3.1.79 \(\int \frac {(e x)^m}{(a+b x) (a c-b c x)^3} \, dx\) [79]

Optimal. Leaf size=151 \[ \frac {(e x)^{1+m}}{4 a^2 c^3 e (a-b x)^2}+\frac {(2-m) (e x)^{1+m}}{4 a^3 c^3 e (a-b x)}+\frac {(e x)^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac {b x}{a}\right )}{8 a^4 c^3 e (1+m)}+\frac {\left (1-4 m+2 m^2\right ) (e x)^{1+m} \, _2F_1\left (1,1+m;2+m;\frac {b x}{a}\right )}{8 a^4 c^3 e (1+m)} \]

[Out]

1/4*(e*x)^(1+m)/a^2/c^3/e/(-b*x+a)^2+1/4*(2-m)*(e*x)^(1+m)/a^3/c^3/e/(-b*x+a)+1/8*(e*x)^(1+m)*hypergeom([1, 1+
m],[2+m],-b*x/a)/a^4/c^3/e/(1+m)+1/8*(2*m^2-4*m+1)*(e*x)^(1+m)*hypergeom([1, 1+m],[2+m],b*x/a)/a^4/c^3/e/(1+m)

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Rubi [A]
time = 0.12, antiderivative size = 151, normalized size of antiderivative = 1.00, 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 = {105, 156, 162, 66} \begin {gather*} \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} \end {gather*}

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 66

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x)^(m + 1)/(b*(m + 1)))*Hypergeometr
ic2F1[-n, m + 1, m + 2, (-d)*(x/c)], 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 105

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] && ILtQ[m, -1] &
& (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])

Rule 156

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] && ILtQ[m, -1]

Rule 162

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]

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*}

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Mathematica [A]
time = 0.14, size = 106, normalized size = 0.70 \begin {gather*} \frac {x (e x)^m \left (-2 a (1+m) (a (-3+m)-b (-2+m) x)+(a-b x)^2 \, _2F_1\left (1,1+m;2+m;-\frac {b x}{a}\right )+\left (1-4 m+2 m^2\right ) (a-b x)^2 \, _2F_1\left (1,1+m;2+m;\frac {b x}{a}\right )\right )}{8 a^4 c^3 (1+m) (a-b x)^2} \end {gather*}

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)

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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (e x \right )^{m}}{\left (b x +a \right ) \left (-b c x +a c \right )^{3}}\, dx\]

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)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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((x*e)^m/((b*c*x - a*c)^3*(b*x + a)), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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(-(x*e)^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)

