3.1735 \(\int (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^p \, dx\)

Optimal. Leaf size=76 \[ \frac{(a+b x) (b d-a e) \left (a^2+2 a b x+b^2 x^2\right )^p}{b^2 (2 p+1)}+\frac{e \left (a^2+2 a b x+b^2 x^2\right )^{p+1}}{2 b^2 (p+1)} \]

[Out]

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

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Rubi [A]  time = 0.0755512, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{(a+b x) (b d-a e) \left (a^2+2 a b x+b^2 x^2\right )^p}{b^2 (2 p+1)}+\frac{e \left (a^2+2 a b x+b^2 x^2\right )^{p+1}}{2 b^2 (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 10.7343, size = 73, normalized size = 0.96 \[ \frac{e \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p + 1}}{2 b^{2} \left (p + 1\right )} - \frac{\left (2 a + 2 b x\right ) \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p}}{2 b^{2} \left (2 p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.049685, size = 54, normalized size = 0.71 \[ \frac{(a+b x) \left ((a+b x)^2\right )^p (-a e+2 b d (p+1)+b e (2 p+1) x)}{2 b^2 (p+1) (2 p+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 65, normalized size = 0.9 \[ -{\frac{ \left ({b}^{2}{x}^{2}+2\,abx+{a}^{2} \right ) ^{p} \left ( -2\,bepx-2\,bdp-bex+ae-2\,bd \right ) \left ( bx+a \right ) }{2\,{b}^{2} \left ( 2\,{p}^{2}+3\,p+1 \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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Maxima [A]  time = 0.769065, size = 105, normalized size = 1.38 \[ \frac{{\left (b x + a\right )}{\left (b x + a\right )}^{2 \, p} d}{b{\left (2 \, p + 1\right )}} + \frac{{\left (b^{2}{\left (2 \, p + 1\right )} x^{2} + 2 \, a b p x - a^{2}\right )}{\left (b x + a\right )}^{2 \, p} e}{2 \,{\left (2 \, p^{2} + 3 \, p + 1\right )} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.221268, size = 130, normalized size = 1.71 \[ \frac{{\left (2 \, a b d p + 2 \, a b d - a^{2} e +{\left (2 \, b^{2} e p + b^{2} e\right )} x^{2} + 2 \,{\left (b^{2} d +{\left (b^{2} d + a b e\right )} p\right )} x\right )}{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}}{2 \,{\left (2 \, b^{2} p^{2} + 3 \, b^{2} p + b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: TypeError

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GIAC/XCAS [A]  time = 0.221572, size = 329, normalized size = 4.33 \[ \frac{2 \, b^{2} p x^{2} e^{\left (p{\rm ln}\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right ) + 1\right )} + 2 \, b^{2} d p x e^{\left (p{\rm ln}\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )\right )} + 2 \, a b p x e^{\left (p{\rm ln}\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right ) + 1\right )} + b^{2} x^{2} e^{\left (p{\rm ln}\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right ) + 1\right )} + 2 \, a b d p e^{\left (p{\rm ln}\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )\right )} + 2 \, b^{2} d x e^{\left (p{\rm ln}\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )\right )} + 2 \, a b d e^{\left (p{\rm ln}\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )\right )} - a^{2} e^{\left (p{\rm ln}\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right ) + 1\right )}}{2 \,{\left (2 \, b^{2} p^{2} + 3 \, b^{2} p + b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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