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

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

[Out]

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

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Rubi [A]  time = 0.236371, 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 \[ \frac{x (b c-a d)^2 \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a b^2}-\frac{d x (a d (n+1)-b (2 c n+c))}{b^2 (n+1)}+\frac{d x \left (c+d x^n\right )}{b (n+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 22.1612, size = 70, normalized size = 0.83 \[ \frac{d x \left (c + d x^{n}\right )}{b \left (n + 1\right )} - \frac{d x \left (a d \left (n + 1\right ) - b c \left (2 n + 1\right )\right )}{b^{2} \left (n + 1\right )} + \frac{x \left (a d - b c\right )^{2}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0705425, size = 82, normalized size = 0.98 \[ \frac{x (b c-a d)^2 \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a b^2}-\frac{x (a d-b c)^2}{a b^2}+\frac{c^2 x}{a}+\frac{d^2 x^{n+1}}{b (n+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c+d*x^n)^2/(a+b*x^n),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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