3.4.34 \(\int \frac {(c+d x^n)^{-1/n}}{(a+b x^n)^2} \, dx\) [334]

Optimal. Leaf size=127 \[ \frac {b x \left (c+d x^n\right )^{-\frac {1-n}{n}}}{a (b c-a d) n \left (a+b x^n\right )}-\frac {(b c (1-n)+a d n) x \left (c+d x^n\right )^{-1/n} \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {(b c-a d) x^n}{a \left (c+d x^n\right )}\right )}{a^2 (b c-a d) n} \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 127, 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 = {390, 387} \begin {gather*} \frac {b x \left (c+d x^n\right )^{-\frac {1-n}{n}}}{a n (b c-a d) \left (a+b x^n\right )}-\frac {x \left (c+d x^n\right )^{-1/n} (a d n+b c (1-n)) \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{a^2 n (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 387

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*(x/(c^(p + 1)*(c + d*x^
n)^(1/n)))*Hypergeometric2F1[1/n, -p, 1 + 1/n, (-(b*c - a*d))*(x^n/(a*(c + d*x^n)))], x] /; FreeQ[{a, b, c, d,
n, q}, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p + q + 1) + 1, 0] && ILtQ[p, 0]

Rule 390

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*
((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Dist[(b*c + n*(p + 1)*(b*c - a*d))/(a*n*(p + 1)*(b*c - a
*d)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && Eq
Q[n*(p + q + 2) + 1, 0] && (LtQ[p, -1] ||  !LtQ[q, -1]) && NeQ[p, -1]

