3.294 \(\int \frac{\sqrt{b x+c x^2}}{(d+e x)^6} \, dx\)

Optimal. Leaf size=337 \[ -\frac{b^2 (2 c d-b e) \left (7 b^2 e^2-16 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{256 d^{9/2} (c d-b e)^{9/2}}+\frac{\sqrt{b x+c x^2} (2 c d-b e) \left (7 b^2 e^2-16 b c d e+16 c^2 d^2\right ) (x (2 c d-b e)+b d)}{128 d^4 (d+e x)^2 (c d-b e)^4}-\frac{e \left (b x+c x^2\right )^{3/2} \left (35 b^2 e^2-108 b c d e+108 c^2 d^2\right )}{240 d^3 (d+e x)^3 (c d-b e)^3}-\frac{7 e \left (b x+c x^2\right )^{3/2} (2 c d-b e)}{40 d^2 (d+e x)^4 (c d-b e)^2}-\frac{e \left (b x+c x^2\right )^{3/2}}{5 d (d+e x)^5 (c d-b e)} \]

[Out]

((2*c*d - b*e)*(16*c^2*d^2 - 16*b*c*d*e + 7*b^2*e^2)*(b*d + (2*c*d - b*e)*x)*Sqr
t[b*x + c*x^2])/(128*d^4*(c*d - b*e)^4*(d + e*x)^2) - (e*(b*x + c*x^2)^(3/2))/(5
*d*(c*d - b*e)*(d + e*x)^5) - (7*e*(2*c*d - b*e)*(b*x + c*x^2)^(3/2))/(40*d^2*(c
*d - b*e)^2*(d + e*x)^4) - (e*(108*c^2*d^2 - 108*b*c*d*e + 35*b^2*e^2)*(b*x + c*
x^2)^(3/2))/(240*d^3*(c*d - b*e)^3*(d + e*x)^3) - (b^2*(2*c*d - b*e)*(16*c^2*d^2
- 16*b*c*d*e + 7*b^2*e^2)*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d -
b*e]*Sqrt[b*x + c*x^2])])/(256*d^(9/2)*(c*d - b*e)^(9/2))

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Rubi [A]  time = 1.20334, antiderivative size = 337, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ -\frac{b^2 (2 c d-b e) \left (7 b^2 e^2-16 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{256 d^{9/2} (c d-b e)^{9/2}}+\frac{\sqrt{b x+c x^2} (2 c d-b e) \left (7 b^2 e^2-16 b c d e+16 c^2 d^2\right ) (x (2 c d-b e)+b d)}{128 d^4 (d+e x)^2 (c d-b e)^4}-\frac{e \left (b x+c x^2\right )^{3/2} \left (35 b^2 e^2-108 b c d e+108 c^2 d^2\right )}{240 d^3 (d+e x)^3 (c d-b e)^3}-\frac{7 e \left (b x+c x^2\right )^{3/2} (2 c d-b e)}{40 d^2 (d+e x)^4 (c d-b e)^2}-\frac{e \left (b x+c x^2\right )^{3/2}}{5 d (d+e x)^5 (c d-b e)} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[b*x + c*x^2]/(d + e*x)^6,x]

[Out]

((2*c*d - b*e)*(16*c^2*d^2 - 16*b*c*d*e + 7*b^2*e^2)*(b*d + (2*c*d - b*e)*x)*Sqr
t[b*x + c*x^2])/(128*d^4*(c*d - b*e)^4*(d + e*x)^2) - (e*(b*x + c*x^2)^(3/2))/(5
*d*(c*d - b*e)*(d + e*x)^5) - (7*e*(2*c*d - b*e)*(b*x + c*x^2)^(3/2))/(40*d^2*(c
*d - b*e)^2*(d + e*x)^4) - (e*(108*c^2*d^2 - 108*b*c*d*e + 35*b^2*e^2)*(b*x + c*
x^2)^(3/2))/(240*d^3*(c*d - b*e)^3*(d + e*x)^3) - (b^2*(2*c*d - b*e)*(16*c^2*d^2
- 16*b*c*d*e + 7*b^2*e^2)*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d -
b*e]*Sqrt[b*x + c*x^2])])/(256*d^(9/2)*(c*d - b*e)^(9/2))

