3.4.23 \(\int (a+b x^n) (c+d x^n)^{-2-\frac {1}{n}} \, dx\) [323]

Optimal. Leaf size=58 \[ \frac {x \left (a+b x^n\right ) \left (c+d x^n\right )^{-1-\frac {1}{n}}}{c (1+n)}+\frac {a n x \left (c+d x^n\right )^{-1/n}}{c^2 (1+n)} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {386, 197} \begin {gather*} \frac {x \left (a+b x^n\right ) \left (c+d x^n\right )^{-\frac {1}{n}-1}}{c (n+1)}+\frac {a n x \left (c+d x^n\right )^{-1/n}}{c^2 (n+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 197

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

Rule 386

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

Rubi steps

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

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.11, size = 82, normalized size = 1.41 \begin {gather*} \frac {x \left (c+d x^n\right )^{-\frac {1+n}{n}} \left (b c x^n+a (1+n) \left (c+d x^n\right ) \left (1+\frac {d x^n}{c}\right )^{\frac {1}{n}} \, _2F_1\left (2+\frac {1}{n},\frac {1}{n};1+\frac {1}{n};-\frac {d x^n}{c}\right )\right )}{c^2 (1+n)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int \left (a +b \,x^{n}\right ) \left (c +d \,x^{n}\right )^{-2-\frac {1}{n}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 3.01, size = 85, normalized size = 1.47 \begin {gather*} \frac {{\left (a d^{2} n + b c d\right )} x x^{2 \, n} + {\left (2 \, a c d n + b c^{2} + a c d\right )} x x^{n} + {\left (a c^{2} n + a c^{2}\right )} x}{{\left (c^{2} n + c^{2}\right )} {\left (d x^{n} + c\right )}^{\frac {2 \, n + 1}{n}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Unable to divide, perhaps due to rounding error%%%{1,[0,0,2,2,1,1,0,1]%%%}+%%%{1,[0,0,2,1,1,1,0,1]%%%}+%%%{
1,[0,0,2,1,

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {a+b\,x^n}{{\left (c+d\,x^n\right )}^{\frac {1}{n}+2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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