3.335 \(\int \frac{(c+d x^n)^{1-\frac{1}{n}}}{(a+b x^n)^3} \, dx\)
Optimal. Leaf size=131 \[ \frac{b x \left (c+d x^n\right )^{2-\frac{1}{n}}}{2 a n (b c-a d) \left (a+b x^n\right )^2}-\frac{c x \left (c+d x^n\right )^{-1/n} (2 a d n+b c (1-2 n)) \, _2F_1\left (2,\frac{1}{n};1+\frac{1}{n};-\frac{(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{2 a^3 n (b c-a d)} \]
[Out]
(b*x*(c + d*x^n)^(2 - n^(-1)))/(2*a*(b*c - a*d)*n*(a + b*x^n)^2) - (c*(b*c*(1 - 2*n) + 2*a*d*n)*x*Hypergeometr
ic2F1[2, n^(-1), 1 + n^(-1), -(((b*c - a*d)*x^n)/(a*(c + d*x^n)))])/(2*a^3*(b*c - a*d)*n*(c + d*x^n)^n^(-1))
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Rubi [A] time = 0.0557339, antiderivative size = 131, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used =
{382, 379} \[ \frac{b x \left (c+d x^n\right )^{2-\frac{1}{n}}}{2 a n (b c-a d) \left (a+b x^n\right )^2}-\frac{c x \left (c+d x^n\right )^{-1/n} (2 a d n+b c (1-2 n)) \, _2F_1\left (2,\frac{1}{n};1+\frac{1}{n};-\frac{(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{2 a^3 n (b c-a d)} \]
Antiderivative was successfully verified.
[In]
Int[(c + d*x^n)^(1 - n^(-1))/(a + b*x^n)^3,x]
[Out]
(b*x*(c + d*x^n)^(2 - n^(-1)))/(2*a*(b*c - a*d)*n*(a + b*x^n)^2) - (c*(b*c*(1 - 2*n) + 2*a*d*n)*x*Hypergeometr
ic2F1[2, n^(-1), 1 + n^(-1), -(((b*c - a*d)*x^n)/(a*(c + d*x^n)))])/(2*a^3*(b*c - a*d)*n*(c + d*x^n)^n^(-1))
Rule 382
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] && EqQ[
n*(p + q + 2) + 1, 0] && (LtQ[p, -1] || !LtQ[q, -1]) && NeQ[p, -1]
Rule 379
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(a^p*x*Hypergeometric2F1[1/
n, -p, 1 + 1/n, -(((b*c - a*d)*x^n)/(a*(c + d*x^n)))])/(c^(p + 1)*(c + d*x^n)^(1/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]
Rubi steps
\begin{align*} \int \frac{\left (c+d x^n\right )^{1-\frac{1}{n}}}{\left (a+b x^n\right )^3} \, dx &=\frac{b x \left (c+d x^n\right )^{2-\frac{1}{n}}}{2 a (b c-a d) n \left (a+b x^n\right )^2}-\frac{(b c-2 (b c-a d) n) \int \frac{\left (c+d x^n\right )^{1-\frac{1}{n}}}{\left (a+b x^n\right )^2} \, dx}{2 a (b c-a d) n}\\ &=\frac{b x \left (c+d x^n\right )^{2-\frac{1}{n}}}{2 a (b c-a d) n \left (a+b x^n\right )^2}-\frac{c (b c (1-2 n)+2 a d n) x \left (c+d x^n\right )^{-1/n} \, _2F_1\left (2,\frac{1}{n};1+\frac{1}{n};-\frac{(b c-a d) x^n}{a \left (c+d x^n\right )}\right )}{2 a^3 (b c-a d) n}\\ \end{align*}
Mathematica [B] time = 42.82, size = 1241, normalized size = 9.47 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[(c + d*x^n)^(1 - n^(-1))/(a + b*x^n)^3,x]
[Out]
-((c^3*(1 + n)*(1 + 2*n)*(1 + 3*n)*x*(c + d*x^n)^(2 - n^(-1))*Gamma[2 + n^(-1)]*(Hypergeometric2F1[1, 3, 1 + n
^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + (d*n*x^n*((c*Hypergeometric2F1[1, 3, 2 + n^(-1), ((b*c - a*d)*x^n)
/(c*(a + b*x^n))])/(1 + n) + (3*(b*c - a*d)*x^n*Gamma[1 + n^(-1)]*Hypergeometric2F1[2, 4, 3 + n^(-1), ((b*c -
a*d)*x^n)/(c*(a + b*x^n))])/((1 + 2*n)*(a + b*x^n)*Gamma[2 + n^(-1)])))/c^2))/(c*d*(1 - 2*n)*(1 + 3*n)*x^n*(a
+ b*x^n)^2*(c^2*(1 + n)*(1 + 2*n)*(a + b*x^n)*Gamma[2 + n^(-1)]*Hypergeometric2F1[1, 3, 1 + n^(-1), ((b*c - a*
d)*x^n)/(c*(a + b*x^n))] + d*n*x^n*(c*(1 + 2*n)*(a + b*x^n)*Gamma[2 + n^(-1)]*Hypergeometric2F1[1, 3, 2 + n^(-
1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + 3*(b*c - a*d)*(1 + n)*x^n*Gamma[1 + n^(-1)]*Hypergeometric2F1[2, 4, 3
