3.1204 \(\int \frac{A+B x}{(d+e x)^2 \left (b x+c x^2\right )^{3/2}} \, dx\)
Optimal. Leaf size=245 \[ -\frac{e \sqrt{b x+c x^2} \left (b^2 (-e) (B d-3 A e)-2 b c d (2 A e+B d)+4 A c^2 d^2\right )}{b^2 d^2 (d+e x) (c d-b e)^2}-\frac{2 (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{b^2 d \sqrt{b x+c x^2} (d+e x) (c d-b e)}+\frac{e (3 A e (2 c d-b e)-B d (4 c d-b e)) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{2 d^{5/2} (c d-b e)^{5/2}} \]
[Out]
(-2*(A*b*(c*d - b*e) + c*(2*A*c*d - b*(B*d + A*e))*x))/(b^2*d*(c*d - b*e)*(d + e
*x)*Sqrt[b*x + c*x^2]) - (e*(4*A*c^2*d^2 - b^2*e*(B*d - 3*A*e) - 2*b*c*d*(B*d +
2*A*e))*Sqrt[b*x + c*x^2])/(b^2*d^2*(c*d - b*e)^2*(d + e*x)) + (e*(3*A*e*(2*c*d
- b*e) - B*d*(4*c*d - b*e))*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d
- b*e]*Sqrt[b*x + c*x^2])])/(2*d^(5/2)*(c*d - b*e)^(5/2))
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Rubi [A] time = 0.772736, antiderivative size = 245, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154
\[ -\frac{e \sqrt{b x+c x^2} \left (b^2 (-e) (B d-3 A e)-2 b c d (2 A e+B d)+4 A c^2 d^2\right )}{b^2 d^2 (d+e x) (c d-b e)^2}-\frac{2 (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{b^2 d \sqrt{b x+c x^2} (d+e x) (c d-b e)}+\frac{e (3 A e (2 c d-b e)-B d (4 c d-b e)) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{2 d^{5/2} (c d-b e)^{5/2}} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/((d + e*x)^2*(b*x + c*x^2)^(3/2)),x]
[Out]
(-2*(A*b*(c*d - b*e) + c*(2*A*c*d - b*(B*d + A*e))*x))/(b^2*d*(c*d - b*e)*(d + e
*x)*Sqrt[b*x + c*x^2]) - (e*(4*A*c^2*d^2 - b^2*e*(B*d - 3*A*e) - 2*b*c*d*(B*d +
2*A*e))*Sqrt[b*x + c*x^2])/(b^2*d^2*(c*d - b*e)^2*(d + e*x)) + (e*(3*A*e*(2*c*d
- b*e) - B*d*(4*c*d - b*e))*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d
- b*e]*Sqrt[b*x + c*x^2])])/(2*d^(5/2)*(c*d - b*e)^(5/2))
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Rubi in Sympy [A] time = 86.4308, size = 236, normalized size = 0.96 \[ - \frac{e \left (3 A b e^{2} - 6 A c d e - B b d e + 4 B c d^{2}\right ) \operatorname{atan}{\left (\frac{- b d + x \left (b e - 2 c d\right )}{2 \sqrt{d} \sqrt{b e - c d} \sqrt{b x + c x^{2}}} \right )}}{2 d^{\frac{5}{2}} \left (b e - c d\right )^{\frac{5}{2}}} - \frac{2 \left (A b \left (b e - c d\right ) + c x \left (A b e - 2 A c d + B b d\right )\right )}{b^{2} d \left (d + e x\right ) \left (b e - c d\right ) \sqrt{b x + c x^{2}}} - \frac{e \sqrt{b x + c x^{2}} \left (b e \left (3 A b e - 2 A c d - B b d\right ) - 2 c d \left (A b e - 2 A c d + B b d\right )\right )}{b^{2} d^{2} \left (d + e x\right ) \left (b e - c d\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/(e*x+d)**2/(c*x**2+b*x)**(3/2),x)
[Out]
-e*(3*A*b*e**2 - 6*A*c*d*e - B*b*d*e + 4*B*c*d**2)*atan((-b*d + x*(b*e - 2*c*d))
/(2*sqrt(d)*sqrt(b*e - c*d)*sqrt(b*x + c*x**2)))/(2*d**(5/2)*(b*e - c*d)**(5/2))
- 2*(A*b*(b*e - c*d) + c*x*(A*b*e - 2*A*c*d + B*b*d))/(b**2*d*(d + e*x)*(b*e -
