3.315 \(\int \frac{(d+e x)^3}{\sqrt{b x+c x^2}} \, dx\)

Optimal. Leaf size=149 \[ \frac{(2 c d-b e) \left (5 b^2 e^2-8 b c d e+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{8 c^{7/2}}+\frac{e \sqrt{b x+c x^2} \left (15 b^2 e^2+10 c e x (2 c d-b e)-54 b c d e+64 c^2 d^2\right )}{24 c^3}+\frac{e \sqrt{b x+c x^2} (d+e x)^2}{3 c} \]

[Out]

(e*(d + e*x)^2*Sqrt[b*x + c*x^2])/(3*c) + (e*(64*c^2*d^2 - 54*b*c*d*e + 15*b^2*e
^2 + 10*c*e*(2*c*d - b*e)*x)*Sqrt[b*x + c*x^2])/(24*c^3) + ((2*c*d - b*e)*(8*c^2
*d^2 - 8*b*c*d*e + 5*b^2*e^2)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(8*c^(7/2)
)

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Rubi [A]  time = 0.387061, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ \frac{(2 c d-b e) \left (5 b^2 e^2-8 b c d e+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{8 c^{7/2}}+\frac{e \sqrt{b x+c x^2} \left (15 b^2 e^2+10 c e x (2 c d-b e)-54 b c d e+64 c^2 d^2\right )}{24 c^3}+\frac{e \sqrt{b x+c x^2} (d+e x)^2}{3 c} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^3/Sqrt[b*x + c*x^2],x]

[Out]

(e*(d + e*x)^2*Sqrt[b*x + c*x^2])/(3*c) + (e*(64*c^2*d^2 - 54*b*c*d*e + 15*b^2*e
^2 + 10*c*e*(2*c*d - b*e)*x)*Sqrt[b*x + c*x^2])/(24*c^3) + ((2*c*d - b*e)*(8*c^2
*d^2 - 8*b*c*d*e + 5*b^2*e^2)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(8*c^(7/2)
)

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Rubi in Sympy [A]  time = 33.2161, size = 148, normalized size = 0.99 \[ \frac{e \left (d + e x\right )^{2} \sqrt{b x + c x^{2}}}{3 c} + \frac{e \sqrt{b x + c x^{2}} \left (\frac{15 b^{2} e^{2}}{4} - \frac{27 b c d e}{2} + 16 c^{2} d^{2} - \frac{5 c e x \left (b e - 2 c d\right )}{2}\right )}{6 c^{3}} - \frac{\left (b e - 2 c d\right ) \left (5 b^{2} e^{2} - 8 b c d e + 8 c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{8 c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**3/(c*x**2+b*x)**(1/2),x)

[Out]

e*(d + e*x)**2*sqrt(b*x + c*x**2)/(3*c) + e*sqrt(b*x + c*x**2)*(15*b**2*e**2/4 -
27*b*c*d*e/2 + 16*c**2*d**2 - 5*c*e*x*(b*e - 2*c*d)/2)/(6*c**3) - (b*e - 2*c*d)
*(5*b**2*e**2 - 8*b*c*d*e + 8*c**2*d**2)*atanh(sqrt(c)*x/sqrt(b*x + c*x**2))/(8*
c**(7/2))

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Mathematica [A]  time = 0.187163, size = 157, normalized size = 1.05 \[ \frac{\sqrt{c} e x (b+c x) \left (15 b^2 e^2-2 b c e (27 d+5 e x)+4 c^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )+3 \sqrt{x} \sqrt{b+c x} \left (-5 b^3 e^3+18 b^2 c d e^2-24 b c^2 d^2 e+16 c^3 d^3\right ) \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{24 c^{7/2} \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^3/Sqrt[b*x + c*x^2],x]

[Out]

(Sqrt[c]*e*x*(b + c*x)*(15*b^2*e^2 - 2*b*c*e*(27*d + 5*e*x) + 4*c^2*(18*d^2 + 9*
d*e*x + 2*e^2*x^2)) + 3*(16*c^3*d^3 - 24*b*c^2*d^2*e + 18*b^2*c*d*e^2 - 5*b^3*e^
3)*Sqrt[x]*Sqrt[b + c*x]*Log[c*Sqrt[x] + Sqrt[c]*Sqrt[b + c*x]])/(24*c^(7/2)*Sqr
t[x*(b + c*x)])

