3.4.52 \(\int (f x)^{-1+m} (d+e x^m)^2 (a+b \log (c x^n)) \, dx\) [352]

Optimal. Leaf size=142 \[ -\frac {b d^2 n x (f x)^{-1+m}}{m^2}-\frac {b d e n x^{1+m} (f x)^{-1+m}}{2 m^2}-\frac {b e^2 n x^{1+2 m} (f x)^{-1+m}}{9 m^2}-\frac {b d^3 n x^{1-m} (f x)^{-1+m} \log (x)}{3 e m}+\frac {x^{1-m} (f x)^{-1+m} \left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right )}{3 e m} \]

[Out]

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

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Rubi [A]
time = 0.14, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2377, 2376, 272, 45} \begin {gather*} \frac {x^{1-m} (f x)^{m-1} \left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right )}{3 e m}-\frac {b d^3 n x^{1-m} \log (x) (f x)^{m-1}}{3 e m}-\frac {b d^2 n x (f x)^{m-1}}{m^2}-\frac {b d e n x^{m+1} (f x)^{m-1}}{2 m^2}-\frac {b e^2 n x^{2 m+1} (f x)^{m-1}}{9 m^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 2376

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_))^(q_.), x_Symbol] :
> Simp[f^m*(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])^p/(e*r*(q + 1))), x] - Dist[b*f^m*n*(p/(e*r*(q + 1))), Int[
(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && EqQ[
m, r - 1] && IGtQ[p, 0] && (IntegerQ[m] || GtQ[f, 0]) && NeQ[r, n] && NeQ[q, -1]

Rule 2377

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

Rubi steps

\begin {align*} \int (f x)^{-1+m} \left (d+e x^m\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx &=\left (x^{1-m} (f x)^{-1+m}\right ) \int x^{-1+m} \left (d+e x^m\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx\\ &=\frac {x^{1-m} (f x)^{-1+m} \left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right )}{3 e m}-\frac {\left (b n x^{1-m} (f x)^{-1+m}\right ) \int \frac {\left (d+e x^m\right )^3}{x} \, dx}{3 e m}\\ &=\frac {x^{1-m} (f x)^{-1+m} \left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right )}{3 e m}-\frac {\left (b n x^{1-m} (f x)^{-1+m}\right ) \text {Subst}\left (\int \frac {(d+e x)^3}{x} \, dx,x,x^m\right )}{3 e m^2}\\ &=\frac {x^{1-m} (f x)^{-1+m} \left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right )}{3 e m}-\frac {\left (b n x^{1-m} (f x)^{-1+m}\right ) \text {Subst}\left (\int \left (3 d^2 e+\frac {d^3}{x}+3 d e^2 x+e^3 x^2\right ) \, dx,x,x^m\right )}{3 e m^2}\\ &=-\frac {b d^2 n x (f x)^{-1+m}}{m^2}-\frac {b d e n x^{1+m} (f x)^{-1+m}}{2 m^2}-\frac {b e^2 n x^{1+2 m} (f x)^{-1+m}}{9 m^2}-\frac {b d^3 n x^{1-m} (f x)^{-1+m} \log (x)}{3 e m}+\frac {x^{1-m} (f x)^{-1+m} \left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right )}{3 e m}\\ \end {align*}

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Mathematica [A]
time = 0.08, size = 101, normalized size = 0.71 \begin {gather*} \frac {(f x)^m \left (6 a m \left (3 d^2+3 d e x^m+e^2 x^{2 m}\right )-b n \left (18 d^2+9 d e x^m+2 e^2 x^{2 m}\right )+6 b m \left (3 d^2+3 d e x^m+e^2 x^{2 m}\right ) \log \left (c x^n\right )\right )}{18 f m^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((f*x)^m*(6*a*m*(3*d^2 + 3*d*e*x^m + e^2*x^(2*m)) - b*n*(18*d^2 + 9*d*e*x^m + 2*e^2*x^(2*m)) + 6*b*m*(3*d^2 +
3*d*e*x^m + e^2*x^(2*m))*Log[c*x^n]))/(18*f*m^2)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.18, size = 616, normalized size = 4.34

