3.196 \(\int \frac{\left (a+b x^n\right )^2}{c+d x^n} \, dx\)

Optimal. Leaf size=84 \[ \frac{x (b c-a d)^2 \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{c d^2}-\frac{b x (b c (n+1)-a d (2 n+1))}{d^2 (n+1)}+\frac{b x \left (a+b x^n\right )}{d (n+1)} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.21881, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{x (b c-a d)^2 \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{c d^2}-\frac{b x (b c (n+1)-a d (2 n+1))}{d^2 (n+1)}+\frac{b x \left (a+b x^n\right )}{d (n+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 22.8338, size = 70, normalized size = 0.83 \[ \frac{b x \left (a + b x^{n}\right )}{d \left (n + 1\right )} + \frac{b x \left (a d \left (2 n + 1\right ) - b c \left (n + 1\right )\right )}{d^{2} \left (n + 1\right )} + \frac{x \left (a d - b c\right )^{2}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{d x^{n}}{c}} \right )}}{c d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*x**n)**2/(c+d*x**n),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.069443, size = 82, normalized size = 0.98 \[ \frac{a^2 x}{c}+\frac{x (a d-b c)^2 \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{c d^2}-\frac{x (b c-a d)^2}{c d^2}+\frac{b^2 x^{n+1}}{d (n+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.062, size = 0, normalized size = 0. \[ \int{\frac{ \left ( a+b{x}^{n} \right ) ^{2}}{c+d{x}^{n}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^n)^2/(c+d*x^n),x)

[Out]

int((a+b*x^n)^2/(c+d*x^n),x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \int \frac{1}{d^{3} x^{n} + c d^{2}}\,{d x} + \frac{b^{2} d x x^{n} -{\left (b^{2} c{\left (n + 1\right )} - 2 \, a b d{\left (n + 1\right )}\right )} x}{d^{2}{\left (n + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*integrate(1/(d^3*x^n + c*d^2), x) + (b^2*d*x*x^n
- (b^2*c*(n + 1) - 2*a*b*d*(n + 1))*x)/(d^2*(n + 1))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b^2*x^(2*n) + 2*a*b*x^n + a^2)/(d*x^n + c), x)

_______________________________________________________________________________________

Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*x**n)**2/(c+d*x**n),x)

[Out]

Exception raised: TypeError

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^n + a)^2/(d*x^n + c), x)