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

Optimal. Leaf size=108 \[ -\frac {\tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )}{c^2 \sqrt {b^2-4 a c}}+\frac {\log \left (a+b x+c x^2\right ) (A c e-b B e+B c d)}{2 c^2}+\frac {B e x}{c} \]

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Rubi [A]  time = 0.13, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {773, 634, 618, 206, 628} \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )}{c^2 \sqrt {b^2-4 a c}}+\frac {\log \left (a+b x+c x^2\right ) (A c e-b B e+B c d)}{2 c^2}+\frac {B e x}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(B*e*x)/c - ((b^2*B*e - b*c*(B*d + A*e) + 2*c*(A*c*d - a*B*e))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c^2*Sq
rt[b^2 - 4*a*c]) + ((B*c*d - b*B*e + A*c*e)*Log[a + b*x + c*x^2])/(2*c^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 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

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

Rule 773

Int[(((d_.) + (e_.)*(x_))*((f_) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*g*x)/
c, x] + Dist[1/c, Int[(c*d*f - a*e*g + (c*e*f + c*d*g - b*e*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]

Rubi steps

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

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Mathematica [A]  time = 0.09, size = 108, normalized size = 1.00 \begin {gather*} \frac {\frac {2 \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right ) \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )}{\sqrt {4 a c-b^2}}+\log (a+x (b+c x)) (A c e-b B e+B c d)+2 B c e x}{2 c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.42, size = 369, normalized size = 3.42 \begin {gather*} \left [\frac {2 \, {\left (B b^{2} c - 4 \, B a c^{2}\right )} e x + \sqrt {b^{2} - 4 \, a c} {\left ({\left (B b c - 2 \, A c^{2}\right )} d - {\left (B b^{2} - {\left (2 \, B a + A b\right )} c\right )} e\right )} \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 ({\left (B b^{2} c - 4 \, B a c^{2}\right )} d - {\left (B b^{3} + 4 \, A a c^{2} - {\left (4 \, B a b + A b^{2}\right )} c\right )} e\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )}}, \frac {2 \, {\left (B b^{2} c - 4 \, B a c^{2}\right )} e x + 2 \, \sqrt {-b^{2} + 4 \, a c} {\left ({\left (B b c - 2 \, A c^{2}\right )} d - {\left (B b^{2} - {\left (2 \, B a + A b\right )} c\right )} e\right )} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + {\left ({\left (B b^{2} c - 4 \, B a c^{2}\right )} d - {\left (B b^{3} + 4 \, A a c^{2} - {\left (4 \, B a b + A b^{2}\right )} c\right )} e\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(2*(B*b^2*c - 4*B*a*c^2)*e*x + sqrt(b^2 - 4*a*c)*((B*b*c - 2*A*c^2)*d - (B*b^2 - (2*B*a + A*b)*c)*e)*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)) + ((B*b^2*c - 4*B*a*c^2
)*d - (B*b^3 + 4*A*a*c^2 - (4*B*a*b + A*b^2)*c)*e)*log(c*x^2 + b*x + a))/(b^2*c^2 - 4*a*c^3), 1/2*(2*(B*b^2*c
- 4*B*a*c^2)*e*x + 2*sqrt(-b^2 + 4*a*c)*((B*b*c - 2*A*c^2)*d - (B*b^2 - (2*B*a + A*b)*c)*e)*arctan(-sqrt(-b^2
+ 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + ((B*b^2*c - 4*B*a*c^2)*d - (B*b^3 + 4*A*a*c^2 - (4*B*a*b + A*b^2)*c)*e)*
log(c*x^2 + b*x + a))/(b^2*c^2 - 4*a*c^3)]

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giac [A]  time = 0.16, size = 112, normalized size = 1.04 \begin {gather*} \frac {B x e}{c} + \frac {{\left (B c d - B b e + A c e\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c^{2}} - \frac {{\left (B b c d - 2 \, A c^{2} d - B b^{2} e + 2 \, B a c e + A b c e\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c} c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B*x*e/c + 1/2*(B*c*d - B*b*e + A*c*e)*log(c*x^2 + b*x + a)/c^2 - (B*b*c*d - 2*A*c^2*d - B*b^2*e + 2*B*a*c*e +
A*b*c*e)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*c^2)

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maple [B]  time = 0.05, size = 261, normalized size = 2.42 \begin {gather*} -\frac {A b e \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c}+\frac {2 A d \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}-\frac {2 B a e \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c}+\frac {B \,b^{2} e \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c^{2}}-\frac {B b d \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c}+\frac {A e \ln \left (c \,x^{2}+b x +a \right )}{2 c}-\frac {B b e \ln \left (c \,x^{2}+b x +a \right )}{2 c^{2}}+\frac {B d \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {B e x}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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((B*x+A)*(e*x+d)/(c*x^2+b*x+a),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 = 2.70, size = 163, normalized size = 1.51 \begin {gather*} \frac {\ln \left (c\,x^2+b\,x+a\right )\,\left (B\,b^3\,e+4\,A\,a\,c^2\,e+4\,B\,a\,c^2\,d-A\,b^2\,c\,e-B\,b^2\,c\,d-4\,B\,a\,b\,c\,e\right )}{2\,\left (4\,a\,c^3-b^2\,c^2\right )}-\frac {\mathrm {atan}\left (\frac {b}{\sqrt {4\,a\,c-b^2}}+\frac {2\,c\,x}{\sqrt {4\,a\,c-b^2}}\right )\,\left (A\,b\,c\,e-B\,b^2\,e-2\,A\,c^2\,d+2\,B\,a\,c\,e+B\,b\,c\,d\right )}{c^2\,\sqrt {4\,a\,c-b^2}}+\frac {B\,e\,x}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log(a + b*x + c*x^2)*(B*b^3*e + 4*A*a*c^2*e + 4*B*a*c^2*d - A*b^2*c*e - B*b^2*c*d - 4*B*a*b*c*e))/(2*(4*a*c^3
- b^2*c^2)) - (atan(b/(4*a*c - b^2)^(1/2) + (2*c*x)/(4*a*c - b^2)^(1/2))*(A*b*c*e - B*b^2*e - 2*A*c^2*d + 2*B
*a*c*e + B*b*c*d))/(c^2*(4*a*c - b^2)^(1/2)) + (B*e*x)/c

