3.1197 \(\int \frac{A+B x}{(d+e x)^3 \sqrt{b x+c x^2}} \, dx\)
Optimal. Leaf size=216 \[ \frac{\left (b^2 e (3 A e+B d)-4 b c d (2 A e+B d)+8 A c^2 d^2\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{8 d^{5/2} (c d-b e)^{5/2}}-\frac{\sqrt{b x+c x^2} (3 A e (2 c d-b e)-B d (b e+2 c d))}{4 d^2 (d+e x) (c d-b e)^2}+\frac{\sqrt{b x+c x^2} (B d-A e)}{2 d (d+e x)^2 (c d-b e)} \]
[Out]
((B*d - A*e)*Sqrt[b*x + c*x^2])/(2*d*(c*d - b*e)*(d + e*x)^2) - ((3*A*e*(2*c*d -
b*e) - B*d*(2*c*d + b*e))*Sqrt[b*x + c*x^2])/(4*d^2*(c*d - b*e)^2*(d + e*x)) +
((8*A*c^2*d^2 - 4*b*c*d*(B*d + 2*A*e) + b^2*e*(B*d + 3*A*e))*ArcTanh[(b*d + (2*c
*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(8*d^(5/2)*(c*d - b
*e)^(5/2))
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Rubi [A] time = 0.644513, antiderivative size = 216, 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
\[ \frac{\left (b^2 e (3 A e+B d)-4 b c d (2 A e+B d)+8 A c^2 d^2\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{8 d^{5/2} (c d-b e)^{5/2}}-\frac{\sqrt{b x+c x^2} (3 A e (2 c d-b e)-B d (b e+2 c d))}{4 d^2 (d+e x) (c d-b e)^2}+\frac{\sqrt{b x+c x^2} (B d-A e)}{2 d (d+e x)^2 (c d-b e)} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/((d + e*x)^3*Sqrt[b*x + c*x^2]),x]
[Out]
((B*d - A*e)*Sqrt[b*x + c*x^2])/(2*d*(c*d - b*e)*(d + e*x)^2) - ((3*A*e*(2*c*d -
b*e) - B*d*(2*c*d + b*e))*Sqrt[b*x + c*x^2])/(4*d^2*(c*d - b*e)^2*(d + e*x)) +
((8*A*c^2*d^2 - 4*b*c*d*(B*d + 2*A*e) + b^2*e*(B*d + 3*A*e))*ArcTanh[(b*d + (2*c
*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(8*d^(5/2)*(c*d - b
*e)^(5/2))
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Rubi in Sympy [A] time = 115.297, size = 206, normalized size = 0.95 \[ \frac{\left (A e - B d\right ) \sqrt{b x + c x^{2}}}{2 d \left (d + e x\right )^{2} \left (b e - c d\right )} + \frac{\sqrt{b x + c x^{2}} \left (3 A b e^{2} - 6 A c d e + B b d e + 2 B c d^{2}\right )}{4 d^{2} \left (d + e x\right ) \left (b e - c d\right )^{2}} + \frac{\left (\frac{3 A b^{2} e^{2}}{8} - A b c d e + A c^{2} d^{2} + \frac{B b^{2} d e}{8} - \frac{B b c d^{2}}{2}\right ) \operatorname{atan}{\left (\frac{- b d + x \left (b e - 2 c d\right )}{2 \sqrt{d} \sqrt{b e - c d} \sqrt{b x + c x^{2}}} \right )}}{d^{\frac{5}{2}} \left (b e - c d\right )^{\frac{5}{2}}} \]
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)**(1/2),x)
[Out]
(A*e - B*d)*sqrt(b*x + c*x**2)/(2*d*(d + e*x)**2*(b*e - c*d)) + sqrt(b*x + c*x**
2)*(3*A*b*e**2 - 6*A*c*d*e + B*b*d*e + 2*B*c*d**2)/(4*d**2*(d + e*x)*(b*e - c*d)
**2) + (3*A*b**2*e**2/8 - A*b*c*d*e + A*c**2*d**2 + B*b**2*d*e/8 - B*b*c*d**2/2)
*atan((-b*d + x*(b*e - 2*c*d))/(2*sqrt(d)*sqrt(b*e - c*d)*sqrt(b*x + c*x**2)))/(
d**(5/2)*(b*e - c*d)**(5/2))
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Mathematica [A] time = 0.767194, size = 200, normalized size = 0.93 \[ \frac{\sqrt{x} \left (\frac{\sqrt{b+c x} \left (b^2 e (3 A e+B d)-4 b c d (2 A e+B d)+8 A c^2 d^2\right ) \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{\sqrt{b e-c d}}+\frac{\sqrt{d} \sqrt{x} (b+c x) ((d+e x) (3 A e (b e-2 c d)+B d (b e+2 c d))+2 d (B d-A e) (c d-b e))}{(d+e x)^2}\right )}{4 d^{5/2} \sqrt{x (b+c x)} (c d-b e)^2} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/((d + e*x)^3*Sqrt[b*x + c*x^2]),x]
[Out]
(Sqrt[x]*((Sqrt[d]*Sqrt[x]*(b + c*x)*(2*d*(B*d - A*e)*(c*d - b*e) + (3*A*e*(-2*c
*d + b*e) + B*d*(2*c*d + b*e))*(d + e*x)))/(d + e*x)^2 + ((8*A*c^2*d^2 - 4*b*c*d
