3.4.18 \(\int (f x)^m (d+e x^2)^3 (a+b \log (c x^n)) \, dx\) [318]

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

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

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Rubi [A]
time = 1.06, antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {276, 2392, 14} \begin {gather*} \frac {d^3 (f x)^{m+1} \left (a+b \log \left (c x^n\right )\right )}{f (m+1)}+\frac {3 d^2 e (f x)^{m+3} \left (a+b \log \left (c x^n\right )\right )}{f^3 (m+3)}+\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 {b d^3 n (f x)^{m+1}}{f (m+1)^2}-\frac {3 b d^2 e n (f x)^{m+3}}{f^3 (m+3)^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} \end {gather*}

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 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 276

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 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 )^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*}

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Mathematica [A]
time = 0.39, size = 235, normalized size = 1.11 \begin {gather*} (f x)^m \left (b n x \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 ) \log (x)+\frac {d^3 x \left (a+a m-b n-b (1+m) n \log (x)+b (1+m) \log \left (c x^n\right )\right )}{(1+m)^2}+\frac {3 d^2 e x^3 \left (3 a+a m-b n-b (3+m) n \log (x)+b (3+m) \log \left (c x^n\right )\right )}{(3+m)^2}+\frac {3 d e^2 x^5 \left (5 a+a m-b n-b (5+m) n \log (x)+b (5+m) \log \left (c x^n\right )\right )}{(5+m)^2}+\frac {e^3 x^7 \left (7 a+a m-b n-b (7+m) n \log (x)+b (7+m) \log \left (c x^n\right )\right )}{(7+m)^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.45, size = 5139, normalized size = 24.36

method	result	size
risch	\(\text {Expression too large to display}\)	\(5139\)

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,method=_RETURNVERBOSE)

[Out]