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Rubi in Sympy [A]  time = 148.961, size = 311, normalized size = 0.92 \[ \frac{b^{2} \left (b e - 2 c d\right ) \left (7 b^{2} e^{2} - 16 b c d e + 16 c^{2} d^{2}\right ) \operatorname{atan}{\left (\frac{- b d + x \left (b e - 2 c d\right )}{2 \sqrt{d} \sqrt{b e - c d} \sqrt{b x + c x^{2}}} \right )}}{256 d^{\frac{9}{2}} \left (b e - c d\right )^{\frac{9}{2}}} + \frac{e \left (b x + c x^{2}\right )^{\frac{3}{2}}}{5 d \left (d + e x\right )^{5} \left (b e - c d\right )} + \frac{7 e \left (b e - 2 c d\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{40 d^{2} \left (d + e x\right )^{4} \left (b e - c d\right )^{2}} + \frac{e \left (b x + c x^{2}\right )^{\frac{3}{2}} \left (35 b^{2} e^{2} - 108 b c d e + 108 c^{2} d^{2}\right )}{240 d^{3} \left (d + e x\right )^{3} \left (b e - c d\right )^{3}} - \frac{\left (b d - x \left (b e - 2 c d\right )\right ) \left (b e - 2 c d\right ) \sqrt{b x + c x^{2}} \left (7 b^{2} e^{2} - 16 b c d e + 16 c^{2} d^{2}\right )}{128 d^{4} \left (d + e x\right )^{2} \left (b e - c d\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x)**(1/2)/(e*x+d)**6,x)

[Out]

b**2*(b*e - 2*c*d)*(7*b**2*e**2 - 16*b*c*d*e + 16*c**2*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)))/(256*d**(9/2)*(b*e -
c*d)**(9/2)) + e*(b*x + c*x**2)**(3/2)/(5*d*(d + e*x)**5*(b*e - c*d)) + 7*e*(b*e
- 2*c*d)*(b*x + c*x**2)**(3/2)/(40*d**2*(d + e*x)**4*(b*e - c*d)**2) + e*(b*x +
c*x**2)**(3/2)*(35*b**2*e**2 - 108*b*c*d*e + 108*c**2*d**2)/(240*d**3*(d + e*x)
**3*(b*e - c*d)**3) - (b*d - x*(b*e - 2*c*d))*(b*e - 2*c*d)*sqrt(b*x + c*x**2)*(
7*b**2*e**2 - 16*b*c*d*e + 16*c**2*d**2)/(128*d**4*(d + e*x)**2*(b*e - c*d)**4)

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Mathematica [A]  time = 1.64295, size = 353, normalized size = 1.05 \[ \frac{\sqrt{x (b+c x)} \left (\frac{15 b^2 (b e-2 c d) \left (7 b^2 e^2-16 b c d e+16 c^2 d^2\right ) \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{\sqrt{b+c x} \sqrt{b e-c d}}-\frac{\sqrt{d} \sqrt{x} \left (-8 d^2 (d+e x)^2 \left (7 b^2 e^2-12 b c d e+12 c^2 d^2\right ) (c d-b e)^2-2 d (d+e x)^3 \left (-35 b^3 e^3+94 b^2 c d e^2-72 b c^2 d^2 e+48 c^3 d^3\right ) (c d-b e)-(d+e x)^4 \left (105 b^4 e^4-380 b^3 c d e^3+476 b^2 c^2 d^2 e^2-192 b c^3 d^3 e+96 c^4 d^4\right )+384 d^4 (c d-b e)^4-48 d^3 (d+e x) (2 c d-b e) (c d-b e)^3\right )}{e (d+e x)^5}\right )}{1920 d^{9/2} \sqrt{x} (c d-b e)^4} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[b*x + c*x^2]/(d + e*x)^6,x]

[Out]

(Sqrt[x*(b + c*x)]*(-((Sqrt[d]*Sqrt[x]*(384*d^4*(c*d - b*e)^4 - 48*d^3*(c*d - b*
e)^3*(2*c*d - b*e)*(d + e*x) - 8*d^2*(c*d - b*e)^2*(12*c^2*d^2 - 12*b*c*d*e + 7*
b^2*e^2)*(d + e*x)^2 - 2*d*(c*d - b*e)*(48*c^3*d^3 - 72*b*c^2*d^2*e + 94*b^2*c*d
*e^2 - 35*b^3*e^3)*(d + e*x)^3 - (96*c^4*d^4 - 192*b*c^3*d^3*e + 476*b^2*c^2*d^2
*e^2 - 380*b^3*c*d*e^3 + 105*b^4*e^4)*(d + e*x)^4))/(e*(d + e*x)^5)) + (15*b^2*(
-2*c*d + b*e)*(16*c^2*d^2 - 16*b*c*d*e + 7*b^2*e^2)*ArcTan[(Sqrt[-(c*d) + b*e]*S
qrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/(Sqrt[-(c*d) + b*e]*Sqrt[b + c*x])))/(1920*d^(
9/2)*(c*d - b*e)^4*Sqrt[x])

