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3.486 \(\int (e x)^m (A+B x) (a+c x^2)^3 \, dx\)

Optimal. Leaf size=169 \[ \frac{3 a^2 A c (e x)^{m+3}}{e^3 (m+3)}+\frac{a^3 A (e x)^{m+1}}{e (m+1)}+\frac{3 a^2 B c (e x)^{m+4}}{e^4 (m+4)}+\frac{a^3 B (e x)^{m+2}}{e^2 (m+2)}+\frac{3 a A c^2 (e x)^{m+5}}{e^5 (m+5)}+\frac{3 a B c^2 (e x)^{m+6}}{e^6 (m+6)}+\frac{A c^3 (e x)^{m+7}}{e^7 (m+7)}+\frac{B c^3 (e x)^{m+8}}{e^8 (m+8)} \]

[Out]

(a^3*A*(e*x)^(1 + m))/(e*(1 + m)) + (a^3*B*(e*x)^(2 + m))/(e^2*(2 + m)) + (3*a^2*A*c*(e*x)^(3 + m))/(e^3*(3 +
m)) + (3*a^2*B*c*(e*x)^(4 + m))/(e^4*(4 + m)) + (3*a*A*c^2*(e*x)^(5 + m))/(e^5*(5 + m)) + (3*a*B*c^2*(e*x)^(6
+ m))/(e^6*(6 + m)) + (A*c^3*(e*x)^(7 + m))/(e^7*(7 + m)) + (B*c^3*(e*x)^(8 + m))/(e^8*(8 + m))

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Rubi [A]  time = 0.102486, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {766} \[ \frac{3 a^2 A c (e x)^{m+3}}{e^3 (m+3)}+\frac{a^3 A (e x)^{m+1}}{e (m+1)}+\frac{3 a^2 B c (e x)^{m+4}}{e^4 (m+4)}+\frac{a^3 B (e x)^{m+2}}{e^2 (m+2)}+\frac{3 a A c^2 (e x)^{m+5}}{e^5 (m+5)}+\frac{3 a B c^2 (e x)^{m+6}}{e^6 (m+6)}+\frac{A c^3 (e x)^{m+7}}{e^7 (m+7)}+\frac{B c^3 (e x)^{m+8}}{e^8 (m+8)} \]

Antiderivative was successfully verified.

[In]

Int[(e*x)^m*(A + B*x)*(a + c*x^2)^3,x]

[Out]

(a^3*A*(e*x)^(1 + m))/(e*(1 + m)) + (a^3*B*(e*x)^(2 + m))/(e^2*(2 + m)) + (3*a^2*A*c*(e*x)^(3 + m))/(e^3*(3 +
m)) + (3*a^2*B*c*(e*x)^(4 + m))/(e^4*(4 + m)) + (3*a*A*c^2*(e*x)^(5 + m))/(e^5*(5 + m)) + (3*a*B*c^2*(e*x)^(6
+ m))/(e^6*(6 + m)) + (A*c^3*(e*x)^(7 + m))/(e^7*(7 + m)) + (B*c^3*(e*x)^(8 + m))/(e^8*(8 + m))

Rule 766

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(e*x
)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int (e x)^m (A+B x) \left (a+c x^2\right )^3 \, dx &=\int \left (a^3 A (e x)^m+\frac{a^3 B (e x)^{1+m}}{e}+\frac{3 a^2 A c (e x)^{2+m}}{e^2}+\frac{3 a^2 B c (e x)^{3+m}}{e^3}+\frac{3 a A c^2 (e x)^{4+m}}{e^4}+\frac{3 a B c^2 (e x)^{5+m}}{e^5}+\frac{A c^3 (e x)^{6+m}}{e^6}+\frac{B c^3 (e x)^{7+m}}{e^7}\right ) \, dx\\ &=\frac{a^3 A (e x)^{1+m}}{e (1+m)}+\frac{a^3 B (e x)^{2+m}}{e^2 (2+m)}+\frac{3 a^2 A c (e x)^{3+m}}{e^3 (3+m)}+\frac{3 a^2 B c (e x)^{4+m}}{e^4 (4+m)}+\frac{3 a A c^2 (e x)^{5+m}}{e^5 (5+m)}+\frac{3 a B c^2 (e x)^{6+m}}{e^6 (6+m)}+\frac{A c^3 (e x)^{7+m}}{e^7 (7+m)}+\frac{B c^3 (e x)^{8+m}}{e^8 (8+m)}\\ \end{align*}