result too large to display

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Maxima [A]
time = 0.30, size = 286, normalized size = 1.36 \begin {gather*} \frac {b f^{m} x^{7} e^{\left (m \log \left (x\right ) + 3\right )} \log \left (c x^{n}\right )}{m + 7} + \frac {a f^{m} x^{7} e^{\left (m \log \left (x\right ) + 3\right )}}{m + 7} - \frac {b f^{m} n x^{7} e^{\left (m \log \left (x\right ) + 3\right )}}{{\left (m + 7\right )}^{2}} + \frac {3 \, b d f^{m} x^{5} e^{\left (m \log \left (x\right ) + 2\right )} \log \left (c x^{n}\right )}{m + 5} + \frac {3 \, a d f^{m} x^{5} e^{\left (m \log \left (x\right ) + 2\right )}}{m + 5} - \frac {3 \, b d f^{m} n x^{5} e^{\left (m \log \left (x\right ) + 2\right )}}{{\left (m + 5\right )}^{2}} + \frac {3 \, b d^{2} f^{m} x^{3} e^{\left (m \log \left (x\right ) + 1\right )} \log \left (c x^{n}\right )}{m + 3} + \frac {3 \, a d^{2} f^{m} x^{3} e^{\left (m \log \left (x\right ) + 1\right )}}{m + 3} - \frac {3 \, b d^{2} f^{m} n x^{3} e^{\left (m \log \left (x\right ) + 1\right )}}{{\left (m + 3\right )}^{2}} - \frac {b d^{3} f^{m} n x x^{m}}{{\left (m + 1\right )}^{2}} + \frac {\left (f x\right )^{m + 1} b d^{3} \log \left (c x^{n}\right )}{f {\left (m + 1\right )}} + \frac {\left (f x\right )^{m + 1} a d^{3}}{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)^3*(a+b*log(c*x^n)),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1023 vs. \(2 (209) = 418\).
time = 0.37, size = 1023, normalized size = 4.85 \begin {gather*} \frac {{\left ({\left (a m^{7} + 25 \, a m^{6} + 253 \, a m^{5} + 1333 \, a m^{4} + 3907 \, a m^{3} + 6283 \, a m^{2} + 5055 \, a m - {\left (b m^{6} + 18 \, b m^{5} + 127 \, b m^{4} + 444 \, b m^{3} + 799 \, b m^{2} + 690 \, b m + 225 \, b\right )} n + 1575 \, a\right )} x^{7} e^{3} + 3 \, {\left (a d m^{7} + 27 \, a d m^{6} + 293 \, a d m^{5} + 1639 \, a d m^{4} + 5043 \, a d m^{3} + 8417 \, a d m^{2} + 6951 \, a d m + 2205 \, a d - {\left (b d m^{6} + 22 \, b d m^{5} + 183 \, b d m^{4} + 724 \, b d m^{3} + 1423 \, b d m^{2} + 1302 \, b d m + 441 \, b d\right )} n\right )} x^{5} e^{2} + 3 \, {\left (a d^{2} m^{7} + 29 \, a d^{2} m^{6} + 341 \, a d^{2} m^{5} + 2081 \, a d^{2} m^{4} + 6995 \, a d^{2} m^{3} + 12647 \, a d^{2} m^{2} + 11095 \, a d^{2} m + 3675 \, a d^{2} - {\left (b d^{2} m^{6} + 26 \, b d^{2} m^{5} + 263 \, b d^{2} m^{4} + 1292 \, b d^{2} m^{3} + 3119 \, b d^{2} m^{2} + 3290 \, b d^{2} m + 1225 \, b d^{2}\right )} n\right )} x^{3} e + {\left (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} - {\left (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}\right )} n\right )} x + {\left ({\left (b m^{7} + 25 \, b m^{6} + 253 \, b m^{5} + 1333 \, b m^{4} + 3907 \, b m^{3} + 6283 \, b m^{2} + 5055 \, b m + 1575 \, b\right )} x^{7} e^{3} + 3 \, {\left (b d m^{7} + 27 \, b d m^{6} + 293 \, b d m^{5} + 1639 \, b d m^{4} + 5043 \, b d m^{3} + 8417 \, b d m^{2} + 6951 \, b d m + 2205 \, b d\right )} x^{5} e^{2} + 3 \, {\left (b d^{2} m^{7} + 29 \, b d^{2} m^{6} + 341 \, b d^{2} m^{5} + 2081 \, b d^{2} m^{4} + 6995 \, b d^{2} m^{3} + 12647 \, b d^{2} m^{2} + 11095 \, b d^{2} m + 3675 \, b d^{2}\right )} x^{3} e + {\left (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}\right )} x\right )} \log \left (c\right ) + {\left ({\left (b m^{7} + 25 \, b m^{6} + 253 \, b m^{5} + 1333 \, b m^{4} + 3907 \, b m^{3} + 6283 \, b m^{2} + 5055 \, b m + 1575 \, b\right )} n x^{7} e^{3} + 3 \, {\left (b d m^{7} + 27 \, b d m^{6} + 293 \, b d m^{5} + 1639 \, b d m^{4} + 5043 \, b d m^{3} + 8417 \, b d m^{2} + 6951 \, b d m + 2205 \, b d\right )} n x^{5} e^{2} + 3 \, {\left (b d^{2} m^{7} + 29 \, b d^{2} m^{6} + 341 \, b d^{2} m^{5} + 2081 \, b d^{2} m^{4} + 6995 \, b d^{2} m^{3} + 12647 \, b d^{2} m^{2} + 11095 \, b d^{2} m + 3675 \, b d^{2}\right )} n x^{3} e + {\left (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}\right )} n x\right )} \log \left (x\right )\right )} e^{\left (m \log \left (f\right ) + m \log \left (x\right )\right )}}{m^{8} + 32 \, m^{7} + 428 \, m^{6} + 3104 \, m^{5} + 13238 \, m^{4} + 33632 \, m^{3} + 49036 \, m^{2} + 36960 \, m + 11025} \end {gather*}

