3.2158 \(\int (a+b x) (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^p \, dx\)

Optimal. Leaf size=183 \[ \frac{3 e^2 (a+b x)^4 (b d-a e) \left (a^2+2 a b x+b^2 x^2\right )^p}{2 b^4 (p+2)}+\frac{3 e (a+b x)^3 (b d-a e)^2 \left (a^2+2 a b x+b^2 x^2\right )^p}{b^4 (2 p+3)}+\frac{(a+b x)^2 (b d-a e)^3 \left (a^2+2 a b x+b^2 x^2\right )^p}{2 b^4 (p+1)}+\frac{e^3 (a+b x)^5 \left (a^2+2 a b x+b^2 x^2\right )^p}{b^4 (2 p+5)} \]

[Out]

((b*d - a*e)^3*(a + b*x)^2*(a^2 + 2*a*b*x + b^2*x^2)^p)/(2*b^4*(1 + p)) + (3*e*(
b*d - a*e)^2*(a + b*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^p)/(b^4*(3 + 2*p)) + (3*e^2*(
b*d - a*e)*(a + b*x)^4*(a^2 + 2*a*b*x + b^2*x^2)^p)/(2*b^4*(2 + p)) + (e^3*(a +
b*x)^5*(a^2 + 2*a*b*x + b^2*x^2)^p)/(b^4*(5 + 2*p))

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Rubi [A]  time = 0.286894, antiderivative size = 183, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097 \[ \frac{3 e^2 (a+b x)^4 (b d-a e) \left (a^2+2 a b x+b^2 x^2\right )^p}{2 b^4 (p+2)}+\frac{3 e (a+b x)^3 (b d-a e)^2 \left (a^2+2 a b x+b^2 x^2\right )^p}{b^4 (2 p+3)}+\frac{(a+b x)^2 (b d-a e)^3 \left (a^2+2 a b x+b^2 x^2\right )^p}{2 b^4 (p+1)}+\frac{e^3 (a+b x)^5 \left (a^2+2 a b x+b^2 x^2\right )^p}{b^4 (2 p+5)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)*(d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^p,x]

[Out]

((b*d - a*e)^3*(a + b*x)^2*(a^2 + 2*a*b*x + b^2*x^2)^p)/(2*b^4*(1 + p)) + (3*e*(
b*d - a*e)^2*(a + b*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^p)/(b^4*(3 + 2*p)) + (3*e^2*(
b*d - a*e)*(a + b*x)^4*(a^2 + 2*a*b*x + b^2*x^2)^p)/(2*b^4*(2 + p)) + (e^3*(a +
b*x)^5*(a^2 + 2*a*b*x + b^2*x^2)^p)/(b^4*(5 + 2*p))

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Rubi in Sympy [A]  time = 66.3307, size = 187, normalized size = 1.02 \[ \frac{\left (d + e x\right )^{3} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p + 1}}{b \left (2 p + 5\right )} - \frac{3 \left (d + e x\right )^{2} \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p + 1}}{2 b^{2} \left (p + 2\right ) \left (2 p + 5\right )} + \frac{3 \left (d + e x\right ) \left (a e - b d\right )^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p + 1}}{b^{3} \left (p + 2\right ) \left (2 p + 3\right ) \left (2 p + 5\right )} - \frac{3 \left (a e - b d\right )^{3} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p + 1}}{2 b^{4} \left (p + 1\right ) \left (p + 2\right ) \left (2 p + 3\right ) \left (2 p + 5\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(e*x+d)**3*(b**2*x**2+2*a*b*x+a**2)**p,x)

[Out]

