3.1170 \(\int \frac{(A+B x) \sqrt{b x+c x^2}}{(d+e x)^5} \, dx\)

Optimal. Leaf size=310 \[ -\frac{b^2 \left (b^2 e (5 A e+3 B d)-8 b c d (2 A e+B d)+16 A c^2 d^2\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{128 d^{7/2} (c d-b e)^{7/2}}+\frac{\sqrt{b x+c x^2} (x (2 c d-b e)+b d) \left (b^2 e (5 A e+3 B d)-8 b c d (2 A e+B d)+16 A c^2 d^2\right )}{64 d^3 (d+e x)^2 (c d-b e)^3}-\frac{\left (b x+c x^2\right )^{3/2} (5 A e (2 c d-b e)-B d (3 b e+2 c d))}{24 d^2 (d+e x)^3 (c d-b e)^2}+\frac{\left (b x+c x^2\right )^{3/2} (B d-A e)}{4 d (d+e x)^4 (c d-b e)} \]

[Out]

((16*A*c^2*d^2 - 8*b*c*d*(B*d + 2*A*e) + b^2*e*(3*B*d + 5*A*e))*(b*d + (2*c*d -
b*e)*x)*Sqrt[b*x + c*x^2])/(64*d^3*(c*d - b*e)^3*(d + e*x)^2) + ((B*d - A*e)*(b*
x + c*x^2)^(3/2))/(4*d*(c*d - b*e)*(d + e*x)^4) - ((5*A*e*(2*c*d - b*e) - B*d*(2
*c*d + 3*b*e))*(b*x + c*x^2)^(3/2))/(24*d^2*(c*d - b*e)^2*(d + e*x)^3) - (b^2*(1
6*A*c^2*d^2 - 8*b*c*d*(B*d + 2*A*e) + b^2*e*(3*B*d + 5*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])])/(128*d^(7/2)*(c*d -
b*e)^(7/2))

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Rubi [A]  time = 0.963672, antiderivative size = 310, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{b^2 \left (b^2 e (5 A e+3 B d)-8 b c d (2 A e+B d)+16 A c^2 d^2\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{128 d^{7/2} (c d-b e)^{7/2}}+\frac{\sqrt{b x+c x^2} (x (2 c d-b e)+b d) \left (b^2 e (5 A e+3 B d)-8 b c d (2 A e+B d)+16 A c^2 d^2\right )}{64 d^3 (d+e x)^2 (c d-b e)^3}-\frac{\left (b x+c x^2\right )^{3/2} (5 A e (2 c d-b e)-B d (3 b e+2 c d))}{24 d^2 (d+e x)^3 (c d-b e)^2}+\frac{\left (b x+c x^2\right )^{3/2} (B d-A e)}{4 d (d+e x)^4 (c d-b e)} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*Sqrt[b*x + c*x^2])/(d + e*x)^5,x]

[Out]

((16*A*c^2*d^2 - 8*b*c*d*(B*d + 2*A*e) + b^2*e*(3*B*d + 5*A*e))*(b*d + (2*c*d -
b*e)*x)*Sqrt[b*x + c*x^2])/(64*d^3*(c*d - b*e)^3*(d + e*x)^2) + ((B*d - A*e)*(b*
x + c*x^2)^(3/2))/(4*d*(c*d - b*e)*(d + e*x)^4) - ((5*A*e*(2*c*d - b*e) - B*d*(2
*c*d + 3*b*e))*(b*x + c*x^2)^(3/2))/(24*d^2*(c*d - b*e)^2*(d + e*x)^3) - (b^2*(1
6*A*c^2*d^2 - 8*b*c*d*(B*d + 2*A*e) + b^2*e*(3*B*d + 5*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])])/(128*d^(7/2)*(c*d -
b*e)^(7/2))

