3.318 \(\int (f x)^m (d+e x^2)^3 (a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=211 \[ \frac{3 d^2 e (f x)^{m+3} \left (a+b \log \left (c x^n\right )\right )}{f^3 (m+3)}+\frac{d^3 (f x)^{m+1} \left (a+b \log \left (c x^n\right )\right )}{f (m+1)}+\frac{3 d e^2 (f x)^{m+5} \left (a+b \log \left (c x^n\right )\right )}{f^5 (m+5)}+\frac{e^3 (f x)^{m+7} \left (a+b \log \left (c x^n\right )\right )}{f^7 (m+7)}-\frac{3 b d^2 e n (f x)^{m+3}}{f^3 (m+3)^2}-\frac{b d^3 n (f x)^{m+1}}{f (m+1)^2}-\frac{3 b d e^2 n (f x)^{m+5}}{f^5 (m+5)^2}-\frac{b e^3 n (f x)^{m+7}}{f^7 (m+7)^2} \]

[Out]

-((b*d^3*n*(f*x)^(1 + m))/(f*(1 + m)^2)) - (3*b*d^2*e*n*(f*x)^(3 + m))/(f^3*(3 + m)^2) - (3*b*d*e^2*n*(f*x)^(5
+ m))/(f^5*(5 + m)^2) - (b*e^3*n*(f*x)^(7 + m))/(f^7*(7 + m)^2) + (d^3*(f*x)^(1 + m)*(a + b*Log[c*x^n]))/(f*(
1 + m)) + (3*d^2*e*(f*x)^(3 + m)*(a + b*Log[c*x^n]))/(f^3*(3 + m)) + (3*d*e^2*(f*x)^(5 + m)*(a + b*Log[c*x^n])
)/(f^5*(5 + m)) + (e^3*(f*x)^(7 + m)*(a + b*Log[c*x^n]))/(f^7*(7 + m))

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Rubi [A]  time = 1.6841, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {270, 2350, 14} \[ \frac{3 d^2 e (f x)^{m+3} \left (a+b \log \left (c x^n\right )\right )}{f^3 (m+3)}+\frac{d^3 (f x)^{m+1} \left (a+b \log \left (c x^n\right )\right )}{f (m+1)}+\frac{3 d e^2 (f x)^{m+5} \left (a+b \log \left (c x^n\right )\right )}{f^5 (m+5)}+\frac{e^3 (f x)^{m+7} \left (a+b \log \left (c x^n\right )\right )}{f^7 (m+7)}-\frac{3 b d^2 e n (f x)^{m+3}}{f^3 (m+3)^2}-\frac{b d^3 n (f x)^{m+1}}{f (m+1)^2}-\frac{3 b d e^2 n (f x)^{m+5}}{f^5 (m+5)^2}-\frac{b e^3 n (f x)^{m+7}}{f^7 (m+7)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*d^3*n*(f*x)^(1 + m))/(f*(1 + m)^2)) - (3*b*d^2*e*n*(f*x)^(3 + m))/(f^3*(3 + m)^2) - (3*b*d*e^2*n*(f*x)^(5
+ m))/(f^5*(5 + m)^2) - (b*e^3*n*(f*x)^(7 + m))/(f^7*(7 + m)^2) + (d^3*(f*x)^(1 + m)*(a + b*Log[c*x^n]))/(f*(
1 + m)) + (3*d^2*e*(f*x)^(3 + m)*(a + b*Log[c*x^n]))/(f^3*(3 + m)) + (3*d*e^2*(f*x)^(5 + m)*(a + b*Log[c*x^n])
)/(f^5*(5 + m)) + (e^3*(f*x)^(7 + m)*(a + b*Log[c*x^n]))/(f^7*(7 + m))

Rule 270

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

Rule 2350

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Wit
h[{u = IntHide[(f*x)^m*(d + e*x^r)^q, x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[SimplifyIntegrand[u/x,
x], x], x] /; ((EqQ[r, 1] || EqQ[r, 2]) && IntegerQ[m] && IntegerQ[q - 1/2]) || InverseFunctionFreeQ[u, x]] /
; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && IntegerQ[2*q] && ((IntegerQ[m] && IntegerQ[r]) || IGtQ[q, 0])