(d + e*x)**3*(a**2 + 2*a*b*x + b**2*x**2)**(p + 1)/(b*(2*p + 5)) - 3*(d + e*x)**
2*(a*e - b*d)*(a**2 + 2*a*b*x + b**2*x**2)**(p + 1)/(2*b**2*(p + 2)*(2*p + 5)) +
3*(d + e*x)*(a*e - b*d)**2*(a**2 + 2*a*b*x + b**2*x**2)**(p + 1)/(b**3*(p + 2)*
(2*p + 3)*(2*p + 5)) - 3*(a*e - b*d)**3*(a**2 + 2*a*b*x + b**2*x**2)**(p + 1)/(2
*b**4*(p + 1)*(p + 2)*(2*p + 3)*(2*p + 5))

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Mathematica [A]  time = 0.430093, size = 222, normalized size = 1.21 \[ \frac{\left ((a+b x)^2\right )^{p+1} \left (-3 a^3 e^3+3 a^2 b e^2 (d (2 p+5)+2 e (p+1) x)-3 a b^2 e \left (d^2 \left (2 p^2+9 p+10\right )+2 d e \left (2 p^2+7 p+5\right ) x+e^2 \left (2 p^2+5 p+3\right ) x^2\right )+b^3 \left (d^3 \left (4 p^3+24 p^2+47 p+30\right )+6 d^2 e \left (2 p^3+11 p^2+19 p+10\right ) x+3 d e^2 \left (4 p^3+20 p^2+31 p+15\right ) x^2+2 e^3 \left (2 p^3+9 p^2+13 p+6\right ) x^3\right )\right )}{2 b^4 (p+1) (p+2) (2 p+3) (2 p+5)} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)*(d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^p,x]

[Out]

(((a + b*x)^2)^(1 + p)*(-3*a^3*e^3 + 3*a^2*b*e^2*(d*(5 + 2*p) + 2*e*(1 + p)*x) -
3*a*b^2*e*(d^2*(10 + 9*p + 2*p^2) + 2*d*e*(5 + 7*p + 2*p^2)*x + e^2*(3 + 5*p +
2*p^2)*x^2) + b^3*(d^3*(30 + 47*p + 24*p^2 + 4*p^3) + 6*d^2*e*(10 + 19*p + 11*p^
2 + 2*p^3)*x + 3*d*e^2*(15 + 31*p + 20*p^2 + 4*p^3)*x^2 + 2*e^3*(6 + 13*p + 9*p^
2 + 2*p^3)*x^3)))/(2*b^4*(1 + p)*(2 + p)*(3 + 2*p)*(5 + 2*p))

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Maple [B]  time = 0.013, size = 407, normalized size = 2.2 \[ -{\frac{ \left ({b}^{2}{x}^{2}+2\,abx+{a}^{2} \right ) ^{p} \left ( -4\,{b}^{3}{e}^{3}{p}^{3}{x}^{3}-12\,{b}^{3}d{e}^{2}{p}^{3}{x}^{2}-18\,{b}^{3}{e}^{3}{p}^{2}{x}^{3}+6\,a{b}^{2}{e}^{3}{p}^{2}{x}^{2}-12\,{b}^{3}{d}^{2}e{p}^{3}x-60\,{b}^{3}d{e}^{2}{p}^{2}{x}^{2}-26\,{b}^{3}{e}^{3}p{x}^{3}+12\,a{b}^{2}d{e}^{2}{p}^{2}x+15\,a{b}^{2}{e}^{3}p{x}^{2}-4\,{b}^{3}{d}^{3}{p}^{3}-66\,{b}^{3}{d}^{2}e{p}^{2}x-93\,{b}^{3}d{e}^{2}p{x}^{2}-12\,{x}^{3}{b}^{3}{e}^{3}-6\,{a}^{2}b{e}^{3}px+6\,a{b}^{2}{d}^{2}e{p}^{2}+42\,a{b}^{2}d{e}^{2}px+9\,{x}^{2}a{b}^{2}{e}^{3}-24\,{b}^{3}{d}^{3}{p}^{2}-114\,{b}^{3}{d}^{2}epx-45\,{x}^{2}{b}^{3}d{e}^{2}-6\,{a}^{2}bd{e}^{2}p-6\,x{a}^{2}b{e}^{3}+27\,a{b}^{2}{d}^{2}ep+30\,xa{b}^{2}d{e}^{2}-47\,{b}^{3}{d}^{3}p-60\,x{b}^{3}{d}^{2}e+3\,{a}^{3}{e}^{3}-15\,{a}^{2}bd{e}^{2}+30\,a{b}^{2}{d}^{2}e-30\,{b}^{3}{d}^{3} \right ) \left ( bx+a \right ) ^{2}}{2\,{b}^{4} \left ( 4\,{p}^{4}+28\,{p}^{3}+71\,{p}^{2}+77\,p+30 \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(e*x+d)^3*(b^2*x^2+2*a*b*x+a^2)^p,x)

