3.62 \(\int (e x)^m (a+b x)^4 (a d-b d x)^3 \, dx\)

Optimal. Leaf size=197 \[ -\frac{3 a^5 b^2 d^3 (e x)^{m+3}}{e^3 (m+3)}-\frac{3 a^4 b^3 d^3 (e x)^{m+4}}{e^4 (m+4)}+\frac{3 a^3 b^4 d^3 (e x)^{m+5}}{e^5 (m+5)}+\frac{3 a^2 b^5 d^3 (e x)^{m+6}}{e^6 (m+6)}+\frac{a^6 b d^3 (e x)^{m+2}}{e^2 (m+2)}+\frac{a^7 d^3 (e x)^{m+1}}{e (m+1)}-\frac{a b^6 d^3 (e x)^{m+7}}{e^7 (m+7)}-\frac{b^7 d^3 (e x)^{m+8}}{e^8 (m+8)} \]

[Out]

(a^7*d^3*(e*x)^(1 + m))/(e*(1 + m)) + (a^6*b*d^3*(e*x)^(2 + m))/(e^2*(2 + m)) - (3*a^5*b^2*d^3*(e*x)^(3 + m))/
(e^3*(3 + m)) - (3*a^4*b^3*d^3*(e*x)^(4 + m))/(e^4*(4 + m)) + (3*a^3*b^4*d^3*(e*x)^(5 + m))/(e^5*(5 + m)) + (3
*a^2*b^5*d^3*(e*x)^(6 + m))/(e^6*(6 + m)) - (a*b^6*d^3*(e*x)^(7 + m))/(e^7*(7 + m)) - (b^7*d^3*(e*x)^(8 + m))/
(e^8*(8 + m))

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Rubi [A]  time = 0.136337, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {88} \[ -\frac{3 a^5 b^2 d^3 (e x)^{m+3}}{e^3 (m+3)}-\frac{3 a^4 b^3 d^3 (e x)^{m+4}}{e^4 (m+4)}+\frac{3 a^3 b^4 d^3 (e x)^{m+5}}{e^5 (m+5)}+\frac{3 a^2 b^5 d^3 (e x)^{m+6}}{e^6 (m+6)}+\frac{a^6 b d^3 (e x)^{m+2}}{e^2 (m+2)}+\frac{a^7 d^3 (e x)^{m+1}}{e (m+1)}-\frac{a b^6 d^3 (e x)^{m+7}}{e^7 (m+7)}-\frac{b^7 d^3 (e x)^{m+8}}{e^8 (m+8)} \]

Antiderivative was successfully verified.

[In]

Int[(e*x)^m*(a + b*x)^4*(a*d - b*d*x)^3,x]

[Out]

(a^7*d^3*(e*x)^(1 + m))/(e*(1 + m)) + (a^6*b*d^3*(e*x)^(2 + m))/(e^2*(2 + m)) - (3*a^5*b^2*d^3*(e*x)^(3 + m))/
(e^3*(3 + m)) - (3*a^4*b^3*d^3*(e*x)^(4 + m))/(e^4*(4 + m)) + (3*a^3*b^4*d^3*(e*x)^(5 + m))/(e^5*(5 + m)) + (3
*a^2*b^5*d^3*(e*x)^(6 + m))/(e^6*(6 + m)) - (a*b^6*d^3*(e*x)^(7 + m))/(e^7*(7 + m)) - (b^7*d^3*(e*x)^(8 + m))/
(e^8*(8 + m))

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

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

Mathematica [A]  time = 0.0768731, size = 122, normalized size = 0.62 \[ d^3 x (e x)^m \left (-\frac{3 a^5 b^2 x^2}{m+3}-\frac{3 a^4 b^3 x^3}{m+4}+\frac{3 a^3 b^4 x^4}{m+5}+\frac{3 a^2 b^5 x^5}{m+6}+\frac{a^6 b x}{m+2}+\frac{a^7}{m+1}-\frac{a b^6 x^6}{m+7}-\frac{b^7 x^7}{m+8}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(e*x)^m*(a + b*x)^4*(a*d - b*d*x)^3,x]

[Out]

d^3*x*(e*x)^m*(a^7/(1 + m) + (a^6*b*x)/(2 + m) - (3*a^5*b^2*x^2)/(3 + m) - (3*a^4*b^3*x^3)/(4 + m) + (3*a^3*b^
4*x^4)/(5 + m) + (3*a^2*b^5*x^5)/(6 + m) - (a*b^6*x^6)/(7 + m) - (b^7*x^7)/(8 + m))