c*d)*sqrt(b*x + c*x**2)) - e*sqrt(b*x + c*x**2)*(b*e*(3*A*b*e - 2*A*c*d - B*b*d)
- 2*c*d*(A*b*e - 2*A*c*d + B*b*d))/(b**2*d**2*(d + e*x)*(b*e - c*d)**2)
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Mathematica [A] time = 1.54928, size = 193, normalized size = 0.79 \[ \frac{x^{3/2} \left (\frac{(b+c x)^2 \left (\frac{2 c^2 x (b B-A c)}{b^2 (b+c x) (c d-b e)^2}-\frac{2 A}{b^2 d^2}+\frac{e^2 x (B d-A e)}{d^2 (d+e x) (c d-b e)^2}\right )}{\sqrt{x}}+\frac{e (b+c x)^{3/2} (B d (b e-4 c d)-3 A e (b e-2 c d)) \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{d^{5/2} (b e-c d)^{5/2}}\right )}{(x (b+c x))^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/((d + e*x)^2*(b*x + c*x^2)^(3/2)),x]
[Out]
(x^(3/2)*(((b + c*x)^2*((-2*A)/(b^2*d^2) + (2*c^2*(b*B - A*c)*x)/(b^2*(c*d - b*e
)^2*(b + c*x)) + (e^2*(B*d - A*e)*x)/(d^2*(c*d - b*e)^2*(d + e*x))))/Sqrt[x] + (
e*(B*d*(-4*c*d + b*e) - 3*A*e*(-2*c*d + b*e))*(b + c*x)^(3/2)*ArcTan[(Sqrt[-(c*d
) + b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/(d^(5/2)*(-(c*d) + b*e)^(5/2))))/(x*
(b + c*x))^(3/2)
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Maple [B] time = 0.019, size = 2069, normalized size = 8.4 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/(e*x+d)^2/(c*x^2+b*x)^(3/2),x)
[Out]
12/e/(b*e-c*d)^2/b^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x
*c^3*B*d-12/(b*e-c*d)^2/b/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1
/2)*x*c^2*B-12/(b*e-c*d)^2/b^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^
2)^(1/2)*x*c^3*A+3*e/d/(b*e-c*d)^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d
)/e^2)^(1/2)*b*B+9*e/d/(b*e-c*d)^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d
)/e^2)^(1/2)*c*A-4*c/d/(b*e-c*d)/b/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d
)/e^2)^(1/2)*A+6*B/e/(b*e-c*d)/b/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/
e^2)^(1/2)*c-3*e^2/d^2/(b*e-c*d)^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d
)/e^2)^(1/2)*b*A+12*B/e/(b*e-c*d)/b^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-
c*d)/e^2)^(1/2)*x*c^2+6/e/(b*e-c*d)^2/b/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*
e-c*d)/e^2)^(1/2)*c^2*B*d+3/2*e^2/d^2/(b*e-c*d)^2/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-
2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+
(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*b*A-3/2*e/d/(b*e-c*d)^2/(
-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-
c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+
x))*b*B-3*e/d/(b*e-c*d)^2/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2
*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*
(b*e-c*d)/e^2)^(1/2))/(d/e+x))*c*A-8*c^2/d/(b*e-c*d)/b^2/(c*(d/e+x)^2+(b*e-2*c*d
)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x*A-2*B/d/(b*e-c*d)/b/(c*(d/e+x)^2+(b*e-2*c*d
)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x*c-3*e^2/d^2/(b*e-c*d)^2/(c*(d/e+x)^2+(b*e-2
*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x*c*A+3*e/d/(b*e-c*d)^2/(c*(d/e+x)^2+(b*e