+ n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))])) + 3*b*c*n*(1 + 3*n)*x^n*(a + b*x^n)*(c + d*x^n)*(c^2*(1 + n)*(
1 + 2*n)*(a + b*x^n)*Gamma[2 + n^(-1)]*Hypergeometric2F1[1, 3, 1 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))]
+ d*n*x^n*(c*(1 + 2*n)*(a + b*x^n)*Gamma[2 + n^(-1)]*Hypergeometric2F1[1, 3, 2 + n^(-1), ((b*c - a*d)*x^n)/(c*
(a + b*x^n))] + 3*(b*c - a*d)*(1 + n)*x^n*Gamma[1 + n^(-1)]*Hypergeometric2F1[2, 4, 3 + n^(-1), ((b*c - a*d)*x
^n)/(c*(a + b*x^n))])) - c*(1 + 3*n)*(a + b*x^n)^2*(c + d*x^n)*(c^2*(1 + n)*(1 + 2*n)*(a + b*x^n)*Gamma[2 + n^
(-1)]*Hypergeometric2F1[1, 3, 1 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + d*n*x^n*(c*(1 + 2*n)*(a + b*x^n
)*Gamma[2 + n^(-1)]*Hypergeometric2F1[1, 3, 2 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + 3*(b*c - a*d)*(1
+ n)*x^n*Gamma[1 + n^(-1)]*Hypergeometric2F1[2, 4, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))])) + n^2*x^n*
(c + d*x^n)*(3*a*c^2*(-(b*c) + a*d)*(1 + 2*n)*(1 + 3*n)*(a + b*x^n)*Gamma[2 + n^(-1)]*Hypergeometric2F1[2, 4,
2 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] - c*d*(1 + 3*n)*(a + b*x^n)^2*(c*(1 + 2*n)*(a + b*x^n)*Gamma[2
+ n^(-1)]*Hypergeometric2F1[1, 3, 2 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + 3*(b*c - a*d)*(1 + n)*x^n*G
amma[1 + n^(-1)]*Hypergeometric2F1[2, 4, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))]) + 3*d*(b*c - a*d)*x^n
*(b*c*(1 + n)*(1 + 3*n)*x^n*(a + b*x^n)*Gamma[1 + n^(-1)]*Hypergeometric2F1[2, 4, 3 + n^(-1), ((b*c - a*d)*x^n
)/(c*(a + b*x^n))] - c*(1 + n)*(1 + 3*n)*(a + b*x^n)^2*Gamma[1 + n^(-1)]*Hypergeometric2F1[2, 4, 3 + n^(-1), (
(b*c - a*d)*x^n)/(c*(a + b*x^n))] - a*c*n*(1 + 3*n)*(a + b*x^n)*Gamma[2 + n^(-1)]*Hypergeometric2F1[2, 4, 3 +
n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + 8*a*(-(b*c) + a*d)*n*(1 + n)*x^n*Gamma[1 + n^(-1)]*Hypergeometric
2F1[3, 5, 4 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))]))))
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Maple [F] time = 0.702, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( a+b{x}^{n} \right ) ^{3}} \left ( c+d{x}^{n} \right ) ^{1-{n}^{-1}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((c+d*x^n)^(1-1/n)/(a+b*x^n)^3,x)
[Out]
int((c+d*x^n)^(1-1/n)/(a+b*x^n)^3,x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x^{n} + c\right )}^{-\frac{1}{n} + 1}}{{\left (b x^{n} + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c+d*x^n)^(1-1/n)/(a+b*x^n)^3,x, algorithm="maxima")
[Out]
integrate((d*x^n + c)^(-1/n + 1)/(b*x^n + a)^3, x)
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (d x^{n} + c\right )}^{\frac{n - 1}{n}}}{b^{3} x^{3 \, n} + 3 \, a b^{2} x^{2 \, n} + 3 \, a^{2} b x^{n} + a^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c+d*x^n)^(1-1/n)/(a+b*x^n)^3,x, algorithm="fricas")
[Out]
integral((d*x^n + c)^((n - 1)/n)/(b^3*x^(3*n) + 3*a*b^2*x^(2*n) + 3*a^2*b*x^n + a^3), x)
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c+d*x**n)**(1-1/n)/(a+b*x**n)**3,x)
[Out]
Timed out
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x^{n} + c\right )}^{-\frac{1}{n} + 1}}{{\left (b x^{n} + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c+d*x^n)^(1-1/n)/(a+b*x^n)^3,x, algorithm="giac")
[Out]
integrate((d*x^n + c)^(-1/n + 1)/(b*x^n + a)^3, x)