3.523 \(\int \frac {(i+j x)^3 (a+b \log (c (d (e+f x)^p)^q))}{g+h x} \, dx\)

Optimal. Leaf size=427 \[ \frac {(h i-g j)^3 \log \left (\frac {f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h^4}+\frac {(i+j x)^2 (h i-g j) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 h^2}+\frac {(i+j x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 h}+\frac {a j x (h i-g j)^2}{h^3}+\frac {b j (e+f x) (h i-g j)^2 \log \left (c \left (d (e+f x)^p\right )^q\right )}{f h^3}-\frac {b p q (f i-e j)^3 \log (e+f x)}{3 f^3 h}-\frac {b p q (f i-e j)^2 \log (e+f x) (h i-g j)}{2 f^2 h^2}-\frac {b j p q x (f i-e j)^2}{3 f^2 h}+\frac {b p q (h i-g j)^3 \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right )}{h^4}-\frac {b j p q x (f i-e j) (h i-g j)}{2 f h^2}-\frac {b p q (i+j x)^2 (f i-e j)}{6 f h}-\frac {b j p q x (h i-g j)^2}{h^3}-\frac {b p q (i+j x)^2 (h i-g j)}{4 h^2}-\frac {b p q (i+j x)^3}{9 h} \]

[Out]

a*j*(-g*j+h*i)^2*x/h^3-1/3*b*j*(-e*j+f*i)^2*p*q*x/f^2/h-1/2*b*j*(-e*j+f*i)*(-g*j+h*i)*p*q*x/f/h^2-b*j*(-g*j+h*
i)^2*p*q*x/h^3-1/6*b*(-e*j+f*i)*p*q*(j*x+i)^2/f/h-1/4*b*(-g*j+h*i)*p*q*(j*x+i)^2/h^2-1/9*b*p*q*(j*x+i)^3/h-1/3
*b*(-e*j+f*i)^3*p*q*ln(f*x+e)/f^3/h-1/2*b*(-e*j+f*i)^2*(-g*j+h*i)*p*q*ln(f*x+e)/f^2/h^2+b*j*(-g*j+h*i)^2*(f*x+
e)*ln(c*(d*(f*x+e)^p)^q)/f/h^3+1/2*(-g*j+h*i)*(j*x+i)^2*(a+b*ln(c*(d*(f*x+e)^p)^q))/h^2+1/3*(j*x+i)^3*(a+b*ln(
c*(d*(f*x+e)^p)^q))/h+(-g*j+h*i)^3*(a+b*ln(c*(d*(f*x+e)^p)^q))*ln(f*(h*x+g)/(-e*h+f*g))/h^4+b*(-g*j+h*i)^3*p*q
*polylog(2,-h*(f*x+e)/(-e*h+f*g))/h^4

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Rubi [A]  time = 0.82, antiderivative size = 427, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 9, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2418, 2389, 2295, 2394, 2393, 2391, 2395, 43, 2445} \[ \frac {b p q (h i-g j)^3 \text {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right )}{h^4}+\frac {(i+j x)^2 (h i-g j) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 h^2}+\frac {(h i-g j)^3 \log \left (\frac {f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h^4}+\frac {(i+j x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 h}+\frac {a j x (h i-g j)^2}{h^3}+\frac {b j (e+f x) (h i-g j)^2 \log \left (c \left (d (e+f x)^p\right )^q\right )}{f h^3}-\frac {b p q (f i-e j)^2 \log (e+f x) (h i-g j)}{2 f^2 h^2}-\frac {b j p q x (f i-e j)^2}{3 f^2 h}-\frac {b p q (f i-e j)^3 \log (e+f x)}{3 f^3 h}-\frac {b j p q x (f i-e j) (h i-g j)}{2 f h^2}-\frac {b p q (i+j x)^2 (f i-e j)}{6 f h}-\frac {b p q (i+j x)^2 (h i-g j)}{4 h^2}-\frac {b j p q x (h i-g j)^2}{h^3}-\frac {b p q (i+j x)^3}{9 h} \]

Antiderivative was successfully verified.

