3.2470 \(\int \frac{A+B x}{(d+e x)^3 \sqrt{a+b x+c x^2}} \, dx\)
Optimal. Leaf size=271 \[ \frac{\left (-4 b \left (a B e^2+2 A c d e+B c d^2\right )+4 c \left (-a A e^2+3 a B d e+2 A c d^2\right )+b^2 e (3 A e+B d)\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{8 \left (a e^2-b d e+c d^2\right )^{5/2}}+\frac{\sqrt{a+b x+c x^2} (B d-A e)}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac{\sqrt{a+b x+c x^2} \left (3 A e (2 c d-b e)-B \left (e (b d-4 a e)+2 c d^2\right )\right )}{4 (d+e x) \left (a e^2-b d e+c d^2\right )^2} \]
[Out]
((B*d - A*e)*Sqrt[a + b*x + c*x^2])/(2*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^2) - ((
3*A*e*(2*c*d - b*e) - B*(2*c*d^2 + e*(b*d - 4*a*e)))*Sqrt[a + b*x + c*x^2])/(4*(
c*d^2 - b*d*e + a*e^2)^2*(d + e*x)) + ((b^2*e*(B*d + 3*A*e) + 4*c*(2*A*c*d^2 + 3
*a*B*d*e - a*A*e^2) - 4*b*(B*c*d^2 + 2*A*c*d*e + a*B*e^2))*ArcTanh[(b*d - 2*a*e
+ (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/(8*(c
*d^2 - b*d*e + a*e^2)^(5/2))
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Rubi [A] time = 0.940925, antiderivative size = 269, normalized size of antiderivative =
0.99, number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148
\[ \frac{\left (-4 b \left (a B e^2+2 A c d e+B c d^2\right )+4 c \left (-a A e^2+3 a B d e+2 A c d^2\right )+b^2 e (3 A e+B d)\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{8 \left (a e^2-b d e+c d^2\right )^{5/2}}+\frac{\sqrt{a+b x+c x^2} (B d-A e)}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}+\frac{\sqrt{a+b x+c x^2} \left (B e (b d-4 a e)-3 A e (2 c d-b e)+2 B c d^2\right )}{4 (d+e x) \left (a e^2-b d e+c d^2\right )^2} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/((d + e*x)^3*Sqrt[a + b*x + c*x^2]),x]
[Out]
((B*d - A*e)*Sqrt[a + b*x + c*x^2])/(2*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^2) + ((
2*B*c*d^2 + B*e*(b*d - 4*a*e) - 3*A*e*(2*c*d - b*e))*Sqrt[a + b*x + c*x^2])/(4*(
c*d^2 - b*d*e + a*e^2)^2*(d + e*x)) + ((b^2*e*(B*d + 3*A*e) + 4*c*(2*A*c*d^2 + 3
*a*B*d*e - a*A*e^2) - 4*b*(B*c*d^2 + 2*A*c*d*e + a*B*e^2))*ArcTanh[(b*d - 2*a*e
+ (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/(8*(c
*d^2 - b*d*e + a*e^2)^(5/2))
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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)**(1/2),x)
[Out]
Timed out
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Mathematica [A] time = 1.37017, size = 327, normalized size = 1.21 \[ \frac{(d+e x)^2 \log (d+e x) \left (-4 b \left (a B e^2+2 A c d e+B c d^2\right )+4 c \left (-a A e^2+3 a B d e+2 A c d^2\right )+b^2 e (3 A e+B d)\right )+(d+e x)^2 \left (4 b \left (a B e^2+2 A c d e+B c d^2\right )+4 c \left (a A e^2-3 a B d e-2 A c d^2\right )+b^2 (-e) (3 A e+B d)\right ) \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )+2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2} \left ((d+e x) \left (B e (b d-4 a e)+3 A e (b e-2 c d)+2 B c d^2\right )+2 (B d-A e) \left (e (a e-b d)+c d^2\right )\right )}{8 (d+e x)^2 \left (e (a e-b d)+c d^2\right )^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/((d + e*x)^3*Sqrt[a + b*x + c*x^2]),x]
[Out]
