3.1746 \(\int (d+e x)^{-3-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p \, dx\)

Optimal. Leaf size=115 \[ \frac{b (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p (d+e x)^{-2 p-1}}{2 (p+1) (2 p+1) (b d-a e)^2}+\frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p (d+e x)^{-2 (p+1)}}{2 (p+1) (b d-a e)} \]

[Out]

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

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Rubi [A]  time = 0.142402, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{b (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p (d+e x)^{-2 p-1}}{2 (p+1) (2 p+1) (b d-a e)^2}+\frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p (d+e x)^{-2 (p+1)}}{2 (p+1) (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 13.9334, size = 104, normalized size = 0.9 \[ \frac{e \left (d + e x\right )^{- 2 p - 2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p + 1}}{2 \left (a e - b d\right )^{2} \left (p \left (2 p + 3\right ) + 1\right )} - \frac{\left (2 a + 2 b x\right ) \left (d + e x\right )^{- 2 p - 2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p}}{2 \left (2 p + 1\right ) \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.158619, size = 72, normalized size = 0.63 \[ \frac{(a+b x) \left ((a+b x)^2\right )^p (d+e x)^{-2 (p+1)} (-a e (2 p+1)+2 b d (p+1)+b e x)}{2 (p+1) (2 p+1) (b d-a e)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 139, normalized size = 1.2 \[ -{\frac{ \left ( bx+a \right ) \left ( ex+d \right ) ^{-2-2\,p} \left ( 2\,aep-2\,bdp-bex+ae-2\,bd \right ) \left ({b}^{2}{x}^{2}+2\,abx+{a}^{2} \right ) ^{p}}{4\,{a}^{2}{e}^{2}{p}^{2}-8\,abde{p}^{2}+4\,{b}^{2}{d}^{2}{p}^{2}+6\,{a}^{2}{e}^{2}p-12\,abdep+6\,{b}^{2}{d}^{2}p+2\,{a}^{2}{e}^{2}-4\,abde+2\,{b}^{2}{d}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b^2*x^2 + 2*a*b*x + a^2)^p*(e*x + d)^(-2*p - 3), x)

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Fricas [A]  time = 0.231264, size = 296, normalized size = 2.57 \[ \frac{{\left (b^{2} e^{2} x^{3} + 2 \, a b d^{2} - a^{2} d e +{\left (3 \, b^{2} d e + 2 \,{\left (b^{2} d e - a b e^{2}\right )} p\right )} x^{2} + 2 \,{\left (a b d^{2} - a^{2} d e\right )} p +{\left (2 \, b^{2} d^{2} + 2 \, a b d e - a^{2} e^{2} + 2 \,{\left (b^{2} d^{2} - a^{2} e^{2}\right )} p\right )} x\right )}{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 3}}{2 \,{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2} + 2 \,{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} p^{2} + 3 \,{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} p\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b^2*x^2 + 2*a*b*x + a^2)^p*(e*x + d)^(-2*p - 3), x)