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

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

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Rubi [A]  time = 0.09, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {416, 388, 245} \[ \frac {x (b c-a d)^2 \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {d x^n}{c}\right )}{c d^2}-\frac {b x (b c (n+1)-a d (2 n+1))}{d^2 (n+1)}+\frac {b x \left (a+b x^n\right )}{d (n+1)} \]

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*Hyperg
eometric2F1[1, n^(-1), 1 + n^(-1), -((d*x^n)/c)])/(c*d^2)

Rule 245

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[-p, 1/n, 1/n + 1, -((b*x^n)/a)],
x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p
] || GtQ[a, 0])

Rule 388

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1))/(b*(n*
(p + 1) + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

Rule 416

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1)*(c
+ d*x^n)^(q - 1))/(b*(n*(p + q) + 1)), x] + Dist[1/(b*(n*(p + q) + 1)), Int[(a + b*x^n)^p*(c + d*x^n)^(q - 2)
*Simp[c*(b*c*(n*(p + q) + 1) - a*d) + d*(b*c*(n*(p + 2*q - 1) + 1) - a*d*(n*(q - 1) + 1))*x^n, x], x], x] /; F
reeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 1] && NeQ[n*(p + q) + 1, 0] &&  !IGtQ[p, 1] && IntB
inomialQ[a, b, c, d, n, p, q, x]

Rubi steps

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

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Mathematica [A]  time = 0.05, 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)

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fricas [F]  time = 1.08, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}}{d x^{n} + c}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{n} + a\right )}^{2}}{d x^{n} + c}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.60, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{n}+a \right )^{2}}{d \,x^{n}+c}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \int \frac {1}{d^{3} x^{n} + c d^{2}}\,{d x} + \frac {b^{2} d x x^{n} - {\left (b^{2} c {\left (n + 1\right )} - 2 \, a b d {\left (n + 1\right )}\right )} x}{d^{2} {\left (n + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,x^n\right )}^2}{c+d\,x^n} \,d x \]

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)

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sympy [C]  time = 6.83, size = 170, normalized size = 2.02 \[ \frac {a^{2} x \Phi \left (\frac {d x^{n} e^{i \pi }}{c}, 1, \frac {1}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{c n^{2} \Gamma \left (1 + \frac {1}{n}\right )} - \frac {2 a b x \Phi \left (\frac {c x^{- n} e^{i \pi }}{d}, 1, \frac {e^{i \pi }}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{d n^{2} \Gamma \left (1 + \frac {1}{n}\right )} + \frac {2 b^{2} x x^{2 n} \Phi \left (\frac {d x^{n} e^{i \pi }}{c}, 1, 2 + \frac {1}{n}\right ) \Gamma \left (2 + \frac {1}{n}\right )}{c n \Gamma \left (3 + \frac {1}{n}\right )} + \frac {b^{2} x x^{2 n} \Phi \left (\frac {d x^{n} e^{i \pi }}{c}, 1, 2 + \frac {1}{n}\right ) \Gamma \left (2 + \frac {1}{n}\right )}{c n^{2} \Gamma \left (3 + \frac {1}{n}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*x*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, 1/n)*gamma(1/n)/(c*n**2*gamma(1 + 1/n)) - 2*a*b*x*lerchphi(c*x**(
-n)*exp_polar(I*pi)/d, 1, exp_polar(I*pi)/n)*gamma(1/n)/(d*n**2*gamma(1 + 1/n)) + 2*b**2*x*x**(2*n)*lerchphi(d
*x**n*exp_polar(I*pi)/c, 1, 2 + 1/n)*gamma(2 + 1/n)/(c*n*gamma(3 + 1/n)) + b**2*x*x**(2*n)*lerchphi(d*x**n*exp
_polar(I*pi)/c, 1, 2 + 1/n)*gamma(2 + 1/n)/(c*n**2*gamma(3 + 1/n))

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