method	result	size
risch	\(\frac {b \left (e^{2} x^{2 m}+3 d e \,x^{m}+3 d^{2}\right ) x \,{\mathrm e}^{\frac {\left (-1+m \right ) \left (-i \pi \mathrm {csgn}\left (i f x \right )^{3}+i \pi \mathrm {csgn}\left (i f x \right )^{2} \mathrm {csgn}\left (i f \right )+i \pi \mathrm {csgn}\left (i f x \right )^{2} \mathrm {csgn}\left (i x \right )-i \pi \,\mathrm {csgn}\left (i f x \right ) \mathrm {csgn}\left (i f \right ) \mathrm {csgn}\left (i x \right )+2 \ln \left (x \right )+2 \ln \left (f \right )\right )}{2}} \ln \left (x^{n}\right )}{3 m}+\frac {\left (3 i \pi b \,e^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x^{2 m} m +9 i \pi b \,d^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} m -3 i \pi b \,e^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3} x^{2 m} m -9 i \pi b d e \mathrm {csgn}\left (i c \,x^{n}\right )^{3} x^{m} m -9 i \pi b \,d^{2} m \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-3 i \pi b \,e^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) x^{2 m} m -9 i \pi b d e \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) x^{m} m +9 i \pi b d e \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x^{m} m +9 i \pi b d e \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x^{m} m +3 i \pi b \,e^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x^{2 m} m +9 i \pi b \,d^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} m -9 i \pi b \,d^{2} m \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+6 \ln \left (c \right ) b \,e^{2} x^{2 m} m +18 \ln \left (c \right ) b d e \,x^{m} m +6 a \,e^{2} x^{2 m} m -2 b \,e^{2} n \,x^{2 m}+18 \ln \left (c \right ) b \,d^{2} m +18 a d e \,x^{m} m -9 b d e n \,x^{m}+18 a \,d^{2} m -18 b \,d^{2} n \right ) x \,{\mathrm e}^{\frac {\left (-1+m \right ) \left (-i \pi \mathrm {csgn}\left (i f x \right )^{3}+i \pi \mathrm {csgn}\left (i f x \right )^{2} \mathrm {csgn}\left (i f \right )+i \pi \mathrm {csgn}\left (i f x \right )^{2} \mathrm {csgn}\left (i x \right )-i \pi \,\mathrm {csgn}\left (i f x \right ) \mathrm {csgn}\left (i f \right ) \mathrm {csgn}\left (i x \right )+2 \ln \left (x \right )+2 \ln \left (f \right )\right )}{2}}}{18 m^{2}}\)	\(616\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x)^(-1+m)*(d+e*x^m)^2*(a+b*ln(c*x^n)),x,method=_RETURNVERBOSE)

[Out]

