3.79 \(\int \frac{(e x)^m}{(a+b x) (a c-b c x)^3} \, dx\)
Optimal. Leaf size=151 \[ \frac{\left (2 m^2-4 m+1\right ) (e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{b x}{a}\right )}{8 a^4 c^3 e (m+1)}+\frac{(e x)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{b x}{a}\right )}{8 a^4 c^3 e (m+1)}+\frac{(2-m) (e x)^{m+1}}{4 a^3 c^3 e (a-b x)}+\frac{(e x)^{m+1}}{4 a^2 c^3 e (a-b x)^2} \]
[Out]
(e*x)^(1 + m)/(4*a^2*c^3*e*(a - b*x)^2) + ((2 - m)*(e*x)^(1 + m))/(4*a^3*c^3*e*(a - b*x)) + ((e*x)^(1 + m)*Hyp
ergeometric2F1[1, 1 + m, 2 + m, -((b*x)/a)])/(8*a^4*c^3*e*(1 + m)) + ((1 - 4*m + 2*m^2)*(e*x)^(1 + m)*Hypergeo
metric2F1[1, 1 + m, 2 + m, (b*x)/a])/(8*a^4*c^3*e*(1 + m))
________________________________________________________________________________________
Rubi [A] time = 0.175591, antiderivative size = 151, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used =
{103, 151, 156, 64} \[ \frac{\left (2 m^2-4 m+1\right ) (e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{b x}{a}\right )}{8 a^4 c^3 e (m+1)}+\frac{(e x)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{b x}{a}\right )}{8 a^4 c^3 e (m+1)}+\frac{(2-m) (e x)^{m+1}}{4 a^3 c^3 e (a-b x)}+\frac{(e x)^{m+1}}{4 a^2 c^3 e (a-b x)^2} \]
Antiderivative was successfully verified.
[In]
Int[(e*x)^m/((a + b*x)*(a*c - b*c*x)^3),x]
[Out]
(e*x)^(1 + m)/(4*a^2*c^3*e*(a - b*x)^2) + ((2 - m)*(e*x)^(1 + m))/(4*a^3*c^3*e*(a - b*x)) + ((e*x)^(1 + m)*Hyp
ergeometric2F1[1, 1 + m, 2 + m, -((b*x)/a)])/(8*a^4*c^3*e*(1 + m)) + ((1 - 4*m + 2*m^2)*(e*x)^(1 + m)*Hypergeo
metric2F1[1, 1 + m, 2 + m, (b*x)/a])/(8*a^4*c^3*e*(1 + m))
Rule 103
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])
Rule 151
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]
Rule 156
Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
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])))
Rubi steps
\begin{align*} \int \frac{(e x)^m}{(a+b x) (a c-b c x)^3} \, dx &=\frac{(e x)^{1+m}}{4 a^2 c^3 e (a-b x)^2}-\frac{\int \frac{(e x)^m \left (-a b c e (3-m)-b^2 c e (1-m) x\right )}{(a+b x) (a c-b c x)^2} \, dx}{4 a^2 b c^2 e}\\ &=\frac{(e x)^{1+m}}{4 a^2 c^3 e (a-b x)^2}+\frac{(2-m) (e x)^{1+m}}{4 a^3 c^3 e (a-b x)}+\frac{\int \frac{(e x)^m \left (2 a^2 b^2 c^2 e^2 (1-m)^2-2 a b^3 c^2 e^2 (2-m) m x\right )}{(a+b x) (a c-b c x)} \, dx}{8 a^4 b^2 c^4 e^2}\\ &=\frac{(e x)^{1+m}}{4 a^2 c^3 e (a-b x)^2}+\frac{(2-m) (e x)^{1+m}}{4 a^3 c^3 e (a-b x)}+\frac{\int \frac{(e x)^m}{a+b x} \, dx}{8 a^3 c^3}+\frac{\left (1-4 m+2 m^2\right ) \int \frac{(e x)^m}{a c-b c x} \, dx}{8 a^3 c^2}\\ &=\frac{(e x)^{1+m}}{4 a^2 c^3 e (a-b x)^2}+\frac{(2-m) (e x)^{1+m}}{4 a^3 c^3 e (a-b x)}+\frac{(e x)^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac{b x}{a}\right )}{8 a^4 c^3 e (1+m)}+\frac{\left (1-4 m+2 m^2\right ) (e x)^{1+m} \, _2F_1\left (1,1+m;2+m;\frac{b x}{a}\right )}{8 a^4 c^3 e (1+m)}\\ \end{align*}
Mathematica [A] time = 0.0909904, size = 106, normalized size = 0.7 \[ \frac{x (e x)^m \left (\left (2 m^2-4 m+1\right ) (a-b x)^2 \, _2F_1\left (1,m+1;m+2;\frac{b x}{a}\right )+(a-b x)^2 \, _2F_1\left (1,m+1;m+2;-\frac{b x}{a}\right )-2 a (m+1) (a (m-3)-b (m-2) x)\right )}{8 a^4 c^3 (m+1) (a-b x)^2} \]
Antiderivative was successfully verified.