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Maple [B]  time = 0.008, size = 786, normalized size = 4. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^m*(b*x+a)^4*(-b*d*x+a*d)^3,x)

[Out]

d^3*(e*x)^m*(-b^7*m^7*x^7-a*b^6*m^7*x^6-28*b^7*m^6*x^7+3*a^2*b^5*m^7*x^5-29*a*b^6*m^6*x^6-322*b^7*m^5*x^7+3*a^
3*b^4*m^7*x^4+90*a^2*b^5*m^6*x^5-343*a*b^6*m^5*x^6-1960*b^7*m^4*x^7-3*a^4*b^3*m^7*x^3+93*a^3*b^4*m^6*x^4+1098*
a^2*b^5*m^5*x^5-2135*a*b^6*m^4*x^6-6769*b^7*m^3*x^7-3*a^5*b^2*m^7*x^2-96*a^4*b^3*m^6*x^3+1173*a^3*b^4*m^5*x^4+
7020*a^2*b^5*m^4*x^5-7504*a*b^6*m^3*x^6-13132*b^7*m^2*x^7+a^6*b*m^7*x-99*a^5*b^2*m^6*x^2-1254*a^4*b^3*m^5*x^3+
7743*a^3*b^4*m^4*x^4+25227*a^2*b^5*m^3*x^5-14756*a*b^6*m^2*x^6-13068*b^7*m*x^7+a^7*m^7+34*a^6*b*m^6*x-1341*a^5
*b^2*m^5*x^2-8592*a^4*b^3*m^4*x^3+28632*a^3*b^4*m^3*x^4+50490*a^2*b^5*m^2*x^5-14832*a*b^6*m*x^6-5040*b^7*x^7+3
5*a^7*m^6+478*a^6*b*m^5*x-9585*a^5*b^2*m^4*x^2-32979*a^4*b^3*m^3*x^3+58692*a^3*b^4*m^2*x^4+51432*a^2*b^5*m*x^5
-5760*a*b^6*x^6+511*a^7*m^5+3580*a^6*b*m^4*x-38592*a^5*b^2*m^3*x^2-69936*a^4*b^3*m^2*x^3+60912*a^3*b^4*m*x^4+2
0160*a^2*b^5*x^5+4025*a^7*m^4+15289*a^6*b*m^3*x-86076*a^5*b^2*m^2*x^2-74628*a^4*b^3*m*x^3+24192*a^3*b^4*x^4+18
424*a^7*m^3+36706*a^6*b*m^2*x-96144*a^5*b^2*m*x^2-30240*a^4*b^3*x^3+48860*a^7*m^2+44712*a^6*b*m*x-40320*a^5*b^
2*x^2+69264*a^7*m+20160*a^6*b*x+40320*a^7)*x/(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)^4*(-b*d*x+a*d)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.12697, size = 1949, normalized size = 9.89 \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)^4*(-b*d*x+a*d)^3,x, algorithm="fricas")

[Out]

