3.73 \(\int \frac {(e x)^m (a+b x)}{(a c-b c x)^2} \, dx\)

Optimal. Leaf size=66 \[ \frac {2 (e x)^{m+1}}{c^2 e (a-b x)}-\frac {(2 m+1) (e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac {b x}{a}\right )}{a c^2 e (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {78, 64} \[ \frac {2 (e x)^{m+1}}{c^2 e (a-b x)}-\frac {(2 m+1) (e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac {b x}{a}\right )}{a c^2 e (m+1)} \]

Antiderivative was successfully verified.

[In]

Int[((e*x)^m*(a + b*x))/(a*c - b*c*x)^2,x]

[Out]

(2*(e*x)^(1 + m))/(c^2*e*(a - b*x)) - ((1 + 2*m)*(e*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (b*x)/a])/(a
*c^2*e*(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 78

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

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 62, normalized size = 0.94 \[ -\frac {x (e x)^m \left (2 a (m+1)-(2 m+1) (a-b x) \, _2F_1\left (1,m+1;m+2;\frac {b x}{a}\right )\right )}{a c^2 (m+1) (b x-a)} \]

Antiderivative was successfully verified.

[In]

Integrate[((e*x)^m*(a + b*x))/(a*c - b*c*x)^2,x]

[Out]

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

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fricas [F]  time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x + a\right )} \left (e x\right )^{m}}{b^{2} c^{2} x^{2} - 2 \, a b c^{2} x + a^{2} c^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*(b*x+a)/(-b*c*x+a*c)^2,x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x + a\right )} \left (e x\right )^{m}}{{\left (b c x - a c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*(b*x+a)/(-b*c*x+a*c)^2,x, algorithm="giac")

[Out]

integrate((b*x + a)*(e*x)^m/(b*c*x - a*c)^2, x)

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maple [F]  time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {\left (b x +a \right ) \left (e x \right )^{m}}{\left (-b c x +a c \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^m*(b*x+a)/(-b*c*x+a*c)^2,x)

[Out]

int((e*x)^m*(b*x+a)/(-b*c*x+a*c)^2,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x + a\right )} \left (e x\right )^{m}}{{\left (b c x - a c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*(b*x+a)/(-b*c*x+a*c)^2,x, algorithm="maxima")

[Out]

integrate((b*x + a)*(e*x)^m/(b*c*x - a*c)^2, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (e\,x\right )}^m\,\left (a+b\,x\right )}{{\left (a\,c-b\,c\,x\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((e*x)^m*(a + b*x))/(a*c - b*c*x)^2,x)

[Out]

int(((e*x)^m*(a + b*x))/(a*c - b*c*x)^2, x)

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sympy [C]  time = 9.21, size = 799, normalized size = 12.11 \[ a \left (\frac {a e^{m} m^{2} x x^{m} \Phi \left (\frac {b x e^{2 i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{- a^{3} c^{2} \Gamma \left (m + 2\right ) + a^{2} b c^{2} x \Gamma \left (m + 2\right )} + \frac {a e^{m} m x x^{m} \Phi \left (\frac {b x e^{2 i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{- a^{3} c^{2} \Gamma \left (m + 2\right ) + a^{2} b c^{2} x \Gamma \left (m + 2\right )} - \frac {a e^{m} m x x^{m} \Gamma \left (m + 1\right )}{- a^{3} c^{2} \Gamma \left (m + 2\right ) + a^{2} b c^{2} x \Gamma \left (m + 2\right )} - \frac {a e^{m} x x^{m} \Gamma \left (m + 1\right )}{- a^{3} c^{2} \Gamma \left (m + 2\right ) + a^{2} b c^{2} x \Gamma \left (m + 2\right )} - \frac {b e^{m} m^{2} x^{2} x^{m} \Phi \left (\frac {b x e^{2 i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{- a^{3} c^{2} \Gamma \left (m + 2\right ) + a^{2} b c^{2} x \Gamma \left (m + 2\right )} - \frac {b e^{m} m x^{2} x^{m} \Phi \left (\frac {b x e^{2 i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{- a^{3} c^{2} \Gamma \left (m + 2\right ) + a^{2} b c^{2} x \Gamma \left (m + 2\right )}\right ) + b \left (\frac {a e^{m} m^{2} x^{2} x^{m} \Phi \left (\frac {b x e^{2 i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{- a^{3} c^{2} \Gamma \left (m + 3\right ) + a^{2} b c^{2} x \Gamma \left (m + 3\right )} + \frac {3 a e^{m} m x^{2} x^{m} \Phi \left (\frac {b x e^{2 i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{- a^{3} c^{2} \Gamma \left (m + 3\right ) + a^{2} b c^{2} x \Gamma \left (m + 3\right )} - \frac {a e^{m} m x^{2} x^{m} \Gamma \left (m + 2\right )}{- a^{3} c^{2} \Gamma \left (m + 3\right ) + a^{2} b c^{2} x \Gamma \left (m + 3\right )} + \frac {2 a e^{m} x^{2} x^{m} \Phi \left (\frac {b x e^{2 i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{- a^{3} c^{2} \Gamma \left (m + 3\right ) + a^{2} b c^{2} x \Gamma \left (m + 3\right )} - \frac {2 a e^{m} x^{2} x^{m} \Gamma \left (m + 2\right )}{- a^{3} c^{2} \Gamma \left (m + 3\right ) + a^{2} b c^{2} x \Gamma \left (m + 3\right )} - \frac {b e^{m} m^{2} x^{3} x^{m} \Phi \left (\frac {b x e^{2 i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{- a^{3} c^{2} \Gamma \left (m + 3\right ) + a^{2} b c^{2} x \Gamma \left (m + 3\right )} - \frac {3 b e^{m} m x^{3} x^{m} \Phi \left (\frac {b x e^{2 i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{- a^{3} c^{2} \Gamma \left (m + 3\right ) + a^{2} b c^{2} x \Gamma \left (m + 3\right )} - \frac {2 b e^{m} x^{3} x^{m} \Phi \left (\frac {b x e^{2 i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{- a^{3} c^{2} \Gamma \left (m + 3\right ) + a^{2} b c^{2} x \Gamma \left (m + 3\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**m*(b*x+a)/(-b*c*x+a*c)**2,x)

