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3.1489 \(\int (A+B x) (d+e x)^m (a+c x^2) \, dx\)

Optimal. Leaf size=126 \[ -\frac{\left (a e^2+c d^2\right ) (B d-A e) (d+e x)^{m+1}}{e^4 (m+1)}+\frac{(d+e x)^{m+2} \left (a B e^2-2 A c d e+3 B c d^2\right )}{e^4 (m+2)}-\frac{c (3 B d-A e) (d+e x)^{m+3}}{e^4 (m+3)}+\frac{B c (d+e x)^{m+4}}{e^4 (m+4)} \]

[Out]

-(((B*d - A*e)*(c*d^2 + a*e^2)*(d + e*x)^(1 + m))/(e^4*(1 + m))) + ((3*B*c*d^2 - 2*A*c*d*e + a*B*e^2)*(d + e*x
)^(2 + m))/(e^4*(2 + m)) - (c*(3*B*d - A*e)*(d + e*x)^(3 + m))/(e^4*(3 + m)) + (B*c*(d + e*x)^(4 + m))/(e^4*(4
 + m))

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Rubi [A]  time = 0.0691842, antiderivative size = 126, 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 = {772} \[ -\frac{\left (a e^2+c d^2\right ) (B d-A e) (d+e x)^{m+1}}{e^4 (m+1)}+\frac{(d+e x)^{m+2} \left (a B e^2-2 A c d e+3 B c d^2\right )}{e^4 (m+2)}-\frac{c (3 B d-A e) (d+e x)^{m+3}}{e^4 (m+3)}+\frac{B c (d+e x)^{m+4}}{e^4 (m+4)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((B*d - A*e)*(c*d^2 + a*e^2)*(d + e*x)^(1 + m))/(e^4*(1 + m))) + ((3*B*c*d^2 - 2*A*c*d*e + a*B*e^2)*(d + e*x
)^(2 + m))/(e^4*(2 + m)) - (c*(3*B*d - A*e)*(d + e*x)^(3 + m))/(e^4*(3 + m)) + (B*c*(d + e*x)^(4 + m))/(e^4*(4
 + m))

Rule 772

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

Rubi steps

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

Mathematica [A]  time = 0.206579, size = 122, normalized size = 0.97 \[ \frac{(d+e x)^{m+1} \left ((A e-B d) \left (\frac{a e^2+c d^2}{m+1}+\frac{c (d+e x)^2}{m+3}-\frac{2 c d (d+e x)}{m+2}\right )+B (d+e x) \left (\frac{a e^2+c d^2}{m+2}+\frac{c (d+e x)^2}{m+4}-\frac{2 c d (d+e x)}{m+3}\right )\right )}{e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((d + e*x)^(1 + m)*((-(B*d) + A*e)*((c*d^2 + a*e^2)/(1 + m) - (2*c*d*(d + e*x))/(2 + m) + (c*(d + e*x)^2)/(3 +
 m)) + B*(d + e*x)*((c*d^2 + a*e^2)/(2 + m) - (2*c*d*(d + e*x))/(3 + m) + (c*(d + e*x)^2)/(4 + m))))/e^4