Mathematica [A]  time = 0.221293, size = 101, normalized size = 0.6 \[ x (e x)^m \left (3 a^2 c x^2 \left (\frac{A}{m+3}+\frac{B x}{m+4}\right )+a^3 \left (\frac{A}{m+1}+\frac{B x}{m+2}\right )+3 a c^2 x^4 \left (\frac{A}{m+5}+\frac{B x}{m+6}\right )+c^3 x^6 \left (\frac{A}{m+7}+\frac{B x}{m+8}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(e*x)^m*(A + B*x)*(a + c*x^2)^3,x]

[Out]

x*(e*x)^m*(a^3*(A/(1 + m) + (B*x)/(2 + m)) + 3*a^2*c*x^2*(A/(3 + m) + (B*x)/(4 + m)) + 3*a*c^2*x^4*(A/(5 + m)
+ (B*x)/(6 + m)) + c^3*x^6*(A/(7 + m) + (B*x)/(8 + m)))

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Maple [B]  time = 0.007, size = 765, normalized size = 4.5 \begin{align*}{\frac{ \left ( B{c}^{3}{m}^{7}{x}^{7}+A{c}^{3}{m}^{7}{x}^{6}+28\,B{c}^{3}{m}^{6}{x}^{7}+29\,A{c}^{3}{m}^{6}{x}^{6}+3\,Ba{c}^{2}{m}^{7}{x}^{5}+322\,B{c}^{3}{m}^{5}{x}^{7}+3\,Aa{c}^{2}{m}^{7}{x}^{4}+343\,A{c}^{3}{m}^{5}{x}^{6}+90\,Ba{c}^{2}{m}^{6}{x}^{5}+1960\,B{c}^{3}{m}^{4}{x}^{7}+93\,Aa{c}^{2}{m}^{6}{x}^{4}+2135\,A{c}^{3}{m}^{4}{x}^{6}+3\,B{a}^{2}c{m}^{7}{x}^{3}+1098\,Ba{c}^{2}{m}^{5}{x}^{5}+6769\,B{c}^{3}{m}^{3}{x}^{7}+3\,A{a}^{2}c{m}^{7}{x}^{2}+1173\,Aa{c}^{2}{m}^{5}{x}^{4}+7504\,A{c}^{3}{m}^{3}{x}^{6}+96\,B{a}^{2}c{m}^{6}{x}^{3}+7020\,Ba{c}^{2}{m}^{4}{x}^{5}+13132\,B{c}^{3}{m}^{2}{x}^{7}+99\,A{a}^{2}c{m}^{6}{x}^{2}+7743\,Aa{c}^{2}{m}^{4}{x}^{4}+14756\,A{c}^{3}{m}^{2}{x}^{6}+B{a}^{3}{m}^{7}x+1254\,B{a}^{2}c{m}^{5}{x}^{3}+25227\,Ba{c}^{2}{m}^{3}{x}^{5}+13068\,B{c}^{3}m{x}^{7}+A{a}^{3}{m}^{7}+1341\,A{a}^{2}c{m}^{5}{x}^{2}+28632\,Aa{c}^{2}{m}^{3}{x}^{4}+14832\,A{c}^{3}m{x}^{6}+34\,B{a}^{3}{m}^{6}x+8592\,B{a}^{2}c{m}^{4}{x}^{3}+50490\,Ba{c}^{2}{m}^{2}{x}^{5}+5040\,B{c}^{3}{x}^{7}+35\,A{a}^{3}{m}^{6}+9585\,A{a}^{2}c{m}^{4}{x}^{2}+58692\,Aa{c}^{2}{m}^{2}{x}^{4}+5760\,A{c}^{3}{x}^{6}+478\,B{a}^{3}{m}^{5}x+32979\,B{a}^{2}c{m}^{3}{x}^{3}+51432\,Ba{c}^{2}m{x}^{5}+511\,A{a}^{3}{m}^{5}+38592\,A{a}^{2}c{m}^{3}{x}^{2}+60912\,Aa{c}^{2}m{x}^{4}+3580\,B{a}^{3}{m}^{4}x+69936\,B{a}^{2}c{m}^{2}{x}^{3}+20160\,aB{c}^{2}{x}^{5}+4025\,A{a}^{3}{m}^{4}+86076\,A{a}^{2}c{m}^{2}{x}^{2}+24192\,aA{c}^{2}{x}^{4}+15289\,B{a}^{3}{m}^{3}x+74628\,B{a}^{2}cm{x}^{3}+18424\,A{a}^{3}{m}^{3}+96144\,A{a}^{2}cm{x}^{2}+36706\,B{a}^{3}{m}^{2}x+30240\,{a}^{2}Bc{x}^{3}+48860\,A{a}^{3}{m}^{2}+40320\,{a}^{2}Ac{x}^{2}+44712\,B{a}^{3}mx+69264\,A{a}^{3}m+20160\,{a}^{3}Bx+40320\,A{a}^{3} \right ) x \left ( ex \right ) ^{m}}{ \left ( 8+m \right ) \left ( 7+m \right ) \left ( 6+m \right ) \left ( 5+m \right ) \left ( 4+m \right ) \left ( 3+m \right ) \left ( 2+m \right ) \left ( 1+m \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^m*(B*x+A)*(c*x^2+a)^3,x)