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Sympy [C] Result contains complex when optimal does not.
time = 2.12, size = 1363, normalized size = 9.03 \begin {gather*} - \frac {2 a^{2} e^{m} m^{3} x^{m} \Phi \left (\frac {a}{b x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{8 a^{5} b c^{3} \Gamma \left (1 - m\right ) - 16 a^{4} b^{2} c^{3} x \Gamma \left (1 - m\right ) + 8 a^{3} b^{3} c^{3} x^{2} \Gamma \left (1 - m\right )} + \frac {4 a^{2} e^{m} m^{2} x^{m} \Phi \left (\frac {a}{b x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{8 a^{5} b c^{3} \Gamma \left (1 - m\right ) - 16 a^{4} b^{2} c^{3} x \Gamma \left (1 - m\right ) + 8 a^{3} b^{3} c^{3} x^{2} \Gamma \left (1 - m\right )} - \frac {a^{2} e^{m} m x^{m} \Phi \left (\frac {a}{b x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{8 a^{5} b c^{3} \Gamma \left (1 - m\right ) - 16 a^{4} b^{2} c^{3} x \Gamma \left (1 - m\right ) + 8 a^{3} b^{3} c^{3} x^{2} \Gamma \left (1 - m\right )} + \frac {a^{2} e^{m} m x^{m} \Phi \left (\frac {a e^{i \pi }}{b x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{8 a^{5} b c^{3} \Gamma \left (1 - m\right ) - 16 a^{4} b^{2} c^{3} x \Gamma \left (1 - m\right ) + 8 a^{3} b^{3} c^{3} x^{2} \Gamma \left (1 - m\right )} + \frac {4 a b e^{m} m^{3} x x^{m} \Phi \left (\frac {a}{b x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{8 a^{5} b c^{3} \Gamma \left (1 - m\right ) - 16 a^{4} b^{2} c^{3} x \Gamma \left (1 - m\right ) + 8 a^{3} b^{3} c^{3} x^{2} \Gamma \left (1 - m\right )} - \frac {8 a b e^{m} m^{2} x x^{m} \Phi \left (\frac {a}{b x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{8 a^{5} b c^{3} \Gamma \left (1 - m\right ) - 16 a^{4} b^{2} c^{3} x \Gamma \left (1 - m\right ) + 8 a^{3} b^{3} c^{3} x^{2} \Gamma \left (1 - m\right )} + \frac {2 a b e^{m} m^{2} x x^{m} \Gamma \left (- m\right )}{8 a^{5} b c^{3} \Gamma \left (1 - m\right ) - 16 a^{4} b^{2} c^{3} x \Gamma \left (1 - m\right ) + 8 a^{3} b^{3} c^{3} x^{2} \Gamma \left (1 - m\right )} + \frac {2 a b e^{m} m x x^{m} \Phi \left (\frac {a}{b x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{8 a^{5} b c^{3} \Gamma \left (1 - m\right ) - 16 a^{4} b^{2} c^{3} x \Gamma \left (1 - m\right ) + 8 a^{3} b^{3} c^{3} x^{2} \Gamma \left (1 - m\right )} - \frac {2 a 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 )}{8 a^{5} b c^{3} \Gamma \left (1 - m\right ) - 16 a^{4} b^{2} c^{3} x \Gamma \left (1 - m\right ) + 8 a^{3} b^{3} c^{3} x^{2} \Gamma \left (1 - m\right )} - \frac {6 a b e^{m} m x x^{m} \Gamma \left (- m\right )}{8 a^{5} b c^{3} \Gamma \left (1 - m\right ) - 16 a^{4} b^{2} c^{3} x \Gamma \left (1 - m\right ) + 8 a^{3} b^{3} c^{3} x^{2} \Gamma \left (1 - m\right )} - \frac {2 b^{2} e^{m} m^{3} x^{2} x^{m} \Phi \left (\frac {a}{b x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{8 a^{5} b c^{3} \Gamma \left (1 - m\right ) - 16 a^{4} b^{2} c^{3} x \Gamma \left (1 - m\right ) + 8 a^{3} b^{3} c^{3} x^{2} \Gamma \left (1 - m\right )} + \frac {4 b^{2} e^{m} m^{2} x^{2} x^{m} \Phi \left (\frac {a}{b x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{8 a^{5} b c^{3} \Gamma \left (1 - m\right ) - 16 a^{4} b^{2} c^{3} x \Gamma \left (1 - m\right ) + 8 a^{3} b^{3} c^{3} x^{2} \Gamma \left (1 - m\right )} - \frac {2 b^{2} e^{m} m^{2} x^{2} x^{m} \Gamma \left (- m\right )}{8 a^{5} b c^{3} \Gamma \left (1 - m\right ) - 16 a^{4} b^{2} c^{3} x \Gamma \left (1 - m\right ) + 8 a^{3} b^{3} c^{3} x^{2} \Gamma \left (1 - m\right )} - \frac {b^{2} e^{m} m x^{2} x^{m} \Phi \left (\frac {a}{b x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{8 a^{5} b c^{3} \Gamma \left (1 - m\right ) - 16 a^{4} b^{2} c^{3} x \Gamma \left (1 - m\right ) + 8 a^{3} b^{3} c^{3} x^{2} \Gamma \left (1 - m\right )} + \frac {b^{2} e^{m} m x^{2} x^{m} \Phi \left (\frac {a e^{i \pi }}{b x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{8 a^{5} b c^{3} \Gamma \left (1 - m\right ) - 16 a^{4} b^{2} c^{3} x \Gamma \left (1 - m\right ) + 8 a^{3} b^{3} c^{3} x^{2} \Gamma \left (1 - m\right )} + \frac {4 b^{2} e^{m} m x^{2} x^{m} \Gamma \left (- m\right )}{8 a^{5} b c^{3} \Gamma \left (1 - m\right ) - 16 a^{4} b^{2} c^{3} x \Gamma \left (1 - m\right ) + 8 a^{3} b^{3} c^{3} x^{2} \Gamma \left (1 - m\right )} \end {gather*}

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))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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(-(x*e)^m/((b*c*x - a*c)^3*(b*x + a)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (e\,x\right )}^m}{{\left (a\,c-b\,c\,x\right )}^3\,\left (a+b\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^m/((a*c - b*c*x)^3*(a + b*x)),x)

[Out]

int((e*x)^m/((a*c - b*c*x)^3*(a + b*x)), x)

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