3.2162 \(\int \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p}{d+e x} \, dx\)

Optimal. Leaf size=74 \[ \frac{(a+b x)^2 \left (a^2+2 a b x+b^2 x^2\right )^p \, _2F_1\left (1,2 (p+1);2 p+3;-\frac{e (a+b x)}{b d-a e}\right )}{2 (p+1) (b d-a e)} \]

[Out]

((a + b*x)^2*(a^2 + 2*a*b*x + b^2*x^2)^p*Hypergeometric2F1[1, 2*(1 + p), 3 + 2*p
, -((e*(a + b*x))/(b*d - a*e))])/(2*(b*d - a*e)*(1 + p))

_______________________________________________________________________________________

Rubi [A]  time = 0.130717, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097 \[ \frac{(a+b x)^2 \left (a^2+2 a b x+b^2 x^2\right )^p \, _2F_1\left (1,2 (p+1);2 p+3;-\frac{e (a+b x)}{b d-a e}\right )}{2 (p+1) (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)^2*(a^2 + 2*a*b*x + b^2*x^2)^p*Hypergeometric2F1[1, 2*(1 + p), 3 + 2*p
, -((e*(a + b*x))/(b*d - a*e))])/(2*(b*d - a*e)*(1 + p))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 29.6982, size = 104, normalized size = 1.41 \[ \frac{\left (a + b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p}}{e \left (2 p + 1\right )} - \frac{\left (a b + b^{2} x\right )^{- 2 p} \left (a b + b^{2} x\right )^{2 p + 1} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} 1, 2 p + 1 \\ 2 p + 2 \end{matrix}\middle |{\frac{e \left (a + b x\right )}{a e - b d}} \right )}}{b e \left (2 p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.236606, size = 97, normalized size = 1.31 \[ \frac{\left ((a+b x)^2\right )^p \left (\frac{2 e (a+b x)}{2 p+1}-\frac{(b d-a e) \left (\frac{e (a+b x)}{b (d+e x)}\right )^{-2 p} \, _2F_1\left (-2 p,-2 p;1-2 p;\frac{b d-a e}{b d+b e x}\right )}{p}\right )}{2 e^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.184, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) \left ({b}^{2}{x}^{2}+2\,abx+{a}^{2} \right ) ^{p}}{ex+d}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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