[Out]

x*(B*c^3*m^7*x^7+A*c^3*m^7*x^6+28*B*c^3*m^6*x^7+29*A*c^3*m^6*x^6+3*B*a*c^2*m^7*x^5+322*B*c^3*m^5*x^7+3*A*a*c^2
*m^7*x^4+343*A*c^3*m^5*x^6+90*B*a*c^2*m^6*x^5+1960*B*c^3*m^4*x^7+93*A*a*c^2*m^6*x^4+2135*A*c^3*m^4*x^6+3*B*a^2
*c*m^7*x^3+1098*B*a*c^2*m^5*x^5+6769*B*c^3*m^3*x^7+3*A*a^2*c*m^7*x^2+1173*A*a*c^2*m^5*x^4+7504*A*c^3*m^3*x^6+9
6*B*a^2*c*m^6*x^3+7020*B*a*c^2*m^4*x^5+13132*B*c^3*m^2*x^7+99*A*a^2*c*m^6*x^2+7743*A*a*c^2*m^4*x^4+14756*A*c^3
*m^2*x^6+B*a^3*m^7*x+1254*B*a^2*c*m^5*x^3+25227*B*a*c^2*m^3*x^5+13068*B*c^3*m*x^7+A*a^3*m^7+1341*A*a^2*c*m^5*x
^2+28632*A*a*c^2*m^3*x^4+14832*A*c^3*m*x^6+34*B*a^3*m^6*x+8592*B*a^2*c*m^4*x^3+50490*B*a*c^2*m^2*x^5+5040*B*c^
3*x^7+35*A*a^3*m^6+9585*A*a^2*c*m^4*x^2+58692*A*a*c^2*m^2*x^4+5760*A*c^3*x^6+478*B*a^3*m^5*x+32979*B*a^2*c*m^3
*x^3+51432*B*a*c^2*m*x^5+511*A*a^3*m^5+38592*A*a^2*c*m^3*x^2+60912*A*a*c^2*m*x^4+3580*B*a^3*m^4*x+69936*B*a^2*
c*m^2*x^3+20160*B*a*c^2*x^5+4025*A*a^3*m^4+86076*A*a^2*c*m^2*x^2+24192*A*a*c^2*x^4+15289*B*a^3*m^3*x+74628*B*a
^2*c*m*x^3+18424*A*a^3*m^3+96144*A*a^2*c*m*x^2+36706*B*a^3*m^2*x+30240*B*a^2*c*x^3+48860*A*a^3*m^2+40320*A*a^2
*c*x^2+44712*B*a^3*m*x+69264*A*a^3*m+20160*B*a^3*x+40320*A*a^3)*(e*x)^m/(8+m)/(7+m)/(6+m)/(5+m)/(4+m)/(3+m)/(2
+m)/(1+m)

