3.4.35 \(\int \frac {(c+d x^n)^{1-\frac {1}{n}}}{(a+b x^n)^3} \, dx\) [335]

Optimal. Leaf size=131 \[ \frac {b x \left (c+d x^n\right )^{2-\frac {1}{n}}}{2 a (b c-a d) n \left (a+b x^n\right )^2}-\frac {c (b c (1-2 n)+2 a d n) x \left (c+d x^n\right )^{-1/n} \, _2F_1\left (2,\frac {1}{n};1+\frac {1}{n};-\frac {(b c-a d) x^n}{a \left (c+d x^n\right )}\right )}{2 a^3 (b c-a d) n} \]

[Out]

1/2*b*x*(c+d*x^n)^(2-1/n)/a/(-a*d+b*c)/n/(a+b*x^n)^2-1/2*c*(b*c*(1-2*n)+2*a*d*n)*x*hypergeom([2, 1/n],[1+1/n],
-(-a*d+b*c)*x^n/a/(c+d*x^n))/a^3/(-a*d+b*c)/n/((c+d*x^n)^(1/n))

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Rubi [A]
time = 0.04, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {390, 387} \begin {gather*} \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)} \end {gather*}

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 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-\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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1241\) vs. \(2(131)=262\).
time = 40.12, size = 1241, normalized size = 9.47 \begin {gather*} -\frac {c^3 (1+n) (1+2 n) (1+3 n) x \left (c+d x^n\right )^{2-\frac {1}{n}} \Gamma \left (2+\frac {1}{n}\right ) \left (\, _2F_1\left (1,3;1+\frac {1}{n};\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )+\frac {d n x^n \left (\frac {c \, _2F_1\left (1,3;2+\frac {1}{n};\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )}{1+n}+\frac {3 (b c-a d) x^n \Gamma \left (1+\frac {1}{n}\right ) \, _2F_1\left (2,4;3+\frac {1}{n};\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )}{(1+2 n) \left (a+b x^n\right ) \Gamma \left (2+\frac {1}{n}\right )}\right )}{c^2}\right )}{c d (1-2 n) (1+3 n) x^n \left (a+b x^n\right )^2 \left (c^2 (1+n) (1+2 n) \left (a+b x^n\right ) \Gamma \left (2+\frac {1}{n}\right ) \, _2F_1\left (1,3;1+\frac {1}{n};\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )+d n x^n \left (c (1+2 n) \left (a+b x^n\right ) \Gamma \left (2+\frac {1}{n}\right ) \, _2F_1\left (1,3;2+\frac {1}{n};\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )+3 (b c-a d) (1+n) x^n \Gamma \left (1+\frac {1}{n}\right ) \, _2F_1\left (2,4;3+\frac {1}{n};\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )\right )\right )+3 b c n (1+3 n) x^n \left (a+b x^n\right ) \left (c+d x^n\right ) \left (c^2 (1+n) (1+2 n) \left (a+b x^n\right ) \Gamma \left (2+\frac {1}{n}\right ) \, _2F_1\left (1,3;1+\frac {1}{n};\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )+d n x^n \left (c (1+2 n) \left (a+b x^n\right ) \Gamma \left (2+\frac {1}{n}\right ) \, _2F_1\left (1,3;2+\frac {1}{n};\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )+3 (b c-a d) (1+n) x^n \Gamma \left (1+\frac {1}{n}\right ) \, _2F_1\left (2,4;3+\frac {1}{n};\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )\right )\right )-c (1+3 n) \left (a+b x^n\right )^2 \left (c+d x^n\right ) \left (c^2 (1+n) (1+2 n) \left (a+b x^n\right ) \Gamma \left (2+\frac {1}{n}\right ) \, _2F_1\left (1,3;1+\frac {1}{n};\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )+d n x^n \left (c (1+2 n) \left (a+b x^n\right ) \Gamma \left (2+\frac {1}{n}\right ) \, _2F_1\left (1,3;2+\frac {1}{n};\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )+3 (b c-a d) (1+n) x^n \Gamma \left (1+\frac {1}{n}\right ) \, _2F_1\left (2,4;3+\frac {1}{n};\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )\right )\right )+n^2 x^n \left (c+d x^n\right ) \left (3 a c^2 (-b c+a d) (1+2 n) (1+3 n) \left (a+b x^n\right ) \Gamma \left (2+\frac {1}{n}\right ) \, _2F_1\left (2,4;2+\frac {1}{n};\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )-c d (1+3 n) \left (a+b x^n\right )^2 \left (c (1+2 n) \left (a+b x^n\right ) \Gamma \left (2+\frac {1}{n}\right ) \, _2F_1\left (1,3;2+\frac {1}{n};\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )+3 (b c-a d) (1+n) x^n \Gamma \left (1+\frac {1}{n}\right ) \, _2F_1\left (2,4;3+\frac {1}{n};\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )\right )+3 d (b c-a d) x^n \left (b c (1+n) (1+3 n) x^n \left (a+b x^n\right ) \Gamma \left (1+\frac {1}{n}\right ) \, _2F_1\left (2,4;3+\frac {1}{n};\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )-c (1+n) (1+3 n) \left (a+b x^n\right )^2 \Gamma \left (1+\frac {1}{n}\right ) \, _2F_1\left (2,4;3+\frac {1}{n};\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )-a c n (1+3 n) \left (a+b x^n\right ) \Gamma \left (2+\frac {1}{n}\right ) \, _2F_1\left (2,4;3+\frac {1}{n};\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )+8 a (-b c+a d) n (1+n) x^n \Gamma \left (1+\frac {1}{n}\right ) \, _2F_1\left (3,5;4+\frac {1}{n};\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )\right )\right )} \end {gather*}

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 = 0.11, size = 0, normalized size = 0.00 \[\int \frac {\left (c +d \,x^{n}\right )^{1-\frac {1}{n}}}{\left (a +b \,x^{n}\right )^{3}}\, dx\]

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

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]

Exception raised: HeuristicGCDFailed >> no luck

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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((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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c+d\,x^n\right )}^{1-\frac {1}{n}}}{{\left (a+b\,x^n\right )}^3} \,d x \end {gather*}

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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