[In]

Int[((i + j*x)^3*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(g + h*x),x]

[Out]

(a*j*(h*i - g*j)^2*x)/h^3 - (b*j*(f*i - e*j)^2*p*q*x)/(3*f^2*h) - (b*j*(f*i - e*j)*(h*i - g*j)*p*q*x)/(2*f*h^2
) - (b*j*(h*i - g*j)^2*p*q*x)/h^3 - (b*(f*i - e*j)*p*q*(i + j*x)^2)/(6*f*h) - (b*(h*i - g*j)*p*q*(i + j*x)^2)/
(4*h^2) - (b*p*q*(i + j*x)^3)/(9*h) - (b*(f*i - e*j)^3*p*q*Log[e + f*x])/(3*f^3*h) - (b*(f*i - e*j)^2*(h*i - g
*j)*p*q*Log[e + f*x])/(2*f^2*h^2) + (b*j*(h*i - g*j)^2*(e + f*x)*Log[c*(d*(e + f*x)^p)^q])/(f*h^3) + ((h*i - g
*j)*(i + j*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(2*h^2) + ((i + j*x)^3*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(3*
h) + ((h*i - g*j)^3*(a + b*Log[c*(d*(e + f*x)^p)^q])*Log[(f*(g + h*x))/(f*g - e*h)])/h^4 + (b*(h*i - g*j)^3*p*
q*PolyLog[2, -((h*(e + f*x))/(f*g - e*h))])/h^4

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2389

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

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 2395

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

Rule 2418

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 2445

Int[((a_.) + Log[(c_.)*((d_.)*((e_.) + (f_.)*(x_))^(m_.))^(n_)]*(b_.))^(p_.)*(u_.), x_Symbol] :> Subst[Int[u*(
a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x], c*d^n*(e + f*x)^(m*n), c*(d*(e + f*x)^m)^n] /; FreeQ[{a, b, c, d, e,
f, m, n, p}, x] &&  !IntegerQ[n] &&  !(EqQ[d, 1] && EqQ[m, 1]) && IntegralFreeQ[IntHide[u*(a + b*Log[c*d^n*(e
+ f*x)^(m*n)])^p, x]]

Rubi steps

\begin {align*} \int \frac {(523+j x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{g+h x} \, dx &=\operatorname {Subst}\left (\int \frac {(523+j x)^3 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{g+h x} \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\operatorname {Subst}\left (\int \left (\frac {j (523 h-g j)^2 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{h^3}+\frac {(523 h-g j)^3 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{h^3 (g+h x)}+\frac {j (523 h-g j) (523+j x) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{h^2}+\frac {j (523+j x)^2 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{h}\right ) \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\operatorname {Subst}\left (\frac {j \int (523+j x)^2 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right ) \, dx}{h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname {Subst}\left (\frac {(j (523 h-g j)) \int (523+j x) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right ) \, dx}{h^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname {Subst}\left (\frac {\left (j (523 h-g j)^2\right ) \int \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right ) \, dx}{h^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname {Subst}\left (\frac {(523 h-g j)^3 \int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{g+h x} \, dx}{h^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {a j (523 h-g j)^2 x}{h^3}+\frac {(523 h-g j) (523+j x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 h^2}+\frac {(523+j x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 h}+\frac {(523 h-g j)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h^4}+\operatorname {Subst}\left (\frac {\left (b j (523 h-g j)^2\right ) \int \log \left (c d^q (e+f x)^{p q}\right ) \, dx}{h^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname {Subst}\left (\frac {(b f p q) \int \frac {(523+j x)^3}{e+f x} \, dx}{3 h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname {Subst}\left (\frac {(b f (523 h-g j) p q) \int \frac {(523+j x)^2}{e+f x} \, dx}{2 h^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname {Subst}\left (\frac {\left (b f (523 h-g j)^3 p q\right ) \int \frac {\log \left (\frac {f (g+h x)}{f g-e h}\right )}{e+f x} \, dx}{h^4},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {a j (523 h-g j)^2 x}{h^3}+\frac {(523 h-g j) (523+j x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 h^2}+\frac {(523+j x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 h}+\frac {(523 h-g j)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h^4}+\operatorname {Subst}\left (\frac {\left (b j (523 h-g j)^2\right ) \operatorname {Subst}\left (\int \log \left (c d^q x^{p q}\right ) \, dx,x,e+f x\right )}{f h^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname {Subst}\left (\frac {(b f p q) \int \left (\frac {j (523 f-e j)^2}{f^3}+\frac {(523 f-e j)^3}{f^3 (e+f x)}+\frac {j (523 f-e j) (523+j x)}{f^2}+\frac {j (523+j x)^2}{f}\right ) \, dx}{3 h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname {Subst}\left (\frac {(b f (523 h-g j) p q) \int \left (\frac {j (523 f-e j)}{f^2}+\frac {(523 f-e j)^2}{f^2 (e+f x)}+\frac {j (523+j x)}{f}\right ) \, dx}{2 h^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname {Subst}\left (\frac {\left (b (523 h-g j)^3 p q\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {h x}{f g-e h}\right )}{x} \, dx,x,e+f x\right )}{h^4},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {a j (523 h-g j)^2 x}{h^3}-\frac {b j (523 f-e j)^2 p q x}{3 f^2 h}-\frac {b j (523 f-e j) (523 h-g j) p q x}{2 f h^2}-\frac {b j (523 h-g j)^2 p q x}{h^3}-\frac {b (523 f-e j) p q (523+j x)^2}{6 f h}-\frac {b (523 h-g j) p q (523+j x)^2}{4 h^2}-\frac {b p q (523+j x)^3}{9 h}-\frac {b (523 f-e j)^3 p q \log (e+f x)}{3 f^3 h}-\frac {b (523 f-e j)^2 (523 h-g j) p q \log (e+f x)}{2 f^2 h^2}+\frac {b j (523 h-g j)^2 (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f h^3}+\frac {(523 h-g j) (523+j x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 h^2}+\frac {(523+j x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 h}+\frac {(523 h-g j)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h^4}+\frac {b (523 h-g j)^3 p q \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right )}{h^4}\\ \end {align*}

