3.1732 \(\int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^p \, dx\)

Optimal. Leaf size=85 \[ \frac{\left (a^2+2 a b x+b^2 x^2\right )^p (d+e x)^{m+1} \left (-\frac{e (a+b x)}{b d-a e}\right )^{-2 p} \, _2F_1\left (m+1,-2 p;m+2;\frac{b (d+e x)}{b d-a e}\right )}{e (m+1)} \]

[Out]

((d + e*x)^(1 + m)*(a^2 + 2*a*b*x + b^2*x^2)^p*Hypergeometric2F1[1 + m, -2*p, 2
+ m, (b*(d + e*x))/(b*d - a*e)])/(e*(1 + m)*(-((e*(a + b*x))/(b*d - a*e)))^(2*p)
)

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Rubi [A]  time = 0.13475, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{\left (a^2+2 a b x+b^2 x^2\right )^p (d+e x)^{m+1} \left (-\frac{e (a+b x)}{b d-a e}\right )^{-2 p} \, _2F_1\left (m+1,-2 p;m+2;\frac{b (d+e x)}{b d-a e}\right )}{e (m+1)} \]

Antiderivative was successfully verified.

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

[Out]

((d + e*x)^(1 + m)*(a^2 + 2*a*b*x + b^2*x^2)^p*Hypergeometric2F1[1 + m, -2*p, 2
+ m, (b*(d + e*x))/(b*d - a*e)])/(e*(1 + m)*(-((e*(a + b*x))/(b*d - a*e)))^(2*p)
)

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Rubi in Sympy [A]  time = 29.6216, size = 71, normalized size = 0.84 \[ \frac{\left (\frac{e \left (a + b x\right )}{a e - b d}\right )^{- 2 p} \left (d + e x\right )^{m + 1} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - 2 p, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{b \left (- d - e x\right )}{a e - b d}} \right )}}{e \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.127946, size = 75, normalized size = 0.88 \[ \frac{\left ((a+b x)^2\right )^p (d+e x)^{m+1} \left (\frac{e (a+b x)}{a e-b d}\right )^{-2 p} \, _2F_1\left (m+1,-2 p;m+2;\frac{b (d+e x)}{b d-a e}\right )}{e (m+1)} \]

Antiderivative was successfully verified.

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

[Out]

(((a + b*x)^2)^p*(d + e*x)^(1 + m)*Hypergeometric2F1[1 + m, -2*p, 2 + m, (b*(d +
e*x))/(b*d - a*e)])/(e*(1 + m)*((e*(a + b*x))/(-(b*d) + a*e))^(2*p))

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Maple [F]  time = 0.233, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) ^{m} \left ({b}^{2}{x}^{2}+2\,abx+{a}^{2} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int((e*x+d)^m*(b^2*x^2+2*a*b*x+a^2)^p,x)

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

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)^m,x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}{\left (e x + d\right )}^{m}, x\right ) \]

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)^m,x, algorithm="fricas")

[Out]

integral((b^2*x^2 + 2*a*b*x + a^2)^p*(e*x + d)^m, x)

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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)**m*(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. \[ \int{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}{\left (e x + d\right )}^{m}\,{d x} \]

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)^m,x, algorithm="giac")

[Out]

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