3.1748 \(\int \frac{(a^2+2 a b x+b^2 x^2)^p}{d+e x} \, dx\)

Optimal. Leaf size=71 \[ \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p \, _2F_1\left (1,2 p+1;2 (p+1);-\frac{e (a+b x)}{b d-a e}\right )}{(2 p+1) (b d-a e)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0323544, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {646, 68} \[ \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p \, _2F_1\left (1,2 p+1;2 (p+1);-\frac{e (a+b x)}{b d-a e}\right )}{(2 p+1) (b d-a e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 646

Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(a + b*x + c*x^2)^Fra
cPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b,
c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p] && NeQ[2*c*d - b*e, 0]

Rule 68

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((b*c - a*d)^n*(a + b*x)^(m + 1)*Hype
rgeometric2F1[-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b^(n + 1)*(m + 1)), x] /; FreeQ[{a, b, c, d, m
}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] && IntegerQ[n]

Rubi steps

\begin{align*} \int \frac{\left (a^2+2 a b x+b^2 x^2\right )^p}{d+e x} \, dx &=\left (\left (a b+b^2 x\right )^{-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p\right ) \int \frac{\left (a b+b^2 x\right )^{2 p}}{d+e x} \, dx\\ &=\frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p \, _2F_1\left (1,1+2 p;2 (1+p);-\frac{e (a+b x)}{b d-a e}\right )}{(b d-a e) (1+2 p)}\\ \end{align*}

Mathematica [A]  time = 0.0169132, size = 62, normalized size = 0.87 \[ -\frac{(a+b x) \left ((a+b x)^2\right )^p \, _2F_1\left (1,2 p+1;2 p+2;\frac{e (a+b x)}{a e-b d}\right )}{(2 p+1) (a e-b d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 1.215, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({b}^{2}{x}^{2}+2\,abx+{a}^{2} \right ) ^{p}}{ex+d}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

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),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

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

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),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\left (a + b x\right )^{2}\right )^{p}}{d + e x}\, dx \end{align*}

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),x)

[Out]

Integral(((a + b*x)**2)**p/(d + e*x), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}}{e x + d}\,{d x} \end{align*}

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),x, algorithm="giac")

[Out]

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