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Maple [B]  time = 0.007, size = 338, normalized size = 2.7 \begin{align*}{\frac{ \left ( ex+d \right ) ^{1+m} \left ( Bc{e}^{3}{m}^{3}{x}^{3}+Ac{e}^{3}{m}^{3}{x}^{2}+6\,Bc{e}^{3}{m}^{2}{x}^{3}+7\,Ac{e}^{3}{m}^{2}{x}^{2}+Ba{e}^{3}{m}^{3}x-3\,Bcd{e}^{2}{m}^{2}{x}^{2}+11\,Bc{e}^{3}m{x}^{3}+Aa{e}^{3}{m}^{3}-2\,Acd{e}^{2}{m}^{2}x+14\,Ac{e}^{3}m{x}^{2}+8\,Ba{e}^{3}{m}^{2}x-9\,Bcd{e}^{2}m{x}^{2}+6\,Bc{x}^{3}{e}^{3}+9\,Aa{e}^{3}{m}^{2}-10\,Acd{e}^{2}mx+8\,Ac{e}^{3}{x}^{2}-Bad{e}^{2}{m}^{2}+19\,Ba{e}^{3}mx+6\,Bc{d}^{2}emx-6\,Bcd{e}^{2}{x}^{2}+26\,Aa{e}^{3}m+2\,Ac{d}^{2}em-8\,Acd{e}^{2}x-7\,Bad{e}^{2}m+12\,Ba{e}^{3}x+6\,Bc{d}^{2}ex+24\,aA{e}^{3}+8\,Ac{d}^{2}e-12\,aBd{e}^{2}-6\,Bc{d}^{3} \right ) }{{e}^{4} \left ({m}^{4}+10\,{m}^{3}+35\,{m}^{2}+50\,m+24 \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(e*x+d)^(1+m)*(B*c*e^3*m^3*x^3+A*c*e^3*m^3*x^2+6*B*c*e^3*m^2*x^3+7*A*c*e^3*m^2*x^2+B*a*e^3*m^3*x-3*B*c*d*e^2*m
^2*x^2+11*B*c*e^3*m*x^3+A*a*e^3*m^3-2*A*c*d*e^2*m^2*x+14*A*c*e^3*m*x^2+8*B*a*e^3*m^2*x-9*B*c*d*e^2*m*x^2+6*B*c
*e^3*x^3+9*A*a*e^3*m^2-10*A*c*d*e^2*m*x+8*A*c*e^3*x^2-B*a*d*e^2*m^2+19*B*a*e^3*m*x+6*B*c*d^2*e*m*x-6*B*c*d*e^2
*x^2+26*A*a*e^3*m+2*A*c*d^2*e*m-8*A*c*d*e^2*x-7*B*a*d*e^2*m+12*B*a*e^3*x+6*B*c*d^2*e*x+24*A*a*e^3+8*A*c*d^2*e-
12*B*a*d*e^2-6*B*c*d^3)/e^4/(m^4+10*m^3+35*m^2+50*m+24)