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*m^7 + 25*a*m^6 + 253*a*m^5 + 1333*a*m^4 + 3907*a*m^3 + 6283*a*m^2 + 5055*a*m - (b*m^6 + 18*b*m^5 + 127*b*m
^4 + 444*b*m^3 + 799*b*m^2 + 690*b*m + 225*b)*n + 1575*a)*x^7*e^3 + 3*(a*d*m^7 + 27*a*d*m^6 + 293*a*d*m^5 + 16
39*a*d*m^4 + 5043*a*d*m^3 + 8417*a*d*m^2 + 6951*a*d*m + 2205*a*d - (b*d*m^6 + 22*b*d*m^5 + 183*b*d*m^4 + 724*b
*d*m^3 + 1423*b*d*m^2 + 1302*b*d*m + 441*b*d)*n)*x^5*e^2 + 3*(a*d^2*m^7 + 29*a*d^2*m^6 + 341*a*d^2*m^5 + 2081*
a*d^2*m^4 + 6995*a*d^2*m^3 + 12647*a*d^2*m^2 + 11095*a*d^2*m + 3675*a*d^2 - (b*d^2*m^6 + 26*b*d^2*m^5 + 263*b*
d^2*m^4 + 1292*b*d^2*m^3 + 3119*b*d^2*m^2 + 3290*b*d^2*m + 1225*b*d^2)*n)*x^3*e + (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*m^7
+ 25*b*m^6 + 253*b*m^5 + 1333*b*m^4 + 3907*b*m^3 + 6283*b*m^2 + 5055*b*m + 1575*b)*x^7*e^3 + 3*(b*d*m^7 + 27*
b*d*m^6 + 293*b*d*m^5 + 1639*b*d*m^4 + 5043*b*d*m^3 + 8417*b*d*m^2 + 6951*b*d*m + 2205*b*d)*x^5*e^2 + 3*(b*d^2
*m^7 + 29*b*d^2*m^6 + 341*b*d^2*m^5 + 2081*b*d^2*m^4 + 6995*b*d^2*m^3 + 12647*b*d^2*m^2 + 11095*b*d^2*m + 3675
*b*d^2)*x^3*e + (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*m^7 + 25*b*m^6 + 253*b*m^5 + 1333*b*m^4 + 3907*b*m^3 + 6283*b*
m^2 + 5055*b*m + 1575*b)*n*x^7*e^3 + 3*(b*d*m^7 + 27*b*d*m^6 + 293*b*d*m^5 + 1639*b*d*m^4 + 5043*b*d*m^3 + 841
7*b*d*m^2 + 6951*b*d*m + 2205*b*d)*n*x^5*e^2 + 3*(b*d^2*m^7 + 29*b*d^2*m^6 + 341*b*d^2*m^5 + 2081*b*d^2*m^4 +
6995*b*d^2*m^3 + 12647*b*d^2*m^2 + 11095*b*d^2*m + 3675*b*d^2)*n*x^3*e + (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)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 6217 vs. \(2 (206) = 412\).
time = 20.01, size = 6217, normalized size = 29.46 \begin {gather*} \text {Too large to display} \end {gather*}

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]