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Maple [B]  time = 0.025, size = 6533, normalized size = 19.4 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^(1/2)/(e*x+d)^6,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)/(e*x + d)^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.252283, 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)/(e*x + d)^6,x, algorithm="fricas")

[Out]

[1/3840*(2*(480*b*c^3*d^7 - 720*b^2*c^2*d^6*e + 450*b^3*c*d^5*e^2 - 105*b^4*d^4*
e^3 + (96*c^4*d^4*e^3 - 192*b*c^3*d^3*e^4 + 476*b^2*c^2*d^2*e^5 - 380*b^3*c*d*e^
6 + 105*b^4*e^7)*x^4 + 2*(240*c^4*d^5*e^2 - 504*b*c^3*d^4*e^3 + 1118*b^2*c^2*d^3
*e^4 - 889*b^3*c*d^2*e^5 + 245*b^4*d*e^6)*x^3 + 2*(480*c^4*d^6*e - 1080*b*c^3*d^
5*e^2 + 2098*b^2*c^2*d^4*e^3 - 1631*b^3*c*d^3*e^4 + 448*b^4*d^2*e^5)*x^2 + 10*(9
6*c^4*d^7 - 240*b*c^3*d^6*e + 402*b^2*c^2*d^5*e^2 - 295*b^3*c*d^4*e^3 + 79*b^4*d
^3*e^4)*x)*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x) - 15*(32*b^2*c^3*d^8 - 48*b^3*c
^2*d^7*e + 30*b^4*c*d^6*e^2 - 7*b^5*d^5*e^3 + (32*b^2*c^3*d^3*e^5 - 48*b^3*c^2*d
^2*e^6 + 30*b^4*c*d*e^7 - 7*b^5*e^8)*x^5 + 5*(32*b^2*c^3*d^4*e^4 - 48*b^3*c^2*d^
3*e^5 + 30*b^4*c*d^2*e^6 - 7*b^5*d*e^7)*x^4 + 10*(32*b^2*c^3*d^5*e^3 - 48*b^3*c^
2*d^4*e^4 + 30*b^4*c*d^3*e^5 - 7*b^5*d^2*e^6)*x^3 + 10*(32*b^2*c^3*d^6*e^2 - 48*
b^3*c^2*d^5*e^3 + 30*b^4*c*d^4*e^4 - 7*b^5*d^3*e^5)*x^2 + 5*(32*b^2*c^3*d^7*e -
48*b^3*c^2*d^6*e^2 + 30*b^4*c*d^5*e^3 - 7*b^5*d^4*e^4)*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^4*d^13 - 4*b*c^3*d^12*e + 6*b^2*c^2*d^11*e^2 - 4*b^3*c*d^10*e^3 + b^4*d^9*e^4
+ (c^4*d^8*e^5 - 4*b*c^3*d^7*e^6 + 6*b^2*c^2*d^6*e^7 - 4*b^3*c*d^5*e^8 + b^4*d^4
*e^9)*x^5 + 5*(c^4*d^9*e^4 - 4*b*c^3*d^8*e^5 + 6*b^2*c^2*d^7*e^6 - 4*b^3*c*d^6*e
^7 + b^4*d^5*e^8)*x^4 + 10*(c^4*d^10*e^3 - 4*b*c^3*d^9*e^4 + 6*b^2*c^2*d^8*e^5 -
4*b^3*c*d^7*e^6 + b^4*d^6*e^7)*x^3 + 10*(c^4*d^11*e^2 - 4*b*c^3*d^10*e^3 + 6*b^
2*c^2*d^9*e^4 - 4*b^3*c*d^8*e^5 + b^4*d^7*e^6)*x^2 + 5*(c^4*d^12*e - 4*b*c^3*d^1
1*e^2 + 6*b^2*c^2*d^10*e^3 - 4*b^3*c*d^9*e^4 + b^4*d^8*e^5)*x)*sqrt(c*d^2 - b*d*