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Mathematica [A]  time = 0.71, size = 386, normalized size = 0.90 \[ \frac {f \left (h j x \left (6 a f^2 \left (6 g^2 j^2-3 g h j (6 i+j x)+h^2 \left (18 i^2+9 i j x+2 j^2 x^2\right )\right )-b p q \left (12 e^2 h^2 j^2-6 e f h j (-3 g j+9 h i+h j x)+f^2 \left (36 g^2 j^2-9 g h j (12 i+j x)+h^2 \left (108 i^2+27 i j x+4 j^2 x^2\right )\right )\right )\right )+36 a f^2 (h i-g j)^3 \log \left (\frac {f (g+h x)}{f g-e h}\right )+6 b f \log \left (c \left (d (e+f x)^p\right )^q\right ) \left (h j \left (6 e \left (g^2 j^2-3 g h i j+3 h^2 i^2\right )+f x \left (6 g^2 j^2-3 g h j (6 i+j x)+h^2 \left (18 i^2+9 i j x+2 j^2 x^2\right )\right )\right )+6 f (h i-g j)^3 \log \left (\frac {f (g+h x)}{f g-e h}\right )\right )\right )+6 b e^2 h^2 j^2 p q \log (e+f x) (2 e h j+3 f g j-9 f h i)+36 b f^3 p q (h i-g j)^3 \text {Li}_2\left (\frac {h (e+f x)}{e h-f g}\right )}{36 f^3 h^4} \]

Antiderivative was successfully verified.

[In]

Integrate[((i + j*x)^3*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(g + h*x),x]

[Out]

(6*b*e^2*h^2*j^2*(-9*f*h*i + 3*f*g*j + 2*e*h*j)*p*q*Log[e + f*x] + f*(h*j*x*(6*a*f^2*(6*g^2*j^2 - 3*g*h*j*(6*i
+ j*x) + h^2*(18*i^2 + 9*i*j*x + 2*j^2*x^2)) - b*p*q*(12*e^2*h^2*j^2 - 6*e*f*h*j*(9*h*i - 3*g*j + h*j*x) + f^
2*(36*g^2*j^2 - 9*g*h*j*(12*i + j*x) + h^2*(108*i^2 + 27*i*j*x + 4*j^2*x^2)))) + 36*a*f^2*(h*i - g*j)^3*Log[(f
*(g + h*x))/(f*g - e*h)] + 6*b*f*Log[c*(d*(e + f*x)^p)^q]*(h*j*(6*e*(3*h^2*i^2 - 3*g*h*i*j + g^2*j^2) + f*x*(6
*g^2*j^2 - 3*g*h*j*(6*i + j*x) + h^2*(18*i^2 + 9*i*j*x + 2*j^2*x^2))) + 6*f*(h*i - g*j)^3*Log[(f*(g + h*x))/(f
*g - e*h)])) + 36*b*f^3*(h*i - g*j)^3*p*q*PolyLog[2, (h*(e + f*x))/(-(f*g) + e*h)])/(36*f^3*h^4)

