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

Optimal. Leaf size=73 \[ \frac {x (b c-a d (1-n)) \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {d x^n}{c}\right )}{c^2 d n}-\frac {x (b c-a d)}{c d n \left (c+d x^n\right )} \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {385, 245} \[ \frac {x (b c-a d (1-n)) \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {d x^n}{c}\right )}{c^2 d n}-\frac {x (b c-a d)}{c d n \left (c+d x^n\right )} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^n)/(c + d*x^n)^2,x]

[Out]

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

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 385

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

Rubi steps

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

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Mathematica [A]  time = 0.05, size = 56, normalized size = 0.77 \[ \frac {x \left (\frac {b}{c+d x^n}-\frac {(a d (n-1)+b c) \, _2F_1\left (2,\frac {1}{n};1+\frac {1}{n};-\frac {d x^n}{c}\right )}{c^2}\right )}{d-d n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.96, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x^{n} + a}{d^{2} x^{2 \, n} + 2 \, c d x^{n} + c^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.59, size = 0, normalized size = 0.00 \[ \int \frac {b \,x^{n}+a}{\left (d \,x^{n}+c \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^n)/(c + d*x^n)^2,x)

[Out]

int((a + b*x^n)/(c + d*x^n)^2, x)

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sympy [C]  time = 8.69, size = 592, normalized size = 8.11 \[ a \left (\frac {n x \Phi \left (\frac {d x^{n} e^{i \pi }}{c}, 1, \frac {1}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{c \left (c n^{3} \Gamma \left (1 + \frac {1}{n}\right ) + d n^{3} x^{n} \Gamma \left (1 + \frac {1}{n}\right )\right )} + \frac {n x \Gamma \left (\frac {1}{n}\right )}{c \left (c n^{3} \Gamma \left (1 + \frac {1}{n}\right ) + d n^{3} x^{n} \Gamma \left (1 + \frac {1}{n}\right )\right )} - \frac {x \Phi \left (\frac {d x^{n} e^{i \pi }}{c}, 1, \frac {1}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{c \left (c n^{3} \Gamma \left (1 + \frac {1}{n}\right ) + d n^{3} x^{n} \Gamma \left (1 + \frac {1}{n}\right )\right )} + \frac {d n x x^{n} \Phi \left (\frac {d x^{n} e^{i \pi }}{c}, 1, \frac {1}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{c^{2} \left (c n^{3} \Gamma \left (1 + \frac {1}{n}\right ) + d n^{3} x^{n} \Gamma \left (1 + \frac {1}{n}\right )\right )} - \frac {d x x^{n} \Phi \left (\frac {d x^{n} e^{i \pi }}{c}, 1, \frac {1}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{c^{2} \left (c n^{3} \Gamma \left (1 + \frac {1}{n}\right ) + d n^{3} x^{n} \Gamma \left (1 + \frac {1}{n}\right )\right )}\right ) + b \left (\frac {n^{2} x x^{n} \Gamma \left (1 + \frac {1}{n}\right )}{c \left (c n^{3} \Gamma \left (2 + \frac {1}{n}\right ) + d n^{3} x^{n} \Gamma \left (2 + \frac {1}{n}\right )\right )} - \frac {n x x^{n} \Phi \left (\frac {d x^{n} e^{i \pi }}{c}, 1, 1 + \frac {1}{n}\right ) \Gamma \left (1 + \frac {1}{n}\right )}{c \left (c n^{3} \Gamma \left (2 + \frac {1}{n}\right ) + d n^{3} x^{n} \Gamma \left (2 + \frac {1}{n}\right )\right )} + \frac {n x x^{n} \Gamma \left (1 + \frac {1}{n}\right )}{c \left (c n^{3} \Gamma \left (2 + \frac {1}{n}\right ) + d n^{3} x^{n} \Gamma \left (2 + \frac {1}{n}\right )\right )} - \frac {x x^{n} \Phi \left (\frac {d x^{n} e^{i \pi }}{c}, 1, 1 + \frac {1}{n}\right ) \Gamma \left (1 + \frac {1}{n}\right )}{c \left (c n^{3} \Gamma \left (2 + \frac {1}{n}\right ) + d n^{3} x^{n} \Gamma \left (2 + \frac {1}{n}\right )\right )} - \frac {d n x x^{2 n} \Phi \left (\frac {d x^{n} e^{i \pi }}{c}, 1, 1 + \frac {1}{n}\right ) \Gamma \left (1 + \frac {1}{n}\right )}{c^{2} \left (c n^{3} \Gamma \left (2 + \frac {1}{n}\right ) + d n^{3} x^{n} \Gamma \left (2 + \frac {1}{n}\right )\right )} - \frac {d x x^{2 n} \Phi \left (\frac {d x^{n} e^{i \pi }}{c}, 1, 1 + \frac {1}{n}\right ) \Gamma \left (1 + \frac {1}{n}\right )}{c^{2} \left (c n^{3} \Gamma \left (2 + \frac {1}{n}\right ) + d n^{3} x^{n} \Gamma \left (2 + \frac {1}{n}\right )\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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