-((b^7*d^3*m^7 + 28*b^7*d^3*m^6 + 322*b^7*d^3*m^5 + 1960*b^7*d^3*m^4 + 6769*b^7*d^3*m^3 + 13132*b^7*d^3*m^2 +
13068*b^7*d^3*m + 5040*b^7*d^3)*x^8 + (a*b^6*d^3*m^7 + 29*a*b^6*d^3*m^6 + 343*a*b^6*d^3*m^5 + 2135*a*b^6*d^3*m
^4 + 7504*a*b^6*d^3*m^3 + 14756*a*b^6*d^3*m^2 + 14832*a*b^6*d^3*m + 5760*a*b^6*d^3)*x^7 - 3*(a^2*b^5*d^3*m^7 +
30*a^2*b^5*d^3*m^6 + 366*a^2*b^5*d^3*m^5 + 2340*a^2*b^5*d^3*m^4 + 8409*a^2*b^5*d^3*m^3 + 16830*a^2*b^5*d^3*m^
2 + 17144*a^2*b^5*d^3*m + 6720*a^2*b^5*d^3)*x^6 - 3*(a^3*b^4*d^3*m^7 + 31*a^3*b^4*d^3*m^6 + 391*a^3*b^4*d^3*m^
5 + 2581*a^3*b^4*d^3*m^4 + 9544*a^3*b^4*d^3*m^3 + 19564*a^3*b^4*d^3*m^2 + 20304*a^3*b^4*d^3*m + 8064*a^3*b^4*d
^3)*x^5 + 3*(a^4*b^3*d^3*m^7 + 32*a^4*b^3*d^3*m^6 + 418*a^4*b^3*d^3*m^5 + 2864*a^4*b^3*d^3*m^4 + 10993*a^4*b^3
*d^3*m^3 + 23312*a^4*b^3*d^3*m^2 + 24876*a^4*b^3*d^3*m + 10080*a^4*b^3*d^3)*x^4 + 3*(a^5*b^2*d^3*m^7 + 33*a^5*
b^2*d^3*m^6 + 447*a^5*b^2*d^3*m^5 + 3195*a^5*b^2*d^3*m^4 + 12864*a^5*b^2*d^3*m^3 + 28692*a^5*b^2*d^3*m^2 + 320
48*a^5*b^2*d^3*m + 13440*a^5*b^2*d^3)*x^3 - (a^6*b*d^3*m^7 + 34*a^6*b*d^3*m^6 + 478*a^6*b*d^3*m^5 + 3580*a^6*b
*d^3*m^4 + 15289*a^6*b*d^3*m^3 + 36706*a^6*b*d^3*m^2 + 44712*a^6*b*d^3*m + 20160*a^6*b*d^3)*x^2 - (a^7*d^3*m^7
+ 35*a^7*d^3*m^6 + 511*a^7*d^3*m^5 + 4025*a^7*d^3*m^4 + 18424*a^7*d^3*m^3 + 48860*a^7*d^3*m^2 + 69264*a^7*d^3
*m + 40320*a^7*d^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 + 10958
4*m + 40320)

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Sympy [A]  time = 4.23502, size = 4888, normalized size = 24.81 \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)**4*(-b*d*x+a*d)**3,x)

[Out]