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fricas [F]  time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a j^{3} x^{3} + 3 \, a i j^{2} x^{2} + 3 \, a i^{2} j x + a i^{3} + {\left (b j^{3} x^{3} + 3 \, b i j^{2} x^{2} + 3 \, b i^{2} j x + b i^{3}\right )} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )}{h x + g}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((j*x+i)^3*(a+b*log(c*(d*(f*x+e)^p)^q))/(h*x+g),x, algorithm="fricas")

[Out]

integral((a*j^3*x^3 + 3*a*i*j^2*x^2 + 3*a*i^2*j*x + a*i^3 + (b*j^3*x^3 + 3*b*i*j^2*x^2 + 3*b*i^2*j*x + b*i^3)*
log(((f*x + e)^p*d)^q*c))/(h*x + g), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (j x + i\right )}^{3} {\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}}{h x + g}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((j*x+i)^3*(a+b*log(c*(d*(f*x+e)^p)^q))/(h*x+g),x, algorithm="giac")

[Out]

integrate((j*x + i)^3*(b*log(((f*x + e)^p*d)^q*c) + a)/(h*x + g), x)

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maple [F]  time = 0.51, size = 0, normalized size = 0.00 \[ \int \frac {\left (j x +i \right )^{3} \left (b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )+a \right )}{h x +g}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((j*x+i)^3*(b*ln(c*(d*(f*x+e)^p)^q)+a)/(h*x+g),x)

[Out]

int((j*x+i)^3*(b*ln(c*(d*(f*x+e)^p)^q)+a)/(h*x+g),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ 3 \, a i^{2} j {\left (\frac {x}{h} - \frac {g \log \left (h x + g\right )}{h^{2}}\right )} - \frac {1}{6} \, a j^{3} {\left (\frac {6 \, g^{3} \log \left (h x + g\right )}{h^{4}} - \frac {2 \, h^{2} x^{3} - 3 \, g h x^{2} + 6 \, g^{2} x}{h^{3}}\right )} + \frac {3}{2} \, a i j^{2} {\left (\frac {2 \, g^{2} \log \left (h x + g\right )}{h^{3}} + \frac {h x^{2} - 2 \, g x}{h^{2}}\right )} + \frac {a i^{3} \log \left (h x + g\right )}{h} + \int \frac {{\left (j^{3} q \log \relax (d) + j^{3} \log \relax (c)\right )} b x^{3} + 3 \, {\left (i j^{2} q \log \relax (d) + i j^{2} \log \relax (c)\right )} b x^{2} + 3 \, {\left (i^{2} j q \log \relax (d) + i^{2} j \log \relax (c)\right )} b x + {\left (i^{3} q \log \relax (d) + i^{3} \log \relax (c)\right )} b + {\left (b j^{3} x^{3} + 3 \, b i j^{2} x^{2} + 3 \, b i^{2} j x + b i^{3}\right )} \log \left ({\left ({\left (f x + e\right )}^{p}\right )}^{q}\right )}{h x + g}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((j*x+i)^3*(a+b*log(c*(d*(f*x+e)^p)^q))/(h*x+g),x, algorithm="maxima")

[Out]

3*a*i^2*j*(x/h - g*log(h*x + g)/h^2) - 1/6*a*j^3*(6*g^3*log(h*x + g)/h^4 - (2*h^2*x^3 - 3*g*h*x^2 + 6*g^2*x)/h
^3) + 3/2*a*i*j^2*(2*g^2*log(h*x + g)/h^3 + (h*x^2 - 2*g*x)/h^2) + a*i^3*log(h*x + g)/h + integrate(((j^3*q*lo
g(d) + j^3*log(c))*b*x^3 + 3*(i*j^2*q*log(d) + i*j^2*log(c))*b*x^2 + 3*(i^2*j*q*log(d) + i^2*j*log(c))*b*x + (
i^3*q*log(d) + i^3*log(c))*b + (b*j^3*x^3 + 3*b*i*j^2*x^2 + 3*b*i^2*j*x + b*i^3)*log(((f*x + e)^p)^q))/(h*x +
g), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (i+j\,x\right )}^3\,\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}{g+h\,x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((i + j*x)^3*(a + b*log(c*(d*(e + f*x)^p)^q)))/(g + h*x),x)

[Out]

int(((i + j*x)^3*(a + b*log(c*(d*(e + f*x)^p)^q)))/(g + h*x), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}\right ) \left (i + j x\right )^{3}}{g + h x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((j*x+i)**3*(a+b*ln(c*(d*(f*x+e)**p)**q))/(h*x+g),x)

[Out]

Integral((a + b*log(c*(d*(e + f*x)**p)**q))*(i + j*x)**3/(g + h*x), x)

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