3.306 \(\int \frac{(c+d x^n)^3}{(a+b x^n)^2} \, dx\)

Optimal. Leaf size=200 \[ -\frac{d x \left (a^2 d^2 \left (2 n^2+3 n+1\right )-a b c d \left (3 n^2+4 n+2\right )+b^2 c^2 (n+1)\right )}{a b^3 n (n+1)}-\frac{x (b c-a d)^2 (b c (1-n)-a d (2 n+1)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2 b^3 n}-\frac{d x \left (c+d x^n\right ) (b c (n+1)-a d (2 n+1))}{a b^2 n (n+1)}+\frac{x (b c-a d) \left (c+d x^n\right )^2}{a b n \left (a+b x^n\right )} \]

[Out]

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

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Rubi [A]  time = 0.259678, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {413, 528, 388, 245} \[ -\frac{d x \left (a^2 d^2 \left (2 n^2+3 n+1\right )-a b c d \left (3 n^2+4 n+2\right )+b^2 c^2 (n+1)\right )}{a b^3 n (n+1)}-\frac{x (b c-a d)^2 (b c (1-n)-a d (2 n+1)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2 b^3 n}-\frac{d x \left (c+d x^n\right ) (b c (n+1)-a d (2 n+1))}{a b^2 n (n+1)}+\frac{x (b c-a d) \left (c+d x^n\right )^2}{a b n \left (a+b x^n\right )} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^n)^3/(a + b*x^n)^2,x]

[Out]

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

Rule 413

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

Rule 528

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

Rule 388

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

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])

Rubi steps

\begin{align*} \int \frac{\left (c+d x^n\right )^3}{\left (a+b x^n\right )^2} \, dx &=\frac{(b c-a d) x \left (c+d x^n\right )^2}{a b n \left (a+b x^n\right )}+\frac{\int \frac{\left (c+d x^n\right ) \left (c (a d-b c (1-n))-d (b c (1+n)-a d (1+2 n)) x^n\right )}{a+b x^n} \, dx}{a b n}\\ &=-\frac{d (b c (1+n)-a d (1+2 n)) x \left (c+d x^n\right )}{a b^2 n (1+n)}+\frac{(b c-a d) x \left (c+d x^n\right )^2}{a b n \left (a+b x^n\right )}+\frac{\int \frac{c \left (2 a b c d (1+n)-a^2 d^2 (1+2 n)-b^2 c^2 \left (1-n^2\right )\right )-d \left (b^2 c^2 (1+n)+a^2 d^2 \left (1+3 n+2 n^2\right )-a b c d \left (2+4 n+3 n^2\right )\right ) x^n}{a+b x^n} \, dx}{a b^2 n (1+n)}\\ &=-\frac{d \left (b^2 c^2 (1+n)+a^2 d^2 \left (1+3 n+2 n^2\right )-a b c d \left (2+4 n+3 n^2\right )\right ) x}{a b^3 n (1+n)}-\frac{d (b c (1+n)-a d (1+2 n)) x \left (c+d x^n\right )}{a b^2 n (1+n)}+\frac{(b c-a d) x \left (c+d x^n\right )^2}{a b n \left (a+b x^n\right )}-\frac{\left ((b c-a d)^2 (b c (1-n)-a d (1+2 n))\right ) \int \frac{1}{a+b x^n} \, dx}{a b^3 n}\\ &=-\frac{d \left (b^2 c^2 (1+n)+a^2 d^2 \left (1+3 n+2 n^2\right )-a b c d \left (2+4 n+3 n^2\right )\right ) x}{a b^3 n (1+n)}-\frac{d (b c (1+n)-a d (1+2 n)) x \left (c+d x^n\right )}{a b^2 n (1+n)}+\frac{(b c-a d) x \left (c+d x^n\right )^2}{a b n \left (a+b x^n\right )}-\frac{(b c-a d)^2 (b c (1-n)-a d (1+2 n)) x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2 b^3 n}\\ \end{align*}

