\(\int \frac {(e x)^m}{(a+b x) (a d-b d x)^2} \, dx\) [65]

Optimal result

Integrand size = 24, antiderivative size = 98 \[ \int \frac {(e x)^m}{(a+b x) (a d-b d x)^2} \, dx=\frac {(e x)^{1+m} \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{2},\frac {3+m}{2},\frac {b^2 x^2}{a^2}\right )}{a^3 d^2 e (1+m)}+\frac {b (e x)^{2+m} \operatorname {Hypergeometric2F1}\left (2,\frac {2+m}{2},\frac {4+m}{2},\frac {b^2 x^2}{a^2}\right )}{a^4 d^2 e^2 (2+m)} \]

[Out]

(e*x)^(1+m)*hypergeom([2, 1/2+1/2*m],[3/2+1/2*m],b^2*x^2/a^2)/a^3/d^2/e/(1+m)+b*(e*x)^(2+m)*hypergeom([2, 1+1/
2*m],[2+1/2*m],b^2*x^2/a^2)/a^4/d^2/e^2/(2+m)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {83, 74, 371} \[ \int \frac {(e x)^m}{(a+b x) (a d-b d x)^2} \, dx=\frac {b (e x)^{m+2} \operatorname {Hypergeometric2F1}\left (2,\frac {m+2}{2},\frac {m+4}{2},\frac {b^2 x^2}{a^2}\right )}{a^4 d^2 e^2 (m+2)}+\frac {(e x)^{m+1} \operatorname {Hypergeometric2F1}\left (2,\frac {m+1}{2},\frac {m+3}{2},\frac {b^2 x^2}{a^2}\right )}{a^3 d^2 e (m+1)} \]

[In]

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

[Out]

((e*x)^(1 + m)*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, (b^2*x^2)/a^2])/(a^3*d^2*e*(1 + m)) + (b*(e*x)^(2 +
m)*Hypergeometric2F1[2, (2 + m)/2, (4 + m)/2, (b^2*x^2)/a^2])/(a^4*d^2*e^2*(2 + m))

Rule 74

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

Rule 83

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

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps \begin{align*} \text {integral}& = a \int \frac {(e x)^m}{(a+b x)^2 (a d-b d x)^2} \, dx+\frac {b \int \frac {(e x)^{1+m}}{(a+b x)^2 (a d-b d x)^2} \, dx}{e} \\ & = a \int \frac {(e x)^m}{\left (a^2 d-b^2 d x^2\right )^2} \, dx+\frac {b \int \frac {(e x)^{1+m}}{\left (a^2 d-b^2 d x^2\right )^2} \, dx}{e} \\ & = \frac {(e x)^{1+m} \, _2F_1\left (2,\frac {1+m}{2};\frac {3+m}{2};\frac {b^2 x^2}{a^2}\right )}{a^3 d^2 e (1+m)}+\frac {b (e x)^{2+m} \, _2F_1\left (2,\frac {2+m}{2};\frac {4+m}{2};\frac {b^2 x^2}{a^2}\right )}{a^4 d^2 e^2 (2+m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.89 \[ \int \frac {(e x)^m}{(a+b x) (a d-b d x)^2} \, dx=\frac {x (e x)^m \left (b (1+m) x \operatorname {Hypergeometric2F1}\left (2,1+\frac {m}{2},2+\frac {m}{2},\frac {b^2 x^2}{a^2}\right )+a (2+m) \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{2},\frac {3+m}{2},\frac {b^2 x^2}{a^2}\right )\right )}{a^4 d^2 (1+m) (2+m)} \]

[In]

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

[Out]

(x*(e*x)^m*(b*(1 + m)*x*Hypergeometric2F1[2, 1 + m/2, 2 + m/2, (b^2*x^2)/a^2] + a*(2 + m)*Hypergeometric2F1[2,
(1 + m)/2, (3 + m)/2, (b^2*x^2)/a^2]))/(a^4*d^2*(1 + m)*(2 + m))

Maple [F]

\[\int \frac {\left (e x \right )^{m}}{\left (b x +a \right ) \left (-b d x +a d \right )^{2}}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {(e x)^m}{(a+b x) (a d-b d x)^2} \, dx=\int { \frac {\left (e x\right )^{m}}{{\left (b d x - a d\right )}^{2} {\left (b x + a\right )}} \,d x } \]

[In]