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.81854, size = 938, normalized size = 7.44 \begin{align*} \frac{{\left (A a d e^{3} m^{3} - 6 \, B c d^{4} + 8 \, A c d^{3} e - 12 \, B a d^{2} e^{2} + 24 \, A a d e^{3} +{\left (B c e^{4} m^{3} + 6 \, B c e^{4} m^{2} + 11 \, B c e^{4} m + 6 \, B c e^{4}\right )} x^{4} +{\left (8 \, A c e^{4} +{\left (B c d e^{3} + A c e^{4}\right )} m^{3} +{\left (3 \, B c d e^{3} + 7 \, A c e^{4}\right )} m^{2} + 2 \,{\left (B c d e^{3} + 7 \, A c e^{4}\right )} m\right )} x^{3} -{\left (B a d^{2} e^{2} - 9 \, A a d e^{3}\right )} m^{2} +{\left (12 \, B a e^{4} +{\left (A c d e^{3} + B a e^{4}\right )} m^{3} -{\left (3 \, B c d^{2} e^{2} - 5 \, A c d e^{3} - 8 \, B a e^{4}\right )} m^{2} -{\left (3 \, B c d^{2} e^{2} - 4 \, A c d e^{3} - 19 \, B a e^{4}\right )} m\right )} x^{2} +{\left (2 \, A c d^{3} e - 7 \, B a d^{2} e^{2} + 26 \, A a d e^{3}\right )} m +{\left (24 \, A a e^{4} +{\left (B a d e^{3} + A a e^{4}\right )} m^{3} -{\left (2 \, A c d^{2} e^{2} - 7 \, B a d e^{3} - 9 \, A a e^{4}\right )} m^{2} + 2 \,{\left (3 \, B c d^{3} e - 4 \, A c d^{2} e^{2} + 6 \, B a d e^{3} + 13 \, A a e^{4}\right )} m\right )} x\right )}{\left (e x + d\right )}^{m}}{e^{4} m^{4} + 10 \, e^{4} m^{3} + 35 \, e^{4} m^{2} + 50 \, e^{4} m + 24 \, e^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(A*a*d*e^3*m^3 - 6*B*c*d^4 + 8*A*c*d^3*e - 12*B*a*d^2*e^2 + 24*A*a*d*e^3 + (B*c*e^4*m^3 + 6*B*c*e^4*m^2 + 11*B
*c*e^4*m + 6*B*c*e^4)*x^4 + (8*A*c*e^4 + (B*c*d*e^3 + A*c*e^4)*m^3 + (3*B*c*d*e^3 + 7*A*c*e^4)*m^2 + 2*(B*c*d*
e^3 + 7*A*c*e^4)*m)*x^3 - (B*a*d^2*e^2 - 9*A*a*d*e^3)*m^2 + (12*B*a*e^4 + (A*c*d*e^3 + B*a*e^4)*m^3 - (3*B*c*d
^2*e^2 - 5*A*c*d*e^3 - 8*B*a*e^4)*m^2 - (3*B*c*d^2*e^2 - 4*A*c*d*e^3 - 19*B*a*e^4)*m)*x^2 + (2*A*c*d^3*e - 7*B
*a*d^2*e^2 + 26*A*a*d*e^3)*m + (24*A*a*e^4 + (B*a*d*e^3 + A*a*e^4)*m^3 - (2*A*c*d^2*e^2 - 7*B*a*d*e^3 - 9*A*a*
e^4)*m^2 + 2*(3*B*c*d^3*e - 4*A*c*d^2*e^2 + 6*B*a*d*e^3 + 13*A*a*e^4)*m)*x)*(e*x + d)^m/(e^4*m^4 + 10*e^4*m^3
+ 35*e^4*m^2 + 50*e^4*m + 24*e^4)

