3.1279 \(\int \frac{(A+B x) (d+e x)^{5/2}}{\left (b x+c x^2\right )^{5/2}} \, dx\)
Optimal. Leaf size=454 \[ -\frac{2 (d+e x)^{3/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}+\frac{2 d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left (-8 b c (A e+B d)+16 A c^2 d+b^2 (-B) e\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{7/2} c^{3/2} \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (b^2 c e (A e+3 B d)-8 b c^2 d (2 A e+B d)+16 A c^3 d^2+2 b^3 B e^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{7/2} c^{3/2} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 \sqrt{d+e x} \left (b d \left (-b c (7 A e+4 B d)+8 A c^2 d+b^2 B e\right )+x \left (b^2 c e (A e+3 B d)-8 b c^2 d (2 A e+B d)+16 A c^3 d^2+2 b^3 B e^2\right )\right )}{3 b^4 c \sqrt{b x+c x^2}} \]
[Out]
(-2*(d + e*x)^(3/2)*(A*b*c*d + (2*A*c^2*d + b^2*B*e - b*c*(B*d + A*e))*x))/(3*b^
2*c*(b*x + c*x^2)^(3/2)) + (2*Sqrt[d + e*x]*(b*d*(8*A*c^2*d + b^2*B*e - b*c*(4*B
*d + 7*A*e)) + (16*A*c^3*d^2 + 2*b^3*B*e^2 + b^2*c*e*(3*B*d + A*e) - 8*b*c^2*d*(
B*d + 2*A*e))*x))/(3*b^4*c*Sqrt[b*x + c*x^2]) - (2*(16*A*c^3*d^2 + 2*b^3*B*e^2 +
b^2*c*e*(3*B*d + A*e) - 8*b*c^2*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)*c^(3/2)*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (2*d*(c*d - b*e)*(16*A*c^2*d
- b^2*B*e - 8*b*c*(B*d + A*e))*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*Ellip
ticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*(-b)^(7/2)*c^(3/2)*Sqr
t[d + e*x]*Sqrt[b*x + c*x^2])
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Rubi [A] time = 1.45585, antiderivative size = 454, 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
\[ -\frac{2 (d+e x)^{3/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}+\frac{2 d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left (-8 b c (A e+B d)+16 A c^2 d+b^2 (-B) e\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{7/2} c^{3/2} \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (b^2 c e (A e+3 B d)-8 b c^2 d (2 A e+B d)+16 A c^3 d^2+2 b^3 B e^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{7/2} c^{3/2} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}-\frac{2 \sqrt{d+e x} \left (b d \left (7 A b c e-8 A c^2 d+b^2 (-B) e+4 b B c d\right )-x \left (b^2 c e (A e+3 B d)-8 b c^2 d (2 A e+B d)+16 A c^3 d^2+2 b^3 B e^2\right )\right )}{3 b^4 c \sqrt{b x+c x^2}} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(d + e*x)^(5/2))/(b*x + c*x^2)^(5/2),x]
[Out]
(-2*(d + e*x)^(3/2)*(A*b*c*d + (2*A*c^2*d + b^2*B*e - b*c*(B*d + A*e))*x))/(3*b^
2*c*(b*x + c*x^2)^(3/2)) - (2*Sqrt[d + e*x]*(b*d*(4*b*B*c*d - 8*A*c^2*d - b^2*B*
e + 7*A*b*c*e) - (16*A*c^3*d^2 + 2*b^3*B*e^2 + b^2*c*e*(3*B*d + A*e) - 8*b*c^2*d
*(B*d + 2*A*e))*x))/(3*b^4*c*Sqrt[b*x + c*x^2]) - (2*(16*A*c^3*d^2 + 2*b^3*B*e^2
+ b^2*c*e*(3*B*d + A*e) - 8*b*c^2*d*(B*d + 2*A*e))*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sq
rt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*(-b)^
(7/2)*c^(3/2)*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (2*d*(c*d - b*e)*(16*A*c^2*
d - b^2*B*e - 8*b*c*(B*d + A*e))*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*Ell
ipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*(-b)^(7/2)*c^(3/2)*S
qrt[d + e*x]*Sqrt[b*x + c*x^2])
