3.1890 \(\int \left (\frac{d (a+b x)}{-b c+a d}\right )^m (c+d x)^n \, dx\)

Optimal. Leaf size=45 \[ \frac{(c+d x)^{n+1} \, _2F_1\left (-m,n+1;n+2;\frac{b (c+d x)}{b c-a d}\right )}{d (n+1)} \]

[Out]

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

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Rubi [A]  time = 0.0582823, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074 \[ \frac{(c+d x)^{n+1} \, _2F_1\left (-m,n+1;n+2;\frac{b (c+d x)}{b c-a d}\right )}{d (n+1)} \]

Antiderivative was successfully verified.

[In]  Int[((d*(a + b*x))/(-(b*c) + a*d))^m*(c + d*x)^n,x]

[Out]

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

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Rubi in Sympy [A]  time = 12.9038, size = 34, normalized size = 0.76 \[ \frac{\left (c + d x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} - m, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{b \left (- c - d x\right )}{a d - b c}} \right )}}{d \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*(b*x+a)/(a*d-b*c))**m*(d*x+c)**n,x)

[Out]

(c + d*x)**(n + 1)*hyper((-m, n + 1), (n + 2,), b*(-c - d*x)/(a*d - b*c))/(d*(n
+ 1))

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Mathematica [A]  time = 0.103873, size = 46, normalized size = 1.02 \[ \frac{(c+d x)^{n+1} \, _2F_1\left (-m,n+1;n+2;\frac{b c+b d x}{b c-a d}\right )}{d n+d} \]

Antiderivative was successfully verified.

[In]  Integrate[((d*(a + b*x))/(-(b*c) + a*d))^m*(c + d*x)^n,x]

[Out]

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

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Maple [F]  time = 0.174, size = 0, normalized size = 0. \[ \int \left ({\frac{d \left ( bx+a \right ) }{ad-bc}} \right ) ^{m} \left ( dx+c \right ) ^{n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*(b*x+a)/(a*d-b*c))^m*(d*x+c)^n,x)

[Out]

int((d*(b*x+a)/(a*d-b*c))^m*(d*x+c)^n,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^n*(-(b*x + a)*d/(b*c - a*d))^m,x, algorithm="maxima")

[Out]

integrate((d*x + c)^n*(-(b*x + a)*d/(b*c - a*d))^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^n*(-(b*x + a)*d/(b*c - a*d))^m,x, algorithm="fricas")

[Out]

integral((d*x + c)^n*(-(b*d*x + a*d)/(b*c - a*d))^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((d*(b*x+a)/(a*d-b*c))**m*(d*x+c)**n,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((d*x + c)^n*(-(b*x + a)*d/(b*c - a*d))^m, x)