} \log{\left (c \right )} + \frac{e^{3} p x^{4} \log{\left (a + b x^{2} \right )}}{4} - \frac{e^{3} p x^{4}}{8} + \frac{e^{3} x^{4} \log{\left (c \right )}}{4} & \text{for}\: b \neq 0 \\\left (d^{3} x + \frac{3 d^{2} e x^{2}}{2} + d e^{2} x^{3} + \frac{e^{3} x^{4}}{4}\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)**3*ln(c*(b*x**2+a)**p),x)

[Out]

Piecewise((-I*a**(3/2)*d*e**2*p*log(a + b*x**2)/(b**2*sqrt(1/b)) + 2*I*a**(3/2)*d*e**2*p*log(-I*sqrt(a)*sqrt(1
/b) + x)/(b**2*sqrt(1/b)) + I*sqrt(a)*d**3*p*log(a + b*x**2)/(b*sqrt(1/b)) - 2*I*sqrt(a)*d**3*p*log(-I*sqrt(a)
*sqrt(1/b) + x)/(b*sqrt(1/b)) - a**2*e**3*p*log(a + b*x**2)/(4*b**2) + 3*a*d**2*e*p*log(a + b*x**2)/(2*b) + 2*
a*d*e**2*p*x/b + a*e**3*p*x**2/(4*b) + d**3*p*x*log(a + b*x**2) - 2*d**3*p*x + d**3*x*log(c) + 3*d**2*e*p*x**2
*log(a + b*x**2)/2 - 3*d**2*e*p*x**2/2 + 3*d**2*e*x**2*log(c)/2 + d*e**2*p*x**3*log(a + b*x**2) - 2*d*e**2*p*x
**3/3 + d*e**2*x**3*log(c) + e**3*p*x**4*log(a + b*x**2)/4 - e**3*p*x**4/8 + e**3*x**4*log(c)/4, Ne(b, 0)), ((
d**3*x + 3*d**2*e*x**2/2 + d*e**2*x**3 + e**3*x**4/4)*log(a**p*c), True))

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Giac [A]  time = 1.26655, size = 383, normalized size = 2.15 \begin{align*} \frac{2 \, a d^{3} p \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b}} - \frac{2 \, a^{2} d p \arctan \left (\frac{b x}{\sqrt{a b}}\right ) e^{2}}{\sqrt{a b} b} + \frac{6 \, b^{2} p x^{4} e^{3} \log \left (b x^{2} + a\right ) + 24 \, b^{2} d p x^{3} e^{2} \log \left (b x^{2} + a\right ) + 36 \, b^{2} d^{2} p x^{2} e \log \left (b x^{2} + a\right ) - 3 \, b^{2} p x^{4} e^{3} - 16 \, b^{2} d p x^{3} e^{2} - 36 \, b^{2} d^{2} p x^{2} e + 24 \, b^{2} d^{3} p x \log \left (b x^{2} + a\right ) + 6 \, b^{2} x^{4} e^{3} \log \left (c\right ) + 24 \, b^{2} d x^{3} e^{2} \log \left (c\right ) + 36 \, b^{2} d^{2} x^{2} e \log \left (c\right ) - 48 \, b^{2} d^{3} p x + 36 \, a b d^{2} p e \log \left (b x^{2} + a\right ) + 24 \, b^{2} d^{3} x \log \left (c\right ) + 6 \, a b p x^{2} e^{3} + 48 \, a b d p x e^{2} - 6 \, a^{2} p e^{3} \log \left (b x^{2} + a\right )}{24 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a*d^3*p*arctan(b*x/sqrt(a*b))/sqrt(a*b) - 2*a^2*d*p*arctan(b*x/sqrt(a*b))*e^2/(sqrt(a*b)*b) + 1/24*(6*b^2*p*
x^4*e^3*log(b*x^2 + a) + 24*b^2*d*p*x^3*e^2*log(b*x^2 + a) + 36*b^2*d^2*p*x^2*e*log(b*x^2 + a) - 3*b^2*p*x^4*e
^3 - 16*b^2*d*p*x^3*e^2 - 36*b^2*d^2*p*x^2*e + 24*b^2*d^3*p*x*log(b*x^2 + a) + 6*b^2*x^4*e^3*log(c) + 24*b^2*d
*x^3*e^2*log(c) + 36*b^2*d^2*x^2*e*log(c) - 48*b^2*d^3*p*x + 36*a*b*d^2*p*e*log(b*x^2 + a) + 24*b^2*d^3*x*log(
c) + 6*a*b*p*x^2*e^3 + 48*a*b*d*p*x*e^2 - 6*a^2*p*e^3*log(b*x^2 + a))/b^2