3.19.62 \(\int \frac {(d+e x)^2}{(a+b x+c x^2)^2} \, dx\)

Optimal. Leaf size=99 \[ \frac {4 \left (a e^2-b d e+c d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {(d+e x) (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \]

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Rubi [A]  time = 0.05, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {722, 618, 206} \begin {gather*} \frac {4 \left (a e^2-b d e+c d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {(d+e x) (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(((d + e*x)*(b*d - 2*a*e + (2*c*d - b*e)*x))/((b^2 - 4*a*c)*(a + b*x + c*x^2))) + (4*(c*d^2 - b*d*e + a*e^2)*
ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(3/2)

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 722

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

Rubi steps

\begin {align*} \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^2} \, dx &=-\frac {(d+e x) (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\left (2 \left (c d^2-b d e+a e^2\right )\right ) \int \frac {1}{a+b x+c x^2} \, dx}{b^2-4 a c}\\ &=-\frac {(d+e x) (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\left (4 \left (c d^2-b d e+a e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{b^2-4 a c}\\ &=-\frac {(d+e x) (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {4 \left (c d^2-b d e+a e^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 128, normalized size = 1.29 \begin {gather*} \frac {a b e^2-2 a c e (2 d+e x)+b^2 e^2 x+b c d (d-2 e x)+2 c^2 d^2 x}{c \left (4 a c-b^2\right ) (a+x (b+c x))}+\frac {4 \left (e (a e-b d)+c d^2\right ) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*b*e^2 + 2*c^2*d^2*x + b^2*e^2*x + b*c*d*(d - 2*e*x) - 2*a*c*e*(2*d + e*x))/(c*(-b^2 + 4*a*c)*(a + x*(b + c*
x))) + (4*(c*d^2 + e*(-(b*d) + a*e))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.41, size = 669, normalized size = 6.76 \begin {gather*} \left [-\frac {{\left (b^{3} c - 4 \, a b c^{2}\right )} d^{2} - 4 \, {\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} d e + {\left (a b^{3} - 4 \, a^{2} b c\right )} e^{2} + 2 \, {\left (a c^{2} d^{2} - a b c d e + a^{2} c e^{2} + {\left (c^{3} d^{2} - b c^{2} d e + a c^{2} e^{2}\right )} x^{2} + {\left (b c^{2} d^{2} - b^{2} c d e + a b c e^{2}\right )} x\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c - \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) + {\left (2 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{2} - 2 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} d e + {\left (b^{4} - 6 \, a b^{2} c + 8 \, a^{2} c^{2}\right )} e^{2}\right )} x}{a b^{4} c - 8 \, a^{2} b^{2} c^{2} + 16 \, a^{3} c^{3} + {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{2} + {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x}, -\frac {{\left (b^{3} c - 4 \, a b c^{2}\right )} d^{2} - 4 \, {\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} d e + {\left (a b^{3} - 4 \, a^{2} b c\right )} e^{2} - 4 \, {\left (a c^{2} d^{2} - a b c d e + a^{2} c e^{2} + {\left (c^{3} d^{2} - b c^{2} d e + a c^{2} e^{2}\right )} x^{2} + {\left (b c^{2} d^{2} - b^{2} c d e + a b c e^{2}\right )} x\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + {\left (2 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{2} - 2 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} d e + {\left (b^{4} - 6 \, a b^{2} c + 8 \, a^{2} c^{2}\right )} e^{2}\right )} x}{a b^{4} c - 8 \, a^{2} b^{2} c^{2} + 16 \, a^{3} c^{3} + {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{2} + {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-((b^3*c - 4*a*b*c^2)*d^2 - 4*(a*b^2*c - 4*a^2*c^2)*d*e + (a*b^3 - 4*a^2*b*c)*e^2 + 2*(a*c^2*d^2 - a*b*c*d*e
+ a^2*c*e^2 + (c^3*d^2 - b*c^2*d*e + a*c^2*e^2)*x^2 + (b*c^2*d^2 - b^2*c*d*e + a*b*c*e^2)*x)*sqrt(b^2 - 4*a*c)
*log((2*c^2*x^2 + 2*b*c*x + b^2 - 2*a*c - sqrt(b^2 - 4*a*c)*(2*c*x + b))/(c*x^2 + b*x + a)) + (2*(b^2*c^2 - 4*
a*c^3)*d^2 - 2*(b^3*c - 4*a*b*c^2)*d*e + (b^4 - 6*a*b^2*c + 8*a^2*c^2)*e^2)*x)/(a*b^4*c - 8*a^2*b^2*c^2 + 16*a
^3*c^3 + (b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^2 + (b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x), -((b^3*c - 4*a*b*
c^2)*d^2 - 4*(a*b^2*c - 4*a^2*c^2)*d*e + (a*b^3 - 4*a^2*b*c)*e^2 - 4*(a*c^2*d^2 - a*b*c*d*e + a^2*c*e^2 + (c^3
*d^2 - b*c^2*d*e + a*c^2*e^2)*x^2 + (b*c^2*d^2 - b^2*c*d*e + a*b*c*e^2)*x)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^
2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + (2*(b^2*c^2 - 4*a*c^3)*d^2 - 2*(b^3*c - 4*a*b*c^2)*d*e + (b^4 - 6*a*b^
2*c + 8*a^2*c^2)*e^2)*x)/(a*b^4*c - 8*a^2*b^2*c^2 + 16*a^3*c^3 + (b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^2 + (b
^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x)]

