3.2369 \(\int \frac{A+B x}{(d+e x)^3 \left (a+b x+c x^2\right )} \, dx\)

Optimal. Leaf size=414 \[ \frac{\log \left (a+b x+c x^2\right ) \left (B \left (a b e^3-3 a c d e^2+c^2 d^3\right )-A e \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right )\right )}{2 \left (a e^2-b d e+c d^2\right )^3}-\frac{\log (d+e x) \left (B \left (a b e^3-3 a c d e^2+c^2 d^3\right )-A e \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right )\right )}{\left (a e^2-b d e+c d^2\right )^3}+\frac{\tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \left (-b^2 e^2 (a B e+3 A c d)+b c \left (-3 a A e^3+3 a B d e^2+3 A c d^2 e+B c d^3\right )-2 c \left (A c d \left (c d^2-3 a e^2\right )+a B e \left (3 c d^2-a e^2\right )\right )+A b^3 e^3\right )}{\sqrt{b^2-4 a c} \left (a e^2-b d e+c d^2\right )^3}+\frac{B d-A e}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac{A e (2 c d-b e)-B \left (c d^2-a e^2\right )}{(d+e x) \left (a e^2-b d e+c d^2\right )^2} \]

[Out]

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

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Rubi [A]  time = 1.72118, antiderivative size = 414, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{\log \left (a+b x+c x^2\right ) \left (B \left (a b e^3-3 a c d e^2+c^2 d^3\right )-A e \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right )\right )}{2 \left (a e^2-b d e+c d^2\right )^3}-\frac{\log (d+e x) \left (B \left (a b e^3-3 a c d e^2+c^2 d^3\right )-A e \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right )\right )}{\left (a e^2-b d e+c d^2\right )^3}+\frac{\tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \left (-b^2 e^2 (a B e+3 A c d)+b c \left (-3 a A e^3+3 a B d e^2+3 A c d^2 e+B c d^3\right )-2 c \left (A c d \left (c d^2-3 a e^2\right )+a B e \left (3 c d^2-a e^2\right )\right )+A b^3 e^3\right )}{\sqrt{b^2-4 a c} \left (a e^2-b d e+c d^2\right )^3}+\frac{B d-A e}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac{A e (2 c d-b e)-B \left (c d^2-a e^2\right )}{(d+e x) \left (a e^2-b d e+c d^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/((d + e*x)^3*(a + b*x + c*x^2)),x]

[Out]

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

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/(e*x+d)**3/(c*x**2+b*x+a),x)

[Out]

Timed out

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Mathematica [A]  time = 1.18206, size = 413, normalized size = 1. \[ -\frac{\log (d+e x) \left (A e \left (c e (a e+3 b d)-b^2 e^2-3 c^2 d^2\right )+B \left (a b e^3-3 a c d e^2+c^2 d^3\right )\right )}{\left (e (a e-b d)+c d^2\right )^3}+\frac{\log (a+x (b+c x)) \left (A e \left (c e (a e+3 b d)-b^2 e^2-3 c^2 d^2\right )+B \left (a b e^3-3 a c d e^2+c^2 d^3\right )\right )}{2 \left (e (a e-b d)+c d^2\right )^3}+\frac{\tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right ) \left (-b^2 e^2 (a B e+3 A c d)+b c \left (-3 a A e^3+3 a B d e^2+3 A c d^2 e+B c d^3\right )+2 c \left (A c d \left (3 a e^2-c d^2\right )+a B e \left (a e^2-3 c d^2\right )\right )+A b^3 e^3\right )}{\sqrt{4 a c-b^2} \left (e (b d-a e)-c d^2\right )^3}+\frac{B \left (c d^2-a e^2\right )+A e (b e-2 c d)}{(d+e x) \left (e (a e-b d)+c d^2\right )^2}+\frac{B d-A e}{2 (d+e x)^2 \left (e (a e-b d)+c d^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/((d + e*x)^3*(a + b*x + c*x^2)),x]

[Out]

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

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Maple [B]  time = 0.021, size = 1339, normalized size = 3.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(e*x+d)^3/(c*x^2+b*x+a),x)

[Out]