(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)]*(2*(B*d - A*e)*(c*d^2 +
e*(-(b*d) + a*e)) + (2*B*c*d^2 + B*e*(b*d - 4*a*e) + 3*A*e*(-2*c*d + b*e))*(d +
e*x)) + (b^2*e*(B*d + 3*A*e) + 4*c*(2*A*c*d^2 + 3*a*B*d*e - a*A*e^2) - 4*b*(B*c*
d^2 + 2*A*c*d*e + a*B*e^2))*(d + e*x)^2*Log[d + e*x] + (-(b^2*e*(B*d + 3*A*e)) +
4*c*(-2*A*c*d^2 - 3*a*B*d*e + a*A*e^2) + 4*b*(B*c*d^2 + 2*A*c*d*e + a*B*e^2))*(
d + e*x)^2*Log[-(b*d) + 2*a*e - 2*c*d*x + b*e*x + 2*Sqrt[c*d^2 + e*(-(b*d) + a*e
)]*Sqrt[a + x*(b + c*x)]])/(8*(c*d^2 + e*(-(b*d) + a*e))^(5/2)*(d + e*x)^2)
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Maple [B] time = 0.025, size = 2204, normalized size = 8.1 \[ \text{result too large to display} \]
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)^(1/2),x)
[Out]
-B/e/(a*e^2-b*d*e+c*d^2)/(x+d/e)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e
+c*d^2)/e^2)^(1/2)+1/2*B/e/(a*e^2-b*d*e+c*d^2)/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*l
n((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(
1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))
*b-3/2*B/e^2/(a*e^2-b*d*e+c*d^2)/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*
d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^
2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))*c*d-1/2/e/(a*
e^2-b*d*e+c*d^2)/(x+d/e)^2*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2
)/e^2)^(1/2)*A+1/2/e^2/(a*e^2-b*d*e+c*d^2)/(x+d/e)^2*((x+d/e)^2*c+(b*e-2*c*d)/e*
(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*B*d+3/4*e/(a*e^2-b*d*e+c*d^2)^2/(x+d/e)*(
(x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b*A-3/4/(a*e^2-
b*d*e+c*d^2)^2/(x+d/e)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^
2)^(1/2)*b*B*d-3/2/(a*e^2-b*d*e+c*d^2)^2/(x+d/e)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d
/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*c*d*A+3/2/e/(a*e^2-b*d*e+c*d^2)^2/(x+d/e)*((x
+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*c*d^2*B-3/8*e/(a*
e^2-b*d*e+c*d^2)^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2
+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d
)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))*b^2*A+3/8/(a*e^2-b*d*e+c*d^
2)^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e
*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a
*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))*b^2*B*d+3/2/(a*e^2-b*d*e+c*d^2)^2/((a*e^2
-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*(
(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c
*d^2)/e^2)^(1/2))/(x+d/e))*b*c*d*A-3/2/e/(a*e^2-b*d*e+c*d^2)^2/((a*e^2-b*d*e+c*d
^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d
*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)
^(1/2))/(x+d/e))*b*c*d^2*B-3/2/e/(a*e^2-b*d*e+c*d^2)^2/((a*e^2-b*d*e+c*d^2)/e^2)
^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d^2
)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/
(x+d/e))*c^2*d^2*A+3/2/e^2/(a*e^2-b*d*e+c*d^2)^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)
*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)
^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e
