3.1734 \(\int (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^p \, dx\)

Optimal. Leaf size=127 \[ \frac{e (a+b x)^2 (b d-a e) \left (a^2+2 a b x+b^2 x^2\right )^p}{b^3 (p+1)}+\frac{(a+b x) (b d-a e)^2 \left (a^2+2 a b x+b^2 x^2\right )^p}{b^3 (2 p+1)}+\frac{e^2 (a+b x)^3 \left (a^2+2 a b x+b^2 x^2\right )^p}{b^3 (2 p+3)} \]

[Out]

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

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Rubi [A]  time = 0.163693, antiderivative size = 127, 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 \[ \frac{e (a+b x)^2 (b d-a e) \left (a^2+2 a b x+b^2 x^2\right )^p}{b^3 (p+1)}+\frac{(a+b x) (b d-a e)^2 \left (a^2+2 a b x+b^2 x^2\right )^p}{b^3 (2 p+1)}+\frac{e^2 (a+b x)^3 \left (a^2+2 a b x+b^2 x^2\right )^p}{b^3 (2 p+3)} \]

Antiderivative was successfully verified.

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

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Rubi in Sympy [A]  time = 25.9489, size = 131, normalized size = 1.03 \[ \frac{\left (2 a + 2 b x\right ) \left (d + e x\right )^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p}}{2 b \left (2 p + 3\right )} - \frac{e \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p + 1}}{b^{3} \left (p + 1\right ) \left (2 p + 3\right )} + \frac{\left (2 a + 2 b x\right ) \left (a e - b d\right )^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p}}{b^{3} \left (2 p + 1\right ) \left (2 p + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.140743, size = 116, normalized size = 0.91 \[ \frac{(a+b x) \left ((a+b x)^2\right )^p \left (a^2 e^2-a b e (d (2 p+3)+e (2 p+1) x)+b^2 \left (d^2 \left (2 p^2+5 p+3\right )+d e \left (4 p^2+8 p+3\right ) x+e^2 \left (2 p^2+3 p+1\right ) x^2\right )\right )}{b^3 (p+1) (2 p+1) (2 p+3)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.014, size = 175, normalized size = 1.4 \[{\frac{ \left ( 2\,{b}^{2}{e}^{2}{p}^{2}{x}^{2}+4\,{b}^{2}de{p}^{2}x+3\,{b}^{2}{e}^{2}p{x}^{2}-2\,ab{e}^{2}px+2\,{b}^{2}{d}^{2}{p}^{2}+8\,{b}^{2}depx+{x}^{2}{b}^{2}{e}^{2}-2\,abdep-xab{e}^{2}+5\,{b}^{2}{d}^{2}p+3\,x{b}^{2}de+{a}^{2}{e}^{2}-3\,abde+3\,{b}^{2}{d}^{2} \right ) \left ( bx+a \right ) \left ({b}^{2}{x}^{2}+2\,abx+{a}^{2} \right ) ^{p}}{{b}^{3} \left ( 4\,{p}^{3}+12\,{p}^{2}+11\,p+3 \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b*x+a)*(2*b^2*e^2*p^2*x^2+4*b^2*d*e*p^2*x+3*b^2*e^2*p*x^2-2*a*b*e^2*p*x+2*b^2*d
^2*p^2+8*b^2*d*e*p*x+b^2*e^2*x^2-2*a*b*d*e*p-a*b*e^2*x+5*b^2*d^2*p+3*b^2*d*e*x+a
^2*e^2-3*a*b*d*e+3*b^2*d^2)*(b^2*x^2+2*a*b*x+a^2)^p/b^3/(4*p^3+12*p^2+11*p+3)