*(B*d + 2*A*e) + b^2*e*(B*d + 3*A*e))*Sqrt[b + c*x]*ArcTan[(Sqrt[-(c*d) + b*e]*S
qrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/Sqrt[-(c*d) + b*e]))/(4*d^(5/2)*(c*d - b*e)^2*
Sqrt[x*(b + c*x)])
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Maple [B] time = 0.016, size = 1821, 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)^3/(c*x^2+b*x)^(1/2),x)
[Out]
B/e/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
)-1/2*B/e/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))*b+3/2*B/e^2/(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))*c+1/2/e/d/(b*e-c*d)/(d/e+x
)^2*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*A-1/2/e^2/(b*e-c*d
)/(d/e+x)^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+3/4*e/d^
2/(b*e-c*d)^2/(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*A-3/4/d/(b*e-c*d)^2/(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*B-3/2/d/(b*e-c*d)^2/(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)*c*A+3/2/e/(b*e-c*d)^2/(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)*c*B-3/8*e/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^2*A+3/8/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^2*B+3/2/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*c*A-3/2/e/(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*c*B-3/2/e/(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^2*A+3/2/e^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))*c^2*B*d-1/2/e*c/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))*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)*(e*x + d)^3),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.299283, 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)*(e*x + d)^3),x, algorithm="fricas")
[Out]
[1/8*(2*(4*B*c*d^3 + 5*A*b*d*e^2 - (B*b + 8*A*c)*d^2*e + (2*B*c*d^2*e + 3*A*b*e^
3 + (B*b - 6*A*c)*d*e^2)*x)*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x) + (3*A*b^2*d^2
*e^2 - 4*(B*b*c - 2*A*c^2)*d^4 + (B*b^2 - 8*A*b*c)*d^3*e + (3*A*b^2*e^4 - 4*(B*b
*c - 2*A*c^2)*d^2*e^2 + (B*b^2 - 8*A*b*c)*d*e^3)*x^2 + 2*(3*A*b^2*d*e^3 - 4*(B*b
*c - 2*A*c^2)*d^3*e + (B*b^2 - 8*A*b*c)*d^2*e^2)*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)))/((c^2*d^
6 - 2*b*c*d^5*e + b^2*d^4*e^2 + (c^2*d^4*e^2 - 2*b*c*d^3*e^3 + b^2*d^2*e^4)*x^2
+ 2*(c^2*d^5*e - 2*b*c*d^4*e^2 + b^2*d^3*e^3)*x)*sqrt(c*d^2 - b*d*e)), 1/4*((4*B
*c*d^3 + 5*A*b*d*e^2 - (B*b + 8*A*c)*d^2*e + (2*B*c*d^2*e + 3*A*b*e^3 + (B*b - 6
*A*c)*d*e^2)*x)*sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x) - (3*A*b^2*d^2*e^2 - 4*(B
*b*c - 2*A*c^2)*d^4 + (B*b^2 - 8*A*b*c)*d^3*e + (3*A*b^2*e^4 - 4*(B*b*c - 2*A*c^
2)*d^2*e^2 + (B*b^2 - 8*A*b*c)*d*e^3)*x^2 + 2*(3*A*b^2*d*e^3 - 4*(B*b*c - 2*A*c^
2)*d^3*e + (B*b^2 - 8*A*b*c)*d^2*e^2)*x)*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2
+ b*x)/((c*d - b*e)*x)))/((c^2*d^6 - 2*b*c*d^5*e + b^2*d^4*e^2 + (c^2*d^4*e^2 -
2*b*c*d^3*e^3 + b^2*d^2*e^4)*x^2 + 2*(c^2*d^5*e - 2*b*c*d^4*e^2 + b^2*d^3*e^3)*
x)*sqrt(-c*d^2 + b*d*e))]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{A + B x}{\sqrt{x \left (b + c x\right )} \left (d + e x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)/(e*x+d)**3/(c*x**2+b*x)**(1/2),x)
[Out]
Integral((A + B*x)/(sqrt(x*(b + c*x))*(d + e*x)**3), x)
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GIAC/XCAS [A] time = 0.654782, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/(sqrt(c*x^2 + b*x)*(e*x + d)^3),x, algorithm="giac")
[Out]
sage0*x