1/3*b*(e^2*(x^m)^2+3*d*e*x^m+3*d^2)*x/m*exp(1/2*(-1+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)+1/18*(3*I*Pi*b*e^2*csgn
(I*c)*csgn(I*c*x^n)^2*(x^m)^2*m+9*I*Pi*b*d^2*csgn(I*c)*csgn(I*c*x^n)^2*m-3*I*Pi*b*e^2*csgn(I*c*x^n)^3*(x^m)^2*
m+3*I*Pi*b*e^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^m)^2*m-9*I*Pi*b*d*e*csgn(I*c*x^n)^3*x^m*m-9*I*Pi*b*d^2*csgn(I*c*
x^n)^3*m-9*I*Pi*b*d*e*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^m*m+9*I*Pi*b*d*e*csgn(I*x^n)*csgn(I*c*x^n)^2*x^m*m
+9*I*Pi*b*d*e*csgn(I*c)*csgn(I*c*x^n)^2*x^m*m-3*I*Pi*b*e^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^m)^2*m+9*I*P
i*b*d^2*csgn(I*x^n)*csgn(I*c*x^n)^2*m-9*I*Pi*b*d^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*m+6*ln(c)*b*e^2*(x^m)^2
*m+18*ln(c)*b*d*e*x^m*m+6*a*e^2*(x^m)^2*m-2*b*e^2*n*(x^m)^2+18*ln(c)*b*d^2*m+18*a*d*e*x^m*m-9*b*d*e*n*x^m+18*a
*d^2*m-18*b*d^2*n)*x/m^2*exp(1/2*(-1+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*c
sgn(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 = 186, normalized size = 1.31 \begin {gather*} -\frac {b d^{2} f^{m - 1} n x^{m}}{m^{2}} + \frac {b d f^{m - 1} e^{\left (2 \, m \log \left (x\right ) + 1\right )} \log \left (c x^{n}\right )}{m} + \frac {a d f^{m - 1} e^{\left (2 \, m \log \left (x\right ) + 1\right )}}{m} - \frac {b d f^{m - 1} n e^{\left (2 \, m \log \left (x\right ) + 1\right )}}{2 \, m^{2}} + \frac {\left (f x\right )^{m} b d^{2} \log \left (c x^{n}\right )}{f m} + \frac {b f^{m - 1} e^{\left (3 \, m \log \left (x\right ) + 2\right )} \log \left (c x^{n}\right )}{3 \, m} + \frac {\left (f x\right )^{m} a d^{2}}{f m} + \frac {a f^{m - 1} e^{\left (3 \, m \log \left (x\right ) + 2\right )}}{3 \, m} - \frac {b f^{m - 1} n e^{\left (3 \, m \log \left (x\right ) + 2\right )}}{9 \, m^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)^(-1+m)*(d+e*x^m)^2*(a+b*log(c*x^n)),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.39, size = 135, normalized size = 0.95 \begin {gather*} \frac {2 \, {\left (3 \, b m n e^{2} \log \left (x\right ) + 3 \, b m e^{2} \log \left (c\right ) + {\left (3 \, a m - b n\right )} e^{2}\right )} f^{m - 1} x^{3 \, m} + 9 \, {\left (2 \, b d m n e \log \left (x\right ) + 2 \, b d m e \log \left (c\right ) + {\left (2 \, a d m - b d n\right )} e\right )} f^{m - 1} x^{2 \, m} + 18 \, {\left (b d^{2} m n \log \left (x\right ) + b d^{2} m \log \left (c\right ) + a d^{2} m - b d^{2} n\right )} f^{m - 1} x^{m}}{18 \, m^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)^(-1+m)*(d+e*x^m)^2*(a+b*log(c*x^n)),x, algorithm="fricas")

[Out]

1/18*(2*(3*b*m*n*e^2*log(x) + 3*b*m*e^2*log(c) + (3*a*m - b*n)*e^2)*f^(m - 1)*x^(3*m) + 9*(2*b*d*m*n*e*log(x)
+ 2*b*d*m*e*log(c) + (2*a*d*m - b*d*n)*e)*f^(m - 1)*x^(2*m) + 18*(b*d^2*m*n*log(x) + b*d^2*m*log(c) + a*d^2*m
- b*d^2*n)*f^(m - 1)*x^m)/m^2

