3.2159 \(\int (a+b x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^p \, dx\)

Optimal. Leaf size=134 \[ \frac{2 e (a+b x)^3 (b d-a e) \left (a^2+2 a b x+b^2 x^2\right )^p}{b^3 (2 p+3)}+\frac{(a+b x)^2 (b d-a e)^2 \left (a^2+2 a b x+b^2 x^2\right )^p}{2 b^3 (p+1)}+\frac{e^2 (a+b x)^4 \left (a^2+2 a b x+b^2 x^2\right )^p}{2 b^3 (p+2)} \]

[Out]

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

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Rubi [A]  time = 0.218643, antiderivative size = 134, 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 \[ \frac{2 e (a+b x)^3 (b d-a e) \left (a^2+2 a b x+b^2 x^2\right )^p}{b^3 (2 p+3)}+\frac{(a+b x)^2 (b d-a e)^2 \left (a^2+2 a b x+b^2 x^2\right )^p}{2 b^3 (p+1)}+\frac{e^2 (a+b x)^4 \left (a^2+2 a b x+b^2 x^2\right )^p}{2 b^3 (p+2)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 39.8199, size = 122, normalized size = 0.91 \[ \frac{\left (d + e x\right )^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p + 1}}{2 b \left (p + 2\right )} - \frac{\left (d + e x\right ) \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p + 1}}{b^{2} \left (p + 2\right ) \left (2 p + 3\right )} + \frac{\left (a e - b d\right )^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p + 1}}{2 b^{3} \left (p + 1\right ) \left (p + 2\right ) \left (2 p + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.152196, size = 109, normalized size = 0.81 \[ \frac{\left ((a+b x)^2\right )^{p+1} \left (a^2 e^2-2 a b e (d (p+2)+e (p+1) x)+b^2 \left (d^2 \left (2 p^2+7 p+6\right )+4 d e \left (p^2+3 p+2\right ) x+e^2 \left (2 p^2+5 p+3\right ) x^2\right )\right )}{2 b^3 (p+1) (p+2) (2 p+3)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.011, size = 179, normalized size = 1.3 \[{\frac{ \left ( 2\,{b}^{2}{e}^{2}{p}^{2}{x}^{2}+4\,{b}^{2}de{p}^{2}x+5\,{b}^{2}{e}^{2}p{x}^{2}-2\,ab{e}^{2}px+2\,{b}^{2}{d}^{2}{p}^{2}+12\,{b}^{2}depx+3\,{x}^{2}{b}^{2}{e}^{2}-2\,abdep-2\,xab{e}^{2}+7\,{b}^{2}{d}^{2}p+8\,x{b}^{2}de+{a}^{2}{e}^{2}-4\,abde+6\,{b}^{2}{d}^{2} \right ) \left ( bx+a \right ) ^{2} \left ({b}^{2}{x}^{2}+2\,abx+{a}^{2} \right ) ^{p}}{2\,{b}^{3} \left ( 2\,{p}^{3}+9\,{p}^{2}+13\,p+6 \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(b*x+a)^2*(2*b^2*e^2*p^2*x^2+4*b^2*d*e*p^2*x+5*b^2*e^2*p*x^2-2*a*b*e^2*p*x+2
*b^2*d^2*p^2+12*b^2*d*e*p*x+3*b^2*e^2*x^2-2*a*b*d*e*p-2*a*b*e^2*x+7*b^2*d^2*p+8*
b^2*d*e*x+a^2*e^2-4*a*b*d*e+6*b^2*d^2)*(b^2*x^2+2*a*b*x+a^2)^p/b^3/(2*p^3+9*p^2+
13*p+6)