Mathematica [C]  time = 4.2095, size = 2050, normalized size = 10.25 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(x*(3*a*(1 + 10*n + 35*n^2 + 50*n^3 + 24*n^4)*(c^3*(1 + n)^4 + 3*c^2*d*(1 + 4*n + 6*n^2 + 2*n^3 + n^4)*x^n + 3
*c*d^2*(1 + n)^4*x^(2*n) + d^3*(1 + n)^4*x^(3*n))*HurwitzLerchPhi[-((b*x^n)/a), 1, 1 + n^(-1)] - 3*a*(1 + 10*n
+ 35*n^2 + 50*n^3 + 24*n^4)*(c^3*(1 + 2*n)^4 + 3*c^2*d*(1 + 2*n)^4*x^n + 3*c*d^2*(1 + 8*n + 24*n^2 + 34*n^3 +
18*n^4)*x^(2*n) + d^3*(1 + 2*n)^4*x^(3*n))*HurwitzLerchPhi[-((b*x^n)/a), 1, 2 + n^(-1)] + a*c^3*HurwitzLerchP
hi[-((b*x^n)/a), 1, 3 + n^(-1)] + 22*a*c^3*n*HurwitzLerchPhi[-((b*x^n)/a), 1, 3 + n^(-1)] + 209*a*c^3*n^2*Hurw
itzLerchPhi[-((b*x^n)/a), 1, 3 + n^(-1)] + 1118*a*c^3*n^3*HurwitzLerchPhi[-((b*x^n)/a), 1, 3 + n^(-1)] + 3675*
a*c^3*n^4*HurwitzLerchPhi[-((b*x^n)/a), 1, 3 + n^(-1)] + 7578*a*c^3*n^5*HurwitzLerchPhi[-((b*x^n)/a), 1, 3 + n
^(-1)] + 9531*a*c^3*n^6*HurwitzLerchPhi[-((b*x^n)/a), 1, 3 + n^(-1)] + 6642*a*c^3*n^7*HurwitzLerchPhi[-((b*x^n
)/a), 1, 3 + n^(-1)] + 1944*a*c^3*n^8*HurwitzLerchPhi[-((b*x^n)/a), 1, 3 + n^(-1)] + 3*a*c^2*d*x^n*HurwitzLerc
hPhi[-((b*x^n)/a), 1, 3 + n^(-1)] + 66*a*c^2*d*n*x^n*HurwitzLerchPhi[-((b*x^n)/a), 1, 3 + n^(-1)] + 627*a*c^2*
d*n^2*x^n*HurwitzLerchPhi[-((b*x^n)/a), 1, 3 + n^(-1)] + 3354*a*c^2*d*n^3*x^n*HurwitzLerchPhi[-((b*x^n)/a), 1,
3 + n^(-1)] + 11025*a*c^2*d*n^4*x^n*HurwitzLerchPhi[-((b*x^n)/a), 1, 3 + n^(-1)] + 22734*a*c^2*d*n^5*x^n*Hurw
itzLerchPhi[-((b*x^n)/a), 1, 3 + n^(-1)] + 28593*a*c^2*d*n^6*x^n*HurwitzLerchPhi[-((b*x^n)/a), 1, 3 + n^(-1)]
+ 19926*a*c^2*d*n^7*x^n*HurwitzLerchPhi[-((b*x^n)/a), 1, 3 + n^(-1)] + 5832*a*c^2*d*n^8*x^n*HurwitzLerchPhi[-(
(b*x^n)/a), 1, 3 + n^(-1)] + 3*a*c*d^2*x^(2*n)*HurwitzLerchPhi[-((b*x^n)/a), 1, 3 + n^(-1)] + 66*a*c*d^2*n*x^(
2*n)*HurwitzLerchPhi[-((b*x^n)/a), 1, 3 + n^(-1)] + 627*a*c*d^2*n^2*x^(2*n)*HurwitzLerchPhi[-((b*x^n)/a), 1, 3
+ n^(-1)] + 3354*a*c*d^2*n^3*x^(2*n)*HurwitzLerchPhi[-((b*x^n)/a), 1, 3 + n^(-1)] + 11025*a*c*d^2*n^4*x^(2*n)
*HurwitzLerchPhi[-((b*x^n)/a), 1, 3 + n^(-1)] + 22734*a*c*d^2*n^5*x^(2*n)*HurwitzLerchPhi[-((b*x^n)/a), 1, 3 +
n^(-1)] + 28593*a*c*d^2*n^6*x^(2*n)*HurwitzLerchPhi[-((b*x^n)/a), 1, 3 + n^(-1)] + 19926*a*c*d^2*n^7*x^(2*n)*
HurwitzLerchPhi[-((b*x^n)/a), 1, 3 + n^(-1)] + 5832*a*c*d^2*n^8*x^(2*n)*HurwitzLerchPhi[-((b*x^n)/a), 1, 3 + n
^(-1)] + a*d^3*x^(3*n)*HurwitzLerchPhi[-((b*x^n)/a), 1, 3 + n^(-1)] + 22*a*d^3*n*x^(3*n)*HurwitzLerchPhi[-((b*
