3.1733 \(\int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^p \, dx\)

Optimal. Leaf size=181 \[ \frac{3 e^2 (a+b x)^3 (b d-a e) \left (a^2+2 a b x+b^2 x^2\right )^p}{b^4 (2 p+3)}+\frac{3 e (a+b x)^2 (b d-a e)^2 \left (a^2+2 a b x+b^2 x^2\right )^p}{2 b^4 (p+1)}+\frac{(a+b x) (b d-a e)^3 \left (a^2+2 a b x+b^2 x^2\right )^p}{b^4 (2 p+1)}+\frac{e^3 (a+b x)^4 \left (a^2+2 a b x+b^2 x^2\right )^p}{2 b^4 (p+2)} \]

[Out]

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

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Rubi [A]  time = 0.235136, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{3 e^2 (a+b x)^3 (b d-a e) \left (a^2+2 a b x+b^2 x^2\right )^p}{b^4 (2 p+3)}+\frac{3 e (a+b x)^2 (b d-a e)^2 \left (a^2+2 a b x+b^2 x^2\right )^p}{2 b^4 (p+1)}+\frac{(a+b x) (b d-a e)^3 \left (a^2+2 a b x+b^2 x^2\right )^p}{b^4 (2 p+1)}+\frac{e^3 (a+b x)^4 \left (a^2+2 a b x+b^2 x^2\right )^p}{2 b^4 (p+2)} \]

Antiderivative was successfully verified.

[In]  Int[(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)*(a^2 + 2*a*b*x + b^2*x^2)^p)/(b^4*(1 + 2*p)) + (3*e*(b*
d - a*e)^2*(a + b*x)^2*(a^2 + 2*a*b*x + b^2*x^2)^p)/(2*b^4*(1 + p)) + (3*e^2*(b*
d - a*e)*(a + b*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^p)/(b^4*(3 + 2*p)) + (e^3*(a + b*
x)^4*(a^2 + 2*a*b*x + b^2*x^2)^p)/(2*b^4*(2 + p))

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Rubi in Sympy [A]  time = 49.2694, size = 201, normalized size = 1.11 \[ \frac{\left (2 a + 2 b x\right ) \left (d + e x\right )^{3} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p}}{4 b \left (p + 2\right )} - \frac{3 \left (2 a + 2 b x\right ) \left (d + e x\right )^{2} \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p}}{4 b^{2} \left (p + 2\right ) \left (2 p + 3\right )} + \frac{3 e \left (a e - b d\right )^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p + 1}}{2 b^{4} \left (p + 1\right ) \left (p + 2\right ) \left (2 p + 3\right )} - \frac{3 \left (2 a + 2 b x\right ) \left (a e - b d\right )^{3} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p}}{2 b^{4} \left (p + 2\right ) \left (2 p + 1\right ) \left (2 p + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.319502, size = 225, normalized size = 1.24 \[ \frac{(a+b x) \left ((a+b x)^2\right )^p \left (-3 a^3 e^3+3 a^2 b e^2 (2 d (p+2)+e (2 p+1) x)-3 a b^2 e \left (d^2 \left (2 p^2+7 p+6\right )+2 d e \left (2 p^2+5 p+2\right ) x+e^2 \left (2 p^2+3 p+1\right ) x^2\right )+b^3 \left (2 d^3 \left (2 p^3+9 p^2+13 p+6\right )+3 d^2 e \left (4 p^3+16 p^2+19 p+6\right ) x+6 d e^2 \left (2 p^3+7 p^2+7 p+2\right ) x^2+e^3 \left (4 p^3+12 p^2+11 p+3\right ) x^3\right )\right )}{2 b^4 (p+1) (p+2) (2 p+1) (2 p+3)} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)*((a + b*x)^2)^p*(-3*a^3*e^3 + 3*a^2*b*e^2*(2*d*(2 + p) + e*(1 + 2*p)*
x) - 3*a*b^2*e*(d^2*(6 + 7*p + 2*p^2) + 2*d*e*(2 + 5*p + 2*p^2)*x + e^2*(1 + 3*p
+ 2*p^2)*x^2) + b^3*(2*d^3*(6 + 13*p + 9*p^2 + 2*p^3) + 3*d^2*e*(6 + 19*p + 16*
p^2 + 4*p^3)*x + 6*d*e^2*(2 + 7*p + 7*p^2 + 2*p^3)*x^2 + e^3*(3 + 11*p + 12*p^2
+ 4*p^3)*x^3)))/(2*b^4*(1 + p)*(2 + p)*(1 + 2*p)*(3 + 2*p))

