3.268 \(\int (f+g x^2)^3 \log (c (d+e x^2)^p) \, dx\)

Optimal. Leaf size=338 \[ f^2 g x^3 \log \left (c \left (d+e x^2\right )^p\right )+f^3 x \log \left (c \left (d+e x^2\right )^p\right )+\frac{3}{5} f g^2 x^5 \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{7} g^3 x^7 \log \left (c \left (d+e x^2\right )^p\right )-\frac{2 d^{3/2} f^2 g p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{e^{3/2}}-\frac{6 d^2 f g^2 p x}{5 e^2}+\frac{6 d^{5/2} f g^2 p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{5 e^{5/2}}-\frac{2 d^2 g^3 p x^3}{21 e^2}+\frac{2 d^3 g^3 p x}{7 e^3}-\frac{2 d^{7/2} g^3 p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{7 e^{7/2}}+\frac{2 d f^2 g p x}{e}+\frac{2 \sqrt{d} f^3 p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}+\frac{2 d f g^2 p x^3}{5 e}+\frac{2 d g^3 p x^5}{35 e}-\frac{2}{3} f^2 g p x^3-2 f^3 p x-\frac{6}{25} f g^2 p x^5-\frac{2}{49} g^3 p x^7 \]

[Out]

-2*f^3*p*x + (2*d*f^2*g*p*x)/e - (6*d^2*f*g^2*p*x)/(5*e^2) + (2*d^3*g^3*p*x)/(7*e^3) - (2*f^2*g*p*x^3)/3 + (2*
d*f*g^2*p*x^3)/(5*e) - (2*d^2*g^3*p*x^3)/(21*e^2) - (6*f*g^2*p*x^5)/25 + (2*d*g^3*p*x^5)/(35*e) - (2*g^3*p*x^7
)/49 + (2*Sqrt[d]*f^3*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e] - (2*d^(3/2)*f^2*g*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])
/e^(3/2) + (6*d^(5/2)*f*g^2*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(5*e^(5/2)) - (2*d^(7/2)*g^3*p*ArcTan[(Sqrt[e]*x)/S
qrt[d]])/(7*e^(7/2)) + f^3*x*Log[c*(d + e*x^2)^p] + f^2*g*x^3*Log[c*(d + e*x^2)^p] + (3*f*g^2*x^5*Log[c*(d + e
*x^2)^p])/5 + (g^3*x^7*Log[c*(d + e*x^2)^p])/7

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Rubi [A]  time = 0.258966, antiderivative size = 338, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {2471, 2448, 321, 205, 2455, 302} \[ f^2 g x^3 \log \left (c \left (d+e x^2\right )^p\right )+f^3 x \log \left (c \left (d+e x^2\right )^p\right )+\frac{3}{5} f g^2 x^5 \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{7} g^3 x^7 \log \left (c \left (d+e x^2\right )^p\right )-\frac{2 d^{3/2} f^2 g p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{e^{3/2}}-\frac{6 d^2 f g^2 p x}{5 e^2}+\frac{6 d^{5/2} f g^2 p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{5 e^{5/2}}-\frac{2 d^2 g^3 p x^3}{21 e^2}+\frac{2 d^3 g^3 p x}{7 e^3}-\frac{2 d^{7/2} g^3 p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{7 e^{7/2}}+\frac{2 d f^2 g p x}{e}+\frac{2 \sqrt{d} f^3 p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}+\frac{2 d f g^2 p x^3}{5 e}+\frac{2 d g^3 p x^5}{35 e}-\frac{2}{3} f^2 g p x^3-2 f^3 p x-\frac{6}{25} f g^2 p x^5-\frac{2}{49} g^3 p x^7 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2*f^3*p*x + (2*d*f^2*g*p*x)/e - (6*d^2*f*g^2*p*x)/(5*e^2) + (2*d^3*g^3*p*x)/(7*e^3) - (2*f^2*g*p*x^3)/3 + (2*
d*f*g^2*p*x^3)/(5*e) - (2*d^2*g^3*p*x^3)/(21*e^2) - (6*f*g^2*p*x^5)/25 + (2*d*g^3*p*x^5)/(35*e) - (2*g^3*p*x^7
)/49 + (2*Sqrt[d]*f^3*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e] - (2*d^(3/2)*f^2*g*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])
/e^(3/2) + (6*d^(5/2)*f*g^2*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(5*e^(5/2)) - (2*d^(7/2)*g^3*p*ArcTan[(Sqrt[e]*x)/S
qrt[d]])/(7*e^(7/2)) + f^3*x*Log[c*(d + e*x^2)^p] + f^2*g*x^3*Log[c*(d + e*x^2)^p] + (3*f*g^2*x^5*Log[c*(d + e
*x^2)^p])/5 + (g^3*x^7*Log[c*(d + e*x^2)^p])/7