6/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*B*
a*c^2*d^2*e+1/2/(a*e^2-b*d*e+c*d^2)^3*c*ln(c*x^2+b*x+a)*a*A*e^3-3/2/(a*e^2-b*d*e
+c*d^2)^3*c^2*ln(c*x^2+b*x+a)*A*d^2*e+1/2/(a*e^2-b*d*e+c*d^2)^3*ln(c*x^2+b*x+a)*
a*b*B*e^3-1/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)
^(1/2))*A*b^3*e^3+2/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*
a*c-b^2)^(1/2))*A*c^3*d^3-1/(a*e^2-b*d*e+c*d^2)^3*ln(e*x+d)*A*a*c*e^3+3/(a*e^2-b
*d*e+c*d^2)^3*ln(e*x+d)*A*c^2*d^2*e-1/(a*e^2-b*d*e+c*d^2)^3*ln(e*x+d)*B*e^3*a*b-
2/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)*A*c*d*e-1/2/(a*e^2-b*d*e+c*d^2)/(e*x+d)^2*A*e+1/
2/(a*e^2-b*d*e+c*d^2)/(e*x+d)^2*B*d+3/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)^(1/2)*ar
ctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*A*a*b*c*e^3-6/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2
)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*A*a*c^2*d*e^2+3/(a*e^2-b*d*e+c*d^2)^
3/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*A*b^2*c*d*e^2-3/(a*e^2-b
*d*e+c*d^2)^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*A*b*c^2*d^2*
e-3/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*
B*a*b*c*d*e^2-1/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)*a*B*e^2+1/(a*e^2-b*d*e+c*d^2)^2/(e
*x+d)*B*c*d^2+1/(a*e^2-b*d*e+c*d^2)^3*ln(e*x+d)*A*b^2*e^3-1/(a*e^2-b*d*e+c*d^2)^
3*ln(e*x+d)*B*c^2*d^3+1/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)*A*b*e^2-1/2/(a*e^2-b*d*e+c
*d^2)^3*ln(c*x^2+b*x+a)*A*b^2*e^3+1/2/(a*e^2-b*d*e+c*d^2)^3*c^2*ln(c*x^2+b*x+a)*
B*d^3-3/(a*e^2-b*d*e+c*d^2)^3*ln(e*x+d)*A*b*c*d*e^2+3/(a*e^2-b*d*e+c*d^2)^3*ln(e
*x+d)*B*a*c*d*e^2+3/2/(a*e^2-b*d*e+c*d^2)^3*c*ln(c*x^2+b*x+a)*A*d*b*e^2-3/2/(a*e
^2-b*d*e+c*d^2)^3*c*ln(c*x^2+b*x+a)*B*d*a*e^2-2/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2
)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*B*a^2*c*e^3+1/(a*e^2-b*d*e+c*d^2)^3/
(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*B*a*b^2*e^3-1/(a*e^2-b*d*e
+c*d^2)^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*B*b*c^2*d^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((c*x^2 + b*x + a)*(e*x + d)^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((c*x^2 + b*x + a)*(e*x + d)^3),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)/(e*x+d)**3/(c*x**2+b*x+a),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.316231, size = 1076, normalized size = 2.6 \[ \frac{{\left (B c^{2} d^{3} - 3 \, A c^{2} d^{2} e - 3 \, B a c d e^{2} + 3 \, A b c d e^{2} + B a b e^{3} - A b^{2} e^{3} + A a c e^{3}\right )}{\rm ln}\left (c x^{2} + b x + a\right )}{2 \,{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} + 3 \, a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} - 6 \, a b c d^{3} e^{3} + 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} c d^{2} e^{4} - 3 \, a^{2} b d e^{5} + a^{3} e^{6}\right )}} - \frac{{\left (B c^{2} d^{3} e - 3 \, A c^{2} d^{2} e^{2} - 3 \, B a c d e^{3} + 3 \, A b c d e^{3} + B a b e^{4} - A b^{2} e^{4} + A a c e^{4}\right )}{\rm ln}\left ({\left | x e + d \right |}\right )}{c^{3} d^{6} e - 3 \, b c^{2} d^{5} e^{2} + 3 \, b^{2} c d^{4} e^{3} + 3 \, a c^{2} d^{4} e^{3} - b^{3} d^{3} e^{4} - 6 \, a b c d^{3} e^{4} + 3 \, a b^{2} d^{2} e^{5} + 3 \, a^{2} c d^{2} e^{5} - 3 \, a^{2} b d e^{6} + a^{3} e^{7}} - \frac{{\left (B b c^{2} d^{3} - 2 \, A c^{3} d^{3} - 6 \, B a c^{2} d^{2} e + 3 \, A b c^{2} d^{2} e + 3 \, B a b c d e^{2} - 3 \, A b^{2} c d e^{2} + 6 \, A a c^{2} d e^{2} - B a b^{2} e^{3} + A b^{3} e^{3} + 2 \, B a^{2} c e^{3} - 3 \, A a b c e^{3}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} + 3 \, a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} - 6 \, a b c d^{3} e^{3} + 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} c d^{2} e^{4} - 3 \, a^{2} b d e^{5} + a^{3} e^{6}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{3 \, B c^{2} d^{5} - 4 \, B b c d^{4} e - 5 \, A c^{2} d^{4} e + B b^{2} d^{3} e^{2} + 2 \, B a c d^{3} e^{2} + 8 \, A b c d^{3} e^{2} - 3 \, A b^{2} d^{2} e^{3} - 6 \, A a c d^{2} e^{3} - B a^{2} d e^{4} + 4 \, A a b d e^{4} - A a^{2} e^{5} + 2 \,{\left (B c^{2} d^{4} e - B b c d^{3} e^{2} - 2 \, A c^{2} d^{3} e^{2} + 3 \, A b c d^{2} e^{3} + B a b d e^{4} - A b^{2} d e^{4} - 2 \, A a c d e^{4} - B a^{2} e^{5} + A a b e^{5}\right )} x}{2 \,{\left (c d^{2} - b d e + a e^{2}\right )}^{3}{\left (x e + d\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((c*x^2 + b*x + a)*(e*x + d)^3),x, algorithm="giac")

