3.1774 \(\int \frac{A+B x}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx\)
Optimal. Leaf size=332 \[ -\frac{b (A b-a B)}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac{b (2 a B e-3 A b e+b B d)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac{e (a+b x) (a B e-3 A b e+2 b B d)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^4}-\frac{e (a+b x) (B d-A e)}{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^3}-\frac{3 b e (a+b x) \log (a+b x) (a B e-2 A b e+b B d)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}+\frac{3 b e (a+b x) \log (d+e x) (a B e-2 A b e+b B d)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5} \]
[Out]
-((b*(b*B*d - 3*A*b*e + 2*a*B*e))/((b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]))
- (b*(A*b - a*B))/(2*(b*d - a*e)^3*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (
e*(B*d - A*e)*(a + b*x))/(2*(b*d - a*e)^3*(d + e*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x
^2]) - (e*(2*b*B*d - 3*A*b*e + a*B*e)*(a + b*x))/((b*d - a*e)^4*(d + e*x)*Sqrt[a
^2 + 2*a*b*x + b^2*x^2]) - (3*b*e*(b*B*d - 2*A*b*e + a*B*e)*(a + b*x)*Log[a + b*
x])/((b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (3*b*e*(b*B*d - 2*A*b*e + a*
B*e)*(a + b*x)*Log[d + e*x])/((b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
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Rubi [A] time = 0.763414, antiderivative size = 332, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061
\[ -\frac{b (A b-a B)}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac{b (2 a B e-3 A b e+b B d)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac{e (a+b x) (a B e-3 A b e+2 b B d)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^4}-\frac{e (a+b x) (B d-A e)}{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^3}-\frac{3 b e (a+b x) \log (a+b x) (a B e-2 A b e+b B d)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}+\frac{3 b e (a+b x) \log (d+e x) (a B e-2 A b e+b B d)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/((d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^(3/2)),x]
[Out]
-((b*(b*B*d - 3*A*b*e + 2*a*B*e))/((b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]))
- (b*(A*b - a*B))/(2*(b*d - a*e)^3*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (
e*(B*d - A*e)*(a + b*x))/(2*(b*d - a*e)^3*(d + e*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x
^2]) - (e*(2*b*B*d - 3*A*b*e + a*B*e)*(a + b*x))/((b*d - a*e)^4*(d + e*x)*Sqrt[a
^2 + 2*a*b*x + b^2*x^2]) - (3*b*e*(b*B*d - 2*A*b*e + a*B*e)*(a + b*x)*Log[a + b*
x])/((b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (3*b*e*(b*B*d - 2*A*b*e + a*