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int (f x)^m \left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{d^3 (f x)^{1+m} \left (a+b \log \left (c x^n\right )\right )}{f (1+m)}+\frac{3 d^2 e (f x)^{3+m} \left (a+b \log \left (c x^n\right )\right )}{f^3 (3+m)}+\frac{3 d e^2 (f x)^{5+m} \left (a+b \log \left (c x^n\right )\right )}{f^5 (5+m)}+\frac{e^3 (f x)^{7+m} \left (a+b \log \left (c x^n\right )\right )}{f^7 (7+m)}-(b n) \int (f x)^m \left (\frac{d^3}{1+m}+\frac{3 d^2 e x^2}{3+m}+\frac{3 d e^2 x^4}{5+m}+\frac{e^3 x^6}{7+m}\right ) \, dx\\ &=\frac{d^3 (f x)^{1+m} \left (a+b \log \left (c x^n\right )\right )}{f (1+m)}+\frac{3 d^2 e (f x)^{3+m} \left (a+b \log \left (c x^n\right )\right )}{f^3 (3+m)}+\frac{3 d e^2 (f x)^{5+m} \left (a+b \log \left (c x^n\right )\right )}{f^5 (5+m)}+\frac{e^3 (f x)^{7+m} \left (a+b \log \left (c x^n\right )\right )}{f^7 (7+m)}-(b n) \int \left (\frac{d^3 (f x)^m}{1+m}+\frac{3 d^2 e (f x)^{2+m}}{f^2 (3+m)}+\frac{3 d e^2 (f x)^{4+m}}{f^4 (5+m)}+\frac{e^3 (f x)^{6+m}}{f^6 (7+m)}\right ) \, dx\\ &=-\frac{b d^3 n (f x)^{1+m}}{f (1+m)^2}-\frac{3 b d^2 e n (f x)^{3+m}}{f^3 (3+m)^2}-\frac{3 b d e^2 n (f x)^{5+m}}{f^5 (5+m)^2}-\frac{b e^3 n (f x)^{7+m}}{f^7 (7+m)^2}+\frac{d^3 (f x)^{1+m} \left (a+b \log \left (c x^n\right )\right )}{f (1+m)}+\frac{3 d^2 e (f x)^{3+m} \left (a+b \log \left (c x^n\right )\right )}{f^3 (3+m)}+\frac{3 d e^2 (f x)^{5+m} \left (a+b \log \left (c x^n\right )\right )}{f^5 (5+m)}+\frac{e^3 (f x)^{7+m} \left (a+b \log \left (c x^n\right )\right )}{f^7 (7+m)}\\ \end{align*}

Mathematica [A]  time = 0.227638, size = 156, normalized size = 0.74 \[ x (f x)^m \left (\frac{3 d^2 e x^2 \left (a+b \log \left (c x^n\right )\right )}{m+3}+\frac{d^3 \left (a+b \log \left (c x^n\right )\right )}{m+1}+\frac{3 d e^2 x^4 \left (a+b \log \left (c x^n\right )\right )}{m+5}+\frac{e^3 x^6 \left (a+b \log \left (c x^n\right )\right )}{m+7}-\frac{3 b d^2 e n x^2}{(m+3)^2}-\frac{b d^3 n}{(m+1)^2}-\frac{3 b d e^2 n x^4}{(m+5)^2}-\frac{b e^3 n x^6}{(m+7)^2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

x*(f*x)^m*(-((b*d^3*n)/(1 + m)^2) - (3*b*d^2*e*n*x^2)/(3 + m)^2 - (3*b*d*e^2*n*x^4)/(5 + m)^2 - (b*e^3*n*x^6)/
(7 + m)^2 + (d^3*(a + b*Log[c*x^n]))/(1 + m) + (3*d^2*e*x^2*(a + b*Log[c*x^n]))/(3 + m) + (3*d*e^2*x^4*(a + b*
Log[c*x^n]))/(5 + m) + (e^3*x^6*(a + b*Log[c*x^n]))/(7 + m))

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Maple [C]  time = 0.423, size = 5139, normalized size = 24.4 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

