3.352 \(\int \frac{x^m}{(a+b x) (c+d x)} \, dx\)

Optimal. Leaf size=82 \[ \frac{b x^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{b x}{a}\right )}{a (m+1) (b c-a d)}-\frac{d x^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{d x}{c}\right )}{c (m+1) (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.0920652, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{b x^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{b x}{a}\right )}{a (m+1) (b c-a d)}-\frac{d x^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{d x}{c}\right )}{c (m+1) (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 11.7752, size = 60, normalized size = 0.73 \[ \frac{d x^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{- \frac{d x}{c}} \right )}}{c \left (m + 1\right ) \left (a d - b c\right )} - \frac{b x^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{- \frac{b x}{a}} \right )}}{a \left (m + 1\right ) \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0659747, size = 65, normalized size = 0.79 \[ \frac{x^{m+1} \left (a d \, _2F_1\left (1,m+1;m+2;-\frac{d x}{c}\right )-b c \, _2F_1\left (1,m+1;m+2;-\frac{b x}{a}\right )\right )}{a c (m+1) (a d-b c)} \]

Antiderivative was successfully verified.

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

[Out]

(x^(1 + m)*(-(b*c*Hypergeometric2F1[1, 1 + m, 2 + m, -((b*x)/a)]) + a*d*Hypergeo
metric2F1[1, 1 + m, 2 + m, -((d*x)/c)]))/(a*c*(-(b*c) + a*d)*(1 + m))

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Maple [F]  time = 0.062, size = 0, normalized size = 0. \[ \int{\frac{{x}^{m}}{ \left ( bx+a \right ) \left ( dx+c \right ) }}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m/(b*x+a)/(d*x+c),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{m}}{b d x^{2} + a c +{\left (b c + a d\right )} x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x^m/(b*d*x^2 + a*c + (b*c + a*d)*x), x)

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Sympy [A]  time = 5.45528, size = 102, normalized size = 1.24 \[ - \frac{b^{m} m x^{m} \Phi \left (\frac{b x e^{i \pi }}{a}, 1, m\right ) \Gamma \left (- m\right )}{a b^{m} d \Gamma \left (- m + 1\right ) - b b^{m} c \Gamma \left (- m + 1\right )} + \frac{b^{m} m x^{m} \Phi \left (\frac{c e^{i \pi }}{d x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{a b^{m} d \Gamma \left (- m + 1\right ) - b b^{m} c \Gamma \left (- m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b**m*m*x**m*lerchphi(b*x*exp_polar(I*pi)/a, 1, m)*gamma(-m)/(a*b**m*d*gamma(-m
+ 1) - b*b**m*c*gamma(-m + 1)) + b**m*m*x**m*lerchphi(c*exp_polar(I*pi)/(d*x), 1
, m*exp_polar(I*pi))*gamma(-m)/(a*b**m*d*gamma(-m + 1) - b*b**m*c*gamma(-m + 1))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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