B*e)*(a + b*x)*Log[d + e*x])/((b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
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Rubi in Sympy [A] time = 79.0609, size = 345, normalized size = 1.04 \[ - \frac{3 b e \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (2 A b e - B a e - B b d\right ) \log{\left (a + b x \right )}}{\left (a + b x\right ) \left (a e - b d\right )^{5}} + \frac{3 b e \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (2 A b e - B a e - B b d\right ) \log{\left (d + e x \right )}}{\left (a + b x\right ) \left (a e - b d\right )^{5}} - \frac{6 e^{2} \left (- A b e + \frac{B \left (a e + b d\right )}{2}\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{\left (d + e x\right ) \left (a e - b d\right )^{5}} + \frac{3 \left (- A b e + \frac{B \left (a e + b d\right )}{2}\right )}{\left (d + e x\right ) \left (a e - b d\right )^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} + \frac{\left (2 a + 2 b x\right ) \left (- A b e + \frac{B \left (a e + b d\right )}{2}\right )}{2 e \left (d + e x\right ) \left (a e - b d\right )^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{\left (2 a + 2 b x\right ) \left (A e - B d\right )}{4 e \left (d + e x\right )^{2} \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/(e*x+d)**3/(b**2*x**2+2*a*b*x+a**2)**(3/2),x)
[Out]
-3*b*e*sqrt(a**2 + 2*a*b*x + b**2*x**2)*(2*A*b*e - B*a*e - B*b*d)*log(a + b*x)/(
(a + b*x)*(a*e - b*d)**5) + 3*b*e*sqrt(a**2 + 2*a*b*x + b**2*x**2)*(2*A*b*e - B*
a*e - B*b*d)*log(d + e*x)/((a + b*x)*(a*e - b*d)**5) - 6*e**2*(-A*b*e + B*(a*e +
b*d)/2)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/((d + e*x)*(a*e - b*d)**5) + 3*(-A*b*e
+ B*(a*e + b*d)/2)/((d + e*x)*(a*e - b*d)**3*sqrt(a**2 + 2*a*b*x + b**2*x**2))
+ (2*a + 2*b*x)*(-A*b*e + B*(a*e + b*d)/2)/(2*e*(d + e*x)*(a*e - b*d)**2*(a**2 +
2*a*b*x + b**2*x**2)**(3/2)) - (2*a + 2*b*x)*(A*e - B*d)/(4*e*(d + e*x)**2*(a*e
- b*d)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2))
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Mathematica [A] time = 0.397708, size = 220, normalized size = 0.66 \[ \frac{(a+b x) \left (\frac{e (a+b x)^2 (b d-a e)^2 (A e-B d)}{(d+e x)^2}-2 b (a+b x) (b d-a e) (2 a B e-3 A b e+b B d)+\frac{2 e (a+b x)^2 (b d-a e) (-a B e+3 A b e-2 b B d)}{d+e x}-6 b e (a+b x)^2 \log (a+b x) (a B e-2 A b e+b B d)+6 b e (a+b x)^2 \log (d+e x) (a B e-2 A b e+b B d)-b (A b-a B) (b d-a e)^2\right )}{2 \left ((a+b x)^2\right )^{3/2} (b d-a e)^5} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/((d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^(3/2)),x]