Piecewise(((-a**7*d**3/(7*x**7) - a**6*b*d**3/(6*x**6) + 3*a**5*b**2*d**3/(5*x**5) + 3*a**4*b**3*d**3/(4*x**4)
- a**3*b**4*d**3/x**3 - 3*a**2*b**5*d**3/(2*x**2) + a*b**6*d**3/x - b**7*d**3*log(x))/e**8, Eq(m, -8)), ((-a*
*7*d**3/(6*x**6) - a**6*b*d**3/(5*x**5) + 3*a**5*b**2*d**3/(4*x**4) + a**4*b**3*d**3/x**3 - 3*a**3*b**4*d**3/(
2*x**2) - 3*a**2*b**5*d**3/x - a*b**6*d**3*log(x) - b**7*d**3*x)/e**7, Eq(m, -7)), ((-a**7*d**3/(5*x**5) - a**
6*b*d**3/(4*x**4) + a**5*b**2*d**3/x**3 + 3*a**4*b**3*d**3/(2*x**2) - 3*a**3*b**4*d**3/x + 3*a**2*b**5*d**3*lo
g(x) - a*b**6*d**3*x - b**7*d**3*x**2/2)/e**6, Eq(m, -6)), ((-a**7*d**3/(4*x**4) - a**6*b*d**3/(3*x**3) + 3*a*
*5*b**2*d**3/(2*x**2) + 3*a**4*b**3*d**3/x + 3*a**3*b**4*d**3*log(x) + 3*a**2*b**5*d**3*x - a*b**6*d**3*x**2/2
- b**7*d**3*x**3/3)/e**5, Eq(m, -5)), ((-a**7*d**3/(3*x**3) - a**6*b*d**3/(2*x**2) + 3*a**5*b**2*d**3/x - 3*a
**4*b**3*d**3*log(x) + 3*a**3*b**4*d**3*x + 3*a**2*b**5*d**3*x**2/2 - a*b**6*d**3*x**3/3 - b**7*d**3*x**4/4)/e
**4, Eq(m, -4)), ((-a**7*d**3/(2*x**2) - a**6*b*d**3/x - 3*a**5*b**2*d**3*log(x) - 3*a**4*b**3*d**3*x + 3*a**3
*b**4*d**3*x**2/2 + a**2*b**5*d**3*x**3 - a*b**6*d**3*x**4/4 - b**7*d**3*x**5/5)/e**3, Eq(m, -3)), ((-a**7*d**
3/x + a**6*b*d**3*log(x) - 3*a**5*b**2*d**3*x - 3*a**4*b**3*d**3*x**2/2 + a**3*b**4*d**3*x**3 + 3*a**2*b**5*d*
*3*x**4/4 - a*b**6*d**3*x**5/5 - b**7*d**3*x**6/6)/e**2, Eq(m, -2)), ((a**7*d**3*log(x) + a**6*b*d**3*x - 3*a*
*5*b**2*d**3*x**2/2 - a**4*b**3*d**3*x**3 + 3*a**3*b**4*d**3*x**4/4 + 3*a**2*b**5*d**3*x**5/5 - a*b**6*d**3*x*
*6/6 - b**7*d**3*x**7/7)/e, Eq(m, -1)), (a**7*d**3*e**m*m**7*x*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) + 35*a**7*d**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**7*d**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**
7*d**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 + 10958
4*m + 40320) + 18424*a**7*d**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**7*d**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**7*d**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**7*d**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) + a**6*b*
d**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 + 1095
84*m + 40320) + 34*a**6*b*d**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*a**6*b*d**3*e**m*m**5*x**2*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) + 3580*a**6*b*d**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*a**6*b*
d**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 + 1095
84*m + 40320) + 36706*a**6*b*d**3*e**m*m**2*x**2*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) + 44712*a**6*b*d**3*e**m*m*x**2*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) + 20160*a**6*b*d**3*e**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) - 3*a**5*b**2*d**
3*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**5*b**2*d**3*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**5*b**2*d**3*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**5*b**2*d**3*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**5*b**2*d**3*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**5*b**2*d**3*e**m*m**2*x**3*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22
449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) - 96144*a**5*b**2*d**3*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**5*b**2*d**3*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**4*b**3*d**3*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 + 11
8124*m**2 + 109584*m + 40320) - 96*a**4*b**3*d**3*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*a**4*b**3*d**3*e**m*m**5*x**4*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) - 8592*a**4*b**3*d**
3*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*a**4*b**3*d**3*e**m*m**3*x**4*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) - 69936*a**4*b**3*d**3*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*a**4*b**3*d**3*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) - 3024
0*a**4*b**3*d**3*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*a**3*b**4*d**3*e**m*m**7*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) + 93*a**3*b**4*d**3*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**3*b**4*d**3*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**3*b**4*d**3*e**m*m**4*x**5*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 11
8124*m**2 + 109584*m + 40320) + 28632*a**3*b**4*d**3*e**m*m**3*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) + 58692*a**3*b**4*d**3*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**3*b**
4*d**3*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 + 10958
4*m + 40320) + 24192*a**3*b**4*d**3*e**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) + 3*a**2*b**5*d**3*e**m*m**7*x**6*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) + 90*a**2*b**5*d**3*e**m*m**6*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) + 1098*a**2*b
**5*d**3*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*a**2*b**5*d**3*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) + 25227*a**2*b**5*d**3*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*a**2*b**5*d**3*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 + 4032
0) + 51432*a**2*b**5*d**3*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*a**2*b**5*d**3*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) - a*b**6*d**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*b**6*d**3*e**m*m**6*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) -
343*a*b**6*d**3*e**m*m**5*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) - 2135*a*b**6*d**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 + 40320) - 7504*a*b**6*d**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*b**6*d**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*b**6*d**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*b**6*d**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**7*d**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**7*d**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**7*d**3*
e**m*m**5*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) - 1960*b**7*d**3*e**m*m**4*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) - 6769*b**7*d**3*e**m*m**3*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) - 13132*b**7*d**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 + 40320) - 13068*b**7*d**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**7*d**3*e**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), True))

________________________________________________________________________________________

Giac [B]  time = 1.26857, size = 1773, normalized size = 9. \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)^4*(-b*d*x+a*d)^3,x, algorithm="giac")

[Out]