-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x*c*B+12*e/d/(b*e-c*d)^2/b/(c*(d/e+x)^2
+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x*c^2*A-6/(b*e-c*d)^2/b/(c*(d/e+x)
^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*c^2*A+3/(b*e-c*d)^2/(-d*(b*e-c*d
)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(
1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*c*B+B/d
/(b*e-c*d)/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)
+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^
(1/2))/(d/e+x))+1/d/(b*e-c*d)/(d/e+x)/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-
c*d)/e^2)^(1/2)*A-1/e/(b*e-c*d)/(d/e+x)/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*
e-c*d)/e^2)^(1/2)*B-2*B/d/(b*e-c*d)/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*
d)/e^2)^(1/2)-9/(b*e-c*d)^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^
(1/2)*c*B
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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)^(3/2)*(e*x + d)^2),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.295611, 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)/((c*x^2 + b*x)^(3/2)*(e*x + d)^2),x, algorithm="fricas")
[Out]
[1/2*((4*B*b^2*c*d^3*e + 3*A*b^3*d*e^3 - (B*b^3 + 6*A*b^2*c)*d^2*e^2 + (4*B*b^2*
c*d^2*e^2 + 3*A*b^3*e^4 - (B*b^3 + 6*A*b^2*c)*d*e^3)*x)*sqrt(c*x^2 + b*x)*log(-(
2*(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x) - sqrt(c*d^2 - b*d*e)*(b*d + (2*c*d - b*e)*x
))/(e*x + d)) - 2*(2*A*b*c^2*d^3 - 4*A*b^2*c*d^2*e + 2*A*b^3*d*e^2 + (3*A*b^2*c*
e^3 - 2*(B*b*c^2 - 2*A*c^3)*d^2*e - (B*b^2*c + 4*A*b*c^2)*d*e^2)*x^2 - (2*A*b*c^
2*d^2*e - 3*A*b^3*e^3 + 2*(B*b*c^2 - 2*A*c^3)*d^3 + (B*b^3 + 2*A*b^2*c)*d*e^2)*x
)*sqrt(c*d^2 - b*d*e))/((b^2*c^2*d^5 - 2*b^3*c*d^4*e + b^4*d^3*e^2 + (b^2*c^2*d^
4*e - 2*b^3*c*d^3*e^2 + b^4*d^2*e^3)*x)*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x)),
((4*B*b^2*c*d^3*e + 3*A*b^3*d*e^3 - (B*b^3 + 6*A*b^2*c)*d^2*e^2 + (4*B*b^2*c*d^2
*e^2 + 3*A*b^3*e^4 - (B*b^3 + 6*A*b^2*c)*d*e^3)*x)*sqrt(c*x^2 + b*x)*arctan(-sqr
t(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/((c*d - b*e)*x)) - (2*A*b*c^2*d^3 - 4*A*b^2*
c*d^2*e + 2*A*b^3*d*e^2 + (3*A*b^2*c*e^3 - 2*(B*b*c^2 - 2*A*c^3)*d^2*e - (B*b^2*
c + 4*A*b*c^2)*d*e^2)*x^2 - (2*A*b*c^2*d^2*e - 3*A*b^3*e^3 + 2*(B*b*c^2 - 2*A*c^
3)*d^3 + (B*b^3 + 2*A*b^2*c)*d*e^2)*x)*sqrt(-c*d^2 + b*d*e))/((b^2*c^2*d^5 - 2*b
^3*c*d^4*e + b^4*d^3*e^2 + (b^2*c^2*d^4*e - 2*b^3*c*d^3*e^2 + b^4*d^2*e^3)*x)*sq
rt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*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)**2/(c*x**2+b*x)**(3/2),x)
[Out]
Timed out
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{B x + A}{{\left (c x^{2} + b x\right )}^{\frac{3}{2}}{\left (e x + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((c*x^2 + b*x)^(3/2)*(e*x + d)^2),x, algorithm="giac")
[Out]
integrate((B*x + A)/((c*x^2 + b*x)^(3/2)*(e*x + d)^2), x)