e)), 1/1920*((480*b*c^3*d^7 - 720*b^2*c^2*d^6*e + 450*b^3*c*d^5*e^2 - 105*b^4*d^
4*e^3 + (96*c^4*d^4*e^3 - 192*b*c^3*d^3*e^4 + 476*b^2*c^2*d^2*e^5 - 380*b^3*c*d*
e^6 + 105*b^4*e^7)*x^4 + 2*(240*c^4*d^5*e^2 - 504*b*c^3*d^4*e^3 + 1118*b^2*c^2*d
^3*e^4 - 889*b^3*c*d^2*e^5 + 245*b^4*d*e^6)*x^3 + 2*(480*c^4*d^6*e - 1080*b*c^3*
d^5*e^2 + 2098*b^2*c^2*d^4*e^3 - 1631*b^3*c*d^3*e^4 + 448*b^4*d^2*e^5)*x^2 + 10*
(96*c^4*d^7 - 240*b*c^3*d^6*e + 402*b^2*c^2*d^5*e^2 - 295*b^3*c*d^4*e^3 + 79*b^4
*d^3*e^4)*x)*sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x) + 15*(32*b^2*c^3*d^8 - 48*b^
3*c^2*d^7*e + 30*b^4*c*d^6*e^2 - 7*b^5*d^5*e^3 + (32*b^2*c^3*d^3*e^5 - 48*b^3*c^
2*d^2*e^6 + 30*b^4*c*d*e^7 - 7*b^5*e^8)*x^5 + 5*(32*b^2*c^3*d^4*e^4 - 48*b^3*c^2
*d^3*e^5 + 30*b^4*c*d^2*e^6 - 7*b^5*d*e^7)*x^4 + 10*(32*b^2*c^3*d^5*e^3 - 48*b^3
*c^2*d^4*e^4 + 30*b^4*c*d^3*e^5 - 7*b^5*d^2*e^6)*x^3 + 10*(32*b^2*c^3*d^6*e^2 -
48*b^3*c^2*d^5*e^3 + 30*b^4*c*d^4*e^4 - 7*b^5*d^3*e^5)*x^2 + 5*(32*b^2*c^3*d^7*e
- 48*b^3*c^2*d^6*e^2 + 30*b^4*c*d^5*e^3 - 7*b^5*d^4*e^4)*x)*arctan(-sqrt(-c*d^2
+ b*d*e)*sqrt(c*x^2 + b*x)/((c*d - b*e)*x)))/((c^4*d^13 - 4*b*c^3*d^12*e + 6*b^
2*c^2*d^11*e^2 - 4*b^3*c*d^10*e^3 + b^4*d^9*e^4 + (c^4*d^8*e^5 - 4*b*c^3*d^7*e^6
+ 6*b^2*c^2*d^6*e^7 - 4*b^3*c*d^5*e^8 + b^4*d^4*e^9)*x^5 + 5*(c^4*d^9*e^4 - 4*b
*c^3*d^8*e^5 + 6*b^2*c^2*d^7*e^6 - 4*b^3*c*d^6*e^7 + b^4*d^5*e^8)*x^4 + 10*(c^4*
d^10*e^3 - 4*b*c^3*d^9*e^4 + 6*b^2*c^2*d^8*e^5 - 4*b^3*c*d^7*e^6 + b^4*d^6*e^7)*
x^3 + 10*(c^4*d^11*e^2 - 4*b*c^3*d^10*e^3 + 6*b^2*c^2*d^9*e^4 - 4*b^3*c*d^8*e^5
+ b^4*d^7*e^6)*x^2 + 5*(c^4*d^12*e - 4*b*c^3*d^11*e^2 + 6*b^2*c^2*d^10*e^3 - 4*b
^3*c*d^9*e^4 + b^4*d^8*e^5)*x)*sqrt(-c*d^2 + b*d*e))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x \left (b + c x\right )}}{\left (d + e x\right )^{6}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x)**(1/2)/(e*x+d)**6,x)

[Out]

Integral(sqrt(x*(b + c*x))/(d + e*x)**6, x)

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GIAC/XCAS [A]  time = 0.593716, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x)/(e*x + d)^6,x, algorithm="giac")

[Out]

sage0*x