3.439 \(\int x^m \left (a x^j+b x^n\right )^p \, dx\)

Optimal. Leaf size=89 \[ \frac{x^{m+1} \left (a+b x^{n-j}\right ) \left (a x^j+b x^n\right )^p \, _2F_1\left (1,p+\frac{m+j p+1}{n-j}+1;\frac{m+j p+1}{n-j}+1;-\frac{b x^{n-j}}{a}\right )}{a (j p+m+1)} \]

[Out]

(x^(1 + m)*(a*x^j + b*x^n)^p*(a + b*x^(-j + n))*Hypergeometric2F1[1, 1 + p + (1
+ m + j*p)/(-j + n), 1 + (1 + m + j*p)/(-j + n), -((b*x^(-j + n))/a)])/(a*(1 + m
+ j*p))

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Rubi [A]  time = 0.139515, antiderivative size = 92, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{x^{m+1} \left (\frac{a x^{j-n}}{b}+1\right )^{-p} \left (a x^j+b x^n\right )^p \, _2F_1\left (-p,\frac{m+n p+1}{j-n};\frac{m+n p+1}{j-n}+1;-\frac{a x^{j-n}}{b}\right )}{m+n p+1} \]

Antiderivative was successfully verified.

[In]  Int[x^m*(a*x^j + b*x^n)^p,x]

[Out]

(x^(1 + m)*(a*x^j + b*x^n)^p*Hypergeometric2F1[-p, (1 + m + n*p)/(j - n), 1 + (1
+ m + n*p)/(j - n), -((a*x^(j - n))/b)])/((1 + m + n*p)*(1 + (a*x^(j - n))/b)^p
)

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Rubi in Sympy [A]  time = 29.4771, size = 82, normalized size = 0.92 \[ \frac{x^{m} x^{- m - n p} x^{m + n p + 1} \left (a x^{j} + b x^{n}\right )^{p} \left (\frac{a x^{j - n}}{b} + 1\right )^{- p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{m + n p + 1}{j - n} \\ 1 + \frac{m + n p + 1}{j - n} \end{matrix}\middle |{- \frac{a x^{j - n}}{b}} \right )}}{m + n p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**m*x**(-m - n*p)*x**(m + n*p + 1)*(a*x**j + b*x**n)**p*(a*x**(j - n)/b + 1)**(
-p)*hyper((-p, (m + n*p + 1)/(j - n)), (1 + (m + n*p + 1)/(j - n),), -a*x**(j -
n)/b)/(m + n*p + 1)

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Mathematica [A]  time = 0.112327, size = 92, normalized size = 1.03 \[ \frac{x^{m+1} \left (\frac{a x^{j-n}}{b}+1\right )^{-p} \left (a x^j+b x^n\right )^p \, _2F_1\left (-p,\frac{m+n p+1}{j-n};\frac{m+n p+1}{j-n}+1;-\frac{a x^{j-n}}{b}\right )}{m+n p+1} \]

Antiderivative was successfully verified.

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

[Out]

(x^(1 + m)*(a*x^j + b*x^n)^p*Hypergeometric2F1[-p, (1 + m + n*p)/(j - n), 1 + (1
+ m + n*p)/(j - n), -((a*x^(j - n))/b)])/((1 + m + n*p)*(1 + (a*x^(j - n))/b)^p
)

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Maple [F]  time = 0.339, size = 0, normalized size = 0. \[ \int{x}^{m} \left ( a{x}^{j}+b{x}^{n} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*(a*x^j+b*x^n)^p,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (a x^{j} + b x^{n}\right )}^{p} x^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((a*x^j + b*x^n)^p*x^m, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (a x^{j} + b x^{n}\right )}^{p} x^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((a*x^j + b*x^n)^p*x^m, x)

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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*x**j+b*x**n)**p,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((a*x^j + b*x^n)^p*x^m, x)