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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((e*x)^m*(B*x+A)*(c*x^2+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.68883, size = 1646, normalized size = 9.74 \begin{align*} \frac{{\left ({\left (B c^{3} m^{7} + 28 \, B c^{3} m^{6} + 322 \, B c^{3} m^{5} + 1960 \, B c^{3} m^{4} + 6769 \, B c^{3} m^{3} + 13132 \, B c^{3} m^{2} + 13068 \, B c^{3} m + 5040 \, B c^{3}\right )} x^{8} +{\left (A c^{3} m^{7} + 29 \, A c^{3} m^{6} + 343 \, A c^{3} m^{5} + 2135 \, A c^{3} m^{4} + 7504 \, A c^{3} m^{3} + 14756 \, A c^{3} m^{2} + 14832 \, A c^{3} m + 5760 \, A c^{3}\right )} x^{7} + 3 \,{\left (B a c^{2} m^{7} + 30 \, B a c^{2} m^{6} + 366 \, B a c^{2} m^{5} + 2340 \, B a c^{2} m^{4} + 8409 \, B a c^{2} m^{3} + 16830 \, B a c^{2} m^{2} + 17144 \, B a c^{2} m + 6720 \, B a c^{2}\right )} x^{6} + 3 \,{\left (A a c^{2} m^{7} + 31 \, A a c^{2} m^{6} + 391 \, A a c^{2} m^{5} + 2581 \, A a c^{2} m^{4} + 9544 \, A a c^{2} m^{3} + 19564 \, A a c^{2} m^{2} + 20304 \, A a c^{2} m + 8064 \, A a c^{2}\right )} x^{5} + 3 \,{\left (B a^{2} c m^{7} + 32 \, B a^{2} c m^{6} + 418 \, B a^{2} c m^{5} + 2864 \, B a^{2} c m^{4} + 10993 \, B a^{2} c m^{3} + 23312 \, B a^{2} c m^{2} + 24876 \, B a^{2} c m + 10080 \, B a^{2} c\right )} x^{4} + 3 \,{\left (A a^{2} c m^{7} + 33 \, A a^{2} c m^{6} + 447 \, A a^{2} c m^{5} + 3195 \, A a^{2} c m^{4} + 12864 \, A a^{2} c m^{3} + 28692 \, A a^{2} c m^{2} + 32048 \, A a^{2} c m + 13440 \, A a^{2} c\right )} x^{3} +{\left (B a^{3} m^{7} + 34 \, B a^{3} m^{6} + 478 \, B a^{3} m^{5} + 3580 \, B a^{3} m^{4} + 15289 \, B a^{3} m^{3} + 36706 \, B a^{3} m^{2} + 44712 \, B a^{3} m + 20160 \, B a^{3}\right )} x^{2} +{\left (A a^{3} m^{7} + 35 \, A a^{3} m^{6} + 511 \, A a^{3} m^{5} + 4025 \, A a^{3} m^{4} + 18424 \, A a^{3} m^{3} + 48860 \, A a^{3} m^{2} + 69264 \, A a^{3} m + 40320 \, A a^{3}\right )} x\right )} \left (e x\right )^{m}}{m^{8} + 36 \, m^{7} + 546 \, m^{6} + 4536 \, m^{5} + 22449 \, m^{4} + 67284 \, m^{3} + 118124 \, m^{2} + 109584 \, m + 40320} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*(B*x+A)*(c*x^2+a)^3,x, algorithm="fricas")

[Out]

