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

Optimal. Leaf size=56 \[ \frac{x^{m+1} (b c-a d) \, _2F_1\left (1,m+1;m+2;-\frac{b x}{a}\right )}{a b (m+1)}+\frac{d x^{m+1}}{b (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.0725885, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{x^{m+1} (b c-a d) \, _2F_1\left (1,m+1;m+2;-\frac{b x}{a}\right )}{a b (m+1)}+\frac{d x^{m+1}}{b (m+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 7.41627, size = 41, normalized size = 0.73 \[ \frac{d x^{m + 1}}{b \left (m + 1\right )} - \frac{x^{m + 1} \left (a d - b c\right ){{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{- \frac{b x}{a}} \right )}}{a b \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0532666, size = 45, normalized size = 0.8 \[ \frac{x^{m+1} \left ((b c-a d) \, _2F_1\left (1,m+1;m+2;-\frac{b x}{a}\right )+a d\right )}{a b (m+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.042, size = 0, normalized size = 0. \[ \int{\frac{{x}^{m} \left ( dx+c \right ) }{bx+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*(d*x+c)/(b*x+a),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((d*x + c)*x^m/(b*x + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((d*x + c)*x^m/(b*x + a), x)

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Sympy [A]  time = 5.92727, size = 136, normalized size = 2.43 \[ \frac{c m x x^{m} \Phi \left (\frac{b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{a \Gamma \left (m + 2\right )} + \frac{c x x^{m} \Phi \left (\frac{b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{a \Gamma \left (m + 2\right )} + \frac{d m x^{2} x^{m} \Phi \left (\frac{b x e^{i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{a \Gamma \left (m + 3\right )} + \frac{2 d x^{2} x^{m} \Phi \left (\frac{b x e^{i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{a \Gamma \left (m + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*m*x*x**m*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 1)*gamma(m + 1)/(a*gamma(m + 2
)) + c*x*x**m*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 1)*gamma(m + 1)/(a*gamma(m
+ 2)) + d*m*x**2*x**m*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 2)*gamma(m + 2)/(a*
gamma(m + 3)) + 2*d*x**2*x**m*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 2)*gamma(m
+ 2)/(a*gamma(m + 3))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((d*x + c)*x^m/(b*x + a), x)