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Rubi in Sympy [A] time = 175.626, size = 481, normalized size = 1.06 \[ - \frac{2 \sqrt{x} \sqrt{1 + \frac{c x}{b}} \sqrt{d + e x} \left (A b^{2} c e^{2} - 16 A b c^{2} d e + 16 A c^{3} d^{2} + 2 B b^{3} e^{2} + 3 B b^{2} c d e - 8 B b c^{2} d^{2}\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{3 c^{\frac{3}{2}} \left (- b\right )^{\frac{7}{2}} \sqrt{1 + \frac{e x}{d}} \sqrt{b x + c x^{2}}} - \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (A b c d + x \left (2 A c^{2} d + B b^{2} e - b c \left (A e + B d\right )\right )\right )}{3 b^{2} c \left (b x + c x^{2}\right )^{\frac{3}{2}}} + \frac{4 \sqrt{d + e x} \left (\frac{b d \left (- 7 A b c e + 8 A c^{2} d + B b^{2} e - 4 B b c d\right )}{2} + x \left (\frac{A b^{2} c e^{2}}{2} - 8 A b c^{2} d e + 8 A c^{3} d^{2} + B b^{3} e^{2} + \frac{3 B b^{2} c d e}{2} - 4 B b c^{2} d^{2}\right )\right )}{3 b^{4} c \sqrt{b x + c x^{2}}} + \frac{2 \sqrt{x} \left (- d\right )^{\frac{3}{2}} \sqrt{1 + \frac{c x}{b}} \sqrt{1 + \frac{e x}{d}} \left (b e - c d\right ) \left (- 8 A b c e + 16 A c^{2} d - B b^{2} e - 8 B b c d\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{e} \sqrt{x}}{\sqrt{- d}} \right )}\middle | \frac{c d}{b e}\right )}{3 b^{4} c \sqrt{e} \sqrt{d + e x} \sqrt{b x + c x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(e*x+d)**(5/2)/(c*x**2+b*x)**(5/2),x)
[Out]
-2*sqrt(x)*sqrt(1 + c*x/b)*sqrt(d + e*x)*(A*b**2*c*e**2 - 16*A*b*c**2*d*e + 16*A
*c**3*d**2 + 2*B*b**3*e**2 + 3*B*b**2*c*d*e - 8*B*b*c**2*d**2)*elliptic_e(asin(s
qrt(c)*sqrt(x)/sqrt(-b)), b*e/(c*d))/(3*c**(3/2)*(-b)**(7/2)*sqrt(1 + e*x/d)*sqr
t(b*x + c*x**2)) - 2*(d + e*x)**(3/2)*(A*b*c*d + x*(2*A*c**2*d + B*b**2*e - b*c*
(A*e + B*d)))/(3*b**2*c*(b*x + c*x**2)**(3/2)) + 4*sqrt(d + e*x)*(b*d*(-7*A*b*c*
e + 8*A*c**2*d + B*b**2*e - 4*B*b*c*d)/2 + x*(A*b**2*c*e**2/2 - 8*A*b*c**2*d*e +
8*A*c**3*d**2 + B*b**3*e**2 + 3*B*b**2*c*d*e/2 - 4*B*b*c**2*d**2))/(3*b**4*c*sq
rt(b*x + c*x**2)) + 2*sqrt(x)*(-d)**(3/2)*sqrt(1 + c*x/b)*sqrt(1 + e*x/d)*(b*e -
c*d)*(-8*A*b*c*e + 16*A*c**2*d - B*b**2*e - 8*B*b*c*d)*elliptic_f(asin(sqrt(e)*
sqrt(x)/sqrt(-d)), c*d/(b*e))/(3*b**4*c*sqrt(e)*sqrt(d + e*x)*sqrt(b*x + c*x**2)
)
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Mathematica [C] time = 6.07156, size = 452, normalized size = 1. \[ -\frac{2 \left (b (d+e x) \left (x^2 (b+c x) (c d-b e) \left (b c (A e+5 B d)-8 A c^2 d+2 b^2 B e\right )+b x^2 (b B-A c) (c d-b e)^2+c d x (b+c x)^2 (7 A b e-8 A c d+3 b B d)+A b c d^2 (b+c x)^2\right )+x \sqrt{\frac{b}{c}} (b+c x) \left (-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (c d-b e) \left (-b c (A e+4 B d)+8 A c^2 d-2 b^2 B e\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (b^2 c e (A e+3 B d)-8 b c^2 d (2 A e+B d)+16 A c^3 d^2+2 b^3 B e^2\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left (b^2 c e (A e+3 B d)-8 b c^2 d (2 A e+B d)+16 A c^3 d^2+2 b^3 B e^2\right )\right )\right )}{3 b^5 c (x (b+c x))^{3/2} \sqrt{d+e x}} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(d + e*x)^(5/2))/(b*x + c*x^2)^(5/2),x]
[Out]
(-2*(b*(d + e*x)*(b*(b*B - A*c)*(c*d - b*e)^2*x^2 + (c*d - b*e)*(-8*A*c^2*d + 2*
b^2*B*e + b*c*(5*B*d + A*e))*x^2*(b + c*x) + A*b*c*d^2*(b + c*x)^2 + c*d*(3*b*B*
d - 8*A*c*d + 7*A*b*e)*x*(b + c*x)^2) + Sqrt[b/c]*x*(b + c*x)*(Sqrt[b/c]*(16*A*c
^3*d^2 + 2*b^3*B*e^2 + b^2*c*e*(3*B*d + A*e) - 8*b*c^2*d*(B*d + 2*A*e))*(b + c*x
)*(d + e*x) + I*b*e*(16*A*c^3*d^2 + 2*b^3*B*e^2 + b^2*c*e*(3*B*d + A*e) - 8*b*c^
2*d*(B*d + 2*A*e))*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSi
nh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*b*e*(c*d - b*e)*(8*A*c^2*d - 2*b^2*B*e -
b*c*(4*B*d + A*e))*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcS
inh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(3*b^5*c*(x*(b + c*x))^(3/2)*Sqrt[d + e*
x])
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Maple [B] time = 0.064, size = 2644, normalized size = 5.8 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(e*x+d)^(5/2)/(c*x^2+b*x)^(5/2),x)
[Out]
2/3*(16*A*x^3*c^6*d^3+2*B*x^4*b^3*c^3*e^3+2*A*x^3*b^3*c^3*e^3+16*A*((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))*x^2*b*c^5*d^3+2*B*((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))*x^2*b^5*c