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Rubi in Sympy [A]  time = 165.015, size = 313, normalized size = 1.01 \[ \frac{b^{2} \left (5 A b^{2} e^{2} - 16 A b c d e + 16 A c^{2} d^{2} + 3 B b^{2} d e - 8 B b c d^{2}\right ) \operatorname{atan}{\left (\frac{- b d + x \left (b e - 2 c d\right )}{2 \sqrt{d} \sqrt{b e - c d} \sqrt{b x + c x^{2}}} \right )}}{128 d^{\frac{7}{2}} \left (b e - c d\right )^{\frac{7}{2}}} + \frac{\left (A e - B d\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{4 d \left (d + e x\right )^{4} \left (b e - c d\right )} + \frac{\left (b x + c x^{2}\right )^{\frac{3}{2}} \left (5 A b e^{2} - 10 A c d e + 3 B b d e + 2 B c d^{2}\right )}{24 d^{2} \left (d + e x\right )^{3} \left (b e - c d\right )^{2}} - \frac{\left (b d - x \left (b e - 2 c d\right )\right ) \sqrt{b x + c x^{2}} \left (5 A b^{2} e^{2} - 16 A b c d e + 16 A c^{2} d^{2} + 3 B b^{2} d e - 8 B b c d^{2}\right )}{64 d^{3} \left (d + e x\right )^{2} \left (b e - c d\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x)**(1/2)/(e*x+d)**5,x)

[Out]

b**2*(5*A*b**2*e**2 - 16*A*b*c*d*e + 16*A*c**2*d**2 + 3*B*b**2*d*e - 8*B*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)))
/(128*d**(7/2)*(b*e - c*d)**(7/2)) + (A*e - B*d)*(b*x + c*x**2)**(3/2)/(4*d*(d +
e*x)**4*(b*e - c*d)) + (b*x + c*x**2)**(3/2)*(5*A*b*e**2 - 10*A*c*d*e + 3*B*b*d
*e + 2*B*c*d**2)/(24*d**2*(d + e*x)**3*(b*e - c*d)**2) - (b*d - x*(b*e - 2*c*d))
*sqrt(b*x + c*x**2)*(5*A*b**2*e**2 - 16*A*b*c*d*e + 16*A*c**2*d**2 + 3*B*b**2*d*
e - 8*B*b*c*d**2)/(64*d**3*(d + e*x)**2*(b*e - c*d)**3)

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Mathematica [A]  time = 2.2541, size = 381, normalized size = 1.23 \[ \frac{\sqrt{x (b+c x)} \left (\frac{\sqrt{d} \sqrt{x} \left (2 d (d+e x)^2 (c d-b e) \left (A e \left (5 b^2 e^2-8 b c d e+8 c^2 d^2\right )+B d \left (3 b^2 e^2-16 b c d e+8 c^2 d^2\right )\right )+(d+e x)^3 \left (A e \left (-15 b^3 e^3+38 b^2 c d e^2-24 b c^2 d^2 e+16 c^3 d^3\right )+B d \left (-9 b^3 e^3+18 b^2 c d e^2-40 b c^2 d^2 e+16 c^3 d^3\right )\right )+48 d^3 (B d-A e) (c d-b e)^3-8 d^2 (d+e x) (c d-b e)^2 (A e (b e-2 c d)+B d (10 c d-9 b e))\right )}{e^2 (d+e x)^4}-\frac{3 b^2 \left (b^2 e (5 A e+3 B d)-8 b c d (2 A e+B d)+16 A c^2 d^2\right ) \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{\sqrt{b+c x} \sqrt{b e-c d}}\right )}{192 d^{7/2} \sqrt{x} (c d-b e)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*Sqrt[b*x + c*x^2])/(d + e*x)^5,x]

[Out]

