3.1278 \(\int \frac{(A+B x) (d+e x)^{7/2}}{\left (b x+c x^2\right )^{5/2}} \, dx\)
Optimal. Leaf size=524 \[ -\frac{2 (d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}+\frac{2 d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left (b^2 c e (B d-A e)-8 b c^2 d (2 A e+B d)+16 A c^3 d^2+4 b^3 B e^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{7/2} c^{5/2} \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (b c d^2 \left (-b c (9 A e+4 B d)+8 A c^2 d+b^2 B e\right )+x \left (b^3 c e^2 (A e+4 B d)+b^2 c^2 d e (6 A e+5 B d)-8 b c^3 d^2 (3 A e+B d)+16 A c^4 d^3-4 b^4 B e^3\right )\right )}{3 b^4 c^2 \sqrt{b x+c x^2}}-\frac{2 \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (b^3 c e^2 (2 A e+5 B d)+b^2 c^2 d e (4 A e+5 B d)-8 b c^3 d^2 (3 A e+B d)+16 A c^4 d^3-8 b^4 B e^3\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{7/2} c^{5/2} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}} \]
[Out]
(-2*(d + e*x)^(5/2)*(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*c*d^2*(8*A*c^2*d + b^2*B*e - b*c*
(4*B*d + 9*A*e)) + (16*A*c^4*d^3 - 4*b^4*B*e^3 + b^3*c*e^2*(4*B*d + A*e) - 8*b*c
^3*d^2*(B*d + 3*A*e) + b^2*c^2*d*e*(5*B*d + 6*A*e))*x))/(3*b^4*c^2*Sqrt[b*x + c*
x^2]) - (2*(16*A*c^4*d^3 - 8*b^4*B*e^3 + b^3*c*e^2*(5*B*d + 2*A*e) - 8*b*c^3*d^2
*(B*d + 3*A*e) + b^2*c^2*d*e*(5*B*d + 4*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^(5/2)*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (2*d*(c*d - b*e)*(16*A*c^3*d^2 +
4*b^3*B*e^2 + b^2*c*e*(B*d - A*e) - 8*b*c^2*d*(B*d + 2*A*e))*Sqrt[x]*Sqrt[1 + (c
*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*
d)])/(3*(-b)^(7/2)*c^(5/2)*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])
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Rubi [A] time = 1.86645, antiderivative size = 524, normalized size of antiderivative = 1.,
number of steps used = 9, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25
\[ -\frac{2 (d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}+\frac{2 d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left (b^2 c e (B d-A e)-8 b c^2 d (2 A e+B d)+16 A c^3 d^2+4 b^3 B e^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{7/2} c^{5/2} \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} \left (b c d^2 \left (9 A b c e-8 A c^2 d+b^2 (-B) e+4 b B c d\right )-x \left (b^3 c e^2 (A e+4 B d)+b^2 c^2 d e (6 A e+5 B d)-8 b c^3 d^2 (3 A e+B d)+16 A c^4 d^3-4 b^4 B e^3\right )\right )}{3 b^4 c^2 \sqrt{b x+c x^2}}-\frac{2 \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (b^3 c e^2 (2 A e+5 B d)+b^2 c^2 d e (4 A e+5 B d)-8 b c^3 d^2 (3 A e+B d)+16 A c^4 d^3-8 b^4 B e^3\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{7/2} c^{5/2} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(d + e*x)^(7/2))/(b*x + c*x^2)^(5/2),x]
[Out]
(-2*(d + e*x)^(5/2)*(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*c*d^2*(4*b*B*c*d - 8*A*c^2*d - b^
2*B*e + 9*A*b*c*e) - (16*A*c^4*d^3 - 4*b^4*B*e^3 + b^3*c*e^2*(4*B*d + A*e) - 8*b
*c^3*d^2*(B*d + 3*A*e) + b^2*c^2*d*e*(5*B*d + 6*A*e))*x))/(3*b^4*c^2*Sqrt[b*x +
c*x^2]) - (2*(16*A*c^4*d^3 - 8*b^4*B*e^3 + b^3*c*e^2*(5*B*d + 2*A*e) - 8*b*c^3*d
^2*(B*d + 3*A*e) + b^2*c^2*d*e*(5*B*d + 4*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^(5/2)*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (2*d*(c*d - b*e)*(16*A*c^3*d^2