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Maple [B]  time = 0.015, size = 405, normalized size = 2.2 \[ -{\frac{ \left ({b}^{2}{x}^{2}+2\,abx+{a}^{2} \right ) ^{p} \left ( -4\,{b}^{3}{e}^{3}{p}^{3}{x}^{3}-12\,{b}^{3}d{e}^{2}{p}^{3}{x}^{2}-12\,{b}^{3}{e}^{3}{p}^{2}{x}^{3}+6\,a{b}^{2}{e}^{3}{p}^{2}{x}^{2}-12\,{b}^{3}{d}^{2}e{p}^{3}x-42\,{b}^{3}d{e}^{2}{p}^{2}{x}^{2}-11\,{b}^{3}{e}^{3}p{x}^{3}+12\,a{b}^{2}d{e}^{2}{p}^{2}x+9\,a{b}^{2}{e}^{3}p{x}^{2}-4\,{b}^{3}{d}^{3}{p}^{3}-48\,{b}^{3}{d}^{2}e{p}^{2}x-42\,{b}^{3}d{e}^{2}p{x}^{2}-3\,{x}^{3}{b}^{3}{e}^{3}-6\,{a}^{2}b{e}^{3}px+6\,a{b}^{2}{d}^{2}e{p}^{2}+30\,a{b}^{2}d{e}^{2}px+3\,{x}^{2}a{b}^{2}{e}^{3}-18\,{b}^{3}{d}^{3}{p}^{2}-57\,{b}^{3}{d}^{2}epx-12\,{x}^{2}{b}^{3}d{e}^{2}-6\,{a}^{2}bd{e}^{2}p-3\,x{a}^{2}b{e}^{3}+21\,a{b}^{2}{d}^{2}ep+12\,xa{b}^{2}d{e}^{2}-26\,{b}^{3}{d}^{3}p-18\,x{b}^{3}{d}^{2}e+3\,{a}^{3}{e}^{3}-12\,{a}^{2}bd{e}^{2}+18\,a{b}^{2}{d}^{2}e-12\,{b}^{3}{d}^{3} \right ) \left ( bx+a \right ) }{2\,{b}^{4} \left ( 4\,{p}^{4}+20\,{p}^{3}+35\,{p}^{2}+25\,p+6 \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((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^2-12*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-42*b^3*d*e^2*p^2*x^2-11*b^3*e^3*
p*x^3+12*a*b^2*d*e^2*p^2*x+9*a*b^2*e^3*p*x^2-4*b^3*d^3*p^3-48*b^3*d^2*e*p^2*x-42
*b^3*d*e^2*p*x^2-3*b^3*e^3*x^3-6*a^2*b*e^3*p*x+6*a*b^2*d^2*e*p^2+30*a*b^2*d*e^2*
p*x+3*a*b^2*e^3*x^2-18*b^3*d^3*p^2-57*b^3*d^2*e*p*x-12*b^3*d*e^2*x^2-6*a^2*b*d*e
^2*p-3*a^2*b*e^3*x+21*a*b^2*d^2*e*p+12*a*b^2*d*e^2*x-26*b^3*d^3*p-18*b^3*d^2*e*x
+3*a^3*e^3-12*a^2*b*d*e^2+18*a*b^2*d^2*e-12*b^3*d^3)*(b*x+a)/b^4/(4*p^4+20*p^3+3
5*p^2+25*p+6)

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Maxima [A]  time = 0.780107, size = 373, normalized size = 2.06 \[ \frac{{\left (b x + a\right )}{\left (b x + a\right )}^{2 \, p} d^{3}}{b{\left (2 \, p + 1\right )}} + \frac{3 \,{\left (b^{2}{\left (2 \, p + 1\right )} x^{2} + 2 \, a b p x - a^{2}\right )}{\left (b x + a\right )}^{2 \, p} d^{2} e}{2 \,{\left (2 \, p^{2} + 3 \, p + 1\right )} b^{2}} + \frac{3 \,{\left ({\left (2 \, p^{2} + 3 \, p + 1\right )} b^{3} x^{3} +{\left (2 \, p^{2} + p\right )} a b^{2} x^{2} - 2 \, a^{2} b p x + a^{3}\right )}{\left (b x + a\right )}^{2 \, p} d e^{2}}{{\left (4 \, p^{3} + 12 \, p^{2} + 11 \, p + 3\right )} b^{3}} + \frac{{\left ({\left (4 \, p^{3} + 12 \, p^{2} + 11 \, p + 3\right )} b^{4} x^{4} + 2 \,{\left (2 \, p^{3} + 3 \, p^{2} + p\right )} a b^{3} x^{3} - 3 \,{\left (2 \, p^{2} + p\right )} a^{2} b^{2} x^{2} + 6 \, a^{3} b p x - 3 \, a^{4}\right )}{\left (b x + a\right )}^{2 \, p} e^{3}}{2 \,{\left (4 \, p^{4} + 20 \, p^{3} + 35 \, p^{2} + 25 \, p + 6\right )} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((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)*d^3/(b*(2*p + 1)) + 3/2*(b^2*(2*p + 1)*x^2 + 2*a*b*p*x
- a^2)*(b*x + a)^(2*p)*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*e^2/((4*p^3
+ 12*p^2 + 11*p + 3)*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)*e^3/((4*p^4 + 20*p^3 + 35*p^2 + 25*p + 6)*b^4)