(Sqrt[x*(b + c*x)]*((Sqrt[d]*Sqrt[x]*(48*d^3*(B*d - A*e)*(c*d - b*e)^3 - 8*d^2*(
c*d - b*e)^2*(B*d*(10*c*d - 9*b*e) + A*e*(-2*c*d + b*e))*(d + e*x) + 2*d*(c*d -
b*e)*(B*d*(8*c^2*d^2 - 16*b*c*d*e + 3*b^2*e^2) + A*e*(8*c^2*d^2 - 8*b*c*d*e + 5*
b^2*e^2))*(d + e*x)^2 + (A*e*(16*c^3*d^3 - 24*b*c^2*d^2*e + 38*b^2*c*d*e^2 - 15*
b^3*e^3) + B*d*(16*c^3*d^3 - 40*b*c^2*d^2*e + 18*b^2*c*d*e^2 - 9*b^3*e^3))*(d +
e*x)^3))/(e^2*(d + e*x)^4) - (3*b^2*(16*A*c^2*d^2 - 8*b*c*d*(B*d + 2*A*e) + b^2*
e*(3*B*d + 5*A*e))*ArcTan[(Sqrt[-(c*d) + b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])
/(Sqrt[-(c*d) + b*e]*Sqrt[b + c*x])))/(192*d^(7/2)*(c*d - b*e)^3*Sqrt[x])

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Maple [B]  time = 0.027, size = 10550, normalized size = 34. \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)^(1/2)/(e*x+d)^5,x)

[Out]

result too large to display

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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(sqrt(c*x^2 + b*x)*(B*x + A)/(e*x + d)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.316229, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x)*(B*x + A)/(e*x + d)^5,x, algorithm="fricas")

[Out]