+ 4*b^3*B*e^2 + b^2*c*e*(B*d - A*e) - 8*b*c^2*d*(B*d + 2*A*e))*Sqrt[x]*Sqrt[1 +
(c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(
c*d)])/(3*(-b)^(7/2)*c^(5/2)*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(e*x+d)**(7/2)/(c*x**2+b*x)**(5/2),x)
[Out]
Timed out
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Mathematica [C] time = 8.37149, size = 530, normalized size = 1.01 \[ -\frac{2 \left (b (d+e x) \left (x^2 (b+c x) (c d-b e)^2 \left (b c (5 B d-2 A e)-8 A c^2 d+5 b^2 B e\right )+c^2 d^2 x (b+c x)^2 (10 A b e-8 A c d+3 b B d)+b x^2 (b B-A c) (c d-b e)^3+A b c^2 d^3 (b+c x)^2\right )+x \sqrt{\frac{b}{c}} (b+c x) \left (-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (c d-b e) \left (-b^2 c e (2 A e+B d)-b c^2 d (5 A e+4 B d)+8 A c^3 d^2+8 b^3 B e^2\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (b^3 c e^2 (2 A e+5 B d)+b^2 c^2 d e (4 A e+5 B d)-8 b c^3 d^2 (3 A e+B d)+16 A c^4 d^3-8 b^4 B e^3\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left (b^3 c e^2 (2 A e+5 B d)+b^2 c^2 d e (4 A e+5 B d)-8 b c^3 d^2 (3 A e+B d)+16 A c^4 d^3-8 b^4 B e^3\right )\right )\right )}{3 b^5 c^2 (x (b+c x))^{3/2} \sqrt{d+e x}} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(d + e*x)^(7/2))/(b*x + c*x^2)^(5/2),x]
[Out]
(-2*(b*(d + e*x)*(b*(b*B - A*c)*(c*d - b*e)^3*x^2 + (c*d - b*e)^2*(-8*A*c^2*d +
5*b^2*B*e + b*c*(5*B*d - 2*A*e))*x^2*(b + c*x) + A*b*c^2*d^3*(b + c*x)^2 + c^2*d
^2*(3*b*B*d - 8*A*c*d + 10*A*b*e)*x*(b + c*x)^2) + Sqrt[b/c]*x*(b + c*x)*(Sqrt[b
/c]*(16*A*c^4*d^3 - 8*b^4*B*e^3 + b^3*c*e^2*(5*B*d + 2*A*e) - 8*b*c^3*d^2*(B*d +
3*A*e) + b^2*c^2*d*e*(5*B*d + 4*A*e))*(b + c*x)*(d + e*x) + I*b*e*(16*A*c^4*d^3
- 8*b^4*B*e^3 + b^3*c*e^2*(5*B*d + 2*A*e) - 8*b*c^3*d^2*(B*d + 3*A*e) + b^2*c^2
*d*e*(5*B*d + 4*A*e))*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*Ar
cSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*b*e*(c*d - b*e)*(8*A*c^3*d^2 + 8*b^3*
B*e^2 - b^2*c*e*(B*d + 2*A*e) - b*c^2*d*(4*B*d + 5*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^5*c^2*(x*(b + c*x))^(3/2)*Sqrt[d + e*x])
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Maple [B] time = 0.067, size = 3247, normalized size = 6.2 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(e*x+d)^(7/2)/(c*x^2+b*x)^(5/2),x)
[Out]
2/3*(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
ticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^5*d^4+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^5*d^4+2*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^6*c*e^4-1
6*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^5*d^4-8*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^4*d^4+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^4*d^4+16*A*x^
4*c^7*d^3*e-8*B*x^3*b*c^6*d^4+24*A*x^2*b*c^6*d^4-4*B*x^3*b^5*c^2*e^4+A*x^3*b^4*c
^3*e^4+6*A*x*b^2*c^5*d^4-12*B*x^2*b^2*c^5*d^4+2*A*x^4*b^3*c^4*e^4-5*B*x^4*b^4*c^
3*e^4+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)*Ell
ipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^6*d^4+2*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^5*c^2*e^4-16*A*x^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))
^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^6*d
^4+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)*Ellipti
cF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*c^3*d^2*e^2+9*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^4*d^3*e+13*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^5