Rule 2471

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*((f_) + (g_.)*(x_)^(s_))^(r_.), x_Symbol]
:> With[{t = ExpandIntegrand[(a + b*Log[c*(d + e*x^n)^p])^q, (f + g*x^s)^r, x]}, Int[t, x] /; SumQ[t]] /; Free
Q[{a, b, c, d, e, f, g, n, p, q, r, s}, x] && IntegerQ[n] && IGtQ[q, 0] && IntegerQ[r] && IntegerQ[s] && (EqQ[
q, 1] || (GtQ[r, 0] && GtQ[s, 1]) || (LtQ[s, 0] && LtQ[r, 0]))

Rule 2448

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)], x_Symbol] :> Simp[x*Log[c*(d + e*x^n)^p], x] - Dist[e*n*p, Int[
x^n/(d + e*x^n), x], x] /; FreeQ[{c, d, e, n, p}, x]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && 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]

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 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rubi steps

\begin{align*} \int \left (f+g x^2\right )^3 \log \left (c \left (d+e x^2\right )^p\right ) \, dx &=\int \left (f^3 \log \left (c \left (d+e x^2\right )^p\right )+3 f^2 g x^2 \log \left (c \left (d+e x^2\right )^p\right )+3 f g^2 x^4 \log \left (c \left (d+e x^2\right )^p\right )+g^3 x^6 \log \left (c \left (d+e x^2\right )^p\right )\right ) \, dx\\ &=f^3 \int \log \left (c \left (d+e x^2\right )^p\right ) \, dx+\left (3 f^2 g\right ) \int x^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx+\left (3 f g^2\right ) \int x^4 \log \left (c \left (d+e x^2\right )^p\right ) \, dx+g^3 \int x^6 \log \left (c \left (d+e x^2\right )^p\right ) \, dx\\ &=f^3 x \log \left (c \left (d+e x^2\right )^p\right )+f^2 g x^3 \log \left (c \left (d+e x^2\right )^p\right )+\frac{3}{5} f g^2 x^5 \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{7} g^3 x^7 \log \left (c \left (d+e x^2\right )^p\right )-\left (2 e f^3 p\right ) \int \frac{x^2}{d+e x^2} \, dx-\left (2 e f^2 g p\right ) \int \frac{x^4}{d+e x^2} \, dx-\frac{1}{5} \left (6 e f g^2 p\right ) \int \frac{x^6}{d+e x^2} \, dx-\frac{1}{7} \left (2 e g^3 p\right ) \int \frac{x^8}{d+e x^2} \, dx\\ &=-2 f^3 p x+f^3 x \log \left (c \left (d+e x^2\right )^p\right )+f^2 g x^3 \log \left (c \left (d+e x^2\right )^p\right )+\frac{3}{5} f g^2 x^5 \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{7} g^3 x^7 \log \left (c \left (d+e x^2\right )^p\right )+\left (2 d f^3 p\right ) \int \frac{1}{d+e x^2} \, dx-\left (2 e f^2 g p\right ) \int \left (-\frac{d}{e^2}+\frac{x^2}{e}+\frac{d^2}{e^2 \left (d+e x^2\right )}\right ) \, dx-\frac{1}{5} \left (6 e f g^2 p\right ) \int \left (\frac{d^2}{e^3}-\frac{d x^2}{e^2}+\frac{x^4}{e}-\frac{d^3}{e^3 \left (d+e x^2\right )}\right ) \, dx-\frac{1}{7} \left (2 e g^3 p\right ) \int \left (-\frac{d^3}{e^4}+\frac{d^2 x^2}{e^3}-\frac{d x^4}{e^2}+\frac{x^6}{e}+\frac{d^4}{e^4 \left (d+e x^2\right )}\right ) \, dx\\ &=-2 f^3 p x+\frac{2 d f^2 g p x}{e}-\frac{6 d^2 f g^2 p x}{5 e^2}+\frac{2 d^3 g^3 p x}{7 e^3}-\frac{2}{3} f^2 g p x^3+\frac{2 d f g^2 p x^3}{5 e}-\frac{2 d^2 g^3 p x^3}{21 e^2}-\frac{6}{25} f g^2 p x^5+\frac{2 d g^3 p x^5}{35 e}-\frac{2}{49} g^3 p x^7+\frac{2 \sqrt{d} f^3 p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}+f^3 x \log \left (c \left (d+e x^2\right )^p\right )+f^2 g x^3 \log \left (c \left (d+e x^2\right )^p\right )+\frac{3}{5} f g^2 x^5 \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{7} g^3 x^7 \log \left (c \left (d+e x^2\right )^p\right )-\frac{\left (2 d^2 f^2 g p\right ) \int \frac{1}{d+e x^2} \, dx}{e}+\frac{\left (6 d^3 f g^2 p\right ) \int \frac{1}{d+e x^2} \, dx}{5 e^2}-\frac{\left (2 d^4 g^3 p\right ) \int \frac{1}{d+e x^2} \, dx}{7 e^3}\\ &=-2 f^3 p x+\frac{2 d f^2 g p x}{e}-\frac{6 d^2 f g^2 p x}{5 e^2}+\frac{2 d^3 g^3 p x}{7 e^3}-\frac{2}{3} f^2 g p x^3+\frac{2 d f g^2 p x^3}{5 e}-\frac{2 d^2 g^3 p x^3}{21 e^2}-\frac{6}{25} f g^2 p x^5+\frac{2 d g^3 p x^5}{35 e}-\frac{2}{49} g^3 p x^7+\frac{2 \sqrt{d} f^3 p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}-\frac{2 d^{3/2} f^2 g p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{e^{3/2}}+\frac{6 d^{5/2} f g^2 p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{5 e^{5/2}}-\frac{2 d^{7/2} g^3 p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{7 e^{7/2}}+f^3 x \log \left (c \left (d+e x^2\right )^p\right )+f^2 g x^3 \log \left (c \left (d+e x^2\right )^p\right )+\frac{3}{5} f g^2 x^5 \log \left (c \left (d+e x^2\right )^p\right )+\frac{1}{7} g^3 x^7 \log \left (c \left (d+e x^2\right )^p\right )\\ \end{align*}

Mathematica [A]  time = 0.273498, size = 215, normalized size = 0.64 \[ \frac{1}{35} x \left (35 f^2 g x^2+35 f^3+21 f g^2 x^4+5 g^3 x^6\right ) \log \left (c \left (d+e x^2\right )^p\right )-\frac{2 p x \left (35 d^2 e g^2 \left (63 f+5 g x^2\right )-525 d^3 g^3-105 d e^2 g \left (35 f^2+7 f g x^2+g^2 x^4\right )+e^3 \left (1225 f^2 g x^2+3675 f^3+441 f g^2 x^4+75 g^3 x^6\right )\right )}{3675 e^3}-\frac{2 \sqrt{d} p \left (-21 d^2 e f g^2+5 d^3 g^3+35 d e^2 f^2 g-35 e^3 f^3\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{35 e^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*p*x*(-525*d^3*g^3 + 35*d^2*e*g^2*(63*f + 5*g*x^2) - 105*d*e^2*g*(35*f^2 + 7*f*g*x^2 + g^2*x^4) + e^3*(3675
*f^3 + 1225*f^2*g*x^2 + 441*f*g^2*x^4 + 75*g^3*x^6)))/(3675*e^3) - (2*Sqrt[d]*(-35*e^3*f^3 + 35*d*e^2*f^2*g -
21*d^2*e*f*g^2 + 5*d^3*g^3)*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(35*e^(7/2)) + (x*(35*f^3 + 35*f^2*g*x^2 + 21*f*g^2
*x^4 + 5*g^3*x^6)*Log[c*(d + e*x^2)^p])/35