[Out]

1/2*(B*c^2*d^3 - 3*A*c^2*d^2*e - 3*B*a*c*d*e^2 + 3*A*b*c*d*e^2 + B*a*b*e^3 - A*b
^2*e^3 + A*a*c*e^3)*ln(c*x^2 + b*x + a)/(c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e
^2 + 3*a*c^2*d^4*e^2 - b^3*d^3*e^3 - 6*a*b*c*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a^2*c
*d^2*e^4 - 3*a^2*b*d*e^5 + a^3*e^6) - (B*c^2*d^3*e - 3*A*c^2*d^2*e^2 - 3*B*a*c*d
*e^3 + 3*A*b*c*d*e^3 + B*a*b*e^4 - A*b^2*e^4 + A*a*c*e^4)*ln(abs(x*e + d))/(c^3*
d^6*e - 3*b*c^2*d^5*e^2 + 3*b^2*c*d^4*e^3 + 3*a*c^2*d^4*e^3 - b^3*d^3*e^4 - 6*a*
b*c*d^3*e^4 + 3*a*b^2*d^2*e^5 + 3*a^2*c*d^2*e^5 - 3*a^2*b*d*e^6 + a^3*e^7) - (B*
b*c^2*d^3 - 2*A*c^3*d^3 - 6*B*a*c^2*d^2*e + 3*A*b*c^2*d^2*e + 3*B*a*b*c*d*e^2 -
3*A*b^2*c*d*e^2 + 6*A*a*c^2*d*e^2 - B*a*b^2*e^3 + A*b^3*e^3 + 2*B*a^2*c*e^3 - 3*
A*a*b*c*e^3)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((c^3*d^6 - 3*b*c^2*d^5*e +
3*b^2*c*d^4*e^2 + 3*a*c^2*d^4*e^2 - b^3*d^3*e^3 - 6*a*b*c*d^3*e^3 + 3*a*b^2*d^2*
e^4 + 3*a^2*c*d^2*e^4 - 3*a^2*b*d*e^5 + a^3*e^6)*sqrt(-b^2 + 4*a*c)) + 1/2*(3*B*
c^2*d^5 - 4*B*b*c*d^4*e - 5*A*c^2*d^4*e + B*b^2*d^3*e^2 + 2*B*a*c*d^3*e^2 + 8*A*
b*c*d^3*e^2 - 3*A*b^2*d^2*e^3 - 6*A*a*c*d^2*e^3 - B*a^2*d*e^4 + 4*A*a*b*d*e^4 -
A*a^2*e^5 + 2*(B*c^2*d^4*e - B*b*c*d^3*e^2 - 2*A*c^2*d^3*e^2 + 3*A*b*c*d^2*e^3 +
B*a*b*d*e^4 - A*b^2*d*e^4 - 2*A*a*c*d*e^4 - B*a^2*e^5 + A*a*b*e^5)*x)/((c*d^2 -
b*d*e + a*e^2)^3*(x*e + d)^2)