[Out]

-1/2*(b^2*x^2+2*a*b*x+a^2)^p*(-4*b^3*e^3*p^3*x^3-12*b^3*d*e^2*p^3*x^2-18*b^3*e^3
*p^2*x^3+6*a*b^2*e^3*p^2*x^2-12*b^3*d^2*e*p^3*x-60*b^3*d*e^2*p^2*x^2-26*b^3*e^3*
p*x^3+12*a*b^2*d*e^2*p^2*x+15*a*b^2*e^3*p*x^2-4*b^3*d^3*p^3-66*b^3*d^2*e*p^2*x-9
3*b^3*d*e^2*p*x^2-12*b^3*e^3*x^3-6*a^2*b*e^3*p*x+6*a*b^2*d^2*e*p^2+42*a*b^2*d*e^
2*p*x+9*a*b^2*e^3*x^2-24*b^3*d^3*p^2-114*b^3*d^2*e*p*x-45*b^3*d*e^2*x^2-6*a^2*b*
d*e^2*p-6*a^2*b*e^3*x+27*a*b^2*d^2*e*p+30*a*b^2*d*e^2*x-47*b^3*d^3*p-60*b^3*d^2*
e*x+3*a^3*e^3-15*a^2*b*d*e^2+30*a*b^2*d^2*e-30*b^3*d^3)*(b*x+a)^2/b^4/(4*p^4+28*
p^3+71*p^2+77*p+30)

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Maxima [A]  time = 0.741238, size = 917, normalized size = 5.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*(e*x + d)^3*(b^2*x^2 + 2*a*b*x + a^2)^p,x, algorithm="maxima")

[Out]

(b*x + a)*(b*x + a)^(2*p)*a*d^3/(b*(2*p + 1)) + 1/2*(b^2*(2*p + 1)*x^2 + 2*a*b*p
*x - a^2)*(b*x + a)^(2*p)*d^3/((2*p^2 + 3*p + 1)*b) + 3/2*(b^2*(2*p + 1)*x^2 + 2
*a*b*p*x - a^2)*(b*x + a)^(2*p)*a*d^2*e/((2*p^2 + 3*p + 1)*b^2) + 3*((2*p^2 + 3*
p + 1)*b^3*x^3 + (2*p^2 + p)*a*b^2*x^2 - 2*a^2*b*p*x + a^3)*(b*x + a)^(2*p)*d^2*
e/((4*p^3 + 12*p^2 + 11*p + 3)*b^2) + 3*((2*p^2 + 3*p + 1)*b^3*x^3 + (2*p^2 + p)
*a*b^2*x^2 - 2*a^2*b*p*x + a^3)*(b*x + a)^(2*p)*a*d*e^2/((4*p^3 + 12*p^2 + 11*p
+ 3)*b^3) + 3/2*((4*p^3 + 12*p^2 + 11*p + 3)*b^4*x^4 + 2*(2*p^3 + 3*p^2 + p)*a*b
^3*x^3 - 3*(2*p^2 + p)*a^2*b^2*x^2 + 6*a^3*b*p*x - 3*a^4)*(b*x + a)^(2*p)*d*e^2/
((4*p^4 + 20*p^3 + 35*p^2 + 25*p + 6)*b^3) + 1/2*((4*p^3 + 12*p^2 + 11*p + 3)*b^
4*x^4 + 2*(2*p^3 + 3*p^2 + p)*a*b^3*x^3 - 3*(2*p^2 + p)*a^2*b^2*x^2 + 6*a^3*b*p*
x - 3*a^4)*(b*x + a)^(2*p)*a*e^3/((4*p^4 + 20*p^3 + 35*p^2 + 25*p + 6)*b^4) + ((
4*p^4 + 20*p^3 + 35*p^2 + 25*p + 6)*b^5*x^5 + (4*p^4 + 12*p^3 + 11*p^2 + 3*p)*a*
b^4*x^4 - 4*(2*p^3 + 3*p^2 + p)*a^2*b^3*x^3 + 6*(2*p^2 + p)*a^3*b^2*x^2 - 12*a^4
*b*p*x + 6*a^5)*(b*x + a)^(2*p)*e^3/((8*p^5 + 60*p^4 + 170*p^3 + 225*p^2 + 137*p
+ 30)*b^4)