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Sympy [A]  time = 5.35685, size = 3958, normalized size = 31.41 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((d**m*(A*a*x + A*c*x**3/3 + B*a*x**2/2 + B*c*x**4/4), Eq(e, 0)), (-2*A*a*e**3/(6*d**3*e**4 + 18*d**2
*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - 2*A*c*d**2*e/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7
*x**3) - 6*A*c*d*e**2*x/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - 6*A*c*e**3*x**2/(6*d**
3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - B*a*d*e**2/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6
*x**2 + 6*e**7*x**3) - 3*B*a*e**3*x/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 6*B*c*d**3
*log(d/e + x)/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 11*B*c*d**3/(6*d**3*e**4 + 18*d*
*2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 18*B*c*d**2*e*x*log(d/e + x)/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*
e**6*x**2 + 6*e**7*x**3) + 27*B*c*d**2*e*x/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 18*
B*c*d*e**2*x**2*log(d/e + x)/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 18*B*c*d*e**2*x**
2/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 6*B*c*e**3*x**3*log(d/e + x)/(6*d**3*e**4 +
18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3), Eq(m, -4)), (-A*a*e**3/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2
) + 2*A*c*d**2*e*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 3*A*c*d**2*e/(2*d**2*e**4 + 4*d*e**5*
x + 2*e**6*x**2) + 4*A*c*d*e**2*x*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 4*A*c*d*e**2*x/(2*d*
*2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 2*A*c*e**3*x**2*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) -
B*a*d*e**2/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 2*B*a*e**3*x/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) -
6*B*c*d**3*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 9*B*c*d**3/(2*d**2*e**4 + 4*d*e**5*x + 2*e*
*6*x**2) - 12*B*c*d**2*e*x*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 12*B*c*d**2*e*x/(2*d**2*e**
4 + 4*d*e**5*x + 2*e**6*x**2) - 6*B*c*d*e**2*x**2*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 2*B*
c*e**3*x**3/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2), Eq(m, -3)), (-2*A*a*e**3/(2*d*e**4 + 2*e**5*x) - 4*A*c*d
**2*e*log(d/e + x)/(2*d*e**4 + 2*e**5*x) - 4*A*c*d**2*e/(2*d*e**4 + 2*e**5*x) - 4*A*c*d*e**2*x*log(d/e + x)/(2
*d*e**4 + 2*e**5*x) + 2*A*c*e**3*x**2/(2*d*e**4 + 2*e**5*x) + 2*B*a*d*e**2*log(d/e + x)/(2*d*e**4 + 2*e**5*x)
+ 2*B*a*d*e**2/(2*d*e**4 + 2*e**5*x) + 2*B*a*e**3*x*log(d/e + x)/(2*d*e**4 + 2*e**5*x) + 6*B*c*d**3*log(d/e +
x)/(2*d*e**4 + 2*e**5*x) + 6*B*c*d**3/(2*d*e**4 + 2*e**5*x) + 6*B*c*d**2*e*x*log(d/e + x)/(2*d*e**4 + 2*e**5*x
) - 3*B*c*d*e**2*x**2/(2*d*e**4 + 2*e**5*x) + B*c*e**3*x**3/(2*d*e**4 + 2*e**5*x), Eq(m, -2)), (A*a*log(d/e +
x)/e + A*c*d**2*log(d/e + x)/e**3 - A*c*d*x/e**2 + A*c*x**2/(2*e) - B*a*d*log(d/e + x)/e**2 + B*a*x/e - B*c*d*
*3*log(d/e + x)/e**4 + B*c*d**2*x/e**3 - B*c*d*x**2/(2*e**2) + B*c*x**3/(3*e), Eq(m, -1)), (A*a*d*e**3*m**3*(d
 + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 9*A*a*d*e**3*m**2*(d + e*x)**m/(e
**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 26*A*a*d*e**3*m*(d + e*x)**m/(e**4*m**4 + 10*e
**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 24*A*a*d*e**3*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**
4*m**2 + 50*e**4*m + 24*e**4) + A*a*e**4*m**3*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**
4*m + 24*e**4) + 9*A*a*e**4*m**2*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4
) + 26*A*a*e**4*m*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 24*A*a*e**4
*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 2*A*c*d**3*e*m*(d + e*x)**m/
(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 8*A*c*d**3*e*(d + e*x)**m/(e**4*m**4 + 10*e*
*4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) - 2*A*c*d**2*e**2*m**2*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3
+ 35*e**4*m**2 + 50*e**4*m + 24*e**4) - 8*A*c*d**2*e**2*m*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m
**2 + 50*e**4*m + 24*e**4) + A*c*d*e**3*m**3*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e
**4*m + 24*e**4) + 5*A*c*d*e**3*m**2*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m +
24*e**4) + 4*A*c*d*e**3*m*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) +
A*c*e**4*m**3*x**3*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 7*A*c*e**4*m
**2*x**3*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 14*A*c*e**4*m*x**3*(d
+ e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 8*A*c*e**4*x**3*(d + e*x)**m/(e**4
*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) - B*a*d**2*e**2*m**2*(d + e*x)**m/(e**4*m**4 + 10*e
**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) - 7*B*a*d**2*e**2*m*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35
*e**4*m**2 + 50*e**4*m + 24*e**4) - 12*B*a*d**2*e**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 5
0*e**4*m + 24*e**4) + B*a*d*e**3*m**3*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24
*e**4) + 7*B*a*d*e**3*m**2*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 12
*B*a*d*e**3*m*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + B*a*e**4*m**3*x
**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 8*B*a*e**4*m**2*x**2*(d + e
*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 19*B*a*e**4*m*x**2*(d + e*x)**m/(e**4
*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 12*B*a*e**4*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**
4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) - 6*B*c*d**4*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**
2 + 50*e**4*m + 24*e**4) + 6*B*c*d**3*e*m*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m
+ 24*e**4) - 3*B*c*d**2*e**2*m**2*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*
e**4) - 3*B*c*d**2*e**2*m*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) +
B*c*d*e**3*m**3*x**3*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 3*B*c*d*e*
*3*m**2*x**3*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 2*B*c*d*e**3*m*x**
3*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + B*c*e**4*m**3*x**4*(d + e*x)*
*m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 6*B*c*e**4*m**2*x**4*(d + e*x)**m/(e**4*m
**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 11*B*c*e**4*m*x**4*(d + e*x)**m/(e**4*m**4 + 10*e**
4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 6*B*c*e**4*x**4*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**
4*m**2 + 50*e**4*m + 24*e**4), True))