[Out]

a*(a*e**m*m**2*x*x**m*lerchphi(b*x*exp_polar(2*I*pi)/a, 1, m + 1)*gamma(m + 1)/(-a**3*c**2*gamma(m + 2) + a**2
*b*c**2*x*gamma(m + 2)) + a*e**m*m*x*x**m*lerchphi(b*x*exp_polar(2*I*pi)/a, 1, m + 1)*gamma(m + 1)/(-a**3*c**2
*gamma(m + 2) + a**2*b*c**2*x*gamma(m + 2)) - a*e**m*m*x*x**m*gamma(m + 1)/(-a**3*c**2*gamma(m + 2) + a**2*b*c
**2*x*gamma(m + 2)) - a*e**m*x*x**m*gamma(m + 1)/(-a**3*c**2*gamma(m + 2) + a**2*b*c**2*x*gamma(m + 2)) - b*e*
*m*m**2*x**2*x**m*lerchphi(b*x*exp_polar(2*I*pi)/a, 1, m + 1)*gamma(m + 1)/(-a**3*c**2*gamma(m + 2) + a**2*b*c
**2*x*gamma(m + 2)) - b*e**m*m*x**2*x**m*lerchphi(b*x*exp_polar(2*I*pi)/a, 1, m + 1)*gamma(m + 1)/(-a**3*c**2*
gamma(m + 2) + a**2*b*c**2*x*gamma(m + 2))) + b*(a*e**m*m**2*x**2*x**m*lerchphi(b*x*exp_polar(2*I*pi)/a, 1, m
+ 2)*gamma(m + 2)/(-a**3*c**2*gamma(m + 3) + a**2*b*c**2*x*gamma(m + 3)) + 3*a*e**m*m*x**2*x**m*lerchphi(b*x*e
xp_polar(2*I*pi)/a, 1, m + 2)*gamma(m + 2)/(-a**3*c**2*gamma(m + 3) + a**2*b*c**2*x*gamma(m + 3)) - a*e**m*m*x
**2*x**m*gamma(m + 2)/(-a**3*c**2*gamma(m + 3) + a**2*b*c**2*x*gamma(m + 3)) + 2*a*e**m*x**2*x**m*lerchphi(b*x
*exp_polar(2*I*pi)/a, 1, m + 2)*gamma(m + 2)/(-a**3*c**2*gamma(m + 3) + a**2*b*c**2*x*gamma(m + 3)) - 2*a*e**m
*x**2*x**m*gamma(m + 2)/(-a**3*c**2*gamma(m + 3) + a**2*b*c**2*x*gamma(m + 3)) - b*e**m*m**2*x**3*x**m*lerchph
i(b*x*exp_polar(2*I*pi)/a, 1, m + 2)*gamma(m + 2)/(-a**3*c**2*gamma(m + 3) + a**2*b*c**2*x*gamma(m + 3)) - 3*b
*e**m*m*x**3*x**m*lerchphi(b*x*exp_polar(2*I*pi)/a, 1, m + 2)*gamma(m + 2)/(-a**3*c**2*gamma(m + 3) + a**2*b*c
**2*x*gamma(m + 3)) - 2*b*e**m*x**3*x**m*lerchphi(b*x*exp_polar(2*I*pi)/a, 1, m + 2)*gamma(m + 2)/(-a**3*c**2*
gamma(m + 3) + a**2*b*c**2*x*gamma(m + 3)))

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