[1/384*(2*(15*A*b^3*d^3*e^2 - 24*(B*b^2*c - 2*A*b*c^2)*d^5 + 3*(3*B*b^3 - 16*A*b
^2*c)*d^4*e + (16*B*c^3*d^4*e - 15*A*b^3*e^5 - 8*(5*B*b*c^2 - 2*A*c^3)*d^3*e^2 +
6*(3*B*b^2*c - 4*A*b*c^2)*d^2*e^3 - (9*B*b^3 - 38*A*b^2*c)*d*e^4)*x^3 + (64*B*c
^3*d^5 - 55*A*b^3*d*e^4 - 8*(21*B*b*c^2 - 8*A*c^3)*d^4*e + 4*(23*B*b^2*c - 26*A*
b*c^2)*d^3*e^2 - (33*B*b^3 - 140*A*b^2*c)*d^2*e^3)*x^2 - (73*A*b^3*d^2*e^3 - 16*
(B*b*c^2 + 6*A*c^3)*d^5 + 2*(47*B*b^2*c + 88*A*b*c^2)*d^4*e - 33*(B*b^3 + 6*A*b^
2*c)*d^3*e^2)*x)*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x) - 3*(5*A*b^4*d^4*e^2 - 8*
(B*b^3*c - 2*A*b^2*c^2)*d^6 + (3*B*b^4 - 16*A*b^3*c)*d^5*e + (5*A*b^4*e^6 - 8*(B
*b^3*c - 2*A*b^2*c^2)*d^2*e^4 + (3*B*b^4 - 16*A*b^3*c)*d*e^5)*x^4 + 4*(5*A*b^4*d
*e^5 - 8*(B*b^3*c - 2*A*b^2*c^2)*d^3*e^3 + (3*B*b^4 - 16*A*b^3*c)*d^2*e^4)*x^3 +
6*(5*A*b^4*d^2*e^4 - 8*(B*b^3*c - 2*A*b^2*c^2)*d^4*e^2 + (3*B*b^4 - 16*A*b^3*c)
*d^3*e^3)*x^2 + 4*(5*A*b^4*d^3*e^3 - 8*(B*b^3*c - 2*A*b^2*c^2)*d^5*e + (3*B*b^4
- 16*A*b^3*c)*d^4*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^3*d^10 - 3*b*c^2*d^9*e + 3*b^2
*c*d^8*e^2 - b^3*d^7*e^3 + (c^3*d^6*e^4 - 3*b*c^2*d^5*e^5 + 3*b^2*c*d^4*e^6 - b^
3*d^3*e^7)*x^4 + 4*(c^3*d^7*e^3 - 3*b*c^2*d^6*e^4 + 3*b^2*c*d^5*e^5 - b^3*d^4*e^
6)*x^3 + 6*(c^3*d^8*e^2 - 3*b*c^2*d^7*e^3 + 3*b^2*c*d^6*e^4 - b^3*d^5*e^5)*x^2 +
4*(c^3*d^9*e - 3*b*c^2*d^8*e^2 + 3*b^2*c*d^7*e^3 - b^3*d^6*e^4)*x)*sqrt(c*d^2 -
b*d*e)), 1/192*((15*A*b^3*d^3*e^2 - 24*(B*b^2*c - 2*A*b*c^2)*d^5 + 3*(3*B*b^3 -
16*A*b^2*c)*d^4*e + (16*B*c^3*d^4*e - 15*A*b^3*e^5 - 8*(5*B*b*c^2 - 2*A*c^3)*d^
3*e^2 + 6*(3*B*b^2*c - 4*A*b*c^2)*d^2*e^3 - (9*B*b^3 - 38*A*b^2*c)*d*e^4)*x^3 +
(64*B*c^3*d^5 - 55*A*b^3*d*e^4 - 8*(21*B*b*c^2 - 8*A*c^3)*d^4*e + 4*(23*B*b^2*c
- 26*A*b*c^2)*d^3*e^2 - (33*B*b^3 - 140*A*b^2*c)*d^2*e^3)*x^2 - (73*A*b^3*d^2*e^
3 - 16*(B*b*c^2 + 6*A*c^3)*d^5 + 2*(47*B*b^2*c + 88*A*b*c^2)*d^4*e - 33*(B*b^3 +
6*A*b^2*c)*d^3*e^2)*x)*sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x) + 3*(5*A*b^4*d^4*
e^2 - 8*(B*b^3*c - 2*A*b^2*c^2)*d^6 + (3*B*b^4 - 16*A*b^3*c)*d^5*e + (5*A*b^4*e^
6 - 8*(B*b^3*c - 2*A*b^2*c^2)*d^2*e^4 + (3*B*b^4 - 16*A*b^3*c)*d*e^5)*x^4 + 4*(5
*A*b^4*d*e^5 - 8*(B*b^3*c - 2*A*b^2*c^2)*d^3*e^3 + (3*B*b^4 - 16*A*b^3*c)*d^2*e^
4)*x^3 + 6*(5*A*b^4*d^2*e^4 - 8*(B*b^3*c - 2*A*b^2*c^2)*d^4*e^2 + (3*B*b^4 - 16*
A*b^3*c)*d^3*e^3)*x^2 + 4*(5*A*b^4*d^3*e^3 - 8*(B*b^3*c - 2*A*b^2*c^2)*d^5*e + (
3*B*b^4 - 16*A*b^3*c)*d^4*e^2)*x)*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)
/((c*d - b*e)*x)))/((c^3*d^10 - 3*b*c^2*d^9*e + 3*b^2*c*d^8*e^2 - b^3*d^7*e^3 +
(c^3*d^6*e^4 - 3*b*c^2*d^5*e^5 + 3*b^2*c*d^4*e^6 - b^3*d^3*e^7)*x^4 + 4*(c^3*d^7
*e^3 - 3*b*c^2*d^6*e^4 + 3*b^2*c*d^5*e^5 - b^3*d^4*e^6)*x^3 + 6*(c^3*d^8*e^2 - 3
*b*c^2*d^7*e^3 + 3*b^2*c*d^6*e^4 - b^3*d^5*e^5)*x^2 + 4*(c^3*d^9*e - 3*b*c^2*d^8
*e^2 + 3*b^2*c*d^7*e^3 - b^3*d^6*e^4)*x)*sqrt(-c*d^2 + b*d*e))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x \left (b + c x\right )} \left (A + B x\right )}{\left (d + e x\right )^{5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x)**(1/2)/(e*x+d)**5,x)

[Out]

Integral(sqrt(x*(b + c*x))*(A + B*x)/(d + e*x)**5, x)

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GIAC/XCAS [A]  time = 0.5113, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x)*(B*x + A)/(e*x + d)^5,x, algorithm="giac")

[Out]

Done