[Out]
((a + b*x)*(-(b*(A*b - a*B)*(b*d - a*e)^2) - 2*b*(b*d - a*e)*(b*B*d - 3*A*b*e +
2*a*B*e)*(a + b*x) + (e*(b*d - a*e)^2*(-(B*d) + A*e)*(a + b*x)^2)/(d + e*x)^2 +
(2*e*(b*d - a*e)*(-2*b*B*d + 3*A*b*e - a*B*e)*(a + b*x)^2)/(d + e*x) - 6*b*e*(b*
B*d - 2*A*b*e + a*B*e)*(a + b*x)^2*Log[a + b*x] + 6*b*e*(b*B*d - 2*A*b*e + a*B*e
)*(a + b*x)^2*Log[d + e*x]))/(2*(b*d - a*e)^5*((a + b*x)^2)^(3/2))
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Maple [B] time = 0.036, size = 1271, normalized size = 3.8 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/(e*x+d)^3/(b^2*x^2+2*a*b*x+a^2)^(3/2),x)
[Out]
-1/2*(-A*d^4*b^4+A*a^4*e^4+14*B*a^3*b*d*e^3*x-14*B*a*b^3*d^3*e*x+8*A*a*b^3*d^3*e
+9*B*a^3*b*d^2*e^2-9*B*a^2*b^2*d^3*e-8*A*a^3*b*d*e^3-24*B*ln(b*x+a)*x^3*a*b^3*d*
e^3+24*B*ln(e*x+d)*x^3*a*b^3*d*e^3+48*A*ln(b*x+a)*x^2*a*b^3*d*e^3-48*A*ln(e*x+d)
*x^2*a*b^3*d*e^3-30*B*ln(b*x+a)*x^2*a^2*b^2*d*e^3-30*B*ln(b*x+a)*x^2*a*b^3*d^2*e
^2+30*B*ln(e*x+d)*x^2*a^2*b^2*d*e^3-B*a*b^3*d^4+B*a^4*d*e^3-2*B*b^4*d^4*x+2*B*e^
4*a^4*x+24*A*ln(b*x+a)*x*a^2*b^2*d*e^3+30*B*ln(e*x+d)*x^2*a*b^3*d^2*e^2-24*A*ln(
e*x+d)*x*a*b^3*d^2*e^2-12*B*ln(b*x+a)*x*a^3*b*d*e^3-24*B*ln(b*x+a)*x*a^2*b^2*d^2
*e^2-12*B*ln(b*x+a)*x*a*b^3*d^3*e+12*B*ln(e*x+d)*x*a^3*b*d*e^3+24*B*ln(e*x+d)*x*
a^2*b^2*d^2*e^2+12*B*ln(e*x+d)*x*a*b^3*d^3*e+24*A*a*b^3*d^2*e^2*x-12*A*ln(e*x+d)
*x^2*b^4*d^2*e^2-6*B*ln(b*x+a)*x^2*a^3*b*e^4-6*B*ln(b*x+a)*x^2*b^4*d^3*e+6*B*ln(
e*x+d)*x^2*a^3*b*e^4+6*B*ln(e*x+d)*x^2*b^4*d^3*e+12*A*ln(b*x+a)*a^2*b^2*d^2*e^2-
12*A*ln(e*x+d)*a^2*b^2*d^2*e^2-6*B*ln(b*x+a)*a^3*b*d^2*e^2-6*B*ln(b*x+a)*a^2*b^2
*d^3*e+6*B*ln(e*x+d)*a^3*b*d^2*e^2+6*B*ln(e*x+d)*a^2*b^2*d^3*e-6*B*ln(b*x+a)*x^4
*a*b^3*e^4-6*B*ln(b*x+a)*x^4*b^4*d*e^3+6*B*ln(e*x+d)*x^4*a*b^3*e^4+6*B*ln(e*x+d)
*x^4*b^4*d*e^3+24*A*ln(b*x+a)*x^3*a*b^3*e^4+24*A*ln(b*x+a)*x^3*b^4*d*e^3-24*A*ln
(e*x+d)*x^3*a*b^3*e^4-24*A*ln(e*x+d)*x^3*b^4*d*e^3-12*B*ln(b*x+a)*x^3*a^2*b^2*e^
4-12*B*ln(b*x+a)*x^3*b^4*d^2*e^2+12*B*ln(e*x+d)*x^3*a^2*b^2*e^4+12*B*ln(e*x+d)*x
^3*b^4*d^2*e^2+12*A*ln(b*x+a)*x^2*a^2*b^2*e^4+12*A*ln(b*x+a)*x^2*b^4*d^2*e^2-12*
A*ln(e*x+d)*x^2*a^2*b^2*e^4-9*B*x^2*a*b^3*d^2*e^2-24*A*a^2*b^2*d*e^3*x+9*B*x^2*a
^2*b^2*d*e^3+24*A*ln(b*x+a)*x*a*b^3*d^2*e^2-6*B*x^3*b^4*d^2*e^2+18*A*x^2*b^4*d^2
*e^2-9*B*x^2*b^4*d^3*e+6*B*x^3*a^2*b^2*e^4-4*A*a^3*b*e^4*x+4*A*b^4*d^3*e*x+9*B*x
^2*a^3*b*e^4-18*A*x^2*a^2*b^2*e^4+12*A*ln(b*x+a)*x^4*b^4*e^4-12*A*ln(e*x+d)*x^4*
b^4*e^4-12*A*x^3*a*b^3*e^4+12*A*x^3*b^4*d*e^3-24*A*ln(e*x+d)*x*a^2*b^2*d*e^3)*(b
*x+a)/(e*x+d)^2/(a*e-b*d)^5/((b*x+a)^2)^(3/2)
