3.1281 \(\int \frac{(A+B x) \sqrt{d+e x}}{\left (b x+c x^2\right )^{5/2}} \, dx\)
Optimal. Leaf size=420 \[ -\frac{2 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (b^2 e (A e+7 B d)-8 b c d (2 A e+B d)+16 A c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{7/2} d \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)}+\frac{2 \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} \left (-8 b c (A e+B d)+16 A c^2 d+3 b^2 B e\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{7/2} \sqrt{c} \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} (A b-x (b B-2 A c))}{3 b^2 \left (b x+c x^2\right )^{3/2}}-\frac{2 \sqrt{d+e x} \left (b (c d-b e) (A b e-8 A c d+4 b B d)-c x \left (b^2 e (A e+7 B d)-8 b c d (2 A e+B d)+16 A c^2 d^2\right )\right )}{3 b^4 d \sqrt{b x+c x^2} (c d-b e)} \]
[Out]
(-2*(A*b - (b*B - 2*A*c)*x)*Sqrt[d + e*x])/(3*b^2*(b*x + c*x^2)^(3/2)) - (2*Sqrt
[d + e*x]*(b*(c*d - b*e)*(4*b*B*d - 8*A*c*d + A*b*e) - c*(16*A*c^2*d^2 + b^2*e*(
7*B*d + A*e) - 8*b*c*d*(B*d + 2*A*e))*x))/(3*b^4*d*(c*d - b*e)*Sqrt[b*x + c*x^2]
) - (2*Sqrt[c]*(16*A*c^2*d^2 + b^2*e*(7*B*d + A*e) - 8*b*c*d*(B*d + 2*A*e))*Sqrt
[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]]
, (b*e)/(c*d)])/(3*(-b)^(7/2)*d*(c*d - b*e)*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2])
+ (2*(16*A*c^2*d + 3*b^2*B*e - 8*b*c*(B*d + A*e))*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqr
t[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*(-
b)^(7/2)*Sqrt[c]*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])
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Rubi [A] time = 1.43563, antiderivative size = 420, normalized size of antiderivative = 1.,
number of steps used = 9, number of rules used = 8, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286
\[ -\frac{2 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (b^2 e (A e+7 B d)-8 b c d (2 A e+B d)+16 A c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{7/2} d \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)}+\frac{2 \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} \left (-8 b c (A e+B d)+16 A c^2 d+3 b^2 B e\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{7/2} \sqrt{c} \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} (A b-x (b B-2 A c))}{3 b^2 \left (b x+c x^2\right )^{3/2}}-\frac{2 \sqrt{d+e x} \left (b (c d-b e) (A b e-8 A c d+4 b B d)-c x \left (b^2 e (A e+7 B d)-8 b c d (2 A e+B d)+16 A c^2 d^2\right )\right )}{3 b^4 d \sqrt{b x+c x^2} (c d-b e)} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*Sqrt[d + e*x])/(b*x + c*x^2)^(5/2),x]
[Out]
(-2*(A*b - (b*B - 2*A*c)*x)*Sqrt[d + e*x])/(3*b^2*(b*x + c*x^2)^(3/2)) - (2*Sqrt
[d + e*x]*(b*(c*d - b*e)*(4*b*B*d - 8*A*c*d + A*b*e) - c*(16*A*c^2*d^2 + b^2*e*(
7*B*d + A*e) - 8*b*c*d*(B*d + 2*A*e))*x))/(3*b^4*d*(c*d - b*e)*Sqrt[b*x + c*x^2]
) - (2*Sqrt[c]*(16*A*c^2*d^2 + b^2*e*(7*B*d + A*e) - 8*b*c*d*(B*d + 2*A*e))*Sqrt
[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]]
, (b*e)/(c*d)])/(3*(-b)^(7/2)*d*(c*d - b*e)*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2])
+ (2*(16*A*c^2*d + 3*b^2*B*e - 8*b*c*(B*d + A*e))*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqr
t[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*(-
b)^(7/2)*Sqrt[c]*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])
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Rubi in Sympy [A] time = 87.6927, size = 415, normalized size = 0.99 \[ \frac{2 \sqrt{c} \sqrt{x} \sqrt{1 + \frac{c x}{b}} \sqrt{d + e x} \left (- 3 b d e \left (2 A c - B b\right ) + \left (A \left (b e - 8 c d\right ) + 4 B b d\right ) \left (b e - 2 c d\right )\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{3 d \left (- b\right )^{\frac{7}{2}} \sqrt{1 + \frac{e x}{d}} \left (b e - c d\right ) \sqrt{b x + c x^{2}}} + \frac{2 \sqrt{x} \sqrt{1 + \frac{c x}{b}} \sqrt{1 + \frac{e x}{d}} \left (- 8 A b c e + 16 A c^{2} d + 3 B b^{2} e - 8 B b c d\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{3 \sqrt{c} \left (- b\right )^{\frac{7}{2}} \sqrt{d + e x} \sqrt{b x + c x^{2}}} - \frac{2 \sqrt{d + e x} \left (A b + x \left (2 A c - B b\right )\right )}{3 b^{2} \left (b x + c x^{2}\right )^{\frac{3}{2}}} - \frac{4 \sqrt{d + e x} \left (\frac{b \left (b e - c d\right ) \left (A b e - 8 A c d + 4 B b d\right )}{2} + \frac{c x \left (A b^{2} e^{2} - 16 A b c d e + 16 A c^{2} d^{2} + 7 B b^{2} d e - 8 B b c d^{2}\right )}{2}\right )}{3 b^{4} d \left (b e - c d\right ) \sqrt{b x + c x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(e*x+d)**(1/2)/(c*x**2+b*x)**(5/2),x)
[Out]
2*sqrt(c)*sqrt(x)*sqrt(1 + c*x/b)*sqrt(d + e*x)*(-3*b*d*e*(2*A*c - B*b) + (A*(b*
e - 8*c*d) + 4*B*b*d)*(b*e - 2*c*d))*elliptic_e(asin(sqrt(c)*sqrt(x)/sqrt(-b)),
b*e/(c*d))/(3*d*(-b)**(7/2)*sqrt(1 + e*x/d)*(b*e - c*d)*sqrt(b*x + c*x**2)) + 2*
sqrt(x)*sqrt(1 + c*x/b)*sqrt(1 + e*x/d)*(-8*A*b*c*e + 16*A*c**2*d + 3*B*b**2*e -
8*B*b*c*d)*elliptic_f(asin(sqrt(c)*sqrt(x)/sqrt(-b)), b*e/(c*d))/(3*sqrt(c)*(-b
)**(7/2)*sqrt(d + e*x)*sqrt(b*x + c*x**2)) - 2*sqrt(d + e*x)*(A*b + x*(2*A*c - B
*b))/(3*b**2*(b*x + c*x**2)**(3/2)) - 4*sqrt(d + e*x)*(b*(b*e - c*d)*(A*b*e - 8*
A*c*d + 4*B*b*d)/2 + c*x*(A*b**2*e**2 - 16*A*b*c*d*e + 16*A*c**2*d**2 + 7*B*b**2
*d*e - 8*B*b*c*d**2)/2)/(3*b**4*d*(b*e - c*d)*sqrt(b*x + c*x**2))
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Mathematica [C] time = 4.00099, size = 441, normalized size = 1.05 \[ -\frac{2 \left (\sqrt{\frac{b}{c}} (d+e x) \left (c d x^2 (b+c x) \left (b c (7 A e+5 B d)-8 A c^2 d-4 b^2 B e\right )+b c d x^2 (b B-A c) (c d-b e)+x (b+c x)^2 (c d-b e) (A b e-8 A c d+3 b B d)+A b d (b+c x)^2 (c d-b e)\right )+x (b+c x) \left (i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (b^2 e (A e+7 B d)-8 b c d (2 A e+B d)+16 A c^2 d^2\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left (b^2 e (A e+7 B d)-8 b c d (2 A e+B d)+16 A c^2 d^2\right )-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (c d-b e) (8 A c d-b (A e+4 B d)) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )\right )\right )}{3 b^4 d \sqrt{\frac{b}{c}} (x (b+c x))^{3/2} \sqrt{d+e x} (c d-b e)} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*Sqrt[d + e*x])/(b*x + c*x^2)^(5/2),x]
[Out]
(-2*(Sqrt[b/c]*(d + e*x)*(b*c*(b*B - A*c)*d*(c*d - b*e)*x^2 + c*d*(-8*A*c^2*d -
4*b^2*B*e + b*c*(5*B*d + 7*A*e))*x^2*(b + c*x) + A*b*d*(c*d - b*e)*(b + c*x)^2 +
(c*d - b*e)*(3*b*B*d - 8*A*c*d + A*b*e)*x*(b + c*x)^2) + x*(b + c*x)*(Sqrt[b/c]
*(16*A*c^2*d^2 + b^2*e*(7*B*d + A*e) - 8*b*c*d*(B*d + 2*A*e))*(b + c*x)*(d + e*x
) + I*b*e*(16*A*c^2*d^2 + b^2*e*(7*B*d + A*e) - 8*b*c*d*(B*d + 2*A*e))*Sqrt[1 +
b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)
/(b*e)] - I*b*e*(c*d - b*e)*(8*A*c*d - b*(4*B*d + A*e))*Sqrt[1 + b/(c*x)]*Sqrt[1
+ d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(3*b
^4*Sqrt[b/c]*d*(c*d - b*e)*(x*(b + c*x))^(3/2)*Sqrt[d + e*x])
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Maple [B] time = 0.064, size = 2503, normalized size = 6. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(e*x+d)^(1/2)/(c*x^2+b*x)^(5/2),x)
[Out]
-2/3/x^2*(-5*A*x^2*b^3*c^2*d*e^2+16*A*x^3*c^5*d^3+16*A*x^4*c^5*d^2*e+2*A*x^3*b^3
*c^2*e^3-8*B*x^3*b*c^4*d^3+24*A*x^2*b*c^4*d^3+6*A*x*b^2*c^3*d^3-12*B*x^2*b^2*c^3
*d^3+A*x^4*b^2*c^3*e^3+A*x^2*b^4*c*e^3+A*x^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-
c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^
4*c*e^3-16*A*x^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*E
llipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^4*d^3+16*A*x^2*((c*x+b)/b)
^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(
b*e/(b*e-c*d))^(1/2))*b*c^4*d^3+8*B*x^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))
^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^3
*d^3-8*B*x^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Ellip
ticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^3*d^3-16*A*x*((c*x+b)/b)^(1/
2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/
(b*e-c*d))^(1/2))*b^2*c^3*d^3+16*A*x*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1
/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^3*d^
3+7*B*x*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(
((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^5*d*e^2+8*B*x*((c*x+b)/b)^(1/2)*(-(e*
x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d
))^(1/2))*b^3*c^2*d^3-3*B*x*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x
/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^5*d*e^2-8*B*x*((c
*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)
^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c^2*d^3-A*b^3*c^2*d^3+11*B*x^2*((c*x+b)/b)^(1/
2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/
(b*e-c*d))^(1/2))*b^3*c^2*d^2*e-17*A*x*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^
(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*c*d*
e^2+32*A*x*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Ellipti
cE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c^2*d^2*e+8*A*x*((c*x+b)/b)^(1/2
)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(
b*e-c*d))^(1/2))*b^4*c*d*e^2-24*A*x*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/
2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c^2*d^2
*e-15*B*x*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Elliptic
E(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*c*d^2*e+11*B*x*((c*x+b)/b)^(1/2)*
(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*
e-c*d))^(1/2))*b^4*c*d^2*e-17*A*x^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/
2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c^2*d*e
^2+3*B*x*b^4*c*d^2*e-16*A*x^4*b*c^4*d*e^2+7*B*x^4*b^2*c^3*d*e^2-8*B*x^4*b*c^4*d^
2*e-24*A*x^3*b^2*c^3*d*e^2+8*A*x^3*b*c^4*d^2*e+2*A*x*b^4*c*d*e^2-8*A*x*b^3*c^2*d
^2*e+11*B*x^3*b^3*c^2*d*e^2-5*B*x^3*b^2*c^3*d^2*e+3*B*x^2*b^4*c*d*e^2-19*A*x^2*b
^2*c^3*d^2*e+8*B*x^2*b^3*c^2*d^2*e-3*B*x*b^3*c^2*d^3+32*A*x^2*((c*x+b)/b)^(1/2)*
(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*
e-c*d))^(1/2))*b^2*c^3*d^2*e+8*A*x^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1
/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c^2*d*
e^2-24*A*x^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Ellip
ticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^3*d^2*e+7*B*x^2*((c*x+b)/b)^
(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b
*e/(b*e-c*d))^(1/2))*b^4*c*d*e^2-15*B*x^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d
))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c
^2*d^2*e-3*B*x^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*E
llipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*c*d*e^2+A*b^4*c*d^2*e+A*x*
((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)
/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^5*e^3)*(x*(c*x+b))^(1/2)/c/b^4/d/(b*e-c*d)/(c
*x+b)^2/(e*x+d)^(1/2)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x + A\right )} \sqrt{e x + d}}{{\left (c x^{2} + b x\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*sqrt(e*x + d)/(c*x^2 + b*x)^(5/2),x, algorithm="maxima")
[Out]
integrate((B*x + A)*sqrt(e*x + d)/(c*x^2 + b*x)^(5/2), x)
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B x + A\right )} \sqrt{e x + d}}{{\left (c^{2} x^{4} + 2 \, b c x^{3} + b^{2} x^{2}\right )} \sqrt{c x^{2} + b x}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*sqrt(e*x + d)/(c*x^2 + b*x)^(5/2),x, algorithm="fricas")
[Out]
integral((B*x + A)*sqrt(e*x + d)/((c^2*x^4 + 2*b*c*x^3 + b^2*x^2)*sqrt(c*x^2 + b
*x)), 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)**(1/2)/(c*x**2+b*x)**(5/2),x)
[Out]
Timed out
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*sqrt(e*x + d)/(c*x^2 + b*x)^(5/2),x, algorithm="giac")
[Out]
Exception raised: NotImplementedError