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Maple [C]  time = 0.546, size = 995, normalized size = 2.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x^2+f)^3*ln(c*(e*x^2+d)^p),x)

[Out]

-2*f^3*p*x+1/7*ln(c)*g^3*x^7+ln(c)*f^3*x-1/7/e^4*(-d*e)^(1/2)*p*ln((-d*e)^(1/2)*x-d)*g^3*d^3+1/7/e^4*(-d*e)^(1
/2)*p*ln(-(-d*e)^(1/2)*x-d)*g^3*d^3-3/10*I*Pi*f*g^2*x^5*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)+(1
/7*g^3*x^7+3/5*f*g^2*x^5+f^2*g*x^3+f^3*x)*ln((e*x^2+d)^p)+2/7*d^3*g^3*p*x/e^3-2/21*d^2*g^3*p*x^3/e^2+2/35*d*g^
3*p*x^5/e-3/5/e^3*(-d*e)^(1/2)*p*ln(-(-d*e)^(1/2)*x-d)*f*g^2*d^2+1/e^2*(-d*e)^(1/2)*p*ln(-(-d*e)^(1/2)*x-d)*f^
2*g*d+3/5/e^3*(-d*e)^(1/2)*p*ln((-d*e)^(1/2)*x-d)*f*g^2*d^2-1/e^2*(-d*e)^(1/2)*p*ln((-d*e)^(1/2)*x-d)*f^2*g*d+
3/10*I*Pi*f*g^2*x^5*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2+3/10*I*Pi*f*g^2*x^5*csgn(I*c*(e*x^2+d)^p)^2*cs
gn(I*c)+ln(c)*f^2*g*x^3+3/5*ln(c)*f*g^2*x^5+1/2*I*Pi*f^2*g*x^3*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2+1/2
*I*Pi*f^2*g*x^3*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)-1/14*I*Pi*g^3*x^7*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*
csgn(I*c)-1/2*I*Pi*f^3*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)*x-2/3*f^2*g*p*x^3-6/25*f*g^2*p*x^5-
2/49*g^3*p*x^7-1/e*(-d*e)^(1/2)*p*ln(-(-d*e)^(1/2)*x-d)*f^3+1/e*(-d*e)^(1/2)*p*ln((-d*e)^(1/2)*x-d)*f^3-1/14*I
*Pi*g^3*x^7*csgn(I*c*(e*x^2+d)^p)^3-1/2*I*Pi*f^3*csgn(I*c*(e*x^2+d)^p)^3*x-6/5*d^2*f*g^2*p*x/e^2+2/5*d*f*g^2*p
*x^3/e+2*d*f^2*g*p*x/e-1/2*I*Pi*f^2*g*x^3*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)-3/10*I*Pi*f*g^2*
x^5*csgn(I*c*(e*x^2+d)^p)^3-1/2*I*Pi*f^2*g*x^3*csgn(I*c*(e*x^2+d)^p)^3+1/2*I*Pi*f^3*csgn(I*(e*x^2+d)^p)*csgn(I
*c*(e*x^2+d)^p)^2*x+1/14*I*Pi*g^3*x^7*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2+1/2*I*Pi*f^3*csgn(I*c*(e*x^2
+d)^p)^2*csgn(I*c)*x+1/14*I*Pi*g^3*x^7*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.10436, size = 1335, normalized size = 3.95 \begin{align*} \left [-\frac{150 \, e^{3} g^{3} p x^{7} + 42 \,{\left (21 \, e^{3} f g^{2} - 5 \, d e^{2} g^{3}\right )} p x^{5} + 70 \,{\left (35 \, e^{3} f^{2} g - 21 \, d e^{2} f g^{2} + 5 \, d^{2} e g^{3}\right )} p x^{3} + 105 \,{\left (35 \, e^{3} f^{3} - 35 \, d e^{2} f^{2} g + 21 \, d^{2} e f g^{2} - 5 \, d^{3} g^{3}\right )} p \sqrt{-\frac{d}{e}} \log \left (\frac{e x^{2} - 2 \, e x \sqrt{-\frac{d}{e}} - d}{e x^{2} + d}\right ) + 210 \,{\left (35 \, e^{3} f^{3} - 35 \, d e^{2} f^{2} g + 21 \, d^{2} e f g^{2} - 5 \, d^{3} g^{3}\right )} p x - 105 \,{\left (5 \, e^{3} g^{3} p x^{7} + 21 \, e^{3} f g^{2} p x^{5} + 35 \, e^{3} f^{2} g p x^{3} + 35 \, e^{3} f^{3} p x\right )} \log \left (e x^{2} + d\right ) - 105 \,{\left (5 \, e^{3} g^{3} x^{7} + 21 \, e^{3} f g^{2} x^{5} + 35 \, e^{3} f^{2} g x^{3} + 35 \, e^{3} f^{3} x\right )} \log \left (c\right )}{3675 \, e^{3}}, -\frac{150 \, e^{3} g^{3} p x^{7} + 42 \,{\left (21 \, e^{3} f g^{2} - 5 \, d e^{2} g^{3}\right )} p x^{5} + 70 \,{\left (35 \, e^{3} f^{2} g - 21 \, d e^{2} f g^{2} + 5 \, d^{2} e g^{3}\right )} p x^{3} - 210 \,{\left (35 \, e^{3} f^{3} - 35 \, d e^{2} f^{2} g + 21 \, d^{2} e f g^{2} - 5 \, d^{3} g^{3}\right )} p \sqrt{\frac{d}{e}} \arctan \left (\frac{e x \sqrt{\frac{d}{e}}}{d}\right ) + 210 \,{\left (35 \, e^{3} f^{3} - 35 \, d e^{2} f^{2} g + 21 \, d^{2} e f g^{2} - 5 \, d^{3} g^{3}\right )} p x - 105 \,{\left (5 \, e^{3} g^{3} p x^{7} + 21 \, e^{3} f g^{2} p x^{5} + 35 \, e^{3} f^{2} g p x^{3} + 35 \, e^{3} f^{3} p x\right )} \log \left (e x^{2} + d\right ) - 105 \,{\left (5 \, e^{3} g^{3} x^{7} + 21 \, e^{3} f g^{2} x^{5} + 35 \, e^{3} f^{2} g x^{3} + 35 \, e^{3} f^{3} x\right )} \log \left (c\right )}{3675 \, e^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/3675*(150*e^3*g^3*p*x^7 + 42*(21*e^3*f*g^2 - 5*d*e^2*g^3)*p*x^5 + 70*(35*e^3*f^2*g - 21*d*e^2*f*g^2 + 5*d^
2*e*g^3)*p*x^3 + 105*(35*e^3*f^3 - 35*d*e^2*f^2*g + 21*d^2*e*f*g^2 - 5*d^3*g^3)*p*sqrt(-d/e)*log((e*x^2 - 2*e*
x*sqrt(-d/e) - d)/(e*x^2 + d)) + 210*(35*e^3*f^3 - 35*d*e^2*f^2*g + 21*d^2*e*f*g^2 - 5*d^3*g^3)*p*x - 105*(5*e
^3*g^3*p*x^7 + 21*e^3*f*g^2*p*x^5 + 35*e^3*f^2*g*p*x^3 + 35*e^3*f^3*p*x)*log(e*x^2 + d) - 105*(5*e^3*g^3*x^7 +
21*e^3*f*g^2*x^5 + 35*e^3*f^2*g*x^3 + 35*e^3*f^3*x)*log(c))/e^3, -1/3675*(150*e^3*g^3*p*x^7 + 42*(21*e^3*f*g^
2 - 5*d*e^2*g^3)*p*x^5 + 70*(35*e^3*f^2*g - 21*d*e^2*f*g^2 + 5*d^2*e*g^3)*p*x^3 - 210*(35*e^3*f^3 - 35*d*e^2*f
^2*g + 21*d^2*e*f*g^2 - 5*d^3*g^3)*p*sqrt(d/e)*arctan(e*x*sqrt(d/e)/d) + 210*(35*e^3*f^3 - 35*d*e^2*f^2*g + 21
*d^2*e*f*g^2 - 5*d^3*g^3)*p*x - 105*(5*e^3*g^3*p*x^7 + 21*e^3*f*g^2*p*x^5 + 35*e^3*f^2*g*p*x^3 + 35*e^3*f^3*p*
x)*log(e*x^2 + d) - 105*(5*e^3*g^3*x^7 + 21*e^3*f*g^2*x^5 + 35*e^3*f^2*g*x^3 + 35*e^3*f^3*x)*log(c))/e^3]