*c^2*d*e^3+9*B*x*((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^3*d^3*e+9*A*x^3*b^3*c^4*
d*e^3+16*A*x^3*c^7*d^4-33*A*x^3*b^2*c^5*d^2*e^2+2*B*x^2*b^4*c^3*d^2*e^2-24*A*x^4
*b*c^6*d^2*e^2-8*B*x^4*b*c^6*d^3*e-11*A*x*b^3*c^4*d^3*e-3*B*x^3*b^4*c^3*d*e^3+13
*B*x^3*b^3*c^4*d^2*e^2-7*B*x^3*b^2*c^5*d^3*e+A*x^2*b^4*c^3*d*e^3-3*A*x^2*b^3*c^4
*d^2*e^2-31*A*x^2*b^2*c^5*d^3*e-4*B*x^2*b^5*c^2*d*e^3+5*B*x^2*b^3*c^4*d^3*e+4*A*
x^4*b^2*c^5*d*e^3+5*B*x^4*b^3*c^4*d*e^3+5*B*x^4*b^2*c^5*d^2*e^2-A*b^3*c^4*d^4+13
*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^6*c*d*e^3-13*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^4*c^3*d^3*e+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^4*c^3*d*e^3+15*
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^4*d^2*e^2-32*A*x^2*((c*x+b)/b)^(1/2
)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(
b*e-c*d))^(1/2))*b^2*c^5*d^3*e+2*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^3*
d*e^3-28*A*x^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Ell
ipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c^4*d^2*e^2+40*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^5*d^3*e-4*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^5*c^2*d*e^3-13*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^4*d^3*e+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^5*c^2*d*e^3+15*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^3*d^2*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)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c^4*d^3*e+2*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*c^2*d*e^3-28*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^3*d^2*e^2+40*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^3*c^4*d^3*e-4*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^6*c*d*e^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*c^2*d^2*e^2-3*B*x*b^3*c^4*d^4-8*B*x^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*
e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*
b^2*c^5*d^4-8*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^6*c*e^4-8*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^7*e^4)/x^2*(x*(c*x+b))^(1/2)/b^4/c^4/(c*x+b)^2/(e*x+d)^(
1/2)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x + A\right )}{\left (e x + d\right )}^{\frac{7}{2}}}{{\left (c x^{2} + b x\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)^(7/2)/(c*x^2 + b*x)^(5/2),x, algorithm="maxima")
[Out]
integrate((B*x + A)*(e*x + d)^(7/2)/(c*x^2 + b*x)^(5/2), x)
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B e^{3} x^{4} + A d^{3} +{\left (3 \, B d e^{2} + A e^{3}\right )} x^{3} + 3 \,{\left (B d^{2} e + A d e^{2}\right )} x^{2} +{\left (B d^{3} + 3 \, A d^{2} e\right )} x\right )} \sqrt{e x + d}}{{\left (c^{2} x^{4} + 2 \, b c x^{3} + b^{2} x^{2}\right )} \sqrt{c x^{2} + b x}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)^(7/2)/(c*x^2 + b*x)^(5/2),x, algorithm="fricas")
[Out]
integral((B*e^3*x^4 + A*d^3 + (3*B*d*e^2 + A*e^3)*x^3 + 3*(B*d^2*e + A*d*e^2)*x^
2 + (B*d^3 + 3*A*d^2*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)**(7/2)/(c*x**2+b*x)**(5/2),x)
[Out]
Timed out
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GIAC/XCAS [A] time = 1.08327, 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)^(7/2)/(c*x^2 + b*x)^(5/2),x, algorithm="giac")
[Out]
Done