*e^3+8*B*((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))*x^2*b^2*c^4*d^3-8*B*((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))*x^2*b^2*c^4*d^3+A*((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))*x*b^5*c*e^3-16
*A*((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))*x*b^2*c^4*d^3+16*A*((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))*x*b^2*c^4*d^3+8*B*((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))*x*b^3*c^3*d^3-8*B*((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))*x*b^3*c^3*d^3+A*((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))*x
^2*b^4*c^2*e^3-16*A*((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))*x^2*b*c^5*d^3+A*x^4*b^2*c^4
*e^3-8*A*x*b^3*c^3*d^2*e-24*A*((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))*x^2*b^2*c^4*d^2*e
+B*((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))*x^2*b^4*c^2*d*e^2-11*B*((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))*x^2*b^3*c^3*d^2*e+B*((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))*x^2*b^4*c^2*d*e
^2+B*((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))*x*b^5*c*d*e^2-16*A*x^4*b*c^5*d*e^2-8*B*x^4
*b*c^5*d^2*e+8*A*x^3*b*c^5*d^2*e-24*A*x^3*b^2*c^4*d*e^2+7*B*x^3*b^3*c^3*d*e^2-9*
B*x^3*b^2*c^4*d^2*e+3*B*x^4*b^2*c^4*d*e^2-5*A*x^2*b^3*c^3*d*e^2-19*A*x^2*b^2*c^4
*d^2*e+B*x^2*b^4*c^2*d*e^2+2*B*x^2*b^3*c^3*d^2*e-A*b^3*c^3*d^3+6*A*x*b^2*c^4*d^3
+B*x^3*b^4*c^2*e^3-3*B*x*b^3*c^3*d^3+16*A*x^4*c^6*d^2*e+24*A*x^2*b*c^5*d^3-8*B*x
^3*b*c^5*d^3-12*B*x^2*b^2*c^4*d^3+8*A*((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))*x*b^4*c^2
*d*e^2-24*A*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Ellipt
icF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^3*c^3*d^2*e+B*((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))*x*b^5*c*d*e^2-11*B*((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))*x*b^4*c^2*d^
2*e+7*B*((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))*x*b^4*c^2*d^2*e+7*B*((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))*x^2*b^3*c^3*d^2*e-17*A*((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))*x*b^4*c^2*
d*e^2+32*A*((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))*x*b^3*c^3*d^2*e-17*A*((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))*x^2*b^3*c^3*d*e^2+32*A*((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))*x^2*b^
2*c^4*d^2*e+8*A*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*El
lipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^3*c^3*d*e^2+2*B*((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))*x*b^6*e^3)/x^2*(x*(c*x+b))^(1/2)/b^4/(c*x+b)^2/c^3/(e*x+
d)^(1/2)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x + A\right )}{\left (e x + d\right )}^{\frac{5}{2}}}{{\left (c x^{2} + b x\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)^(5/2)/(c*x^2 + b*x)^(5/2),x, algorithm="maxima")
[Out]
integrate((B*x + A)*(e*x + d)^(5/2)/(c*x^2 + b*x)^(5/2), x)
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B e^{2} x^{3} + A d^{2} +{\left (2 \, B d e + A e^{2}\right )} x^{2} +{\left (B d^{2} + 2 \, A d e\right )} x\right )} \sqrt{e x + d}}{{\left (c^{2} x^{4} + 2 \, b c x^{3} + b^{2} x^{2}\right )} \sqrt{c x^{2} + b x}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)^(5/2)/(c*x^2 + b*x)^(5/2),x, algorithm="fricas")
[Out]
integral((B*e^2*x^3 + A*d^2 + (2*B*d*e + A*e^2)*x^2 + (B*d^2 + 2*A*d*e)*x)*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)**(5/2)/(c*x**2+b*x)**(5/2),x)
[Out]
Timed out
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GIAC/XCAS [A] time = 0.923756, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)^(5/2)/(c*x^2 + b*x)^(5/2),x, algorithm="giac")
[Out]
Done