3.314 \(\int (a+b x^n)^p (c+d x^n)^2 \, dx\)

Optimal. Leaf size=202 \[ -\frac {x \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} (b c (n p+n+1) (a d-b c (n (p+2)+1))-a d (a d (n+1)-b c (n (p+3)+1))) \, _2F_1\left (\frac {1}{n},-p;1+\frac {1}{n};-\frac {b x^n}{a}\right )}{b^2 (n p+n+1) (n (p+2)+1)}-\frac {d x \left (a+b x^n\right )^{p+1} (a d (n+1)-b c (n (p+3)+1))}{b^2 (n p+n+1) (n (p+2)+1)}+\frac {d x \left (c+d x^n\right ) \left (a+b x^n\right )^{p+1}}{b (n p+2 n+1)} \]

[Out]

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

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Rubi [A]  time = 0.27, antiderivative size = 197, normalized size of antiderivative = 0.98, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {416, 388, 246, 245} \[ -\frac {d x \left (a+b x^n\right )^{p+1} (a d (n+1)-b (c n (p+3)+c))}{b^2 (n p+n+1) (n (p+2)+1)}-\frac {x \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \left (c (a d-b (c n (p+2)+c))-\frac {a d (a d (n+1)-b (c n (p+3)+c))}{b (n p+n+1)}\right ) \, _2F_1\left (\frac {1}{n},-p;1+\frac {1}{n};-\frac {b x^n}{a}\right )}{b (n (p+2)+1)}+\frac {d x \left (c+d x^n\right ) \left (a+b x^n\right )^{p+1}}{b (n (p+2)+1)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^n)^p*(c + d*x^n)^2,x]

[Out]

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

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])

Rule 246

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])/(1 + (b*x^n)/a)^Fr
acPart[p], Int[(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILt
Q[Simplify[1/n + p], 0] &&  !(IntegerQ[p] || GtQ[a, 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 416

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

Rubi steps

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

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Mathematica [A]  time = 5.20, size = 140, normalized size = 0.69 \[ \frac {x \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \left ((n+1) \left (c^2 (2 n+1) \, _2F_1\left (\frac {1}{n},-p;1+\frac {1}{n};-\frac {b x^n}{a}\right )+d^2 x^{2 n} \, _2F_1\left (2+\frac {1}{n},-p;3+\frac {1}{n};-\frac {b x^n}{a}\right )\right )+2 c d (2 n+1) x^n \, _2F_1\left (1+\frac {1}{n},-p;2+\frac {1}{n};-\frac {b x^n}{a}\right )\right )}{(n+1) (2 n+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.84, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (d^{2} x^{2 \, n} + 2 \, c d x^{n} + c^{2}\right )} {\left (b x^{n} + a\right )}^{p}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unab
le to divide, perhaps due to rounding error%%%{-1,[1,0,4,3,1,3,3,2,0]%%%}+%%%{-3,[1,0,4,3,1,2,3,2,0]%%%}+%%%{-
3,[1,0,4,3,1,1,3,2,0]%%%}+%%%{-1,[1,0,4,3,1,0,3,2,0]%%%}+%%%{-1,[0,0,4,3,2,3,2,0,2]%%%}+%%%{-3,[0,0,4,3,2,2,2,
0,2]%%%}+%%%{-3,[0,0,4,3,2,1,2,0,2]%%%}+%%%{-1,[0,0,4,3,2,0,2,0,2]%%%}+%%%{-1,[0,0,4,2,2,2,2,0,2]%%%}+%%%{-2,[
0,0,4,2,2,1,2,0,2]%%%}+%%%{-1,[0,0,4,2,2,0,2,0,2]%%%}+%%%{2,[0,0,4,2,1,2,3,1,1]%%%}+%%%{4,[0,0,4,2,1,1,3,1,1]%
%%}+%%%{2,[0,0,4,2,1,0,3,1,1]%%%}+%%%{-1,[0,0,4,2,0,2,4,2,0]%%%}+%%%{-2,[0,0,4,2,0,1,4,2,0]%%%}+%%%{-1,[0,0,4,
2,0,0,4,2,0]%%%} / %%%{-1,[0,0,5,3,2,3,3,0,0]%%%}+%%%{-3,[0,0,5,3,2,2,3,0,0]%%%}+%%%{-3,[0,0,5,3,2,1,3,0,0]%%%
}+%%%{-1,[0,0,5,3,2,0,3,0,0]%%%} Error: Bad Argument Value

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maple [F]  time = 0.72, size = 0, normalized size = 0.00 \[ \int \left (d \,x^{n}+c \right )^{2} \left (b \,x^{n}+a \right )^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x^{n} + c\right )}^{2} {\left (b x^{n} + a\right )}^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^n)^p*(c + d*x^n)^2,x)

[Out]

int((a + b*x^n)^p*(c + d*x^n)^2, x)

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sympy [C]  time = 42.75, size = 143, normalized size = 0.71 \[ \frac {a^{p} c^{2} x \Gamma \left (\frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{n}, - p \\ 1 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (1 + \frac {1}{n}\right )} + \frac {2 a^{p} c d x x^{n} \Gamma \left (1 + \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, 1 + \frac {1}{n} \\ 2 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (2 + \frac {1}{n}\right )} + \frac {a^{p} d^{2} x x^{2 n} \Gamma \left (2 + \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, 2 + \frac {1}{n} \\ 3 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (3 + \frac {1}{n}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**n)**p*(c+d*x**n)**2,x)

[Out]

a**p*c**2*x*gamma(1/n)*hyper((1/n, -p), (1 + 1/n,), b*x**n*exp_polar(I*pi)/a)/(n*gamma(1 + 1/n)) + 2*a**p*c*d*
x*x**n*gamma(1 + 1/n)*hyper((-p, 1 + 1/n), (2 + 1/n,), b*x**n*exp_polar(I*pi)/a)/(n*gamma(2 + 1/n)) + a**p*d**
2*x*x**(2*n)*gamma(2 + 1/n)*hyper((-p, 2 + 1/n), (3 + 1/n,), b*x**n*exp_polar(I*pi)/a)/(n*gamma(3 + 1/n))

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