x^n)/a), 1, 3 + n^(-1)] + 209*a*d^3*n^2*x^(3*n)*HurwitzLerchPhi[-((b*x^n)/a), 1, 3 + n^(-1)] + 1112*a*d^3*n^3*
x^(3*n)*HurwitzLerchPhi[-((b*x^n)/a), 1, 3 + n^(-1)] + 3603*a*d^3*n^4*x^(3*n)*HurwitzLerchPhi[-((b*x^n)/a), 1,
3 + n^(-1)] + 7248*a*d^3*n^5*x^(3*n)*HurwitzLerchPhi[-((b*x^n)/a), 1, 3 + n^(-1)] + 8811*a*d^3*n^6*x^(3*n)*Hu
rwitzLerchPhi[-((b*x^n)/a), 1, 3 + n^(-1)] + 5898*a*d^3*n^7*x^(3*n)*HurwitzLerchPhi[-((b*x^n)/a), 1, 3 + n^(-1
)] + 1656*a*d^3*n^8*x^(3*n)*HurwitzLerchPhi[-((b*x^n)/a), 1, 3 + n^(-1)] - a*c^3*HurwitzLerchPhi[-((b*x^n)/a),
1, n^(-1)] - 10*a*c^3*n*HurwitzLerchPhi[-((b*x^n)/a), 1, n^(-1)] - 35*a*c^3*n^2*HurwitzLerchPhi[-((b*x^n)/a),
1, n^(-1)] - 56*a*c^3*n^3*HurwitzLerchPhi[-((b*x^n)/a), 1, n^(-1)] - 78*a*c^3*n^4*HurwitzLerchPhi[-((b*x^n)/a
), 1, n^(-1)] - 150*a*c^3*n^5*HurwitzLerchPhi[-((b*x^n)/a), 1, n^(-1)] - 90*a*c^3*n^6*HurwitzLerchPhi[-((b*x^n
)/a), 1, n^(-1)] + 156*a*c^3*n^7*HurwitzLerchPhi[-((b*x^n)/a), 1, n^(-1)] + 144*a*c^3*n^8*HurwitzLerchPhi[-((b
*x^n)/a), 1, n^(-1)] - 3*a*c^2*d*x^n*HurwitzLerchPhi[-((b*x^n)/a), 1, n^(-1)] - 30*a*c^2*d*n*x^n*HurwitzLerchP
hi[-((b*x^n)/a), 1, n^(-1)] - 105*a*c^2*d*n^2*x^n*HurwitzLerchPhi[-((b*x^n)/a), 1, n^(-1)] - 150*a*c^2*d*n^3*x
^n*HurwitzLerchPhi[-((b*x^n)/a), 1, n^(-1)] - 72*a*c^2*d*n^4*x^n*HurwitzLerchPhi[-((b*x^n)/a), 1, n^(-1)] - 3*
a*c*d^2*x^(2*n)*HurwitzLerchPhi[-((b*x^n)/a), 1, n^(-1)] - 30*a*c*d^2*n*x^(2*n)*HurwitzLerchPhi[-((b*x^n)/a),
1, n^(-1)] - 105*a*c*d^2*n^2*x^(2*n)*HurwitzLerchPhi[-((b*x^n)/a), 1, n^(-1)] - 150*a*c*d^2*n^3*x^(2*n)*Hurwit
zLerchPhi[-((b*x^n)/a), 1, n^(-1)] - 72*a*c*d^2*n^4*x^(2*n)*HurwitzLerchPhi[-((b*x^n)/a), 1, n^(-1)] - a*d^3*x
^(3*n)*HurwitzLerchPhi[-((b*x^n)/a), 1, n^(-1)] - 10*a*d^3*n*x^(3*n)*HurwitzLerchPhi[-((b*x^n)/a), 1, n^(-1)]
- 35*a*d^3*n^2*x^(3*n)*HurwitzLerchPhi[-((b*x^n)/a), 1, n^(-1)] - 50*a*d^3*n^3*x^(3*n)*HurwitzLerchPhi[-((b*x^
n)/a), 1, n^(-1)] - 24*a*d^3*n^4*x^(3*n)*HurwitzLerchPhi[-((b*x^n)/a), 1, n^(-1)] - 6*b*c^3*n^8*x^n*Hypergeome
tricPFQ[{2, 2, 2, 2, 1 + n^(-1)}, {1, 1, 1, 5 + n^(-1)}, -((b*x^n)/a)] - 18*b*c^2*d*n^8*x^(2*n)*Hypergeometric
PFQ[{2, 2, 2, 2, 1 + n^(-1)}, {1, 1, 1, 5 + n^(-1)}, -((b*x^n)/a)] - 18*b*c*d^2*n^8*x^(3*n)*HypergeometricPFQ[
{2, 2, 2, 2, 1 + n^(-1)}, {1, 1, 1, 5 + n^(-1)}, -((b*x^n)/a)] - 6*b*d^3*n^8*x^(4*n)*HypergeometricPFQ[{2, 2,
2, 2, 1 + n^(-1)}, {1, 1, 1, 5 + n^(-1)}, -((b*x^n)/a)]))/(6*a^3*n^5*(1 + n)*(1 + 2*n)*(1 + 3*n)*(1 + 4*n))