Piecewise(((-a*d**3/(6*x**6) - 3*a*d**2*e/(4*x**4) - 3*a*d*e**2/(2*x**2) + a*e**3*log(x) + b*d**3*(-n/(36*x**6
) - log(c*x**n)/(6*x**6)) + 3*b*d**2*e*(-n/(16*x**4) - log(c*x**n)/(4*x**4)) + 3*b*d*e**2*(-n/(4*x**2) - log(c
*x**n)/(2*x**2)) - b*e**3*Piecewise((-log(c)*log(x), Eq(n, 0)), (-log(c*x**n)**2/(2*n), True)))/f**7, Eq(m, -7
)), ((-a*d**3/(4*x**4) - 3*a*d**2*e/(2*x**2) + 3*a*d*e**2*log(c*x**n)/n + a*e**3*x**2/2 - b*d**3*n/(16*x**4) -
b*d**3*log(c*x**n)/(4*x**4) - 3*b*d**2*e*n/(4*x**2) - 3*b*d**2*e*log(c*x**n)/(2*x**2) + 3*b*d*e**2*log(c*x**n
)**2/(2*n) - b*e**3*n*x**2/4 + b*e**3*x**2*log(c*x**n)/2)/f**5, Eq(m, -5)), ((-a*d**3/(2*x**2) + 3*a*d**2*e*lo
g(c*x**n)/n + 3*a*d*e**2*x**2/2 + a*e**3*x**4/4 - b*d**3*n/(4*x**2) - b*d**3*log(c*x**n)/(2*x**2) + 3*b*d**2*e
*log(c*x**n)**2/(2*n) - 3*b*d*e**2*n*x**2/4 + 3*b*d*e**2*x**2*log(c*x**n)/2 - b*e**3*n*x**4/16 + b*e**3*x**4*l
og(c*x**n)/4)/f**3, Eq(m, -3)), ((a*d**3*log(c*x**n)/n + 3*a*d**2*e*x**2/2 + 3*a*d*e**2*x**4/4 + a*e**3*x**6/6
+ b*d**3*log(c*x**n)**2/(2*n) - 3*b*d**2*e*n*x**2/4 + 3*b*d**2*e*x**2*log(c*x**n)/2 - 3*b*d*e**2*n*x**4/16 +
3*b*d*e**2*x**4*log(c*x**n)/4 - b*e**3*n*x**6/36 + b*e**3*x**6*log(c*x**n)/6)/f, Eq(m, -1)), (a*d**3*m**7*x*(f
*x)**m/(m**8 + 32*m**7 + 428*m**6 + 3104*m**5 + 13238*m**4 + 33632*m**3 + 49036*m**2 + 36960*m + 11025) + 31*a
*d**3*m**6*x*(f*x)**m/(m**8 + 32*m**7 + 428*m**6 + 3104*m**5 + 13238*m**4 + 33632*m**3 + 49036*m**2 + 36960*m
+ 11025) + 397*a*d**3*m**5*x*(f*x)**m/(m**8 + 32*m**7 + 428*m**6 + 3104*m**5 + 13238*m**4 + 33632*m**3 + 49036
*m**2 + 36960*m + 11025) + 2707*a*d**3*m**4*x*(f*x)**m/(m**8 + 32*m**7 + 428*m**6 + 3104*m**5 + 13238*m**4 + 3
3632*m**3 + 49036*m**2 + 36960*m + 11025) + 10531*a*d**3*m**3*x*(f*x)**m/(m**8 + 32*m**7 + 428*m**6 + 3104*m**
5 + 13238*m**4 + 33632*m**3 + 49036*m**2 + 36960*m + 11025) + 23101*a*d**3*m**2*x*(f*x)**m/(m**8 + 32*m**7 + 4
28*m**6 + 3104*m**5 + 13238*m**4 + 33632*m**3 + 49036*m**2 + 36960*m + 11025) + 25935*a*d**3*m*x*(f*x)**m/(m**
8 + 32*m**7 + 428*m**6 + 3104*m**5 + 13238*m**4 + 33632*m**3 + 49036*m**2 + 36960*m + 11025) + 11025*a*d**3*x*
(f*x)**m/(m**8 + 32*m**7 + 428*m**6 + 3104*m**5 + 13238*m**4 + 33632*m**3 + 49036*m**2 + 36960*m + 11025) + 3*
a*d**2*e*m**7*x**3*(f*x)**m/(m**8 + 32*m**7 + 428*m**6 + 3104*m**5 + 13238*m**4 + 33632*m**3 + 49036*m**2 + 36
960*m + 11025) + 87*a*d**2*e*m**6*x**3*(f*x)**m/(m**8 + 32*m**7 + 428*m**6 + 3104*m**5 + 13238*m**4 + 33632*m*
*3 + 49036*m**2 + 36960*m + 11025) + 1023*a*d**2*e*m**5*x**3*(f*x)**m/(m**8 + 32*m**7 + 428*m**6 + 3104*m**5 +
13238*m**4 + 33632*m**3 + 49036*m**2 + 36960*m + 11025) + 6243*a*d**2*e*m**4*x**3*(f*x)**m/(m**8 + 32*m**7 +
428*m**6 + 3104*m**5 + 13238*m**4 + 33632*m**3 + 49036*m**2 + 36960*m + 11025) + 20985*a*d**2*e*m**3*x**3*(f*x
