\(\int \frac {(d+e x)^m}{(a+b x+c x^2)^2} \, dx\) [2554]

Optimal result

Integrand size = 20, antiderivative size = 425 \[ \int \frac {(d+e x)^m}{\left (a+b x+c x^2\right )^2} \, dx=-\frac {(d+e x)^{1+m} \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}+\frac {c \left (4 c^2 d^2+b \left (b+\sqrt {b^2-4 a c}\right ) e^2 m-2 c e \left (2 b d-2 a e (1-m)+\sqrt {b^2-4 a c} d m\right )\right ) (d+e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}\right )}{\left (b^2-4 a c\right )^{3/2} \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) \left (c d^2-b d e+a e^2\right ) (1+m)}-\frac {c \left (e (2 c d-b e) m+\frac {4 c^2 d^2-4 c e (b d-a e (1-m))+b^2 e^2 m}{\sqrt {b^2-4 a c}}\right ) (d+e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\left (b^2-4 a c\right ) \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \left (c d^2-b d e+a e^2\right ) (1+m)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {754, 844, 70} \[ \int \frac {(d+e x)^m}{\left (a+b x+c x^2\right )^2} \, dx=\frac {c (d+e x)^{m+1} \left (-2 c e \left (d m \sqrt {b^2-4 a c}-2 a e (1-m)+2 b d\right )+b e^2 m \left (\sqrt {b^2-4 a c}+b\right )+4 c^2 d^2\right ) \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}\right )}{(m+1) \left (b^2-4 a c\right )^{3/2} \left (2 c d-e \left (b-\sqrt {b^2-4 a c}\right )\right ) \left (a e^2-b d e+c d^2\right )}-\frac {c (d+e x)^{m+1} \left (\frac {-4 c e (b d-a e (1-m))+b^2 e^2 m+4 c^2 d^2}{\sqrt {b^2-4 a c}}+e m (2 c d-b e)\right ) \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{(m+1) \left (b^2-4 a c\right ) \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right ) \left (a e^2-b d e+c d^2\right )}-\frac {(d+e x)^{m+1} \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )} \]

[In]

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

[Out]

-(((d + e*x)^(1 + m)*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a
+ b*x + c*x^2))) + (c*(4*c^2*d^2 + b*(b + Sqrt[b^2 - 4*a*c])*e^2*m - 2*c*e*(2*b*d - 2*a*e*(1 - m) + Sqrt[b^2 -
4*a*c]*d*m))*(d + e*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (2*c*(d + e*x))/(2*c*d - (b - Sqrt[b^2 - 4*
a*c])*e)])/((b^2 - 4*a*c)^(3/2)*(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e)*(c*d^2 - b*d*e + a*e^2)*(1 + m)) - (c*(e*(
2*c*d - b*e)*m + (4*c^2*d^2 - 4*c*e*(b*d - a*e*(1 - m)) + b^2*e^2*m)/Sqrt[b^2 - 4*a*c])*(d + e*x)^(1 + m)*Hype
rgeometric2F1[1, 1 + m, 2 + m, (2*c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/((b^2 - 4*a*c)*(2*c*d - (
b + Sqrt[b^2 - 4*a*c])*e)*(c*d^2 - b*d*e + a*e^2)*(1 + m))

Rule 70

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

Rule 754

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(b
*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e +
a*e^2))), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m
+ 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x
, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b
*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 844

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[(d + e*x)^m, (f + g*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] &&  !RationalQ[m]

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

Mathematica [A] (verified)

Time = 0.82 (sec) , antiderivative size = 339, normalized size of antiderivative = 0.80 \[ \int \frac {(d+e x)^m}{\left (a+b x+c x^2\right )^2} \, dx=\frac {(d+e x)^{1+m} \left (\frac {b^2 e-2 c (a e+c d x)+b c (-d+e x)}{a+x (b+c x)}-\frac {c \left (e (2 c d-b e) m+\frac {-4 c^2 d^2+4 c e (b d+a e (-1+m))-b^2 e^2 m}{\sqrt {b^2-4 a c}}\right ) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {2 c (d+e x)}{2 c d+\left (-b+\sqrt {b^2-4 a c}\right ) e}\right )}{\left (2 c d+\left (-b+\sqrt {b^2-4 a c}\right ) e\right ) (1+m)}-\frac {c \left (e (2 c d-b e) m+\frac {4 c^2 d^2-4 c e (b d+a e (-1+m))+b^2 e^2 m}{\sqrt {b^2-4 a c}}\right ) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) (1+m)}\right )}{\left (b^2-4 a c\right ) \left (c d^2+e (-b d+a e)\right )} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral((e*x + d)^m/(c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2), x)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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