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1593 vs. \(2 (129) = 258\).
time = 40.51, size = 1593, normalized size = 11.22 \begin {gather*} \begin {cases} \tilde {\infty } \left (d + e\right )^{2} \left (a x - b n x + b x \log {\left (c x^{n} \right )}\right ) & \text {for}\: f = 0 \wedge m = 0 \\\frac {\left (d + e\right )^{2} \left (\begin {cases} a \log {\left (x \right )} & \text {for}\: b = 0 \\- \left (- a - b \log {\left (c \right )}\right ) \log {\left (x \right )} & \text {for}\: n = 0 \\\frac {\left (- a - b \log {\left (c x^{n} \right )}\right )^{2}}{2 b n} & \text {otherwise} \end {cases}\right )}{f} & \text {for}\: m = 0 \\0^{m - 1} \cdot \left (\frac {4 a d^{2} m^{4} x}{4 m^{4} + 12 m^{3} + 13 m^{2} + 6 m + 1} + \frac {12 a d^{2} m^{3} x}{4 m^{4} + 12 m^{3} + 13 m^{2} + 6 m + 1} + \frac {13 a d^{2} m^{2} x}{4 m^{4} + 12 m^{3} + 13 m^{2} + 6 m + 1} + \frac {6 a d^{2} m x}{4 m^{4} + 12 m^{3} + 13 m^{2} + 6 m + 1} + \frac {a d^{2} x}{4 m^{4} + 12 m^{3} + 13 m^{2} + 6 m + 1} + \frac {8 a d e m^{3} x x^{m}}{4 m^{4} + 12 m^{3} + 13 m^{2} + 6 m + 1} + \frac {16 a d e m^{2} x x^{m}}{4 m^{4} + 12 m^{3} + 13 m^{2} + 6 m + 1} + \frac {10 a d e m x x^{m}}{4 m^{4} + 12 m^{3} + 13 m^{2} + 6 m + 1} + \frac {2 a d e x x^{m}}{4 m^{4} + 12 m^{3} + 13 m^{2} + 6 m + 1} + \frac {2 a e^{2} m^{3} x x^{2 m}}{4 m^{4} + 12 m^{3} + 13 m^{2} + 6 m + 1} + \frac {5 a e^{2} m^{2} x x^{2 m}}{4 m^{4} + 12 m^{3} + 13 m^{2} + 6 m + 1} + \frac {4 a e^{2} m x x^{2 m}}{4 m^{4} + 12 m^{3} + 13 m^{2} + 6 m + 1} + \frac {a e^{2} x x^{2 m}}{4 m^{4} + 12 m^{3} + 13 m^{2} + 6 m + 1} - \frac {4 b d^{2} m^{4} n x}{4 m^{4} + 12 m^{3} + 13 m^{2} + 6 m + 1} + \frac {4 b d^{2} m^{4} x \log {\left (c x^{n} \right )}}{4 m^{4} + 12 m^{3} + 13 m^{2} + 6 m + 1} - \frac {12 b d^{2} m^{3} n x}{4 m^{4} + 12 m^{3} + 13 m^{2} + 6 m + 1} + \frac {12 b d^{2} m^{3} x \log {\left (c x^{n} \right )}}{4 m^{4} + 12 m^{3} + 13 m^{2} + 6 m + 1} - \frac {13 b d^{2} m^{2} n x}{4 m^{4} + 12 m^{3} + 13 m^{2} + 6 m + 1} + \frac {13 b d^{2} m^{2} x \log {\left (c x^{n} \right )}}{4 m^{4} + 12 m^{3} + 13 m^{2} + 6 m + 1} - \frac {6 b d^{2} m n x}{4 m^{4} + 12 m^{3} + 13 m^{2} + 6 m + 1} + \frac {6 b d^{2} m x \log {\left (c x^{n} \right )}}{4 m^{4} + 12 m^{3} + 13 m^{2} + 6 m + 1} - \frac {b d^{2} n x}{4 m^{4} + 12 m^{3} + 13 m^{2} + 6 m + 1} + \frac {b d^{2} x \log {\left (c x^{n} \right )}}{4 m^{4} + 12 m^{3} + 13 m^{2} + 6 m + 1} + \frac {8 b d e m^{3} x x^{m} \log {\left (c x^{n} \right )}}{4 m^{4} + 12 m^{3} + 13 m^{2} + 6 m + 1} - \frac {8 b d e m^{2} n x x^{m}}{4 m^{4} + 12 m^{3} + 13 m^{2} + 6 m + 1} + \frac {16 b d e m^{2} x x^{m} \log {\left (c x^{n} \right )}}{4 m^{4} + 12 m^{3} + 13 m^{2} + 6 m + 1} - \frac {8 b d e m n x x^{m}}{4 m^{4} + 12 m^{3} + 13 m^{2} + 6 m + 1} + \frac {10 b d e m x x^{m} \log {\left (c x^{n} \right )}}{4 m^{4} + 12 m^{3} + 13 m^{2} + 6 m + 1} - \frac {2 b d e n x x^{m}}{4 m^{4} + 12 m^{3} + 13 m^{2} + 6 m + 1} + \frac {2 b d e x x^{m} \log {\left (c x^{n} \right )}}{4 m^{4} + 12 m^{3} + 13 m^{2} + 6 m + 1} + \frac {2 b e^{2} m^{3} x x^{2 m} \log {\left (c x^{n} \right )}}{4 m^{4} + 12 m^{3} + 13 m^{2} + 6 m + 1} - \frac {b e^{2} m^{2} n x x^{2 m}}{4 m^{4} + 12 m^{3} + 13 m^{2} + 6 m + 1} + \frac {5 b e^{2} m^{2} x x^{2 m} \log {\left (c x^{n} \right )}}{4 m^{4} + 12 m^{3} + 13 m^{2} + 6 m + 1} - \frac {2 b e^{2} m n x x^{2 m}}{4 m^{4} + 12 m^{3} + 13 m^{2} + 6 m + 1} + \frac {4 b e^{2} m x x^{2 m} \log {\left (c x^{n} \right )}}{4 m^{4} + 12 m^{3} + 13 m^{2} + 6 m + 1} - \frac {b e^{2} n x x^{2 m}}{4 m^{4} + 12 m^{3} + 13 m^{2} + 6 m + 1} + \frac {b e^{2} x x^{2 m} \log {\left (c x^{n} \right )}}{4 m^{4} + 12 m^{3} + 13 m^{2} + 6 m + 1}\right ) & \text {for}\: f = 0 \\\frac {a d^{2} \left (f x\right )^{m}}{f m} + \frac {a d e x^{m} \left (f x\right )^{m}}{f m} + \frac {a e^{2} x^{2 m} \left (f x\right )^{m}}{3 f m} + \frac {b d^{2} \left (f x\right )^{m} \log {\left (c x^{n} \right )}}{f m} - \frac {b d^{2} n \left (f x\right )^{m}}{f m^{2}} + \frac {b d e x^{m} \left (f x\right )^{m} \log {\left (c x^{n} \right )}}{f m} - \frac {b d e n x^{m} \left (f x\right )^{m}}{2 f m^{2}} + \frac {b e^{2} x^{2 m} \left (f x\right )^{m} \log {\left (c x^{n} \right )}}{3 f m} - \frac {b e^{2} n x^{2 m} \left (f x\right )^{m}}{9 f m^{2}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)**(-1+m)*(d+e*x**m)**2*(a+b*ln(c*x**n)),x)