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Maxima [A]  time = 0.727977, size = 545, normalized size = 4.07 \[ \frac{{\left (b x + a\right )}{\left (b x + a\right )}^{2 \, p} a d^{2}}{b{\left (2 \, p + 1\right )}} + \frac{{\left (b^{2}{\left (2 \, p + 1\right )} x^{2} + 2 \, a b p x - a^{2}\right )}{\left (b x + a\right )}^{2 \, p} d^{2}}{2 \,{\left (2 \, p^{2} + 3 \, p + 1\right )} b} + \frac{{\left (b^{2}{\left (2 \, p + 1\right )} x^{2} + 2 \, a b p x - a^{2}\right )}{\left (b x + a\right )}^{2 \, p} a d e}{{\left (2 \, p^{2} + 3 \, p + 1\right )} b^{2}} + \frac{2 \,{\left ({\left (2 \, p^{2} + 3 \, p + 1\right )} b^{3} x^{3} +{\left (2 \, p^{2} + p\right )} a b^{2} x^{2} - 2 \, a^{2} b p x + a^{3}\right )}{\left (b x + a\right )}^{2 \, p} d e}{{\left (4 \, p^{3} + 12 \, p^{2} + 11 \, p + 3\right )} b^{2}} + \frac{{\left ({\left (2 \, p^{2} + 3 \, p + 1\right )} b^{3} x^{3} +{\left (2 \, p^{2} + p\right )} a b^{2} x^{2} - 2 \, a^{2} b p x + a^{3}\right )}{\left (b x + a\right )}^{2 \, p} a e^{2}}{{\left (4 \, p^{3} + 12 \, p^{2} + 11 \, p + 3\right )} b^{3}} + \frac{{\left ({\left (4 \, p^{3} + 12 \, p^{2} + 11 \, p + 3\right )} b^{4} x^{4} + 2 \,{\left (2 \, p^{3} + 3 \, p^{2} + p\right )} a b^{3} x^{3} - 3 \,{\left (2 \, p^{2} + p\right )} a^{2} b^{2} x^{2} + 6 \, a^{3} b p x - 3 \, a^{4}\right )}{\left (b x + a\right )}^{2 \, p} e^{2}}{2 \,{\left (4 \, p^{4} + 20 \, p^{3} + 35 \, p^{2} + 25 \, p + 6\right )} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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Fricas [A]  time = 0.323038, size = 477, normalized size = 3.56 \[ \frac{{\left (2 \, a^{2} b^{2} d^{2} p^{2} + 6 \, a^{2} b^{2} d^{2} - 4 \, a^{3} b d e + a^{4} e^{2} +{\left (2 \, b^{4} e^{2} p^{2} + 5 \, b^{4} e^{2} p + 3 \, b^{4} e^{2}\right )} x^{4} + 4 \,{\left (2 \, b^{4} d e + a b^{3} e^{2} +{\left (b^{4} d e + a b^{3} e^{2}\right )} p^{2} +{\left (3 \, b^{4} d e + 2 \, a b^{3} e^{2}\right )} p\right )} x^{3} +{\left (6 \, b^{4} d^{2} + 12 \, a b^{3} d e + 2 \,{\left (b^{4} d^{2} + 4 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )} p^{2} +{\left (7 \, b^{4} d^{2} + 22 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )} p\right )} x^{2} +{\left (7 \, a^{2} b^{2} d^{2} - 2 \, a^{3} b d e\right )} p + 2 \,{\left (6 \, a b^{3} d^{2} + 2 \,{\left (a b^{3} d^{2} + a^{2} b^{2} d e\right )} p^{2} +{\left (7 \, a b^{3} d^{2} + 4 \, a^{2} b^{2} d e - a^{3} b e^{2}\right )} p\right )} x\right )}{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}}{2 \,{\left (2 \, b^{3} p^{3} + 9 \, b^{3} p^{2} + 13 \, b^{3} p + 6 \, b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(2*a^2*b^2*d^2*p^2 + 6*a^2*b^2*d^2 - 4*a^3*b*d*e + a^4*e^2 + (2*b^4*e^2*p^2
+ 5*b^4*e^2*p + 3*b^4*e^2)*x^4 + 4*(2*b^4*d*e + a*b^3*e^2 + (b^4*d*e + a*b^3*e^2
)*p^2 + (3*b^4*d*e + 2*a*b^3*e^2)*p)*x^3 + (6*b^4*d^2 + 12*a*b^3*d*e + 2*(b^4*d^
2 + 4*a*b^3*d*e + a^2*b^2*e^2)*p^2 + (7*b^4*d^2 + 22*a*b^3*d*e + a^2*b^2*e^2)*p)
*x^2 + (7*a^2*b^2*d^2 - 2*a^3*b*d*e)*p + 2*(6*a*b^3*d^2 + 2*(a*b^3*d^2 + a^2*b^2
*d*e)*p^2 + (7*a*b^3*d^2 + 4*a^2*b^2*d*e - a^3*b*e^2)*p)*x)*(b^2*x^2 + 2*a*b*x +
a^2)^p/(2*b^3*p^3 + 9*b^3*p^2 + 13*b^3*p + 6*b^3)