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giac [A]  time = 0.16, size = 139, normalized size = 1.40 \begin {gather*} -\frac {4 \, {\left (c d^{2} - b d e + a e^{2}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} - 4 \, a c\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {2 \, c^{2} d^{2} x - 2 \, b c d x e + b c d^{2} + b^{2} x e^{2} - 2 \, a c x e^{2} - 4 \, a c d e + a b e^{2}}{{\left (b^{2} c - 4 \, a c^{2}\right )} {\left (c x^{2} + b x + a\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*(c*d^2 - b*d*e + a*e^2)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^2 - 4*a*c)*sqrt(-b^2 + 4*a*c)) - (2*c^2*
d^2*x - 2*b*c*d*x*e + b*c*d^2 + b^2*x*e^2 - 2*a*c*x*e^2 - 4*a*c*d*e + a*b*e^2)/((b^2*c - 4*a*c^2)*(c*x^2 + b*x
+ a))

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maple [B]  time = 0.05, size = 212, normalized size = 2.14 \begin {gather*} \frac {4 a \,e^{2} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}}}-\frac {4 b d e \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}}}+\frac {4 c \,d^{2} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}}}+\frac {-\frac {\left (2 a c \,e^{2}-b^{2} e^{2}+2 b c d e -2 c^{2} d^{2}\right ) x}{\left (4 a c -b^{2}\right ) c}+\frac {a b \,e^{2}-4 a c d e +b c \,d^{2}}{\left (4 a c -b^{2}\right ) c}}{c \,x^{2}+b x +a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-(2*a*c*e^2-b^2*e^2+2*b*c*d*e-2*c^2*d^2)/c/(4*a*c-b^2)*x+1/c*(a*b*e^2-4*a*c*d*e+b*c*d^2)/(4*a*c-b^2))/(c*x^2+
b*x+a)+4/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*e^2-4/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a
*c-b^2)^(1/2))*b*d*e+4/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*c*d^2

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more details)Is 4*a*c-b^2 positive or negative?

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mupad [B]  time = 0.16, size = 230, normalized size = 2.32 \begin {gather*} \frac {\frac {b\,c\,d^2-4\,a\,c\,d\,e+a\,b\,e^2}{c\,\left (4\,a\,c-b^2\right )}+\frac {x\,\left (b^2\,e^2-2\,b\,c\,d\,e+2\,c^2\,d^2-2\,a\,c\,e^2\right )}{c\,\left (4\,a\,c-b^2\right )}}{c\,x^2+b\,x+a}-\frac {4\,\mathrm {atan}\left (\frac {\left (\frac {2\,\left (b^3-4\,a\,b\,c\right )\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}{{\left (4\,a\,c-b^2\right )}^{5/2}}-\frac {4\,c\,x\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}{{\left (4\,a\,c-b^2\right )}^{3/2}}\right )\,\left (4\,a\,c-b^2\right )}{2\,c\,d^2-2\,b\,d\,e+2\,a\,e^2}\right )\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}{{\left (4\,a\,c-b^2\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((a*b*e^2 + b*c*d^2 - 4*a*c*d*e)/(c*(4*a*c - b^2)) + (x*(b^2*e^2 + 2*c^2*d^2 - 2*a*c*e^2 - 2*b*c*d*e))/(c*(4*a
*c - b^2)))/(a + b*x + c*x^2) - (4*atan((((2*(b^3 - 4*a*b*c)*(a*e^2 + c*d^2 - b*d*e))/(4*a*c - b^2)^(5/2) - (4
*c*x*(a*e^2 + c*d^2 - b*d*e))/(4*a*c - b^2)^(3/2))*(4*a*c - b^2))/(2*a*e^2 + 2*c*d^2 - 2*b*d*e))*(a*e^2 + c*d^
2 - b*d*e))/(4*a*c - b^2)^(3/2)