[Out]

Piecewise((zoo*(d + e)**2*(a*x - b*n*x + b*x*log(c*x**n)), Eq(f, 0) & Eq(m, 0)), ((d + e)**2*Piecewise((a*log(
x), Eq(b, 0)), (-(-a - b*log(c))*log(x), Eq(n, 0)), ((-a - b*log(c*x**n))**2/(2*b*n), True))/f, Eq(m, 0)), (0*
*(m - 1)*(4*a*d**2*m**4*x/(4*m**4 + 12*m**3 + 13*m**2 + 6*m + 1) + 12*a*d**2*m**3*x/(4*m**4 + 12*m**3 + 13*m**
2 + 6*m + 1) + 13*a*d**2*m**2*x/(4*m**4 + 12*m**3 + 13*m**2 + 6*m + 1) + 6*a*d**2*m*x/(4*m**4 + 12*m**3 + 13*m
**2 + 6*m + 1) + a*d**2*x/(4*m**4 + 12*m**3 + 13*m**2 + 6*m + 1) + 8*a*d*e*m**3*x*x**m/(4*m**4 + 12*m**3 + 13*
m**2 + 6*m + 1) + 16*a*d*e*m**2*x*x**m/(4*m**4 + 12*m**3 + 13*m**2 + 6*m + 1) + 10*a*d*e*m*x*x**m/(4*m**4 + 12
*m**3 + 13*m**2 + 6*m + 1) + 2*a*d*e*x*x**m/(4*m**4 + 12*m**3 + 13*m**2 + 6*m + 1) + 2*a*e**2*m**3*x*x**(2*m)/
(4*m**4 + 12*m**3 + 13*m**2 + 6*m + 1) + 5*a*e**2*m**2*x*x**(2*m)/(4*m**4 + 12*m**3 + 13*m**2 + 6*m + 1) + 4*a
*e**2*m*x*x**(2*m)/(4*m**4 + 12*m**3 + 13*m**2 + 6*m + 1) + a*e**2*x*x**(2*m)/(4*m**4 + 12*m**3 + 13*m**2 + 6*
m + 1) - 4*b*d**2*m**4*n*x/(4*m**4 + 12*m**3 + 13*m**2 + 6*m + 1) + 4*b*d**2*m**4*x*log(c*x**n)/(4*m**4 + 12*m
**3 + 13*m**2 + 6*m + 1) - 12*b*d**2*m**3*n*x/(4*m**4 + 12*m**3 + 13*m**2 + 6*m + 1) + 12*b*d**2*m**3*x*log(c*
x**n)/(4*m**4 + 12*m**3 + 13*m**2 + 6*m + 1) - 13*b*d**2*m**2*n*x/(4*m**4 + 12*m**3 + 13*m**2 + 6*m + 1) + 13*
b*d**2*m**2*x*log(c*x**n)/(4*m**4 + 12*m**3 + 13*m**2 + 6*m + 1) - 6*b*d**2*m*n*x/(4*m**4 + 12*m**3 + 13*m**2
+ 6*m + 1) + 6*b*d**2*m*x*log(c*x**n)/(4*m**4 + 12*m**3 + 13*m**2 + 6*m + 1) - b*d**2*n*x/(4*m**4 + 12*m**3 +
13*m**2 + 6*m + 1) + b*d**2*x*log(c*x**n)/(4*m**4 + 12*m**3 + 13*m**2 + 6*m + 1) + 8*b*d*e*m**3*x*x**m*log(c*x
**n)/(4*m**4 + 12*m**3 + 13*m**2 + 6*m + 1) - 8*b*d*e*m**2*n*x*x**m/(4*m**4 + 12*m**3 + 13*m**2 + 6*m + 1) + 1
6*b*d*e*m**2*x*x**m*log(c*x**n)/(4*m**4 + 12*m**3 + 13*m**2 + 6*m + 1) - 8*b*d*e*m*n*x*x**m/(4*m**4 + 12*m**3
+ 13*m**2 + 6*m + 1) + 10*b*d*e*m*x*x**m*log(c*x**n)/(4*m**4 + 12*m**3 + 13*m**2 + 6*m + 1) - 2*b*d*e*n*x*x**m
/(4*m**4 + 12*m**3 + 13*m**2 + 6*m + 1) + 2*b*d*e*x*x**m*log(c*x**n)/(4*m**4 + 12*m**3 + 13*m**2 + 6*m + 1) +
2*b*e**2*m**3*x*x**(2*m)*log(c*x**n)/(4*m**4 + 12*m**3 + 13*m**2 + 6*m + 1) - b*e**2*m**2*n*x*x**(2*m)/(4*m**4
+ 12*m**3 + 13*m**2 + 6*m + 1) + 5*b*e**2*m**2*x*x**(2*m)*log(c*x**n)/(4*m**4 + 12*m**3 + 13*m**2 + 6*m + 1)
- 2*b*e**2*m*n*x*x**(2*m)/(4*m**4 + 12*m**3 + 13*m**2 + 6*m + 1) + 4*b*e**2*m*x*x**(2*m)*log(c*x**n)/(4*m**4 +
12*m**3 + 13*m**2 + 6*m + 1) - b*e**2*n*x*x**(2*m)/(4*m**4 + 12*m**3 + 13*m**2 + 6*m + 1) + b*e**2*x*x**(2*m)
*log(c*x**n)/(4*m**4 + 12*m**3 + 13*m**2 + 6*m + 1)), Eq(f, 0)), (a*d**2*(f*x)**m/(f*m) + a*d*e*x**m*(f*x)**m/
(f*m) + a*e**2*x**(2*m)*(f*x)**m/(3*f*m) + b*d**2*(f*x)**m*log(c*x**n)/(f*m) - b*d**2*n*(f*x)**m/(f*m**2) + b*
d*e*x**m*(f*x)**m*log(c*x**n)/(f*m) - b*d*e*n*x**m*(f*x)**m/(2*f*m**2) + b*e**2*x**(2*m)*(f*x)**m*log(c*x**n)/
(3*f*m) - b*e**2*n*x**(2*m)*(f*x)**m/(9*f*m**2), True))