)**m/(m**8 + 32*m**7 + 428*m**6 + 3104*m**5 + 13238*m**4 + 33632*m**3 + 49036*m**2 + 36960*m + 11025) + 37941*
a*d**2*e*m**2*x**3*(f*x)**m/(m**8 + 32*m**7 + 428*m**6 + 3104*m**5 + 13238*m**4 + 33632*m**3 + 49036*m**2 + 36
960*m + 11025) + 33285*a*d**2*e*m*x**3*(f*x)**m/(m**8 + 32*m**7 + 428*m**6 + 3104*m**5 + 13238*m**4 + 33632*m*
*3 + 49036*m**2 + 36960*m + 11025) + 11025*a*d**2*e*x**3*(f*x)**m/(m**8 + 32*m**7 + 428*m**6 + 3104*m**5 + 132
38*m**4 + 33632*m**3 + 49036*m**2 + 36960*m + 11025) + 3*a*d*e**2*m**7*x**5*(f*x)**m/(m**8 + 32*m**7 + 428*m**
6 + 3104*m**5 + 13238*m**4 + 33632*m**3 + 49036*m**2 + 36960*m + 11025) + 81*a*d*e**2*m**6*x**5*(f*x)**m/(m**8
+ 32*m**7 + 428*m**6 + 3104*m**5 + 13238*m**4 + 33632*m**3 + 49036*m**2 + 36960*m + 11025) + 879*a*d*e**2*m**
5*x**5*(f*x)**m/(m**8 + 32*m**7 + 428*m**6 + 3104*m**5 + 13238*m**4 + 33632*m**3 + 49036*m**2 + 36960*m + 1102
5) + 4917*a*d*e**2*m**4*x**5*(f*x)**m/(m**8 + 32*m**7 + 428*m**6 + 3104*m**5 + 13238*m**4 + 33632*m**3 + 49036
*m**2 + 36960*m + 11025) + 15129*a*d*e**2*m**3*x**5*(f*x)**m/(m**8 + 32*m**7 + 428*m**6 + 3104*m**5 + 13238*m*
*4 + 33632*m**3 + 49036*m**2 + 36960*m + 11025) + 25251*a*d*e**2*m**2*x**5*(f*x)**m/(m**8 + 32*m**7 + 428*m**6
+ 3104*m**5 + 13238*m**4 + 33632*m**3 + 49036*m**2 + 36960*m + 11025) + 20853*a*d*e**2*m*x**5*(f*x)**m/(m**8
+ 32*m**7 + 428*m**6 + 3104*m**5 + 13238*m**4 + 33632*m**3 + 49036*m**2 + 36960*m + 11025) + 6615*a*d*e**2*x**
5*(f*x)**m/(m**8 + 32*m**7 + 428*m**6 + 3104*m**5 + 13238*m**4 + 33632*m**3 + 49036*m**2 + 36960*m + 11025) +
a*e**3*m**7*x**7*(f*x)**m/(m**8 + 32*m**7 + 428*m**6 + 3104*m**5 + 13238*m**4 + 33632*m**3 + 49036*m**2 + 3696
0*m + 11025) + 25*a*e**3*m**6*x**7*(f*x)**m/(m**8 + 32*m**7 + 428*m**6 + 3104*m**5 + 13238*m**4 + 33632*m**3 +
49036*m**2 + 36960*m + 11025) + 253*a*e**3*m**5*x**7*(f*x)**m/(m**8 + 32*m**7 + 428*m**6 + 3104*m**5 + 13238*
m**4 + 33632*m**3 + 49036*m**2 + 36960*m + 11025) + 1333*a*e**3*m**4*x**7*(f*x)**m/(m**8 + 32*m**7 + 428*m**6
+ 3104*m**5 + 13238*m**4 + 33632*m**3 + 49036*m**2 + 36960*m + 11025) + 3907*a*e**3*m**3*x**7*(f*x)**m/(m**8 +
32*m**7 + 428*m**6 + 3104*m**5 + 13238*m**4 + 33632*m**3 + 49036*m**2 + 36960*m + 11025) + 6283*a*e**3*m**2*x
**7*(f*x)**m/(m**8 + 32*m**7 + 428*m**6 + 3104*...

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 553 vs. \(2 (209) = 418\).
time = 2.60, size = 553, normalized size = 2.62 \begin {gather*} \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 {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (f\,x\right )}^m\,{\left (e\,x^2+d\right )}^3\,\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)^3*(a + b*log(c*x^n)),x)

[Out]

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

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