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sympy [B]  time = 1.79, size = 517, normalized size = 5.22 \begin {gather*} - 2 \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (a e^{2} - b d e + c d^{2}\right ) \log {\left (x + \frac {- 32 a^{2} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (a e^{2} - b d e + c d^{2}\right ) + 16 a b^{2} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (a e^{2} - b d e + c d^{2}\right ) + 2 a b e^{2} - 2 b^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (a e^{2} - b d e + c d^{2}\right ) - 2 b^{2} d e + 2 b c d^{2}}{4 a c e^{2} - 4 b c d e + 4 c^{2} d^{2}} \right )} + 2 \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (a e^{2} - b d e + c d^{2}\right ) \log {\left (x + \frac {32 a^{2} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (a e^{2} - b d e + c d^{2}\right ) - 16 a b^{2} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (a e^{2} - b d e + c d^{2}\right ) + 2 a b e^{2} + 2 b^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (a e^{2} - b d e + c d^{2}\right ) - 2 b^{2} d e + 2 b c d^{2}}{4 a c e^{2} - 4 b c d e + 4 c^{2} d^{2}} \right )} + \frac {a b e^{2} - 4 a c d e + b c d^{2} + x \left (- 2 a c e^{2} + b^{2} e^{2} - 2 b c d e + 2 c^{2} d^{2}\right )}{4 a^{2} c^{2} - a b^{2} c + x^{2} \left (4 a c^{3} - b^{2} c^{2}\right ) + x \left (4 a b c^{2} - b^{3} c\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(-1/(4*a*c - b**2)**3)*(a*e**2 - b*d*e + c*d**2)*log(x + (-32*a**2*c**2*sqrt(-1/(4*a*c - b**2)**3)*(a*e
**2 - b*d*e + c*d**2) + 16*a*b**2*c*sqrt(-1/(4*a*c - b**2)**3)*(a*e**2 - b*d*e + c*d**2) + 2*a*b*e**2 - 2*b**4
*sqrt(-1/(4*a*c - b**2)**3)*(a*e**2 - b*d*e + c*d**2) - 2*b**2*d*e + 2*b*c*d**2)/(4*a*c*e**2 - 4*b*c*d*e + 4*c
**2*d**2)) + 2*sqrt(-1/(4*a*c - b**2)**3)*(a*e**2 - b*d*e + c*d**2)*log(x + (32*a**2*c**2*sqrt(-1/(4*a*c - b**
2)**3)*(a*e**2 - b*d*e + c*d**2) - 16*a*b**2*c*sqrt(-1/(4*a*c - b**2)**3)*(a*e**2 - b*d*e + c*d**2) + 2*a*b*e*
*2 + 2*b**4*sqrt(-1/(4*a*c - b**2)**3)*(a*e**2 - b*d*e + c*d**2) - 2*b**2*d*e + 2*b*c*d**2)/(4*a*c*e**2 - 4*b*
c*d*e + 4*c**2*d**2)) + (a*b*e**2 - 4*a*c*d*e + b*c*d**2 + x*(-2*a*c*e**2 + b**2*e**2 - 2*b*c*d*e + 2*c**2*d**
2))/(4*a**2*c**2 - a*b**2*c + x**2*(4*a*c**3 - b**2*c**2) + x*(4*a*b*c**2 - b**3*c))

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