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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)/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(e*x + d)^3),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.311696, size = 1640, normalized size = 4.94 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(e*x + d)^3),x, algorithm="fricas")
[Out]
1/2*(9*B*a^3*b*d^2*e^2 + A*a^4*e^4 - (B*a*b^3 + A*b^4)*d^4 - (9*B*a^2*b^2 - 8*A*
a*b^3)*d^3*e + (B*a^4 - 8*A*a^3*b)*d*e^3 - 6*(B*b^4*d^2*e^2 - 2*A*b^4*d*e^3 - (B
*a^2*b^2 - 2*A*a*b^3)*e^4)*x^3 - 9*(B*b^4*d^3*e - B*a^2*b^2*d*e^3 + (B*a*b^3 - 2
*A*b^4)*d^2*e^2 - (B*a^3*b - 2*A*a^2*b^2)*e^4)*x^2 - 2*(B*b^4*d^4 - 12*A*a*b^3*d
^2*e^2 + (7*B*a*b^3 - 2*A*b^4)*d^3*e - (7*B*a^3*b - 12*A*a^2*b^2)*d*e^3 - (B*a^4
- 2*A*a^3*b)*e^4)*x - 6*(B*a^2*b^2*d^3*e + (B*a^3*b - 2*A*a^2*b^2)*d^2*e^2 + (B
*b^4*d*e^3 + (B*a*b^3 - 2*A*b^4)*e^4)*x^4 + 2*(B*b^4*d^2*e^2 + 2*(B*a*b^3 - A*b^
4)*d*e^3 + (B*a^2*b^2 - 2*A*a*b^3)*e^4)*x^3 + (B*b^4*d^3*e + (5*B*a*b^3 - 2*A*b^
4)*d^2*e^2 + (5*B*a^2*b^2 - 8*A*a*b^3)*d*e^3 + (B*a^3*b - 2*A*a^2*b^2)*e^4)*x^2
+ 2*(B*a*b^3*d^3*e + 2*(B*a^2*b^2 - A*a*b^3)*d^2*e^2 + (B*a^3*b - 2*A*a^2*b^2)*d
*e^3)*x)*log(b*x + a) + 6*(B*a^2*b^2*d^3*e + (B*a^3*b - 2*A*a^2*b^2)*d^2*e^2 + (
B*b^4*d*e^3 + (B*a*b^3 - 2*A*b^4)*e^4)*x^4 + 2*(B*b^4*d^2*e^2 + 2*(B*a*b^3 - A*b
^4)*d*e^3 + (B*a^2*b^2 - 2*A*a*b^3)*e^4)*x^3 + (B*b^4*d^3*e + (5*B*a*b^3 - 2*A*b
^4)*d^2*e^2 + (5*B*a^2*b^2 - 8*A*a*b^3)*d*e^3 + (B*a^3*b - 2*A*a^2*b^2)*e^4)*x^2
+ 2*(B*a*b^3*d^3*e + 2*(B*a^2*b^2 - A*a*b^3)*d^2*e^2 + (B*a^3*b - 2*A*a^2*b^2)*
d*e^3)*x)*log(e*x + d))/(a^2*b^5*d^7 - 5*a^3*b^4*d^6*e + 10*a^4*b^3*d^5*e^2 - 10
*a^5*b^2*d^4*e^3 + 5*a^6*b*d^3*e^4 - a^7*d^2*e^5 + (b^7*d^5*e^2 - 5*a*b^6*d^4*e^
3 + 10*a^2*b^5*d^3*e^4 - 10*a^3*b^4*d^2*e^5 + 5*a^4*b^3*d*e^6 - a^5*b^2*e^7)*x^4
+ 2*(b^7*d^6*e - 4*a*b^6*d^5*e^2 + 5*a^2*b^5*d^4*e^3 - 5*a^4*b^3*d^2*e^5 + 4*a^
5*b^2*d*e^6 - a^6*b*e^7)*x^3 + (b^7*d^7 - a*b^6*d^6*e - 9*a^2*b^5*d^5*e^2 + 25*a
^3*b^4*d^4*e^3 - 25*a^4*b^3*d^3*e^4 + 9*a^5*b^2*d^2*e^5 + a^6*b*d*e^6 - a^7*e^7)
*x^2 + 2*(a*b^6*d^7 - 4*a^2*b^5*d^6*e + 5*a^3*b^4*d^5*e^2 - 5*a^5*b^2*d^3*e^4 +
4*a^6*b*d^2*e^5 - a^7*d*e^6)*x)
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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/(b**2*x**2+2*a*b*x+a**2)**(3/2),x)
[Out]
Timed out
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GIAC/XCAS [A] time = 0.624025, 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)/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(e*x + d)^3),x, algorithm="giac")
[Out]
sage0*x