-(b^7*d^3*m^7*x^8*x^m*e^m + a*b^6*d^3*m^7*x^7*x^m*e^m + 28*b^7*d^3*m^6*x^8*x^m*e^m - 3*a^2*b^5*d^3*m^7*x^6*x^m
*e^m + 29*a*b^6*d^3*m^6*x^7*x^m*e^m + 322*b^7*d^3*m^5*x^8*x^m*e^m - 3*a^3*b^4*d^3*m^7*x^5*x^m*e^m - 90*a^2*b^5
*d^3*m^6*x^6*x^m*e^m + 343*a*b^6*d^3*m^5*x^7*x^m*e^m + 1960*b^7*d^3*m^4*x^8*x^m*e^m + 3*a^4*b^3*d^3*m^7*x^4*x^
m*e^m - 93*a^3*b^4*d^3*m^6*x^5*x^m*e^m - 1098*a^2*b^5*d^3*m^5*x^6*x^m*e^m + 2135*a*b^6*d^3*m^4*x^7*x^m*e^m + 6
769*b^7*d^3*m^3*x^8*x^m*e^m + 3*a^5*b^2*d^3*m^7*x^3*x^m*e^m + 96*a^4*b^3*d^3*m^6*x^4*x^m*e^m - 1173*a^3*b^4*d^
3*m^5*x^5*x^m*e^m - 7020*a^2*b^5*d^3*m^4*x^6*x^m*e^m + 7504*a*b^6*d^3*m^3*x^7*x^m*e^m + 13132*b^7*d^3*m^2*x^8*
x^m*e^m - a^6*b*d^3*m^7*x^2*x^m*e^m + 99*a^5*b^2*d^3*m^6*x^3*x^m*e^m + 1254*a^4*b^3*d^3*m^5*x^4*x^m*e^m - 7743
*a^3*b^4*d^3*m^4*x^5*x^m*e^m - 25227*a^2*b^5*d^3*m^3*x^6*x^m*e^m + 14756*a*b^6*d^3*m^2*x^7*x^m*e^m + 13068*b^7
*d^3*m*x^8*x^m*e^m - a^7*d^3*m^7*x*x^m*e^m - 34*a^6*b*d^3*m^6*x^2*x^m*e^m + 1341*a^5*b^2*d^3*m^5*x^3*x^m*e^m +
8592*a^4*b^3*d^3*m^4*x^4*x^m*e^m - 28632*a^3*b^4*d^3*m^3*x^5*x^m*e^m - 50490*a^2*b^5*d^3*m^2*x^6*x^m*e^m + 14
832*a*b^6*d^3*m*x^7*x^m*e^m + 5040*b^7*d^3*x^8*x^m*e^m - 35*a^7*d^3*m^6*x*x^m*e^m - 478*a^6*b*d^3*m^5*x^2*x^m*
e^m + 9585*a^5*b^2*d^3*m^4*x^3*x^m*e^m + 32979*a^4*b^3*d^3*m^3*x^4*x^m*e^m - 58692*a^3*b^4*d^3*m^2*x^5*x^m*e^m
- 51432*a^2*b^5*d^3*m*x^6*x^m*e^m + 5760*a*b^6*d^3*x^7*x^m*e^m - 511*a^7*d^3*m^5*x*x^m*e^m - 3580*a^6*b*d^3*m
^4*x^2*x^m*e^m + 38592*a^5*b^2*d^3*m^3*x^3*x^m*e^m + 69936*a^4*b^3*d^3*m^2*x^4*x^m*e^m - 60912*a^3*b^4*d^3*m*x
^5*x^m*e^m - 20160*a^2*b^5*d^3*x^6*x^m*e^m - 4025*a^7*d^3*m^4*x*x^m*e^m - 15289*a^6*b*d^3*m^3*x^2*x^m*e^m + 86
076*a^5*b^2*d^3*m^2*x^3*x^m*e^m + 74628*a^4*b^3*d^3*m*x^4*x^m*e^m - 24192*a^3*b^4*d^3*x^5*x^m*e^m - 18424*a^7*
d^3*m^3*x*x^m*e^m - 36706*a^6*b*d^3*m^2*x^2*x^m*e^m + 96144*a^5*b^2*d^3*m*x^3*x^m*e^m + 30240*a^4*b^3*d^3*x^4*
x^m*e^m - 48860*a^7*d^3*m^2*x*x^m*e^m - 44712*a^6*b*d^3*m*x^2*x^m*e^m + 40320*a^5*b^2*d^3*x^3*x^m*e^m - 69264*
a^7*d^3*m*x*x^m*e^m - 20160*a^6*b*d^3*x^2*x^m*e^m - 40320*a^7*d^3*x*x^m*e^m)/(m^8 + 36*m^7 + 546*m^6 + 4536*m^
5 + 22449*m^4 + 67284*m^3 + 118124*m^2 + 109584*m + 40320)