3.4.20 \(\int (f x)^m (d+e x^2) (a+b \log (c x^n)) \, dx\) [320]

Optimal. Leaf size=95 \[ -\frac {b d n (f x)^{1+m}}{f (1+m)^2}-\frac {b e n (f x)^{3+m}}{f^3 (3+m)^2}+\frac {d (f x)^{1+m} \left (a+b \log \left (c x^n\right )\right )}{f (1+m)}+\frac {e (f x)^{3+m} \left (a+b \log \left (c x^n\right )\right )}{f^3 (3+m)} \]

[Out]

-b*d*n*(f*x)^(1+m)/f/(1+m)^2-b*e*n*(f*x)^(3+m)/f^3/(3+m)^2+d*(f*x)^(1+m)*(a+b*ln(c*x^n))/f/(1+m)+e*(f*x)^(3+m)
*(a+b*ln(c*x^n))/f^3/(3+m)

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Rubi [A]
time = 0.06, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {14, 2392} \begin {gather*} \frac {d (f x)^{m+1} \left (a+b \log \left (c x^n\right )\right )}{f (m+1)}+\frac {e (f x)^{m+3} \left (a+b \log \left (c x^n\right )\right )}{f^3 (m+3)}-\frac {b d n (f x)^{m+1}}{f (m+1)^2}-\frac {b e n (f x)^{m+3}}{f^3 (m+3)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]]

Rule 2392

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])

Rubi steps

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

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Mathematica [A]
time = 0.10, size = 94, normalized size = 0.99 \begin {gather*} \frac {x (f x)^m \left (a \left (3+4 m+m^2\right ) \left (d (3+m)+e (1+m) x^2\right )-b n \left (d (3+m)^2+e (1+m)^2 x^2\right )+b \left (3+4 m+m^2\right ) \left (d (3+m)+e (1+m) x^2\right ) \log \left (c x^n\right )\right )}{(1+m)^2 (3+m)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.10, size = 1180, normalized size = 12.42