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Maxima [A]  time = 0.763034, size = 212, normalized size = 1.67 \[ \frac{{\left (b x + a\right )}{\left (b x + a\right )}^{2 \, p} d^{2}}{b{\left (2 \, p + 1\right )}} + \frac{{\left (b^{2}{\left (2 \, p + 1\right )} x^{2} + 2 \, a b p x - a^{2}\right )}{\left (b x + a\right )}^{2 \, p} d e}{{\left (2 \, p^{2} + 3 \, p + 1\right )} b^{2}} + \frac{{\left ({\left (2 \, p^{2} + 3 \, p + 1\right )} b^{3} x^{3} +{\left (2 \, p^{2} + p\right )} a b^{2} x^{2} - 2 \, a^{2} b p x + a^{3}\right )}{\left (b x + a\right )}^{2 \, p} e^{2}}{{\left (4 \, p^{3} + 12 \, p^{2} + 11 \, p + 3\right )} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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Fricas [A]  time = 0.225114, size = 336, normalized size = 2.65 \[ \frac{{\left (2 \, a b^{2} d^{2} p^{2} + 3 \, a b^{2} d^{2} - 3 \, a^{2} b d e + a^{3} e^{2} +{\left (2 \, b^{3} e^{2} p^{2} + 3 \, b^{3} e^{2} p + b^{3} e^{2}\right )} x^{3} +{\left (3 \, b^{3} d e + 2 \,{\left (2 \, b^{3} d e + a b^{2} e^{2}\right )} p^{2} +{\left (8 \, b^{3} d e + a b^{2} e^{2}\right )} p\right )} x^{2} +{\left (5 \, a b^{2} d^{2} - 2 \, a^{2} b d e\right )} p +{\left (3 \, b^{3} d^{2} + 2 \,{\left (b^{3} d^{2} + 2 \, a b^{2} d e\right )} p^{2} +{\left (5 \, b^{3} d^{2} + 6 \, a b^{2} d e - 2 \, a^{2} b e^{2}\right )} p\right )} x\right )}{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}}{4 \, b^{3} p^{3} + 12 \, b^{3} p^{2} + 11 \, b^{3} p + 3 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(2*a*b^2*d^2*p^2 + 3*a*b^2*d^2 - 3*a^2*b*d*e + a^3*e^2 + (2*b^3*e^2*p^2 + 3*b^3*
e^2*p + b^3*e^2)*x^3 + (3*b^3*d*e + 2*(2*b^3*d*e + a*b^2*e^2)*p^2 + (8*b^3*d*e +
a*b^2*e^2)*p)*x^2 + (5*a*b^2*d^2 - 2*a^2*b*d*e)*p + (3*b^3*d^2 + 2*(b^3*d^2 + 2
*a*b^2*d*e)*p^2 + (5*b^3*d^2 + 6*a*b^2*d*e - 2*a^2*b*e^2)*p)*x)*(b^2*x^2 + 2*a*b
*x + a^2)^p/(4*b^3*p^3 + 12*b^3*p^2 + 11*b^3*p + 3*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((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 = 0.225645, size = 875, normalized size = 6.89 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(2*b^3*p^2*x^3*e^(p*ln(b^2*x^2 + 2*a*b*x + a^2) + 2) + 4*b^3*d*p^2*x^2*e^(p*ln(b
^2*x^2 + 2*a*b*x + a^2) + 1) + 2*b^3*d^2*p^2*x*e^(p*ln(b^2*x^2 + 2*a*b*x + a^2))
+ 2*a*b^2*p^2*x^2*e^(p*ln(b^2*x^2 + 2*a*b*x + a^2) + 2) + 3*b^3*p*x^3*e^(p*ln(b
^2*x^2 + 2*a*b*x + a^2) + 2) + 4*a*b^2*d*p^2*x*e^(p*ln(b^2*x^2 + 2*a*b*x + a^2)
+ 1) + 8*b^3*d*p*x^2*e^(p*ln(b^2*x^2 + 2*a*b*x + a^2) + 1) + 2*a*b^2*d^2*p^2*e^(
p*ln(b^2*x^2 + 2*a*b*x + a^2)) + 5*b^3*d^2*p*x*e^(p*ln(b^2*x^2 + 2*a*b*x + a^2))
+ a*b^2*p*x^2*e^(p*ln(b^2*x^2 + 2*a*b*x + a^2) + 2) + b^3*x^3*e^(p*ln(b^2*x^2 +
2*a*b*x + a^2) + 2) + 6*a*b^2*d*p*x*e^(p*ln(b^2*x^2 + 2*a*b*x + a^2) + 1) + 3*b
^3*d*x^2*e^(p*ln(b^2*x^2 + 2*a*b*x + a^2) + 1) + 5*a*b^2*d^2*p*e^(p*ln(b^2*x^2 +
2*a*b*x + a^2)) + 3*b^3*d^2*x*e^(p*ln(b^2*x^2 + 2*a*b*x + a^2)) - 2*a^2*b*p*x*e
^(p*ln(b^2*x^2 + 2*a*b*x + a^2) + 2) - 2*a^2*b*d*p*e^(p*ln(b^2*x^2 + 2*a*b*x + a
^2) + 1) + 3*a*b^2*d^2*e^(p*ln(b^2*x^2 + 2*a*b*x + a^2)) - 3*a^2*b*d*e^(p*ln(b^2
*x^2 + 2*a*b*x + a^2) + 1) + a^3*e^(p*ln(b^2*x^2 + 2*a*b*x + a^2) + 2))/(4*b^3*p
^3 + 12*b^3*p^2 + 11*b^3*p + 3*b^3)