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Giac [A]
time = 4.43, size = 241, normalized size = 1.70 \begin {gather*} \frac {b d^{2} f^{m} n x^{m} \log \left (x\right )}{f m} + \frac {b d f^{m} n x^{2 \, m} e \log \left (x\right )}{f m} + \frac {b d^{2} f^{m} x^{m} \log \left (c\right )}{f m} + \frac {b d f^{m} x^{2 \, m} e \log \left (c\right )}{f m} + \frac {b f^{m} n x^{3 \, m} e^{2} \log \left (x\right )}{3 \, f m} + \frac {a d^{2} f^{m} x^{m}}{f m} - \frac {b d^{2} f^{m} n x^{m}}{f m^{2}} + \frac {a d f^{m} x^{2 \, m} e}{f m} - \frac {b d f^{m} n x^{2 \, m} e}{2 \, f m^{2}} + \frac {b f^{m} x^{3 \, m} e^{2} \log \left (c\right )}{3 \, f m} + \frac {a f^{m} x^{3 \, m} e^{2}}{3 \, f m} - \frac {b f^{m} n x^{3 \, m} e^{2}}{9 \, f m^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)^(-1+m)*(d+e*x^m)^2*(a+b*log(c*x^n)),x, algorithm="giac")

[Out]

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

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

[Out]

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

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