[In]
Integrate[(e*x)^m/((a + b*x)*(a*c - b*c*x)^3),x]
[Out]
(x*(e*x)^m*(-2*a*(1 + m)*(a*(-3 + m) - b*(-2 + m)*x) + (a - b*x)^2*Hypergeometric2F1[1, 1 + m, 2 + m, -((b*x)/
a)] + (1 - 4*m + 2*m^2)*(a - b*x)^2*Hypergeometric2F1[1, 1 + m, 2 + m, (b*x)/a]))/(8*a^4*c^3*(1 + m)*(a - b*x)
^2)
________________________________________________________________________________________
Maple [F] time = 0.062, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex \right ) ^{m}}{ \left ( bx+a \right ) \left ( -bcx+ac \right ) ^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x)^m/(b*x+a)/(-b*c*x+a*c)^3,x)
[Out]
int((e*x)^m/(b*x+a)/(-b*c*x+a*c)^3,x)
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{\left (e x\right )^{m}}{{\left (b c x - a c\right )}^{3}{\left (b x + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)^m/(b*x+a)/(-b*c*x+a*c)^3,x, algorithm="maxima")
[Out]
-integrate((e*x)^m/((b*c*x - a*c)^3*(b*x + a)), x)
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\left (e x\right )^{m}}{b^{4} c^{3} x^{4} - 2 \, a b^{3} c^{3} x^{3} + 2 \, a^{3} b c^{3} x - a^{4} c^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)^m/(b*x+a)/(-b*c*x+a*c)^3,x, algorithm="fricas")
[Out]
integral(-(e*x)^m/(b^4*c^3*x^4 - 2*a*b^3*c^3*x^3 + 2*a^3*b*c^3*x - a^4*c^3), x)
________________________________________________________________________________________
Sympy [C] time = 4.43417, size = 1363, normalized size = 9.03 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)**m/(b*x+a)/(-b*c*x+a*c)**3,x)
[Out]
-2*a**2*e**m*m**3*x**m*lerchphi(a/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(8*a**5*b*c**3*gamma(1 - m) - 16*a**4
*b**2*c**3*x*gamma(1 - m) + 8*a**3*b**3*c**3*x**2*gamma(1 - m)) + 4*a**2*e**m*m**2*x**m*lerchphi(a/(b*x), 1, m
*exp_polar(I*pi))*gamma(-m)/(8*a**5*b*c**3*gamma(1 - m) - 16*a**4*b**2*c**3*x*gamma(1 - m) + 8*a**3*b**3*c**3*
x**2*gamma(1 - m)) - a**2*e**m*m*x**m*lerchphi(a/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(8*a**5*b*c**3*gamma(1
- m) - 16*a**4*b**2*c**3*x*gamma(1 - m) + 8*a**3*b**3*c**3*x**2*gamma(1 - m)) + a**2*e**m*m*x**m*lerchphi(a*e
xp_polar(I*pi)/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(8*a**5*b*c**3*gamma(1 - m) - 16*a**4*b**2*c**3*x*gamma(
1 - m) + 8*a**3*b**3*c**3*x**2*gamma(1 - m)) + 4*a*b*e**m*m**3*x*x**m*lerchphi(a/(b*x), 1, m*exp_polar(I*pi))*