((B*c^3*m^7 + 28*B*c^3*m^6 + 322*B*c^3*m^5 + 1960*B*c^3*m^4 + 6769*B*c^3*m^3 + 13132*B*c^3*m^2 + 13068*B*c^3*m
 + 5040*B*c^3)*x^8 + (A*c^3*m^7 + 29*A*c^3*m^6 + 343*A*c^3*m^5 + 2135*A*c^3*m^4 + 7504*A*c^3*m^3 + 14756*A*c^3
*m^2 + 14832*A*c^3*m + 5760*A*c^3)*x^7 + 3*(B*a*c^2*m^7 + 30*B*a*c^2*m^6 + 366*B*a*c^2*m^5 + 2340*B*a*c^2*m^4
+ 8409*B*a*c^2*m^3 + 16830*B*a*c^2*m^2 + 17144*B*a*c^2*m + 6720*B*a*c^2)*x^6 + 3*(A*a*c^2*m^7 + 31*A*a*c^2*m^6
 + 391*A*a*c^2*m^5 + 2581*A*a*c^2*m^4 + 9544*A*a*c^2*m^3 + 19564*A*a*c^2*m^2 + 20304*A*a*c^2*m + 8064*A*a*c^2)
*x^5 + 3*(B*a^2*c*m^7 + 32*B*a^2*c*m^6 + 418*B*a^2*c*m^5 + 2864*B*a^2*c*m^4 + 10993*B*a^2*c*m^3 + 23312*B*a^2*
c*m^2 + 24876*B*a^2*c*m + 10080*B*a^2*c)*x^4 + 3*(A*a^2*c*m^7 + 33*A*a^2*c*m^6 + 447*A*a^2*c*m^5 + 3195*A*a^2*
c*m^4 + 12864*A*a^2*c*m^3 + 28692*A*a^2*c*m^2 + 32048*A*a^2*c*m + 13440*A*a^2*c)*x^3 + (B*a^3*m^7 + 34*B*a^3*m
^6 + 478*B*a^3*m^5 + 3580*B*a^3*m^4 + 15289*B*a^3*m^3 + 36706*B*a^3*m^2 + 44712*B*a^3*m + 20160*B*a^3)*x^2 + (
A*a^3*m^7 + 35*A*a^3*m^6 + 511*A*a^3*m^5 + 4025*A*a^3*m^4 + 18424*A*a^3*m^3 + 48860*A*a^3*m^2 + 69264*A*a^3*m
+ 40320*A*a^3)*x)*(e*x)^m/(m^8 + 36*m^7 + 546*m^6 + 4536*m^5 + 22449*m^4 + 67284*m^3 + 118124*m^2 + 109584*m +
 40320)

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Sympy [A]  time = 3.26218, size = 4507, normalized size = 26.67 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**m*(B*x+A)*(c*x**2+a)**3,x)

[Out]