Rubi steps

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1070\) vs. \(2(127)=254\).
time = 36.73, size = 1070, normalized size = 8.43 \begin {gather*} \frac {c^2 (1+2 n) (1+3 n) x \left (a+b x^n\right ) \left (c+d x^n\right )^{-1/n} \left (1+\frac {d x^n}{c}\right ) \Gamma \left (2+\frac {1}{n}\right ) \Gamma \left (3+\frac {1}{n}\right ) \left (\frac {c \left (c+c n+d n x^n\right ) \, _2F_1\left (1,2;2+\frac {1}{n};\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )}{\Gamma \left (2+\frac {1}{n}\right )}+\frac {2 (b c-a d) n x^n \left (c+d x^n\right ) \, _2F_1\left (2,3;3+\frac {1}{n};\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )}{\left (a+b x^n\right ) \Gamma \left (3+\frac {1}{n}\right )}\right )}{-c d (1-n) (1+2 n) (1+3 n) x^n \left (a+b x^n\right )^2 \left (c \left (a+b x^n\right ) \left (c+c n+d n x^n\right ) \Gamma \left (3+\frac {1}{n}\right ) \, _2F_1\left (1,2;2+\frac {1}{n};\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )+2 (b c-a d) n x^n \left (c+d x^n\right ) \Gamma \left (2+\frac {1}{n}\right ) \, _2F_1\left (2,3;3+\frac {1}{n};\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )\right )-2 b c n (1+2 n) (1+3 n) x^n \left (a+b x^n\right ) \left (c+d x^n\right ) \left (c \left (a+b x^n\right ) \left (c+c n+d n x^n\right ) \Gamma \left (3+\frac {1}{n}\right ) \, _2F_1\left (1,2;2+\frac {1}{n};\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )+2 (b c-a d) n x^n \left (c+d x^n\right ) \Gamma \left (2+\frac {1}{n}\right ) \, _2F_1\left (2,3;3+\frac {1}{n};\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )\right )+c (1+2 n) (1+3 n) \left (a+b x^n\right )^2 \left (c+d x^n\right ) \left (c \left (a+b x^n\right ) \left (c+c n+d n x^n\right ) \Gamma \left (3+\frac {1}{n}\right ) \, _2F_1\left (1,2;2+\frac {1}{n};\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )+2 (b c-a d) n x^n \left (c+d x^n\right ) \Gamma \left (2+\frac {1}{n}\right ) \, _2F_1\left (2,3;3+\frac {1}{n};\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )\right )+n^2 x^n \left (c+d x^n\right ) \left (c^2 d (1+2 n) (1+3 n) \left (a+b x^n\right )^3 \Gamma \left (3+\frac {1}{n}\right ) \, _2F_1\left (1,2;2+\frac {1}{n};\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )+2 c d (b c-a d) (1+2 n) (1+3 n) x^n \left (a+b x^n\right )^2 \Gamma \left (2+\frac {1}{n}\right ) \, _2F_1\left (2,3;3+\frac {1}{n};\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )-2 b c (b c-a d) (1+2 n) (1+3 n) x^n \left (a+b x^n\right ) \left (c+d x^n\right ) \Gamma \left (2+\frac {1}{n}\right ) \, _2F_1\left (2,3;3+\frac {1}{n};\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )+2 c (b c-a d) (1+2 n) (1+3 n) \left (a+b x^n\right )^2 \left (c+d x^n\right ) \Gamma \left (2+\frac {1}{n}\right ) \, _2F_1\left (2,3;3+\frac {1}{n};\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )+2 a c (b c-a d) (1+3 n) \left (a+b x^n\right ) \left (c+c n+d n x^n\right ) \Gamma \left (3+\frac {1}{n}\right ) \, _2F_1\left (2,3;3+\frac {1}{n};\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )+12 a (b c-a d)^2 n (1+2 n) x^n \left (c+d x^n\right ) \Gamma \left (2+\frac {1}{n}\right ) \, _2F_1\left (3,4;4+\frac {1}{n};\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^2*(1 + 2*n)*(1 + 3*n)*x*(a + b*x^n)*(1 + (d*x^n)/c)*Gamma[2 + n^(-1)]*Gamma[3 + n^(-1)]*((c*(c + c*n + d*n*
x^n)*Hypergeometric2F1[1, 2, 2 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))])/Gamma[2 + n^(-1)] + (2*(b*c - a*d
)*n*x^n*(c + d*x^n)*Hypergeometric2F1[2, 3, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))])/((a + b*x^n)*Gamma
[3 + n^(-1)])))/((c + d*x^n)^n^(-1)*(-(c*d*(1 - n)*(1 + 2*n)*(1 + 3*n)*x^n*(a + b*x^n)^2*(c*(a + b*x^n)*(c + c
*n + d*n*x^n)*Gamma[3 + n^(-1)]*Hypergeometric2F1[1, 2, 2 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + 2*(b*
c - a*d)*n*x^n*(c + d*x^n)*Gamma[2 + n^(-1)]*Hypergeometric2F1[2, 3, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x
^n))])) - 2*b*c*n*(1 + 2*n)*(1 + 3*n)*x^n*(a + b*x^n)*(c + d*x^n)*(c*(a + b*x^n)*(c + c*n + d*n*x^n)*Gamma[3 +
n^(-1)]*Hypergeometric2F1[1, 2, 2 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + 2*(b*c - a*d)*n*x^n*(c + d*x
^n)*Gamma[2 + n^(-1)]*Hypergeometric2F1[2, 3, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))]) + c*(1 + 2*n)*(1
+ 3*n)*(a + b*x^n)^2*(c + d*x^n)*(c*(a + b*x^n)*(c + c*n + d*n*x^n)*Gamma[3 + n^(-1)]*Hypergeometric2F1[1, 2,
2 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + 2*(b*c - a*d)*n*x^n*(c + d*x^n)*Gamma[2 + n^(-1)]*Hypergeome
tric2F1[2, 3, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))]) + n^2*x^n*(c + d*x^n)*(c^2*d*(1 + 2*n)*(1 + 3*n)
*(a + b*x^n)^3*Gamma[3 + n^(-1)]*Hypergeometric2F1[1, 2, 2 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + 2*c*
d*(b*c - a*d)*(1 + 2*n)*(1 + 3*n)*x^n*(a + b*x^n)^2*Gamma[2 + n^(-1)]*Hypergeometric2F1[2, 3, 3 + n^(-1), ((b*
c - a*d)*x^n)/(c*(a + b*x^n))] - 2*b*c*(b*c - a*d)*(1 + 2*n)*(1 + 3*n)*x^n*(a + b*x^n)*(c + d*x^n)*Gamma[2 + n
^(-1)]*Hypergeometric2F1[2, 3, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + 2*c*(b*c - a*d)*(1 + 2*n)*(1 +
3*n)*(a + b*x^n)^2*(c + d*x^n)*Gamma[2 + n^(-1)]*Hypergeometric2F1[2, 3, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a
+ b*x^n))] + 2*a*c*(b*c - a*d)*(1 + 3*n)*(a + b*x^n)*(c + c*n + d*n*x^n)*Gamma[3 + n^(-1)]*Hypergeometric2F1[2
, 3, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + 12*a*(b*c - a*d)^2*n*(1 + 2*n)*x^n*(c + d*x^n)*Gamma[2 +
n^(-1)]*Hypergeometric2F1[3, 4, 4 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))])))

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Maple [F]
time = 0.10, size = 0, normalized size = 0.00 \[\int \frac {\left (c +d \,x^{n}\right )^{-\frac {1}{n}}}{\left (a +b \,x^{n}\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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