))*c^2*d^3*B+1/2/e*c/(a*e^2-b*d*e+c*d^2)/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(
a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(
(x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))*A
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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)/(sqrt(c*x^2 + b*x + a)*(e*x + d)^3),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 9.40559, size = 1, normalized size = 0. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/(sqrt(c*x^2 + b*x + a)*(e*x + d)^3),x, algorithm="fricas")
[Out]
[1/16*(4*(4*B*c*d^3 - 2*A*a*e^3 - (B*b + 8*A*c)*d^2*e - (2*B*a - 5*A*b)*d*e^2 +
(2*B*c*d^2*e + (B*b - 6*A*c)*d*e^2 - (4*B*a - 3*A*b)*e^3)*x)*sqrt(c*d^2 - b*d*e
+ a*e^2)*sqrt(c*x^2 + b*x + a) + (4*(B*b*c - 2*A*c^2)*d^4 - (B*b^2 + 4*(3*B*a -
2*A*b)*c)*d^3*e + (4*B*a*b - 3*A*b^2 + 4*A*a*c)*d^2*e^2 + (4*(B*b*c - 2*A*c^2)*d
^2*e^2 - (B*b^2 + 4*(3*B*a - 2*A*b)*c)*d*e^3 + (4*B*a*b - 3*A*b^2 + 4*A*a*c)*e^4
)*x^2 + 2*(4*(B*b*c - 2*A*c^2)*d^3*e - (B*b^2 + 4*(3*B*a - 2*A*b)*c)*d^2*e^2 + (
4*B*a*b - 3*A*b^2 + 4*A*a*c)*d*e^3)*x)*log(((8*a*b*d*e - 8*a^2*e^2 - (b^2 + 4*a*
c)*d^2 - (8*c^2*d^2 - 8*b*c*d*e + (b^2 + 4*a*c)*e^2)*x^2 - 2*(4*b*c*d^2 + 4*a*b*
e^2 - (3*b^2 + 4*a*c)*d*e)*x)*sqrt(c*d^2 - b*d*e + a*e^2) + 4*(b*c*d^3 + 3*a*b*d
*e^2 - 2*a^2*e^3 - (b^2 + 2*a*c)*d^2*e + (2*c^2*d^3 - 3*b*c*d^2*e - a*b*e^3 + (b
^2 + 2*a*c)*d*e^2)*x)*sqrt(c*x^2 + b*x + a))/(e^2*x^2 + 2*d*e*x + d^2)))/((c^2*d
^6 - 2*b*c*d^5*e - 2*a*b*d^3*e^3 + a^2*d^2*e^4 + (b^2 + 2*a*c)*d^4*e^2 + (c^2*d^
4*e^2 - 2*b*c*d^3*e^3 - 2*a*b*d*e^5 + a^2*e^6 + (b^2 + 2*a*c)*d^2*e^4)*x^2 + 2*(
c^2*d^5*e - 2*b*c*d^4*e^2 - 2*a*b*d^2*e^4 + a^2*d*e^5 + (b^2 + 2*a*c)*d^3*e^3)*x
)*sqrt(c*d^2 - b*d*e + a*e^2)), 1/8*(2*(4*B*c*d^3 - 2*A*a*e^3 - (B*b + 8*A*c)*d^
2*e - (2*B*a - 5*A*b)*d*e^2 + (2*B*c*d^2*e + (B*b - 6*A*c)*d*e^2 - (4*B*a - 3*A*
b)*e^3)*x)*sqrt(-c*d^2 + b*d*e - a*e^2)*sqrt(c*x^2 + b*x + a) + (4*(B*b*c - 2*A*
c^2)*d^4 - (B*b^2 + 4*(3*B*a - 2*A*b)*c)*d^3*e + (4*B*a*b - 3*A*b^2 + 4*A*a*c)*d
^2*e^2 + (4*(B*b*c - 2*A*c^2)*d^2*e^2 - (B*b^2 + 4*(3*B*a - 2*A*b)*c)*d*e^3 + (4
*B*a*b - 3*A*b^2 + 4*A*a*c)*e^4)*x^2 + 2*(4*(B*b*c - 2*A*c^2)*d^3*e - (B*b^2 + 4
*(3*B*a - 2*A*b)*c)*d^2*e^2 + (4*B*a*b - 3*A*b^2 + 4*A*a*c)*d*e^3)*x)*arctan(-1/
2*sqrt(-c*d^2 + b*d*e - a*e^2)*(b*d - 2*a*e + (2*c*d - b*e)*x)/((c*d^2 - b*d*e +
a*e^2)*sqrt(c*x^2 + b*x + a))))/((c^2*d^6 - 2*b*c*d^5*e - 2*a*b*d^3*e^3 + a^2*d
^2*e^4 + (b^2 + 2*a*c)*d^4*e^2 + (c^2*d^4*e^2 - 2*b*c*d^3*e^3 - 2*a*b*d*e^5 + a^
2*e^6 + (b^2 + 2*a*c)*d^2*e^4)*x^2 + 2*(c^2*d^5*e - 2*b*c*d^4*e^2 - 2*a*b*d^2*e^
4 + a^2*d*e^5 + (b^2 + 2*a*c)*d^3*e^3)*x)*sqrt(-c*d^2 + b*d*e - a*e^2))]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{A + B x}{\left (d + e x\right )^{3} \sqrt{a + b x + c x^{2}}}\, dx \]
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)**(1/2),x)
[Out]
Integral((A + B*x)/((d + e*x)**3*sqrt(a + b*x + c*x**2)), x)
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GIAC/XCAS [A] time = 0.59177, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/(sqrt(c*x^2 + b*x + a)*(e*x + d)^3),x, algorithm="giac")
[Out]
sage0*x