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Fricas [A]  time = 0.333561, size = 965, normalized size = 5.27 \[ \frac{{\left (4 \, a^{2} b^{3} d^{3} p^{3} + 30 \, a^{2} b^{3} d^{3} - 30 \, a^{3} b^{2} d^{2} e + 15 \, a^{4} b d e^{2} - 3 \, a^{5} e^{3} + 2 \,{\left (2 \, b^{5} e^{3} p^{3} + 9 \, b^{5} e^{3} p^{2} + 13 \, b^{5} e^{3} p + 6 \, b^{5} e^{3}\right )} x^{5} +{\left (45 \, b^{5} d e^{2} + 15 \, a b^{4} e^{3} + 4 \,{\left (3 \, b^{5} d e^{2} + 2 \, a b^{4} e^{3}\right )} p^{3} + 30 \,{\left (2 \, b^{5} d e^{2} + a b^{4} e^{3}\right )} p^{2} +{\left (93 \, b^{5} d e^{2} + 37 \, a b^{4} e^{3}\right )} p\right )} x^{4} + 2 \,{\left (30 \, b^{5} d^{2} e + 30 \, a b^{4} d e^{2} + 2 \,{\left (3 \, b^{5} d^{2} e + 6 \, a b^{4} d e^{2} + a^{2} b^{3} e^{3}\right )} p^{3} + 3 \,{\left (11 \, b^{5} d^{2} e + 18 \, a b^{4} d e^{2} + a^{2} b^{3} e^{3}\right )} p^{2} +{\left (57 \, b^{5} d^{2} e + 72 \, a b^{4} d e^{2} + a^{2} b^{3} e^{3}\right )} p\right )} x^{3} + 6 \,{\left (4 \, a^{2} b^{3} d^{3} - a^{3} b^{2} d^{2} e\right )} p^{2} +{\left (30 \, b^{5} d^{3} + 90 \, a b^{4} d^{2} e + 4 \,{\left (b^{5} d^{3} + 6 \, a b^{4} d^{2} e + 3 \, a^{2} b^{3} d e^{2}\right )} p^{3} + 6 \,{\left (4 \, b^{5} d^{3} + 21 \, a b^{4} d^{2} e + 6 \, a^{2} b^{3} d e^{2} - a^{3} b^{2} e^{3}\right )} p^{2} +{\left (47 \, b^{5} d^{3} + 201 \, a b^{4} d^{2} e + 15 \, a^{2} b^{3} d e^{2} - 3 \, a^{3} b^{2} e^{3}\right )} p\right )} x^{2} +{\left (47 \, a^{2} b^{3} d^{3} - 27 \, a^{3} b^{2} d^{2} e + 6 \, a^{4} b d e^{2}\right )} p + 2 \,{\left (30 \, a b^{4} d^{3} + 2 \,{\left (2 \, a b^{4} d^{3} + 3 \, a^{2} b^{3} d^{2} e\right )} p^{3} + 3 \,{\left (8 \, a b^{4} d^{3} + 9 \, a^{2} b^{3} d^{2} e - 2 \, a^{3} b^{2} d e^{2}\right )} p^{2} +{\left (47 \, a b^{4} d^{3} + 30 \, a^{2} b^{3} d^{2} e - 15 \, a^{3} b^{2} d e^{2} + 3 \, a^{4} b e^{3}\right )} p\right )} x\right )}{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}}{2 \,{\left (4 \, b^{4} p^{4} + 28 \, b^{4} p^{3} + 71 \, b^{4} p^{2} + 77 \, b^{4} p + 30 \, b^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*(e*x + d)^3*(b^2*x^2 + 2*a*b*x + a^2)^p,x, algorithm="fricas")