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

[Out]

integral((e*x)^m/(b^3*d^2*x^3 - a*b^2*d^2*x^2 - a^2*b*d^2*x + a^3*d^2), x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.59 (sec) , antiderivative size = 440, normalized size of antiderivative = 4.49 \[ \int \frac {(e x)^m}{(a+b x) (a d-b d x)^2} \, dx=- \frac {2 a e^{m} m^{2} x^{m} \Phi \left (\frac {a}{b x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{- 4 a^{3} b d^{2} \Gamma \left (1 - m\right ) + 4 a^{2} b^{2} d^{2} x \Gamma \left (1 - m\right )} + \frac {a e^{m} m x^{m} \Phi \left (\frac {a}{b x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{- 4 a^{3} b d^{2} \Gamma \left (1 - m\right ) + 4 a^{2} b^{2} d^{2} x \Gamma \left (1 - m\right )} - \frac {a e^{m} m x^{m} \Phi \left (\frac {a e^{i \pi }}{b x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{- 4 a^{3} b d^{2} \Gamma \left (1 - m\right ) + 4 a^{2} b^{2} d^{2} x \Gamma \left (1 - m\right )} + \frac {2 b e^{m} m^{2} x x^{m} \Phi \left (\frac {a}{b x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{- 4 a^{3} b d^{2} \Gamma \left (1 - m\right ) + 4 a^{2} b^{2} d^{2} x \Gamma \left (1 - m\right )} - \frac {b e^{m} m x x^{m} \Phi \left (\frac {a}{b x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{- 4 a^{3} b d^{2} \Gamma \left (1 - m\right ) + 4 a^{2} b^{2} d^{2} x \Gamma \left (1 - m\right )} + \frac {b e^{m} m x x^{m} \Phi \left (\frac {a e^{i \pi }}{b x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{- 4 a^{3} b d^{2} \Gamma \left (1 - m\right ) + 4 a^{2} b^{2} d^{2} x \Gamma \left (1 - m\right )} + \frac {2 b e^{m} m x x^{m} \Gamma \left (- m\right )}{- 4 a^{3} b d^{2} \Gamma \left (1 - m\right ) + 4 a^{2} b^{2} d^{2} x \Gamma \left (1 - m\right )} \]

[In]

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

[Out]

-2*a*e**m*m**2*x**m*lerchphi(a/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(-4*a**3*b*d**2*gamma(1 - m) + 4*a**2*b*
*2*d**2*x*gamma(1 - m)) + a*e**m*m*x**m*lerchphi(a/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(-4*a**3*b*d**2*gamm
a(1 - m) + 4*a**2*b**2*d**2*x*gamma(1 - m)) - a*e**m*m*x**m*lerchphi(a*exp_polar(I*pi)/(b*x), 1, m*exp_polar(I
*pi))*gamma(-m)/(-4*a**3*b*d**2*gamma(1 - m) + 4*a**2*b**2*d**2*x*gamma(1 - m)) + 2*b*e**m*m**2*x*x**m*lerchph
i(a/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(-4*a**3*b*d**2*gamma(1 - m) + 4*a**2*b**2*d**2*x*gamma(1 - m)) - b
*e**m*m*x*x**m*lerchphi(a/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(-4*a**3*b*d**2*gamma(1 - m) + 4*a**2*b**2*d*
*2*x*gamma(1 - m)) + b*e**m*m*x*x**m*lerchphi(a*exp_polar(I*pi)/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(-4*a**
3*b*d**2*gamma(1 - m) + 4*a**2*b**2*d**2*x*gamma(1 - m)) + 2*b*e**m*m*x*x**m*gamma(-m)/(-4*a**3*b*d**2*gamma(1
- m) + 4*a**2*b**2*d**2*x*gamma(1 - m))

Maxima [F]

\[ \int \frac {(e x)^m}{(a+b x) (a d-b d x)^2} \, dx=\int { \frac {\left (e x\right )^{m}}{{\left (b d x - a d\right )}^{2} {\left (b x + a\right )}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {(e x)^m}{(a+b x) (a d-b d x)^2} \, dx=\int { \frac {\left (e x\right )^{m}}{{\left (b d x - a d\right )}^{2} {\left (b x + a\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^m}{(a+b x) (a d-b d x)^2} \, dx=\int \frac {{\left (e\,x\right )}^m}{{\left (a\,d-b\,d\,x\right )}^2\,\left (a+b\,x\right )} \,d x \]

[In]

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

[Out]

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