method	result	size
risch	\(\text {Expression too large to display}\)	\(1180\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*x*(e*m*x^2+e*x^2+d*m+3*d)/(1+m)/(3+m)*exp(1/2*m*(-I*Pi*csgn(I*f*x)^3+I*Pi*csgn(I*f*x)^2*csgn(I*f)+I*Pi*csgn(
I*f*x)^2*csgn(I*x)-I*Pi*csgn(I*f*x)*csgn(I*f)*csgn(I*x)+2*ln(x)+2*ln(f)))*ln(x^n)-1/2*x*(-30*a*d*m-6*a*e*x^2+1
8*b*d*n-14*a*e*m*x^2-18*a*d-7*I*Pi*b*e*m*x^2*csgn(I*c)*csgn(I*c*x^n)^2-7*I*Pi*b*e*m*x^2*csgn(I*x^n)*csgn(I*c*x
^n)^2-3*I*Pi*b*e*x^2*csgn(I*c)*csgn(I*c*x^n)^2-2*a*d*m^3-5*I*Pi*b*e*m^2*x^2*csgn(I*c)*csgn(I*c*x^n)^2+9*I*Pi*b
*d*csgn(I*c*x^n)^3-18*d*b*ln(c)+2*b*e*m^2*n*x^2+I*Pi*b*e*m^3*x^2*csgn(I*c*x^n)^3-15*I*Pi*b*d*m*csgn(I*x^n)*csg
n(I*c*x^n)^2-2*a*e*m^3*x^2-14*ln(c)*b*d*m^2-30*ln(c)*b*d*m-2*ln(c)*b*d*m^3+2*b*d*m^2*n-2*ln(c)*b*e*m^3*x^2-10*
ln(c)*b*e*m^2*x^2-14*ln(c)*b*e*m*x^2-14*a*d*m^2+I*Pi*b*d*m^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+12*b*d*m*n-6*
ln(c)*b*e*x^2-10*a*e*m^2*x^2-7*I*Pi*b*d*m^2*csgn(I*x^n)*csgn(I*c*x^n)^2+4*b*e*m*n*x^2+9*I*Pi*b*d*csgn(I*c)*csg
n(I*x^n)*csgn(I*c*x^n)-3*I*Pi*b*e*x^2*csgn(I*x^n)*csgn(I*c*x^n)^2-I*Pi*b*d*m^3*csgn(I*x^n)*csgn(I*c*x^n)^2+I*P
i*b*d*m^3*csgn(I*c*x^n)^3+I*Pi*b*e*m^3*x^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+2*b*e*n*x^2-15*I*Pi*b*d*m*csgn(
I*c)*csgn(I*c*x^n)^2+7*I*Pi*b*e*m*x^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+5*I*Pi*b*e*m^2*x^2*csgn(I*c*x^n)^3+7
*I*Pi*b*e*m*x^2*csgn(I*c*x^n)^3-7*I*Pi*b*d*m^2*csgn(I*c)*csgn(I*c*x^n)^2-9*I*Pi*b*d*csgn(I*c)*csgn(I*c*x^n)^2-
I*Pi*b*d*m^3*csgn(I*c)*csgn(I*c*x^n)^2+15*I*Pi*b*d*m*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+7*I*Pi*b*d*m^2*csgn(I
*c)*csgn(I*x^n)*csgn(I*c*x^n)+3*I*Pi*b*e*x^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+5*I*Pi*b*e*m^2*x^2*csgn(I*c)*
csgn(I*x^n)*csgn(I*c*x^n)-5*I*Pi*b*e*m^2*x^2*csgn(I*x^n)*csgn(I*c*x^n)^2-I*Pi*b*e*m^3*x^2*csgn(I*c)*csgn(I*c*x
^n)^2-I*Pi*b*e*m^3*x^2*csgn(I*x^n)*csgn(I*c*x^n)^2-9*I*Pi*b*d*csgn(I*x^n)*csgn(I*c*x^n)^2+15*I*Pi*b*d*m*csgn(I
*c*x^n)^3+7*I*Pi*b*d*m^2*csgn(I*c*x^n)^3+3*I*Pi*b*e*x^2*csgn(I*c*x^n)^3)/(3+m)^2/(1+m)^2*exp(1/2*m*(-I*Pi*csgn
(I*f*x)^3+I*Pi*csgn(I*f*x)^2*csgn(I*f)+I*Pi*csgn(I*f*x)^2*csgn(I*x)-I*Pi*csgn(I*f*x)*csgn(I*f)*csgn(I*x)+2*ln(
x)+2*ln(f)))