Piecewise(((-A*a**3/(7*x**7) - 3*A*a**2*c/(5*x**5) - A*a*c**2/x**3 - A*c**3/x - B*a**3/(6*x**6) - 3*B*a**2*c/(
4*x**4) - 3*B*a*c**2/(2*x**2) + B*c**3*log(x))/e**8, Eq(m, -8)), ((-A*a**3/(6*x**6) - 3*A*a**2*c/(4*x**4) - 3*
A*a*c**2/(2*x**2) + A*c**3*log(x) - B*a**3/(5*x**5) - B*a**2*c/x**3 - 3*B*a*c**2/x + B*c**3*x)/e**7, Eq(m, -7)
), ((-A*a**3/(5*x**5) - A*a**2*c/x**3 - 3*A*a*c**2/x + A*c**3*x - B*a**3/(4*x**4) - 3*B*a**2*c/(2*x**2) + 3*B*
a*c**2*log(x) + B*c**3*x**2/2)/e**6, Eq(m, -6)), ((-A*a**3/(4*x**4) - 3*A*a**2*c/(2*x**2) + 3*A*a*c**2*log(x)
+ A*c**3*x**2/2 - B*a**3/(3*x**3) - 3*B*a**2*c/x + 3*B*a*c**2*x + B*c**3*x**3/3)/e**5, Eq(m, -5)), ((-A*a**3/(
3*x**3) - 3*A*a**2*c/x + 3*A*a*c**2*x + A*c**3*x**3/3 - B*a**3/(2*x**2) + 3*B*a**2*c*log(x) + 3*B*a*c**2*x**2/
2 + B*c**3*x**4/4)/e**4, Eq(m, -4)), ((-A*a**3/(2*x**2) + 3*A*a**2*c*log(x) + 3*A*a*c**2*x**2/2 + A*c**3*x**4/
4 - B*a**3/x + 3*B*a**2*c*x + B*a*c**2*x**3 + B*c**3*x**5/5)/e**3, Eq(m, -3)), ((-A*a**3/x + 3*A*a**2*c*x + A*
a*c**2*x**3 + A*c**3*x**5/5 + B*a**3*log(x) + 3*B*a**2*c*x**2/2 + 3*B*a*c**2*x**4/4 + B*c**3*x**6/6)/e**2, Eq(
m, -2)), ((A*a**3*log(x) + 3*A*a**2*c*x**2/2 + 3*A*a*c**2*x**4/4 + A*c**3*x**6/6 + B*a**3*x + B*a**2*c*x**3 +
3*B*a*c**2*x**5/5 + B*c**3*x**7/7)/e, Eq(m, -1)), (A*a**3*e**m*m**7*x*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m
**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 35*A*a**3*e**m*m**6*x*x**m/(m**8 + 36*m**7 +
 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 511*A*a**3*e**m*m**5*x*x**
m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 4025*A*
a**3*e**m*m**4*x*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*
m + 40320) + 18424*A*a**3*e**m*m**3*x*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 +
118124*m**2 + 109584*m + 40320) + 48860*A*a**3*e**m*m**2*x*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449
*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 69264*A*a**3*e**m*m*x*x**m/(m**8 + 36*m**7 + 546*m**6 +
 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 40320*A*a**3*e**m*x*x**m/(m**8 + 36*m
**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 3*A*a**2*c*e**m*m**7*
x**3*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) +
 99*A*a**2*c*e**m*m**6*x**3*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**
2 + 109584*m + 40320) + 1341*A*a**2*c*e**m*m**5*x**3*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4
+ 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 9585*A*a**2*c*e**m*m**4*x**3*x**m/(m**8 + 36*m**7 + 546*m**6
+ 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 38592*A*a**2*c*e**m*m**3*x**3*x**m/(
m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 86076*A*a*
*2*c*e**m*m**2*x**3*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 1095
84*m + 40320) + 96144*A*a**2*c*e**m*m*x**3*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m*
*3 + 118124*m**2 + 109584*m + 40320) + 40320*A*a**2*c*e**m*x**3*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 +
22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 3*A*a*c**2*e**m*m**7*x**5*x**m/(m**8 + 36*m**7 + 5
46*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 93*A*a*c**2*e**m*m**6*x**5*x
**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 1173*
A*a*c**2*e**m*m**5*x**5*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 +
109584*m + 40320) + 7743*A*a*c**2*e**m*m**4*x**5*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67
284*m**3 + 118124*m**2 + 109584*m + 40320) + 28632*A*a*c**2*e**m*m**3*x**5*x**m/(m**8 + 36*m**7 + 546*m**6 + 4
536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 58692*A*a*c**2*e**m*m**2*x**5*x**m/(m**
8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 60912*A*a*c**
2*e**m*m*x**5*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m +
 40320) + 24192*A*a*c**2*e**m*x**5*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118
124*m**2 + 109584*m + 40320) + A*c**3*e**m*m**7*x**7*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4
+ 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 29*A*c**3*e**m*m**6*x**7*x**m/(m**8 + 36*m**7 + 546*m**6 + 45
36*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 343*A*c**3*e**m*m**5*x**7*x**m/(m**8 + 3