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sympy [B]  time = 3.44, size = 677, normalized size = 6.27 \begin {gather*} \frac {B e x}{c} + \left (- \frac {\sqrt {- 4 a c + b^{2}} \left (A b c e - 2 A c^{2} d + 2 B a c e - B b^{2} e + B b c d\right )}{2 c^{2} \left (4 a c - b^{2}\right )} - \frac {- A c e + B b e - B c d}{2 c^{2}}\right ) \log {\left (x + \frac {2 A a c e - A b c d - B a b e + 2 B a c d - 4 a c^{2} \left (- \frac {\sqrt {- 4 a c + b^{2}} \left (A b c e - 2 A c^{2} d + 2 B a c e - B b^{2} e + B b c d\right )}{2 c^{2} \left (4 a c - b^{2}\right )} - \frac {- A c e + B b e - B c d}{2 c^{2}}\right ) + b^{2} c \left (- \frac {\sqrt {- 4 a c + b^{2}} \left (A b c e - 2 A c^{2} d + 2 B a c e - B b^{2} e + B b c d\right )}{2 c^{2} \left (4 a c - b^{2}\right )} - \frac {- A c e + B b e - B c d}{2 c^{2}}\right )}{A b c e - 2 A c^{2} d + 2 B a c e - B b^{2} e + B b c d} \right )} + \left (\frac {\sqrt {- 4 a c + b^{2}} \left (A b c e - 2 A c^{2} d + 2 B a c e - B b^{2} e + B b c d\right )}{2 c^{2} \left (4 a c - b^{2}\right )} - \frac {- A c e + B b e - B c d}{2 c^{2}}\right ) \log {\left (x + \frac {2 A a c e - A b c d - B a b e + 2 B a c d - 4 a c^{2} \left (\frac {\sqrt {- 4 a c + b^{2}} \left (A b c e - 2 A c^{2} d + 2 B a c e - B b^{2} e + B b c d\right )}{2 c^{2} \left (4 a c - b^{2}\right )} - \frac {- A c e + B b e - B c d}{2 c^{2}}\right ) + b^{2} c \left (\frac {\sqrt {- 4 a c + b^{2}} \left (A b c e - 2 A c^{2} d + 2 B a c e - B b^{2} e + B b c d\right )}{2 c^{2} \left (4 a c - b^{2}\right )} - \frac {- A c e + B b e - B c d}{2 c^{2}}\right )}{A b c e - 2 A c^{2} d + 2 B a c e - B b^{2} e + B b c d} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B*e*x/c + (-sqrt(-4*a*c + b**2)*(A*b*c*e - 2*A*c**2*d + 2*B*a*c*e - B*b**2*e + B*b*c*d)/(2*c**2*(4*a*c - b**2)
) - (-A*c*e + B*b*e - B*c*d)/(2*c**2))*log(x + (2*A*a*c*e - A*b*c*d - B*a*b*e + 2*B*a*c*d - 4*a*c**2*(-sqrt(-4
*a*c + b**2)*(A*b*c*e - 2*A*c**2*d + 2*B*a*c*e - B*b**2*e + B*b*c*d)/(2*c**2*(4*a*c - b**2)) - (-A*c*e + B*b*e
- B*c*d)/(2*c**2)) + b**2*c*(-sqrt(-4*a*c + b**2)*(A*b*c*e - 2*A*c**2*d + 2*B*a*c*e - B*b**2*e + B*b*c*d)/(2*
c**2*(4*a*c - b**2)) - (-A*c*e + B*b*e - B*c*d)/(2*c**2)))/(A*b*c*e - 2*A*c**2*d + 2*B*a*c*e - B*b**2*e + B*b*
c*d)) + (sqrt(-4*a*c + b**2)*(A*b*c*e - 2*A*c**2*d + 2*B*a*c*e - B*b**2*e + B*b*c*d)/(2*c**2*(4*a*c - b**2)) -
(-A*c*e + B*b*e - B*c*d)/(2*c**2))*log(x + (2*A*a*c*e - A*b*c*d - B*a*b*e + 2*B*a*c*d - 4*a*c**2*(sqrt(-4*a*c
+ b**2)*(A*b*c*e - 2*A*c**2*d + 2*B*a*c*e - B*b**2*e + B*b*c*d)/(2*c**2*(4*a*c - b**2)) - (-A*c*e + B*b*e - B
*c*d)/(2*c**2)) + b**2*c*(sqrt(-4*a*c + b**2)*(A*b*c*e - 2*A*c**2*d + 2*B*a*c*e - B*b**2*e + B*b*c*d)/(2*c**2*
(4*a*c - b**2)) - (-A*c*e + B*b*e - B*c*d)/(2*c**2)))/(A*b*c*e - 2*A*c**2*d + 2*B*a*c*e - B*b**2*e + B*b*c*d))

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