3.6.23 \(\int \frac {(i+j x)^3 (a+b \log (c (d (e+f x)^p)^q))}{g+h x} \, dx\) [523]

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

[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 = 0.55, 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 = {2465, 2436, 2332, 2441, 2440, 2438, 2442, 45, 2495} \begin {gather*} \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 {(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 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} \end {gather*}

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 45

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 2332

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

Rule 2436

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 2438

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 2440

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 2441

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 2442

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 2465

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 2495

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 &=\text {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 )\\ &=\text {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 )\\ &=\text {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 )+\text {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 )+\text {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 )+\text {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}+\text {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 )-\text {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 )-\text {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 )-\text {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}+\text {Subst}\left (\frac {\left (b j (523 h-g j)^2\right ) \text {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 )-\text {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 )-\text {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 )-\text {Subst}\left (\frac {\left (b (523 h-g j)^3 p q\right ) \text {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 = 0.55, size = 451, normalized size = 1.06 \begin {gather*} \frac {f h j \left (6 a f^2 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 )-b p q \left (12 e^2 h^2 j^2 x+6 e f \left (6 g^2 j^2+3 g h j (-6 i+j x)+h^2 \left (18 i^2-9 i j x-j^2 x^2\right )\right )+f^2 x \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^3 (h i-g j)^3 \log (g+h x)+6 b f^3 \log \left (c \left (d (e+f x)^p\right )^q\right ) \left (h j 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 )+6 (h i-g j)^3 \log (g+h x)\right )+6 b p q \log (e+f x) \left (e h j \left (2 e^2 h^2 j^2+3 e f h j (-3 h i+g j)+6 f^2 \left (3 h^2 i^2-3 g h i j+g^2 j^2\right )\right )-6 f^3 (h i-g j)^3 \log (g+h x)+6 f^3 (h i-g j)^3 \log \left (\frac {f (g+h x)}{f g-e h}\right )\right )+36 b f^3 (h i-g j)^3 p q \text {Li}_2\left (\frac {h (e+f x)}{-f g+e h}\right )}{36 f^3 h^4} \end {gather*}

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]

(f*h*j*(6*a*f^2*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)) - b*p*q*(12*e^2*h^2*j
^2*x + 6*e*f*(6*g^2*j^2 + 3*g*h*j*(-6*i + j*x) + h^2*(18*i^2 - 9*i*j*x - j^2*x^2)) + f^2*x*(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^3*(h*i - g*j)^3*Log[g + h*x] + 6*b*f^3*Log[
c*(d*(e + f*x)^p)^q]*(h*j*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*(h*i -
g*j)^3*Log[g + h*x]) + 6*b*p*q*Log[e + f*x]*(e*h*j*(2*e^2*h^2*j^2 + 3*e*f*h*j*(-3*h*i + g*j) + 6*f^2*(3*h^2*i^
2 - 3*g*h*i*j + g^2*j^2)) - 6*f^3*(h*i - g*j)^3*Log[g + h*x] + 6*f^3*(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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Maple [F]
time = 0.34, size = 0, normalized size = 0.00 \[\int \frac {\left (j x +i \right )^{3} \left (a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )\right )}{h x +g}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \end {gather*}

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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