6*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 2135*A*c**3*e**m*m
**4*x**7*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 4032
0) + 7504*A*c**3*e**m*m**3*x**7*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124
*m**2 + 109584*m + 40320) + 14756*A*c**3*e**m*m**2*x**7*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m*
*4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 14832*A*c**3*e**m*m*x**7*x**m/(m**8 + 36*m**7 + 546*m**6 +
 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 5760*A*c**3*e**m*x**7*x**m/(m**8 + 36
*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + B*a**3*e**m*m**7*x*
*2*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 3
4*B*a**3*e**m*m**6*x**2*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 +
109584*m + 40320) + 478*B*a**3*e**m*m**5*x**2*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284
*m**3 + 118124*m**2 + 109584*m + 40320) + 3580*B*a**3*e**m*m**4*x**2*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m*
*5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 15289*B*a**3*e**m*m**3*x**2*x**m/(m**8 + 36*m
**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 36706*B*a**3*e**m*m**
2*x**2*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320)
 + 44712*B*a**3*e**m*m*x**2*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**
2 + 109584*m + 40320) + 20160*B*a**3*e**m*x**2*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 6728
4*m**3 + 118124*m**2 + 109584*m + 40320) + 3*B*a**2*c*e**m*m**7*x**4*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m*
*5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 96*B*a**2*c*e**m*m**6*x**4*x**m/(m**8 + 36*m*
*7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 1254*B*a**2*c*e**m*m**
5*x**4*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320)
 + 8592*B*a**2*c*e**m*m**4*x**4*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124
*m**2 + 109584*m + 40320) + 32979*B*a**2*c*e**m*m**3*x**4*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*
m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 69936*B*a**2*c*e**m*m**2*x**4*x**m/(m**8 + 36*m**7 + 546
*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 74628*B*a**2*c*e**m*m*x**4*x**
m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 30240*B
*a**2*c*e**m*x**4*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584
*m + 40320) + 3*B*a*c**2*e**m*m**7*x**6*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3
+ 118124*m**2 + 109584*m + 40320) + 90*B*a*c**2*e**m*m**6*x**6*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 2
2449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 1098*B*a*c**2*e**m*m**5*x**6*x**m/(m**8 + 36*m**7 +
 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 7020*B*a*c**2*e**m*m**4*x*
*6*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 2
5227*B*a*c**2*e**m*m**3*x**6*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m*
*2 + 109584*m + 40320) + 50490*B*a*c**2*e**m*m**2*x**6*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**
4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 51432*B*a*c**2*e**m*m*x**6*x**m/(m**8 + 36*m**7 + 546*m**6
+ 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 20160*B*a*c**2*e**m*x**6*x**m/(m**8
+ 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + B*c**3*e**m*m**
7*x**8*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320)
 + 28*B*c**3*e**m*m**6*x**8*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**
2 + 109584*m + 40320) + 322*B*c**3*e**m*m**5*x**8*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 6
7284*m**3 + 118124*m**2 + 109584*m + 40320) + 1960*B*c**3*e**m*m**4*x**8*x**m/(m**8 + 36*m**7 + 546*m**6 + 453
6*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 6769*B*c**3*e**m*m**3*x**8*x**m/(m**8 + 3
6*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 13132*B*c**3*e**m*
m**2*x**8*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 403
20) + 13068*B*c**3*e**m*m*x**8*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*
m**2 + 109584*m + 40320) + 5040*B*c**3*e**m*x**8*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67
284*m**3 + 118124*m**2 + 109584*m + 40320), True))