________________________________________________________________________________________

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((f*x)^m*(e*x^2+d)^3*(a+b*log(c*x^n)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 1.44943, size = 3047, normalized size = 14.44 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((a*e^3*m^7 + 25*a*e^3*m^6 + 253*a*e^3*m^5 + 1333*a*e^3*m^4 + 3907*a*e^3*m^3 + 6283*a*e^3*m^2 + 5055*a*e^3*m +
1575*a*e^3 - (b*e^3*m^6 + 18*b*e^3*m^5 + 127*b*e^3*m^4 + 444*b*e^3*m^3 + 799*b*e^3*m^2 + 690*b*e^3*m + 225*b*
e^3)*n)*x^7 + 3*(a*d*e^2*m^7 + 27*a*d*e^2*m^6 + 293*a*d*e^2*m^5 + 1639*a*d*e^2*m^4 + 5043*a*d*e^2*m^3 + 8417*a
*d*e^2*m^2 + 6951*a*d*e^2*m + 2205*a*d*e^2 - (b*d*e^2*m^6 + 22*b*d*e^2*m^5 + 183*b*d*e^2*m^4 + 724*b*d*e^2*m^3
+ 1423*b*d*e^2*m^2 + 1302*b*d*e^2*m + 441*b*d*e^2)*n)*x^5 + 3*(a*d^2*e*m^7 + 29*a*d^2*e*m^6 + 341*a*d^2*e*m^5
+ 2081*a*d^2*e*m^4 + 6995*a*d^2*e*m^3 + 12647*a*d^2*e*m^2 + 11095*a*d^2*e*m + 3675*a*d^2*e - (b*d^2*e*m^6 + 2
6*b*d^2*e*m^5 + 263*b*d^2*e*m^4 + 1292*b*d^2*e*m^3 + 3119*b*d^2*e*m^2 + 3290*b*d^2*e*m + 1225*b*d^2*e)*n)*x^3
+ (a*d^3*m^7 + 31*a*d^3*m^6 + 397*a*d^3*m^5 + 2707*a*d^3*m^4 + 10531*a*d^3*m^3 + 23101*a*d^3*m^2 + 25935*a*d^3
*m + 11025*a*d^3 - (b*d^3*m^6 + 30*b*d^3*m^5 + 367*b*d^3*m^4 + 2340*b*d^3*m^3 + 8191*b*d^3*m^2 + 14910*b*d^3*m
+ 11025*b*d^3)*n)*x + ((b*e^3*m^7 + 25*b*e^3*m^6 + 253*b*e^3*m^5 + 1333*b*e^3*m^4 + 3907*b*e^3*m^3 + 6283*b*e
^3*m^2 + 5055*b*e^3*m + 1575*b*e^3)*x^7 + 3*(b*d*e^2*m^7 + 27*b*d*e^2*m^6 + 293*b*d*e^2*m^5 + 1639*b*d*e^2*m^4
+ 5043*b*d*e^2*m^3 + 8417*b*d*e^2*m^2 + 6951*b*d*e^2*m + 2205*b*d*e^2)*x^5 + 3*(b*d^2*e*m^7 + 29*b*d^2*e*m^6
+ 341*b*d^2*e*m^5 + 2081*b*d^2*e*m^4 + 6995*b*d^2*e*m^3 + 12647*b*d^2*e*m^2 + 11095*b*d^2*e*m + 3675*b*d^2*e)*
x^3 + (b*d^3*m^7 + 31*b*d^3*m^6 + 397*b*d^3*m^5 + 2707*b*d^3*m^4 + 10531*b*d^3*m^3 + 23101*b*d^3*m^2 + 25935*b
*d^3*m + 11025*b*d^3)*x)*log(c) + ((b*e^3*m^7 + 25*b*e^3*m^6 + 253*b*e^3*m^5 + 1333*b*e^3*m^4 + 3907*b*e^3*m^3
+ 6283*b*e^3*m^2 + 5055*b*e^3*m + 1575*b*e^3)*n*x^7 + 3*(b*d*e^2*m^7 + 27*b*d*e^2*m^6 + 293*b*d*e^2*m^5 + 163
9*b*d*e^2*m^4 + 5043*b*d*e^2*m^3 + 8417*b*d*e^2*m^2 + 6951*b*d*e^2*m + 2205*b*d*e^2)*n*x^5 + 3*(b*d^2*e*m^7 +
29*b*d^2*e*m^6 + 341*b*d^2*e*m^5 + 2081*b*d^2*e*m^4 + 6995*b*d^2*e*m^3 + 12647*b*d^2*e*m^2 + 11095*b*d^2*e*m +
3675*b*d^2*e)*n*x^3 + (b*d^3*m^7 + 31*b*d^3*m^6 + 397*b*d^3*m^5 + 2707*b*d^3*m^4 + 10531*b*d^3*m^3 + 23101*b*
d^3*m^2 + 25935*b*d^3*m + 11025*b*d^3)*n*x)*log(x))*e^(m*log(f) + m*log(x))/(m^8 + 32*m^7 + 428*m^6 + 3104*m^5
+ 13238*m^4 + 33632*m^3 + 49036*m^2 + 36960*m + 11025)

