3.275 \(\int x^{-1+n-j p} (c+d x^n) (a x^j+b x^{j+n})^p \, dx\)

Optimal. Leaf size=95 \[ \frac {d x^{n-j (p+1)} \left (a x^j+b x^{j+n}\right )^{p+1}}{b n (p+2)}-\frac {x^{-j (p+1)} (a d-b c (p+2)) \left (a x^j+b x^{j+n}\right )^{p+1}}{b^2 n (p+1) (p+2)} \]

[Out]

-(a*d-b*c*(2+p))*(a*x^j+b*x^(j+n))^(1+p)/b^2/n/(p^2+3*p+2)/(x^(j*(1+p)))+d*x^(n-j*(1+p))*(a*x^j+b*x^(j+n))^(1+
p)/b/n/(2+p)

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Rubi [A]  time = 0.16, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2039, 2014} \[ \frac {d x^{n-j (p+1)} \left (a x^j+b x^{j+n}\right )^{p+1}}{b n (p+2)}-\frac {x^{-j (p+1)} (a d-b c (p+2)) \left (a x^j+b x^{j+n}\right )^{p+1}}{b^2 n (p+1) (p+2)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((a*d - b*c*(2 + p))*(a*x^j + b*x^(j + n))^(1 + p))/(b^2*n*(1 + p)*(2 + p)*x^(j*(1 + p)))) + (d*x^(n - j*(1
+ p))*(a*x^j + b*x^(j + n))^(1 + p))/(b*n*(2 + p))

Rule 2014

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> -Simp[(c^(j - 1)*(c*x)^(m - j
+ 1)*(a*x^j + b*x^n)^(p + 1))/(a*(n - j)*(p + 1)), x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && N
eQ[n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])

Rule 2039

Int[((e_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(jn_.))^(p_)*((c_) + (d_.)*(x_)^(n_.)), x_Symbol] :> Sim
p[(d*e^(j - 1)*(e*x)^(m - j + 1)*(a*x^j + b*x^(j + n))^(p + 1))/(b*(m + n + p*(j + n) + 1)), x] - Dist[(a*d*(m
+ j*p + 1) - b*c*(m + n + p*(j + n) + 1))/(b*(m + n + p*(j + n) + 1)), Int[(e*x)^m*(a*x^j + b*x^(j + n))^p, x
], x] /; FreeQ[{a, b, c, d, e, j, m, n, p}, x] && EqQ[jn, j + n] &&  !IntegerQ[p] && NeQ[b*c - a*d, 0] && NeQ[
m + n + p*(j + n) + 1, 0] && (GtQ[e, 0] || IntegerQ[j])

Rubi steps

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

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Mathematica [A]  time = 0.08, size = 63, normalized size = 0.66 \[ \frac {x^{-j p} \left (a+b x^n\right ) \left (x^j \left (a+b x^n\right )\right )^p \left (-a d+b c (p+2)+b d (p+1) x^n\right )}{b^2 n (p+1) (p+2)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x^n)*(x^j*(a + b*x^n))^p*(-(a*d) + b*c*(2 + p) + b*d*(1 + p)*x^n))/(b^2*n*(1 + p)*(2 + p)*x^(j*p))

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fricas [A]  time = 0.92, size = 140, normalized size = 1.47 \[ \frac {{\left ({\left (b^{2} d p + b^{2} d\right )} x x^{-j p + n - 1} x^{2 \, n} + {\left (2 \, b^{2} c + {\left (b^{2} c + a b d\right )} p\right )} x x^{-j p + n - 1} x^{n} + {\left (a b c p + 2 \, a b c - a^{2} d\right )} x x^{-j p + n - 1}\right )} \left (\frac {{\left (b x^{n} + a\right )} x^{j + n}}{x^{n}}\right )^{p}}{{\left (b^{2} n p^{2} + 3 \, b^{2} n p + 2 \, b^{2} n\right )} x^{n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-j*p+n-1)*(c+d*x^n)*(a*x^j+b*x^(j+n))^p,x, algorithm="fricas")

[Out]

((b^2*d*p + b^2*d)*x*x^(-j*p + n - 1)*x^(2*n) + (2*b^2*c + (b^2*c + a*b*d)*p)*x*x^(-j*p + n - 1)*x^n + (a*b*c*
p + 2*a*b*c - a^2*d)*x*x^(-j*p + n - 1))*((b*x^n + a)*x^(j + n)/x^n)^p/((b^2*n*p^2 + 3*b^2*n*p + 2*b^2*n)*x^n)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x^{n} + c\right )} {\left (b x^{j + n} + a x^{j}\right )}^{p} x^{-j p + n - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-j*p+n-1)*(c+d*x^n)*(a*x^j+b*x^(j+n))^p,x, algorithm="giac")

[Out]

integrate((d*x^n + c)*(b*x^(j + n) + a*x^j)^p*x^(-j*p + n - 1), x)

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maple [F]  time = 0.80, size = 0, normalized size = 0.00 \[ \int \left (d \,x^{n}+c \right ) x^{-j p +n -1} \left (a \,x^{j}+b \,x^{j +n}\right )^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-j*p+n-1)*(d*x^n+c)*(a*x^j+b*x^(j+n))^p,x)

[Out]

int(x^(-j*p+n-1)*(d*x^n+c)*(a*x^j+b*x^(j+n))^p,x)

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maxima [A]  time = 2.04, size = 112, normalized size = 1.18 \[ \frac {{\left (b x^{n} + a\right )} c e^{\left (-j p \log \relax (x) + p \log \left (b x^{n} + a\right ) + p \log \left (x^{j}\right )\right )}}{b n {\left (p + 1\right )}} + \frac {{\left (b^{2} {\left (p + 1\right )} x^{2 \, n} + a b p x^{n} - a^{2}\right )} d e^{\left (-j p \log \relax (x) + p \log \left (b x^{n} + a\right ) + p \log \left (x^{j}\right )\right )}}{{\left (p^{2} + 3 \, p + 2\right )} b^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-j*p+n-1)*(c+d*x^n)*(a*x^j+b*x^(j+n))^p,x, algorithm="maxima")

[Out]

(b*x^n + a)*c*e^(-j*p*log(x) + p*log(b*x^n + a) + p*log(x^j))/(b*n*(p + 1)) + (b^2*(p + 1)*x^(2*n) + a*b*p*x^n
- a^2)*d*e^(-j*p*log(x) + p*log(b*x^n + a) + p*log(x^j))/((p^2 + 3*p + 2)*b^2*n)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x^{n-j\,p-1}\,{\left (a\,x^j+b\,x^{j+n}\right )}^p\,\left (c+d\,x^n\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(n - j*p - 1)*(a*x^j + b*x^(j + n))^p*(c + d*x^n),x)

[Out]

int(x^(n - j*p - 1)*(a*x^j + b*x^(j + n))^p*(c + d*x^n), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-j*p+n-1)*(c+d*x**n)*(a*x**j+b*x**(j+n))**p,x)

[Out]

Timed out

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