________________________________________________________________________________________

Giac [B]  time = 1.32558, size = 1488, normalized size = 8.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*(B*x+A)*(c*x^2+a)^3,x, algorithm="giac")

[Out]

(B*c^3*m^7*x^8*x^m*e^m + A*c^3*m^7*x^7*x^m*e^m + 28*B*c^3*m^6*x^8*x^m*e^m + 3*B*a*c^2*m^7*x^6*x^m*e^m + 29*A*c
^3*m^6*x^7*x^m*e^m + 322*B*c^3*m^5*x^8*x^m*e^m + 3*A*a*c^2*m^7*x^5*x^m*e^m + 90*B*a*c^2*m^6*x^6*x^m*e^m + 343*
A*c^3*m^5*x^7*x^m*e^m + 1960*B*c^3*m^4*x^8*x^m*e^m + 3*B*a^2*c*m^7*x^4*x^m*e^m + 93*A*a*c^2*m^6*x^5*x^m*e^m +
1098*B*a*c^2*m^5*x^6*x^m*e^m + 2135*A*c^3*m^4*x^7*x^m*e^m + 6769*B*c^3*m^3*x^8*x^m*e^m + 3*A*a^2*c*m^7*x^3*x^m
*e^m + 96*B*a^2*c*m^6*x^4*x^m*e^m + 1173*A*a*c^2*m^5*x^5*x^m*e^m + 7020*B*a*c^2*m^4*x^6*x^m*e^m + 7504*A*c^3*m
^3*x^7*x^m*e^m + 13132*B*c^3*m^2*x^8*x^m*e^m + B*a^3*m^7*x^2*x^m*e^m + 99*A*a^2*c*m^6*x^3*x^m*e^m + 1254*B*a^2
*c*m^5*x^4*x^m*e^m + 7743*A*a*c^2*m^4*x^5*x^m*e^m + 25227*B*a*c^2*m^3*x^6*x^m*e^m + 14756*A*c^3*m^2*x^7*x^m*e^
m + 13068*B*c^3*m*x^8*x^m*e^m + A*a^3*m^7*x*x^m*e^m + 34*B*a^3*m^6*x^2*x^m*e^m + 1341*A*a^2*c*m^5*x^3*x^m*e^m
+ 8592*B*a^2*c*m^4*x^4*x^m*e^m + 28632*A*a*c^2*m^3*x^5*x^m*e^m + 50490*B*a*c^2*m^2*x^6*x^m*e^m + 14832*A*c^3*m
*x^7*x^m*e^m + 5040*B*c^3*x^8*x^m*e^m + 35*A*a^3*m^6*x*x^m*e^m + 478*B*a^3*m^5*x^2*x^m*e^m + 9585*A*a^2*c*m^4*
x^3*x^m*e^m + 32979*B*a^2*c*m^3*x^4*x^m*e^m + 58692*A*a*c^2*m^2*x^5*x^m*e^m + 51432*B*a*c^2*m*x^6*x^m*e^m + 57
60*A*c^3*x^7*x^m*e^m + 511*A*a^3*m^5*x*x^m*e^m + 3580*B*a^3*m^4*x^2*x^m*e^m + 38592*A*a^2*c*m^3*x^3*x^m*e^m +
69936*B*a^2*c*m^2*x^4*x^m*e^m + 60912*A*a*c^2*m*x^5*x^m*e^m + 20160*B*a*c^2*x^6*x^m*e^m + 4025*A*a^3*m^4*x*x^m
*e^m + 15289*B*a^3*m^3*x^2*x^m*e^m + 86076*A*a^2*c*m^2*x^3*x^m*e^m + 74628*B*a^2*c*m*x^4*x^m*e^m + 24192*A*a*c
^2*x^5*x^m*e^m + 18424*A*a^3*m^3*x*x^m*e^m + 36706*B*a^3*m^2*x^2*x^m*e^m + 96144*A*a^2*c*m*x^3*x^m*e^m + 30240
*B*a^2*c*x^4*x^m*e^m + 48860*A*a^3*m^2*x*x^m*e^m + 44712*B*a^3*m*x^2*x^m*e^m + 40320*A*a^2*c*x^3*x^m*e^m + 692
64*A*a^3*m*x*x^m*e^m + 20160*B*a^3*x^2*x^m*e^m + 40320*A*a^3*x*x^m*e^m)/(m^8 + 36*m^7 + 546*m^6 + 4536*m^5 + 2
2449*m^4 + 67284*m^3 + 118124*m^2 + 109584*m + 40320)

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