3.1880 \(\int (a+b x)^m (c+d x)^{1+2 n-2 (1+n)} \, dx\)

Optimal. Leaf size=51 \[ \frac{(a+b x)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{(m+1) (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.034466, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{(a+b x)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{(m+1) (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 5.27879, size = 37, normalized size = 0.73 \[ - \frac{\left (a + b x\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{d \left (a + b x\right )}{a d - b c}} \right )}}{\left (m + 1\right ) \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0377666, size = 66, normalized size = 1.29 \[ \frac{(a+b x)^m \left (\frac{d (a+b x)}{b (c+d x)}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{b c-a d}{b c+b d x}\right )}{d m} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^m/(d*x+c),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x + a)^m/(d*x + c), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x + a)^m/(d*x + c), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**m/(d*x+c),x)

[Out]

Integral((a + b*x)**m/(c + d*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x + a)^m/(d*x + c), x)