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Maxima [A]
time = 0.30, size = 128, normalized size = 1.35 \begin {gather*} \frac {b f^{m} x^{3} e^{\left (m \log \left (x\right ) + 1\right )} \log \left (c x^{n}\right )}{m + 3} + \frac {a f^{m} x^{3} e^{\left (m \log \left (x\right ) + 1\right )}}{m + 3} - \frac {b f^{m} n x^{3} e^{\left (m \log \left (x\right ) + 1\right )}}{{\left (m + 3\right )}^{2}} - \frac {b d f^{m} n x x^{m}}{{\left (m + 1\right )}^{2}} + \frac {\left (f x\right )^{m + 1} b d \log \left (c x^{n}\right )}{f {\left (m + 1\right )}} + \frac {\left (f x\right )^{m + 1} a d}{f {\left (m + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*f^m*x^3*e^(m*log(x) + 1)*log(c*x^n)/(m + 3) + a*f^m*x^3*e^(m*log(x) + 1)/(m + 3) - b*f^m*n*x^3*e^(m*log(x) +
1)/(m + 3)^2 - b*d*f^m*n*x*x^m/(m + 1)^2 + (f*x)^(m + 1)*b*d*log(c*x^n)/(f*(m + 1)) + (f*x)^(m + 1)*a*d/(f*(m
+ 1))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 225 vs. \(2 (97) = 194\).
time = 0.36, size = 225, normalized size = 2.37 \begin {gather*} \frac {{\left ({\left (a m^{3} + 5 \, a m^{2} + 7 \, a m - {\left (b m^{2} + 2 \, b m + b\right )} n + 3 \, a\right )} x^{3} e + {\left (a d m^{3} + 7 \, a d m^{2} + 15 \, a d m + 9 \, a d - {\left (b d m^{2} + 6 \, b d m + 9 \, b d\right )} n\right )} x + {\left ({\left (b m^{3} + 5 \, b m^{2} + 7 \, b m + 3 \, b\right )} x^{3} e + {\left (b d m^{3} + 7 \, b d m^{2} + 15 \, b d m + 9 \, b d\right )} x\right )} \log \left (c\right ) + {\left ({\left (b m^{3} + 5 \, b m^{2} + 7 \, b m + 3 \, b\right )} n x^{3} e + {\left (b d m^{3} + 7 \, b d m^{2} + 15 \, b d m + 9 \, b d\right )} n x\right )} \log \left (x\right )\right )} e^{\left (m \log \left (f\right ) + m \log \left (x\right )\right )}}{m^{4} + 8 \, m^{3} + 22 \, m^{2} + 24 \, m + 9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((a*m^3 + 5*a*m^2 + 7*a*m - (b*m^2 + 2*b*m + b)*n + 3*a)*x^3*e + (a*d*m^3 + 7*a*d*m^2 + 15*a*d*m + 9*a*d - (b*
d*m^2 + 6*b*d*m + 9*b*d)*n)*x + ((b*m^3 + 5*b*m^2 + 7*b*m + 3*b)*x^3*e + (b*d*m^3 + 7*b*d*m^2 + 15*b*d*m + 9*b
*d)*x)*log(c) + ((b*m^3 + 5*b*m^2 + 7*b*m + 3*b)*n*x^3*e + (b*d*m^3 + 7*b*d*m^2 + 15*b*d*m + 9*b*d)*n*x)*log(x
))*e^(m*log(f) + m*log(x))/(m^4 + 8*m^3 + 22*m^2 + 24*m + 9)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 920 vs. \(2 (87) = 174\).
time = 4.25, size = 920, normalized size = 9.68 \begin {gather*} \begin {cases} \frac {- \frac {a d}{2 x^{2}} + a e \log {\left (x \right )} + b d \left (- \frac {n}{4 x^{2}} - \frac {\log {\left (c x^{n} \right )}}{2 x^{2}}\right ) - b e \left (\begin {cases} - \log {\left (c \right )} \log {\left (x \right )} & \text {for}\: n = 0 \\- \frac {\log {\left (c x^{n} \right )}^{2}}{2 n} & \text {otherwise} \end {cases}\right )}{f^{3}} & \text {for}\: m = -3 \\\frac {\frac {a d \log {\left (c x^{n} \right )}}{n} + \frac {a e x^{2}}{2} + \frac {b d \log {\left (c x^{n} \right )}^{2}}{2 n} - \frac {b e n x^{2}}{4} + \frac {b e x^{2} \log {\left (c x^{n} \right )}}{2}}{f} & \text {for}\: m = -1 \\\frac {a d m^{3} x \left (f x\right )^{m}}{m^{4} + 8 m^{3} + 22 m^{2} + 24 m + 9} + \frac {7 a d m^{2} x \left (f x\right )^{m}}{m^{4} + 8 m^{3} + 22 m^{2} + 24 m + 9} + \frac {15 a d m x \left (f x\right )^{m}}{m^{4} + 8 m^{3} + 22 m^{2} + 24 m + 9} + \frac {9 a d x \left (f x\right )^{m}}{m^{4} + 8 m^{3} + 22 m^{2} + 24 m + 9} + \frac {a e m^{3} x^{3} \left (f x\right )^{m}}{m^{4} + 8 m^{3} + 22 m^{2} + 24 m + 9} + \frac {5 a e m^{2} x^{3} \left (f x\right )^{m}}{m^{4} + 8 m^{3} + 22 m^{2} + 24 m + 9} + \frac {7 a e m x^{3} \left (f x\right )^{m}}{m^{4} + 8 m^{3} + 22 m^{2} + 24 m + 9} + \frac {3 a e x^{3} \left (f x\right )^{m}}{m^{4} + 8 m^{3} + 22 m^{2} + 24 m + 9} + \frac {b d m^{3} x \left (f x\right )^{m} \log {\left (c x^{n} \right )}}{m^{4} + 8 m^{3} + 22 m^{2} + 24 m + 9} - \frac {b d m^{2} n x \left (f x\right )^{m}}{m^{4} + 8 m^{3} + 22 m^{2} + 24 m + 9} + \frac {7 b d m^{2} x \left (f x\right )^{m} \log {\left (c x^{n} \right )}}{m^{4} + 8 m^{3} + 22 m^{2} + 24 m + 9} - \frac {6 b d m n x \left (f x\right )^{m}}{m^{4} + 8 m^{3} + 22 m^{2} + 24 m + 9} + \frac {15 b d m x \left (f x\right )^{m} \log {\left (c x^{n} \right )}}{m^{4} + 8 m^{3} + 22 m^{2} + 24 m + 9} - \frac {9 b d n x \left (f x\right )^{m}}{m^{4} + 8 m^{3} + 22 m^{2} + 24 m + 9} + \frac {9 b d x \left (f x\right )^{m} \log {\left (c x^{n} \right )}}{m^{4} + 8 m^{3} + 22 m^{2} + 24 m + 9} + \frac {b e m^{3} x^{3} \left (f x\right )^{m} \log {\left (c x^{n} \right )}}{m^{4} + 8 m^{3} + 22 m^{2} + 24 m + 9} - \frac {b e m^{2} n x^{3} \left (f x\right )^{m}}{m^{4} + 8 m^{3} + 22 m^{2} + 24 m + 9} + \frac {5 b e m^{2} x^{3} \left (f x\right )^{m} \log {\left (c x^{n} \right )}}{m^{4} + 8 m^{3} + 22 m^{2} + 24 m + 9} - \frac {2 b e m n x^{3} \left (f x\right )^{m}}{m^{4} + 8 m^{3} + 22 m^{2} + 24 m + 9} + \frac {7 b e m x^{3} \left (f x\right )^{m} \log {\left (c x^{n} \right )}}{m^{4} + 8 m^{3} + 22 m^{2} + 24 m + 9} - \frac {b e n x^{3} \left (f x\right )^{m}}{m^{4} + 8 m^{3} + 22 m^{2} + 24 m + 9} + \frac {3 b e x^{3} \left (f x\right )^{m} \log {\left (c x^{n} \right )}}{m^{4} + 8 m^{3} + 22 m^{2} + 24 m + 9} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise(((-a*d/(2*x**2) + a*e*log(x) + b*d*(-n/(4*x**2) - log(c*x**n)/(2*x**2)) - b*e*Piecewise((-log(c)*log
(x), Eq(n, 0)), (-log(c*x**n)**2/(2*n), True)))/f**3, Eq(m, -3)), ((a*d*log(c*x**n)/n + a*e*x**2/2 + b*d*log(c
*x**n)**2/(2*n) - b*e*n*x**2/4 + b*e*x**2*log(c*x**n)/2)/f, Eq(m, -1)), (a*d*m**3*x*(f*x)**m/(m**4 + 8*m**3 +