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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)**2*(b**2*x**2+2*a*b*x+a**2)**p,x)

[Out]

Exception raised: TypeError

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GIAC/XCAS [A]  time = 0.323946, size = 1297, normalized size = 9.68 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(2*b^4*p^2*x^4*e^(p*ln(b^2*x^2 + 2*a*b*x + a^2) + 2) + 4*b^4*d*p^2*x^3*e^(p*
ln(b^2*x^2 + 2*a*b*x + a^2) + 1) + 2*b^4*d^2*p^2*x^2*e^(p*ln(b^2*x^2 + 2*a*b*x +
a^2)) + 4*a*b^3*p^2*x^3*e^(p*ln(b^2*x^2 + 2*a*b*x + a^2) + 2) + 5*b^4*p*x^4*e^(
p*ln(b^2*x^2 + 2*a*b*x + a^2) + 2) + 8*a*b^3*d*p^2*x^2*e^(p*ln(b^2*x^2 + 2*a*b*x
+ a^2) + 1) + 12*b^4*d*p*x^3*e^(p*ln(b^2*x^2 + 2*a*b*x + a^2) + 1) + 4*a*b^3*d^
2*p^2*x*e^(p*ln(b^2*x^2 + 2*a*b*x + a^2)) + 7*b^4*d^2*p*x^2*e^(p*ln(b^2*x^2 + 2*
a*b*x + a^2)) + 2*a^2*b^2*p^2*x^2*e^(p*ln(b^2*x^2 + 2*a*b*x + a^2) + 2) + 8*a*b^
3*p*x^3*e^(p*ln(b^2*x^2 + 2*a*b*x + a^2) + 2) + 3*b^4*x^4*e^(p*ln(b^2*x^2 + 2*a*
b*x + a^2) + 2) + 4*a^2*b^2*d*p^2*x*e^(p*ln(b^2*x^2 + 2*a*b*x + a^2) + 1) + 22*a
*b^3*d*p*x^2*e^(p*ln(b^2*x^2 + 2*a*b*x + a^2) + 1) + 8*b^4*d*x^3*e^(p*ln(b^2*x^2
+ 2*a*b*x + a^2) + 1) + 2*a^2*b^2*d^2*p^2*e^(p*ln(b^2*x^2 + 2*a*b*x + a^2)) + 1
4*a*b^3*d^2*p*x*e^(p*ln(b^2*x^2 + 2*a*b*x + a^2)) + 6*b^4*d^2*x^2*e^(p*ln(b^2*x^
2 + 2*a*b*x + a^2)) + a^2*b^2*p*x^2*e^(p*ln(b^2*x^2 + 2*a*b*x + a^2) + 2) + 4*a*
b^3*x^3*e^(p*ln(b^2*x^2 + 2*a*b*x + a^2) + 2) + 8*a^2*b^2*d*p*x*e^(p*ln(b^2*x^2
+ 2*a*b*x + a^2) + 1) + 12*a*b^3*d*x^2*e^(p*ln(b^2*x^2 + 2*a*b*x + a^2) + 1) + 7
*a^2*b^2*d^2*p*e^(p*ln(b^2*x^2 + 2*a*b*x + a^2)) + 12*a*b^3*d^2*x*e^(p*ln(b^2*x^
2 + 2*a*b*x + a^2)) - 2*a^3*b*p*x*e^(p*ln(b^2*x^2 + 2*a*b*x + a^2) + 2) - 2*a^3*
b*d*p*e^(p*ln(b^2*x^2 + 2*a*b*x + a^2) + 1) + 6*a^2*b^2*d^2*e^(p*ln(b^2*x^2 + 2*
a*b*x + a^2)) - 4*a^3*b*d*e^(p*ln(b^2*x^2 + 2*a*b*x + a^2) + 1) + a^4*e^(p*ln(b^
2*x^2 + 2*a*b*x + a^2) + 2))/(2*b^3*p^3 + 9*b^3*p^2 + 13*b^3*p + 6*b^3)