3.30 \(\int \left (b x+d x^3\right )^n \, dx\)

Optimal. Leaf size=53 \[ \frac{x \left (b+d x^2\right ) \left (b x+d x^3\right )^n \, _2F_1\left (1,\frac{3 (n+1)}{2};\frac{n+3}{2};-\frac{d x^2}{b}\right )}{b (n+1)} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0579224, antiderivative size = 59, normalized size of antiderivative = 1.11, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{x \left (\frac{d x^2}{b}+1\right )^{-n} \left (b x+d x^3\right )^n \, _2F_1\left (-n,\frac{n+1}{2};\frac{n+3}{2};-\frac{d x^2}{b}\right )}{n+1} \]

Antiderivative was successfully verified.

[In]  Int[(b*x + d*x^3)^n,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 9.85145, size = 53, normalized size = 1. \[ \frac{x^{- n} x^{n + 1} \left (1 + \frac{d x^{2}}{b}\right )^{- n} \left (b x + d x^{3}\right )^{n}{{}_{2}F_{1}\left (\begin{matrix} - n, \frac{n}{2} + \frac{1}{2} \\ \frac{n}{2} + \frac{3}{2} \end{matrix}\middle |{- \frac{d x^{2}}{b}} \right )}}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x**3+b*x)**n,x)

[Out]

x**(-n)*x**(n + 1)*(1 + d*x**2/b)**(-n)*(b*x + d*x**3)**n*hyper((-n, n/2 + 1/2),
(n/2 + 3/2,), -d*x**2/b)/(n + 1)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0469738, size = 61, normalized size = 1.15 \[ \frac{x \left (x \left (b+d x^2\right )\right )^n \left (\frac{d x^2}{b}+1\right )^{-n} \, _2F_1\left (-n,\frac{n+1}{2};\frac{n+1}{2}+1;-\frac{d x^2}{b}\right )}{n+1} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x + d*x^3)^n,x]

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.053, size = 0, normalized size = 0. \[ \int \left ( d{x}^{3}+bx \right ) ^{n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^3+b*x)^n,x)

[Out]

int((d*x^3+b*x)^n,x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^3 + b*x)^n,x, algorithm="maxima")

[Out]

integrate((d*x^3 + b*x)^n, x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (d x^{3} + b x\right )}^{n}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^3 + b*x)^n,x, algorithm="fricas")

[Out]

integral((d*x^3 + b*x)^n, x)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (b x + d x^{3}\right )^{n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x**3+b*x)**n,x)

[Out]

Integral((b*x + d*x**3)**n, x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^3 + b*x)^n,x, algorithm="giac")

[Out]

integrate((d*x^3 + b*x)^n, x)