gamma(-m)/(8*a**5*b*c**3*gamma(1 - m) - 16*a**4*b**2*c**3*x*gamma(1 - m) + 8*a**3*b**3*c**3*x**2*gamma(1 - m))
- 8*a*b*e**m*m**2*x*x**m*lerchphi(a/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(8*a**5*b*c**3*gamma(1 - m) - 16*a
**4*b**2*c**3*x*gamma(1 - m) + 8*a**3*b**3*c**3*x**2*gamma(1 - m)) + 2*a*b*e**m*m**2*x*x**m*gamma(-m)/(8*a**5*
b*c**3*gamma(1 - m) - 16*a**4*b**2*c**3*x*gamma(1 - m) + 8*a**3*b**3*c**3*x**2*gamma(1 - m)) + 2*a*b*e**m*m*x*
x**m*lerchphi(a/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(8*a**5*b*c**3*gamma(1 - m) - 16*a**4*b**2*c**3*x*gamma
(1 - m) + 8*a**3*b**3*c**3*x**2*gamma(1 - m)) - 2*a*b*e**m*m*x*x**m*lerchphi(a*exp_polar(I*pi)/(b*x), 1, m*exp
_polar(I*pi))*gamma(-m)/(8*a**5*b*c**3*gamma(1 - m) - 16*a**4*b**2*c**3*x*gamma(1 - m) + 8*a**3*b**3*c**3*x**2
*gamma(1 - m)) - 6*a*b*e**m*m*x*x**m*gamma(-m)/(8*a**5*b*c**3*gamma(1 - m) - 16*a**4*b**2*c**3*x*gamma(1 - m)
+ 8*a**3*b**3*c**3*x**2*gamma(1 - m)) - 2*b**2*e**m*m**3*x**2*x**m*lerchphi(a/(b*x), 1, m*exp_polar(I*pi))*gam
ma(-m)/(8*a**5*b*c**3*gamma(1 - m) - 16*a**4*b**2*c**3*x*gamma(1 - m) + 8*a**3*b**3*c**3*x**2*gamma(1 - m)) +
4*b**2*e**m*m**2*x**2*x**m*lerchphi(a/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(8*a**5*b*c**3*gamma(1 - m) - 16*
a**4*b**2*c**3*x*gamma(1 - m) + 8*a**3*b**3*c**3*x**2*gamma(1 - m)) - 2*b**2*e**m*m**2*x**2*x**m*gamma(-m)/(8*
a**5*b*c**3*gamma(1 - m) - 16*a**4*b**2*c**3*x*gamma(1 - m) + 8*a**3*b**3*c**3*x**2*gamma(1 - m)) - b**2*e**m*
m*x**2*x**m*lerchphi(a/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(8*a**5*b*c**3*gamma(1 - m) - 16*a**4*b**2*c**3*
x*gamma(1 - m) + 8*a**3*b**3*c**3*x**2*gamma(1 - m)) + b**2*e**m*m*x**2*x**m*lerchphi(a*exp_polar(I*pi)/(b*x),
1, m*exp_polar(I*pi))*gamma(-m)/(8*a**5*b*c**3*gamma(1 - m) - 16*a**4*b**2*c**3*x*gamma(1 - m) + 8*a**3*b**3*
c**3*x**2*gamma(1 - m)) + 4*b**2*e**m*m*x**2*x**m*gamma(-m)/(8*a**5*b*c**3*gamma(1 - m) - 16*a**4*b**2*c**3*x*
gamma(1 - m) + 8*a**3*b**3*c**3*x**2*gamma(1 - m))
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{\left (e x\right )^{m}}{{\left (b c x - a c\right )}^{3}{\left (b x + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)^m/(b*x+a)/(-b*c*x+a*c)^3,x, algorithm="giac")
[Out]
integrate(-(e*x)^m/((b*c*x - a*c)^3*(b*x + a)), x)