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

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*(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))

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Rubi [A]  time = 0.276863, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{d x^{n-j (p+1)} \left (a x^j+b x^{j+n}\right )^{p+1}}{b n (p+2)}-\frac{x^{-j (p+1)} (a d-b c (p+2)) \left (a x^j+b x^{j+n}\right )^{p+1}}{b^2 n (p+1) (p+2)} \]

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))

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Rubi in Sympy [A]  time = 26.7154, size = 80, normalized size = 0.84 \[ \frac{d x^{- j p - j + n} \left (a x^{j} + b x^{j + n}\right )^{p + 1}}{b n \left (p + 2\right )} - \frac{x^{- j \left (p + 1\right )} \left (a x^{j} + b x^{j + n}\right )^{p + 1} \left (a d - b c p - 2 b c\right )}{b^{2} n \left (p^{2} + 3 p + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.126242, 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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Maple [F]  time = 0.247, size = 0, normalized size = 0. \[ \int{x}^{-jp+n-1} \left ( c+d{x}^{n} \right ) \left ( a{x}^{j}+b{x}^{j+n} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.4599, size = 151, normalized size = 1.59 \[ \frac{{\left (b x^{n} + a\right )} c e^{\left (-j p \log \left (x\right ) + p \log \left (b x^{n} + a\right ) + p \log \left (x^{j}\right )\right )}}{b n{\left (p + 1\right )}} + \frac{{\left (b^{2}{\left (p + 1\right )} x^{2 \, n} + a b p x^{n} - a^{2}\right )} d e^{\left (-j p \log \left (x\right ) + p \log \left (b x^{n} + a\right ) + p \log \left (x^{j}\right )\right )}}{{\left (p^{2} + 3 \, p + 2\right )} b^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^n + c)*(b*x^(j + n) + a*x^j)^p*x^(-j*p + n - 1),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*lo
g(x^j))/((p^2 + 3*p + 2)*b^2*n)

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Fricas [A]  time = 0.245168, size = 189, normalized size = 1.99 \[ \frac{{\left ({\left (b^{2} d p + b^{2} d\right )} x x^{-j p + n - 1} x^{2 \, n} +{\left (2 \, b^{2} c +{\left (b^{2} c + a b d\right )} p\right )} x x^{-j p + n - 1} x^{n} +{\left (a b c p + 2 \, a b c - a^{2} d\right )} x x^{-j p + n - 1}\right )} \left (\frac{{\left (b x^{n} + a\right )} x^{j + n}}{x^{n}}\right )^{p}}{{\left (b^{2} n p^{2} + 3 \, b^{2} n p + 2 \, b^{2} n\right )} x^{n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^n + c)*(b*x^(j + n) + a*x^j)^p*x^(-j*p + n - 1),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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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