22*m**2 + 24*m + 9) + 7*a*d*m**2*x*(f*x)**m/(m**4 + 8*m**3 + 22*m**2 + 24*m + 9) + 15*a*d*m*x*(f*x)**m/(m**4 +
8*m**3 + 22*m**2 + 24*m + 9) + 9*a*d*x*(f*x)**m/(m**4 + 8*m**3 + 22*m**2 + 24*m + 9) + a*e*m**3*x**3*(f*x)**m
/(m**4 + 8*m**3 + 22*m**2 + 24*m + 9) + 5*a*e*m**2*x**3*(f*x)**m/(m**4 + 8*m**3 + 22*m**2 + 24*m + 9) + 7*a*e*
m*x**3*(f*x)**m/(m**4 + 8*m**3 + 22*m**2 + 24*m + 9) + 3*a*e*x**3*(f*x)**m/(m**4 + 8*m**3 + 22*m**2 + 24*m + 9
) + b*d*m**3*x*(f*x)**m*log(c*x**n)/(m**4 + 8*m**3 + 22*m**2 + 24*m + 9) - b*d*m**2*n*x*(f*x)**m/(m**4 + 8*m**
3 + 22*m**2 + 24*m + 9) + 7*b*d*m**2*x*(f*x)**m*log(c*x**n)/(m**4 + 8*m**3 + 22*m**2 + 24*m + 9) - 6*b*d*m*n*x
*(f*x)**m/(m**4 + 8*m**3 + 22*m**2 + 24*m + 9) + 15*b*d*m*x*(f*x)**m*log(c*x**n)/(m**4 + 8*m**3 + 22*m**2 + 24
*m + 9) - 9*b*d*n*x*(f*x)**m/(m**4 + 8*m**3 + 22*m**2 + 24*m + 9) + 9*b*d*x*(f*x)**m*log(c*x**n)/(m**4 + 8*m**
3 + 22*m**2 + 24*m + 9) + b*e*m**3*x**3*(f*x)**m*log(c*x**n)/(m**4 + 8*m**3 + 22*m**2 + 24*m + 9) - b*e*m**2*n
*x**3*(f*x)**m/(m**4 + 8*m**3 + 22*m**2 + 24*m + 9) + 5*b*e*m**2*x**3*(f*x)**m*log(c*x**n)/(m**4 + 8*m**3 + 22
*m**2 + 24*m + 9) - 2*b*e*m*n*x**3*(f*x)**m/(m**4 + 8*m**3 + 22*m**2 + 24*m + 9) + 7*b*e*m*x**3*(f*x)**m*log(c
*x**n)/(m**4 + 8*m**3 + 22*m**2 + 24*m + 9) - b*e*n*x**3*(f*x)**m/(m**4 + 8*m**3 + 22*m**2 + 24*m + 9) + 3*b*e
*x**3*(f*x)**m*log(c*x**n)/(m**4 + 8*m**3 + 22*m**2 + 24*m + 9), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 239 vs. \(2 (97) = 194\).
time = 2.05, size = 239, normalized size = 2.52 \begin {gather*} \frac {b f^{2} f^{m} x^{3} x^{m} e \log \left (c\right )}{f^{2} m + 3 \, f^{2}} + \frac {b f^{m} m n x^{3} x^{m} e \log \left (x\right )}{m^{2} + 6 \, m + 9} + \frac {a f^{2} f^{m} x^{3} x^{m} e}{f^{2} m + 3 \, f^{2}} + \frac {3 \, b f^{m} n x^{3} x^{m} e \log \left (x\right )}{m^{2} + 6 \, m + 9} - \frac {b f^{m} n x^{3} x^{m} e}{m^{2} + 6 \, m + 9} + \frac {b d f^{m} m n x x^{m} \log \left (x\right )}{m^{2} + 2 \, m + 1} + \frac {b d f^{m} n x x^{m} \log \left (x\right )}{m^{2} + 2 \, m + 1} - \frac {b d f^{m} n x x^{m}}{m^{2} + 2 \, m + 1} + \frac {\left (f x\right )^{m} b d x \log \left (c\right )}{m + 1} + \frac {\left (f x\right )^{m} a d x}{m + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*f^2*f^m*x^3*x^m*e*log(c)/(f^2*m + 3*f^2) + b*f^m*m*n*x^3*x^m*e*log(x)/(m^2 + 6*m + 9) + a*f^2*f^m*x^3*x^m*e/
(f^2*m + 3*f^2) + 3*b*f^m*n*x^3*x^m*e*log(x)/(m^2 + 6*m + 9) - b*f^m*n*x^3*x^m*e/(m^2 + 6*m + 9) + b*d*f^m*m*n
*x*x^m*log(x)/(m^2 + 2*m + 1) + b*d*f^m*n*x*x^m*log(x)/(m^2 + 2*m + 1) - b*d*f^m*n*x*x^m/(m^2 + 2*m + 1) + (f*
x)^m*b*d*x*log(c)/(m + 1) + (f*x)^m*a*d*x/(m + 1)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (f\,x\right )}^m\,\left (e\,x^2+d\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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