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Maple [F]  time = 0.368, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( c+d{x}^{n} \right ) ^{3}}{ \left ( a+b{x}^{n} \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c+d*x^n)^3/(a+b*x^n)^2,x)

[Out]

int((c+d*x^n)^3/(a+b*x^n)^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\left (a^{3} d^{3}{\left (2 \, n + 1\right )} - 3 \, a^{2} b c d^{2}{\left (n + 1\right )} + b^{3} c^{3}{\left (n - 1\right )} + 3 \, a b^{2} c^{2} d\right )} \int \frac{1}{a b^{4} n x^{n} + a^{2} b^{3} n}\,{d x} + \frac{a b^{2} d^{3} n x x^{2 \, n} +{\left (3 \,{\left (n^{2} + n\right )} a b^{2} c d^{2} -{\left (2 \, n^{2} + n\right )} a^{2} b d^{3}\right )} x x^{n} +{\left (3 \,{\left (n^{2} + 2 \, n + 1\right )} a^{2} b c d^{2} -{\left (2 \, n^{2} + 3 \, n + 1\right )} a^{3} d^{3} + b^{3} c^{3}{\left (n + 1\right )} - 3 \, a b^{2} c^{2} d{\left (n + 1\right )}\right )} x}{{\left (n^{2} + n\right )} a b^{4} x^{n} +{\left (n^{2} + n\right )} a^{2} b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*x^n)^3/(a+b*x^n)^2,x, algorithm="maxima")

[Out]

(a^3*d^3*(2*n + 1) - 3*a^2*b*c*d^2*(n + 1) + b^3*c^3*(n - 1) + 3*a*b^2*c^2*d)*integrate(1/(a*b^4*n*x^n + a^2*b
^3*n), x) + (a*b^2*d^3*n*x*x^(2*n) + (3*(n^2 + n)*a*b^2*c*d^2 - (2*n^2 + n)*a^2*b*d^3)*x*x^n + (3*(n^2 + 2*n +
1)*a^2*b*c*d^2 - (2*n^2 + 3*n + 1)*a^3*d^3 + b^3*c^3*(n + 1) - 3*a*b^2*c^2*d*(n + 1))*x)/((n^2 + n)*a*b^4*x^n
+ (n^2 + n)*a^2*b^3)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{d^{3} x^{3 \, n} + 3 \, c d^{2} x^{2 \, n} + 3 \, c^{2} d x^{n} + c^{3}}{b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*x^n)^3/(a+b*x^n)^2,x, algorithm="fricas")

[Out]

integral((d^3*x^(3*n) + 3*c*d^2*x^(2*n) + 3*c^2*d*x^n + c^3)/(b^2*x^(2*n) + 2*a*b*x^n + a^2), 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)**3/(a+b*x**n)**2,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 )}^{3}}{{\left (b x^{n} + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*x^n)^3/(a+b*x^n)^2,x, algorithm="giac")

[Out]

integrate((d*x^n + c)^3/(b*x^n + a)^2, x)