________________________________________________________________________________________

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((f*x)**m*(e*x**2+d)**3*(a+b*ln(c*x**n)),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.33161, size = 747, normalized size = 3.54 \begin{align*} \frac{b f^{6} f^{m} x^{7} x^{m} e^{3} \log \left (c\right )}{f^{6} m + 7 \, f^{6}} + \frac{a f^{6} f^{m} x^{7} x^{m} e^{3}}{f^{6} m + 7 \, f^{6}} + \frac{3 \, b d f^{4} f^{m} x^{5} x^{m} e^{2} \log \left (c\right )}{f^{4} m + 5 \, f^{4}} + \frac{3 \, a d f^{4} f^{m} x^{5} x^{m} e^{2}}{f^{4} m + 5 \, f^{4}} + \frac{b f^{m} m n x^{7} x^{m} e^{3} \log \left (x\right )}{m^{2} + 14 \, m + 49} + \frac{7 \, b f^{m} n x^{7} x^{m} e^{3} \log \left (x\right )}{m^{2} + 14 \, m + 49} + \frac{3 \, b d f^{m} m n x^{5} x^{m} e^{2} \log \left (x\right )}{m^{2} + 10 \, m + 25} - \frac{b f^{m} n x^{7} x^{m} e^{3}}{m^{2} + 14 \, m + 49} + \frac{3 \, b d^{2} f^{2} f^{m} x^{3} x^{m} e \log \left (c\right )}{f^{2} m + 3 \, f^{2}} + \frac{15 \, b d f^{m} n x^{5} x^{m} e^{2} \log \left (x\right )}{m^{2} + 10 \, m + 25} + \frac{3 \, b d^{2} f^{m} m n x^{3} x^{m} e \log \left (x\right )}{m^{2} + 6 \, m + 9} - \frac{3 \, b d f^{m} n x^{5} x^{m} e^{2}}{m^{2} + 10 \, m + 25} + \frac{3 \, a d^{2} f^{2} f^{m} x^{3} x^{m} e}{f^{2} m + 3 \, f^{2}} + \frac{9 \, b d^{2} f^{m} n x^{3} x^{m} e \log \left (x\right )}{m^{2} + 6 \, m + 9} - \frac{3 \, b d^{2} f^{m} n x^{3} x^{m} e}{m^{2} + 6 \, m + 9} + \frac{b d^{3} f^{m} m n x x^{m} \log \left (x\right )}{m^{2} + 2 \, m + 1} + \frac{b d^{3} f^{m} n x x^{m} \log \left (x\right )}{m^{2} + 2 \, m + 1} - \frac{b d^{3} f^{m} n x x^{m}}{m^{2} + 2 \, m + 1} + \frac{\left (f x\right )^{m} b d^{3} x \log \left (c\right )}{m + 1} + \frac{\left (f x\right )^{m} a d^{3} x}{m + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*f^6*f^m*x^7*x^m*e^3*log(c)/(f^6*m + 7*f^6) + a*f^6*f^m*x^7*x^m*e^3/(f^6*m + 7*f^6) + 3*b*d*f^4*f^m*x^5*x^m*e
^2*log(c)/(f^4*m + 5*f^4) + 3*a*d*f^4*f^m*x^5*x^m*e^2/(f^4*m + 5*f^4) + b*f^m*m*n*x^7*x^m*e^3*log(x)/(m^2 + 14
*m + 49) + 7*b*f^m*n*x^7*x^m*e^3*log(x)/(m^2 + 14*m + 49) + 3*b*d*f^m*m*n*x^5*x^m*e^2*log(x)/(m^2 + 10*m + 25)
- b*f^m*n*x^7*x^m*e^3/(m^2 + 14*m + 49) + 3*b*d^2*f^2*f^m*x^3*x^m*e*log(c)/(f^2*m + 3*f^2) + 15*b*d*f^m*n*x^5
*x^m*e^2*log(x)/(m^2 + 10*m + 25) + 3*b*d^2*f^m*m*n*x^3*x^m*e*log(x)/(m^2 + 6*m + 9) - 3*b*d*f^m*n*x^5*x^m*e^2
/(m^2 + 10*m + 25) + 3*a*d^2*f^2*f^m*x^3*x^m*e/(f^2*m + 3*f^2) + 9*b*d^2*f^m*n*x^3*x^m*e*log(x)/(m^2 + 6*m + 9
) - 3*b*d^2*f^m*n*x^3*x^m*e/(m^2 + 6*m + 9) + b*d^3*f^m*m*n*x*x^m*log(x)/(m^2 + 2*m + 1) + b*d^3*f^m*n*x*x^m*l
og(x)/(m^2 + 2*m + 1) - b*d^3*f^m*n*x*x^m/(m^2 + 2*m + 1) + (f*x)^m*b*d^3*x*log(c)/(m + 1) + (f*x)^m*a*d^3*x/(
m + 1)