________________________________________________________________________________________

Giac [B]  time = 1.18566, size = 1040, normalized size = 8.25 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x*e + d)^m*B*c*m^3*x^4*e^4 + (x*e + d)^m*B*c*d*m^3*x^3*e^3 + (x*e + d)^m*A*c*m^3*x^3*e^4 + 6*(x*e + d)^m*B*c
*m^2*x^4*e^4 + (x*e + d)^m*A*c*d*m^3*x^2*e^3 + 3*(x*e + d)^m*B*c*d*m^2*x^3*e^3 - 3*(x*e + d)^m*B*c*d^2*m^2*x^2
*e^2 + (x*e + d)^m*B*a*m^3*x^2*e^4 + 7*(x*e + d)^m*A*c*m^2*x^3*e^4 + 11*(x*e + d)^m*B*c*m*x^4*e^4 + (x*e + d)^
m*B*a*d*m^3*x*e^3 + 5*(x*e + d)^m*A*c*d*m^2*x^2*e^3 + 2*(x*e + d)^m*B*c*d*m*x^3*e^3 - 2*(x*e + d)^m*A*c*d^2*m^
2*x*e^2 - 3*(x*e + d)^m*B*c*d^2*m*x^2*e^2 + 6*(x*e + d)^m*B*c*d^3*m*x*e + (x*e + d)^m*A*a*m^3*x*e^4 + 8*(x*e +
 d)^m*B*a*m^2*x^2*e^4 + 14*(x*e + d)^m*A*c*m*x^3*e^4 + 6*(x*e + d)^m*B*c*x^4*e^4 + (x*e + d)^m*A*a*d*m^3*e^3 +
 7*(x*e + d)^m*B*a*d*m^2*x*e^3 + 4*(x*e + d)^m*A*c*d*m*x^2*e^3 - (x*e + d)^m*B*a*d^2*m^2*e^2 - 8*(x*e + d)^m*A
*c*d^2*m*x*e^2 + 2*(x*e + d)^m*A*c*d^3*m*e - 6*(x*e + d)^m*B*c*d^4 + 9*(x*e + d)^m*A*a*m^2*x*e^4 + 19*(x*e + d
)^m*B*a*m*x^2*e^4 + 8*(x*e + d)^m*A*c*x^3*e^4 + 9*(x*e + d)^m*A*a*d*m^2*e^3 + 12*(x*e + d)^m*B*a*d*m*x*e^3 - 7
*(x*e + d)^m*B*a*d^2*m*e^2 + 8*(x*e + d)^m*A*c*d^3*e + 26*(x*e + d)^m*A*a*m*x*e^4 + 12*(x*e + d)^m*B*a*x^2*e^4
 + 26*(x*e + d)^m*A*a*d*m*e^3 - 12*(x*e + d)^m*B*a*d^2*e^2 + 24*(x*e + d)^m*A*a*x*e^4 + 24*(x*e + d)^m*A*a*d*e
^3)/(m^4*e^4 + 10*m^3*e^4 + 35*m^2*e^4 + 50*m*e^4 + 24*e^4)

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