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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((g*x**2+f)**3*ln(c*(e*x**2+d)**p),x)

[Out]

Timed out

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Giac [A]  time = 1.78728, size = 417, normalized size = 1.23 \begin{align*} -\frac{2 \,{\left (5 \, d^{4} g^{3} p - 21 \, d^{3} f g^{2} p e + 35 \, d^{2} f^{2} g p e^{2} - 35 \, d f^{3} p e^{3}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{7}{2}\right )}}{35 \, \sqrt{d}} + \frac{1}{3675} \,{\left (525 \, g^{3} p x^{7} e^{3} \log \left (x^{2} e + d\right ) - 150 \, g^{3} p x^{7} e^{3} + 525 \, g^{3} x^{7} e^{3} \log \left (c\right ) + 210 \, d g^{3} p x^{5} e^{2} + 2205 \, f g^{2} p x^{5} e^{3} \log \left (x^{2} e + d\right ) - 882 \, f g^{2} p x^{5} e^{3} - 350 \, d^{2} g^{3} p x^{3} e + 2205 \, f g^{2} x^{5} e^{3} \log \left (c\right ) + 1470 \, d f g^{2} p x^{3} e^{2} + 3675 \, f^{2} g p x^{3} e^{3} \log \left (x^{2} e + d\right ) + 1050 \, d^{3} g^{3} p x - 2450 \, f^{2} g p x^{3} e^{3} - 4410 \, d^{2} f g^{2} p x e + 3675 \, f^{2} g x^{3} e^{3} \log \left (c\right ) + 7350 \, d f^{2} g p x e^{2} + 3675 \, f^{3} p x e^{3} \log \left (x^{2} e + d\right ) - 7350 \, f^{3} p x e^{3} + 3675 \, f^{3} x e^{3} \log \left (c\right )\right )} e^{\left (-3\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/35*(5*d^4*g^3*p - 21*d^3*f*g^2*p*e + 35*d^2*f^2*g*p*e^2 - 35*d*f^3*p*e^3)*arctan(x*e^(1/2)/sqrt(d))*e^(-7/2
)/sqrt(d) + 1/3675*(525*g^3*p*x^7*e^3*log(x^2*e + d) - 150*g^3*p*x^7*e^3 + 525*g^3*x^7*e^3*log(c) + 210*d*g^3*
p*x^5*e^2 + 2205*f*g^2*p*x^5*e^3*log(x^2*e + d) - 882*f*g^2*p*x^5*e^3 - 350*d^2*g^3*p*x^3*e + 2205*f*g^2*x^5*e
^3*log(c) + 1470*d*f*g^2*p*x^3*e^2 + 3675*f^2*g*p*x^3*e^3*log(x^2*e + d) + 1050*d^3*g^3*p*x - 2450*f^2*g*p*x^3
*e^3 - 4410*d^2*f*g^2*p*x*e + 3675*f^2*g*x^3*e^3*log(c) + 7350*d*f^2*g*p*x*e^2 + 3675*f^3*p*x*e^3*log(x^2*e +
d) - 7350*f^3*p*x*e^3 + 3675*f^3*x*e^3*log(c))*e^(-3)