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

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

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Rubi [A]  time = 0.02, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {80, 64} \[ \frac {x^{m+1} (b c-a d) \, _2F_1\left (1,m+1;m+2;-\frac {b x}{a}\right )}{a b (m+1)}+\frac {d x^{m+1}}{b (m+1)} \]

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
))

Rule 64

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c^n*(b*x)^(m + 1)*Hypergeometric2F1[-n, m +
1, m + 2, -((d*x)/c)])/(b*(m + 1)), x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && (IntegerQ[n] || (GtQ[
c, 0] &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-(d/(b*c)), 0])))

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rubi steps

\begin {align*} \int \frac {x^m (c+d x)}{a+b x} \, dx &=\frac {d x^{1+m}}{b (1+m)}+\frac {(b c (1+m)-a d (1+m)) \int \frac {x^m}{a+b x} \, dx}{b (1+m)}\\ &=\frac {d x^{1+m}}{b (1+m)}+\frac {(b c-a d) x^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac {b x}{a}\right )}{a b (1+m)}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 45, normalized size = 0.80 \[ \frac {x^{m+1} \left ((b c-a d) \, _2F_1\left (1,m+1;m+2;-\frac {b x}{a}\right )+a d\right )}{a b (m+1)} \]

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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fricas [F]  time = 0.84, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (d x + c\right )} x^{m}}{b x + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((d*x + c)*x^m/(b*x + a), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (d x + c\right )} x^{m}}{b x + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((d*x + c)*x^m/(b*x + a), x)

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maple [F]  time = 0.12, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x +c \right ) x^{m}}{b x +a}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (d x + c\right )} x^{m}}{b x + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((d*x + c)*x^m/(b*x + a), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^m\,\left (c+d\,x\right )}{a+b\,x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^m*(c + d*x))/(a + b*x),x)

[Out]

int((x^m*(c + d*x))/(a + b*x), x)

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sympy [C]  time = 3.37, 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*ga
mma(m + 3))

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