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Fricas [A]  time = 0.222545, size = 697, normalized size = 3.85 \[ \frac{{\left (4 \, a b^{3} d^{3} p^{3} + 12 \, a b^{3} d^{3} - 18 \, a^{2} b^{2} d^{2} e + 12 \, a^{3} b d e^{2} - 3 \, a^{4} e^{3} +{\left (4 \, b^{4} e^{3} p^{3} + 12 \, b^{4} e^{3} p^{2} + 11 \, b^{4} e^{3} p + 3 \, b^{4} e^{3}\right )} x^{4} + 2 \,{\left (6 \, b^{4} d e^{2} + 2 \,{\left (3 \, b^{4} d e^{2} + a b^{3} e^{3}\right )} p^{3} + 3 \,{\left (7 \, b^{4} d e^{2} + a b^{3} e^{3}\right )} p^{2} +{\left (21 \, b^{4} d e^{2} + a b^{3} e^{3}\right )} p\right )} x^{3} + 6 \,{\left (3 \, a b^{3} d^{3} - a^{2} b^{2} d^{2} e\right )} p^{2} + 3 \,{\left (6 \, b^{4} d^{2} e + 4 \,{\left (b^{4} d^{2} e + a b^{3} d e^{2}\right )} p^{3} + 2 \,{\left (8 \, b^{4} d^{2} e + 5 \, a b^{3} d e^{2} - a^{2} b^{2} e^{3}\right )} p^{2} +{\left (19 \, b^{4} d^{2} e + 4 \, a b^{3} d e^{2} - a^{2} b^{2} e^{3}\right )} p\right )} x^{2} +{\left (26 \, a b^{3} d^{3} - 21 \, a^{2} b^{2} d^{2} e + 6 \, a^{3} b d e^{2}\right )} p + 2 \,{\left (6 \, b^{4} d^{3} + 2 \,{\left (b^{4} d^{3} + 3 \, a b^{3} d^{2} e\right )} p^{3} + 3 \,{\left (3 \, b^{4} d^{3} + 7 \, a b^{3} d^{2} e - 2 \, a^{2} b^{2} d e^{2}\right )} p^{2} +{\left (13 \, b^{4} d^{3} + 18 \, a b^{3} d^{2} e - 12 \, a^{2} b^{2} d e^{2} + 3 \, a^{3} b e^{3}\right )} p\right )} x\right )}{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}}{2 \,{\left (4 \, b^{4} p^{4} + 20 \, b^{4} p^{3} + 35 \, b^{4} p^{2} + 25 \, b^{4} p + 6 \, b^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(4*a*b^3*d^3*p^3 + 12*a*b^3*d^3 - 18*a^2*b^2*d^2*e + 12*a^3*b*d*e^2 - 3*a^4*
e^3 + (4*b^4*e^3*p^3 + 12*b^4*e^3*p^2 + 11*b^4*e^3*p + 3*b^4*e^3)*x^4 + 2*(6*b^4
*d*e^2 + 2*(3*b^4*d*e^2 + a*b^3*e^3)*p^3 + 3*(7*b^4*d*e^2 + a*b^3*e^3)*p^2 + (21
*b^4*d*e^2 + a*b^3*e^3)*p)*x^3 + 6*(3*a*b^3*d^3 - a^2*b^2*d^2*e)*p^2 + 3*(6*b^4*
d^2*e + 4*(b^4*d^2*e + a*b^3*d*e^2)*p^3 + 2*(8*b^4*d^2*e + 5*a*b^3*d*e^2 - a^2*b
^2*e^3)*p^2 + (19*b^4*d^2*e + 4*a*b^3*d*e^2 - a^2*b^2*e^3)*p)*x^2 + (26*a*b^3*d^
3 - 21*a^2*b^2*d^2*e + 6*a^3*b*d*e^2)*p + 2*(6*b^4*d^3 + 2*(b^4*d^3 + 3*a*b^3*d^
2*e)*p^3 + 3*(3*b^4*d^3 + 7*a*b^3*d^2*e - 2*a^2*b^2*d*e^2)*p^2 + (13*b^4*d^3 + 1
8*a*b^3*d^2*e - 12*a^2*b^2*d*e^2 + 3*a^3*b*e^3)*p)*x)*(b^2*x^2 + 2*a*b*x + a^2)^
p/(4*b^4*p^4 + 20*b^4*p^3 + 35*b^4*p^2 + 25*b^4*p + 6*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((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 = 0.222395, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done