3.283 \(\int x^m \left (a+b x^n\right )^p \left (a (1+m+q) x^q+b (1+m+n (1+p)+q) x^{n+q}\right ) \, dx\)

Optimal. Leaf size=18 \[ x^{m+q+1} \left (a+b x^n\right )^{p+1} \]

[Out]

x^(1 + m + q)*(a + b*x^n)^(1 + p)

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Rubi [A]  time = 0.0892685, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.051 \[ x^{m+q+1} \left (a+b x^n\right )^{p+1} \]

Antiderivative was successfully verified.

[In]  Int[x^m*(a + b*x^n)^p*(a*(1 + m + q)*x^q + b*(1 + m + n*(1 + p) + q)*x^(n + q)),x]

[Out]

x^(1 + m + q)*(a + b*x^n)^(1 + p)

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Rubi in Sympy [A]  time = 11.3823, size = 15, normalized size = 0.83 \[ x^{m + q + 1} \left (a + b x^{n}\right )^{p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m*(a+b*x**n)**p*(a*(1+m+q)*x**q+b*(1+m+n*(1+p)+q)*x**(n+q)),x)

[Out]

x**(m + q + 1)*(a + b*x**n)**(p + 1)

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Mathematica [A]  time = 0.122284, size = 18, normalized size = 1. \[ x^{m+q+1} \left (a+b x^n\right )^{p+1} \]

Antiderivative was successfully verified.

[In]  Integrate[x^m*(a + b*x^n)^p*(a*(1 + m + q)*x^q + b*(1 + m + n*(1 + p) + q)*x^(n + q)),x]

[Out]

x^(1 + m + q)*(a + b*x^n)^(1 + p)

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Maple [F]  time = 0.354, size = 0, normalized size = 0. \[ \int{x}^{m} \left ( a+b{x}^{n} \right ) ^{p} \left ( a \left ( 1+m+q \right ){x}^{q}+b \left ( 1+m+n \left ( 1+p \right ) +q \right ){x}^{n+q} \right ) \, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*(a+b*x^n)^p*(a*(1+m+q)*x^q+b*(1+m+n*(1+p)+q)*x^(n+q)),x)

[Out]

int(x^m*(a+b*x^n)^p*(a*(1+m+q)*x^q+b*(1+m+n*(1+p)+q)*x^(n+q)),x)

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Maxima [A]  time = 1.85863, size = 50, normalized size = 2.78 \[{\left (a x x^{m} + b x e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}\right )} e^{\left (p \log \left (b x^{n} + a\right ) + q \log \left (x\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((n*(p + 1) + m + q + 1)*b*x^(n + q) + a*(m + q + 1)*x^q)*(b*x^n + a)^p*x^m,x, algorithm="maxima")

[Out]

(a*x*x^m + b*x*e^(m*log(x) + n*log(x)))*e^(p*log(b*x^n + a) + q*log(x))

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Fricas [A]  time = 0.236825, size = 58, normalized size = 3.22 \[{\left (b x x^{m} x^{n + q} + a x x^{m} x^{q}\right )} \left (\frac{b x^{n + q} + a x^{q}}{x^{q}}\right )^{p} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((n*(p + 1) + m + q + 1)*b*x^(n + q) + a*(m + q + 1)*x^q)*(b*x^n + a)^p*x^m,x, algorithm="fricas")

[Out]

(b*x*x^m*x^(n + q) + a*x*x^m*x^q)*((b*x^(n + q) + a*x^q)/x^q)^p

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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**m*(a+b*x**n)**p*(a*(1+m+q)*x**q+b*(1+m+n*(1+p)+q)*x**(n+q)),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.220585, size = 74, normalized size = 4.11 \[ b x e^{\left (p{\rm ln}\left (b e^{\left (n{\rm ln}\left (x\right )\right )} + a\right ) + m{\rm ln}\left (x\right ) + n{\rm ln}\left (x\right ) + q{\rm ln}\left (x\right )\right )} + a x e^{\left (p{\rm ln}\left (b e^{\left (n{\rm ln}\left (x\right )\right )} + a\right ) + m{\rm ln}\left (x\right ) + q{\rm ln}\left (x\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((n*(p + 1) + m + q + 1)*b*x^(n + q) + a*(m + q + 1)*x^q)*(b*x^n + a)^p*x^m,x, algorithm="giac")

[Out]

b*x*e^(p*ln(b*e^(n*ln(x)) + a) + m*ln(x) + n*ln(x) + q*ln(x)) + a*x*e^(p*ln(b*e^
(n*ln(x)) + a) + m*ln(x) + q*ln(x))