[Out]

1/2*(4*a^2*b^3*d^3*p^3 + 30*a^2*b^3*d^3 - 30*a^3*b^2*d^2*e + 15*a^4*b*d*e^2 - 3*
a^5*e^3 + 2*(2*b^5*e^3*p^3 + 9*b^5*e^3*p^2 + 13*b^5*e^3*p + 6*b^5*e^3)*x^5 + (45
*b^5*d*e^2 + 15*a*b^4*e^3 + 4*(3*b^5*d*e^2 + 2*a*b^4*e^3)*p^3 + 30*(2*b^5*d*e^2
+ a*b^4*e^3)*p^2 + (93*b^5*d*e^2 + 37*a*b^4*e^3)*p)*x^4 + 2*(30*b^5*d^2*e + 30*a
*b^4*d*e^2 + 2*(3*b^5*d^2*e + 6*a*b^4*d*e^2 + a^2*b^3*e^3)*p^3 + 3*(11*b^5*d^2*e
+ 18*a*b^4*d*e^2 + a^2*b^3*e^3)*p^2 + (57*b^5*d^2*e + 72*a*b^4*d*e^2 + a^2*b^3*
e^3)*p)*x^3 + 6*(4*a^2*b^3*d^3 - a^3*b^2*d^2*e)*p^2 + (30*b^5*d^3 + 90*a*b^4*d^2
*e + 4*(b^5*d^3 + 6*a*b^4*d^2*e + 3*a^2*b^3*d*e^2)*p^3 + 6*(4*b^5*d^3 + 21*a*b^4
*d^2*e + 6*a^2*b^3*d*e^2 - a^3*b^2*e^3)*p^2 + (47*b^5*d^3 + 201*a*b^4*d^2*e + 15
*a^2*b^3*d*e^2 - 3*a^3*b^2*e^3)*p)*x^2 + (47*a^2*b^3*d^3 - 27*a^3*b^2*d^2*e + 6*
a^4*b*d*e^2)*p + 2*(30*a*b^4*d^3 + 2*(2*a*b^4*d^3 + 3*a^2*b^3*d^2*e)*p^3 + 3*(8*
a*b^4*d^3 + 9*a^2*b^3*d^2*e - 2*a^3*b^2*d*e^2)*p^2 + (47*a*b^4*d^3 + 30*a^2*b^3*
d^2*e - 15*a^3*b^2*d*e^2 + 3*a^4*b*e^3)*p)*x)*(b^2*x^2 + 2*a*b*x + a^2)^p/(4*b^4
*p^4 + 28*b^4*p^3 + 71*b^4*p^2 + 77*b^4*p + 30*b^4)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)*(e*x+d)**3*(b**2*x**2+2*a*b*x+a**2)**p,x)

[Out]

Exception raised: TypeError

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GIAC/XCAS [A]  time = 0.308229, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*(e*x + d)^3*(b^2*x^2 + 2*a*b*x + a^2)^p,x, algorithm="giac")

[Out]

Done