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Maple [A]  time = 0.013, size = 265, normalized size = 1.8 \[{{d}^{3}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){\frac{1}{\sqrt{c}}}}+{\frac{{e}^{3}{x}^{2}}{3\,c}\sqrt{c{x}^{2}+bx}}-{\frac{5\,b{e}^{3}x}{12\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{5\,{e}^{3}{b}^{2}}{8\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{5\,{b}^{3}{e}^{3}}{16}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{7}{2}}}}+{\frac{3\,d{e}^{2}x}{2\,c}\sqrt{c{x}^{2}+bx}}-{\frac{9\,bd{e}^{2}}{4\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{9\,{b}^{2}d{e}^{2}}{8}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{5}{2}}}}+3\,{\frac{{d}^{2}e\sqrt{c{x}^{2}+bx}}{c}}-{\frac{3\,b{d}^{2}e}{2}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^3/(c*x^2+b*x)^(1/2),x)

[Out]

d^3*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))/c^(1/2)+1/3*e^3*x^2/c*(c*x^2+b*x)^
(1/2)-5/12*e^3*b/c^2*x*(c*x^2+b*x)^(1/2)+5/8*e^3*b^2/c^3*(c*x^2+b*x)^(1/2)-5/16*
e^3*b^3/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))+3/2*d*e^2*x/c*(c*x^2+b
*x)^(1/2)-9/4*d*e^2*b/c^2*(c*x^2+b*x)^(1/2)+9/8*d*e^2*b^2/c^(5/2)*ln((1/2*b+c*x)
/c^(1/2)+(c*x^2+b*x)^(1/2))+3*d^2*e/c*(c*x^2+b*x)^(1/2)-3/2*d^2*e*b/c^(3/2)*ln((
1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))

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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((e*x + d)^3/sqrt(c*x^2 + b*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.254291, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (8 \, c^{2} e^{3} x^{2} + 72 \, c^{2} d^{2} e - 54 \, b c d e^{2} + 15 \, b^{2} e^{3} + 2 \,{\left (18 \, c^{2} d e^{2} - 5 \, b c e^{3}\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{c} - 3 \,{\left (16 \, c^{3} d^{3} - 24 \, b c^{2} d^{2} e + 18 \, b^{2} c d e^{2} - 5 \, b^{3} e^{3}\right )} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} - 2 \, \sqrt{c x^{2} + b x} c\right )}{48 \, c^{\frac{7}{2}}}, \frac{{\left (8 \, c^{2} e^{3} x^{2} + 72 \, c^{2} d^{2} e - 54 \, b c d e^{2} + 15 \, b^{2} e^{3} + 2 \,{\left (18 \, c^{2} d e^{2} - 5 \, b c e^{3}\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{-c} + 3 \,{\left (16 \, c^{3} d^{3} - 24 \, b c^{2} d^{2} e + 18 \, b^{2} c d e^{2} - 5 \, b^{3} e^{3}\right )} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right )}{24 \, \sqrt{-c} c^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3/sqrt(c*x^2 + b*x),x, algorithm="fricas")

[Out]

[1/48*(2*(8*c^2*e^3*x^2 + 72*c^2*d^2*e - 54*b*c*d*e^2 + 15*b^2*e^3 + 2*(18*c^2*d
*e^2 - 5*b*c*e^3)*x)*sqrt(c*x^2 + b*x)*sqrt(c) - 3*(16*c^3*d^3 - 24*b*c^2*d^2*e
+ 18*b^2*c*d*e^2 - 5*b^3*e^3)*log((2*c*x + b)*sqrt(c) - 2*sqrt(c*x^2 + b*x)*c))/
c^(7/2), 1/24*((8*c^2*e^3*x^2 + 72*c^2*d^2*e - 54*b*c*d*e^2 + 15*b^2*e^3 + 2*(18
*c^2*d*e^2 - 5*b*c*e^3)*x)*sqrt(c*x^2 + b*x)*sqrt(-c) + 3*(16*c^3*d^3 - 24*b*c^2
*d^2*e + 18*b^2*c*d*e^2 - 5*b^3*e^3)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)))/(
sqrt(-c)*c^3)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{3}}{\sqrt{x \left (b + c x\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**3/(c*x**2+b*x)**(1/2),x)

[Out]

Integral((d + e*x)**3/sqrt(x*(b + c*x)), x)

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GIAC/XCAS [A]  time = 0.236069, size = 198, normalized size = 1.33 \[ \frac{1}{24} \, \sqrt{c x^{2} + b x}{\left (2 \, x{\left (\frac{4 \, x e^{3}}{c} + \frac{18 \, c^{2} d e^{2} - 5 \, b c e^{3}}{c^{3}}\right )} + \frac{3 \,{\left (24 \, c^{2} d^{2} e - 18 \, b c d e^{2} + 5 \, b^{2} e^{3}\right )}}{c^{3}}\right )} - \frac{{\left (16 \, c^{3} d^{3} - 24 \, b c^{2} d^{2} e + 18 \, b^{2} c d e^{2} - 5 \, b^{3} e^{3}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{16 \, c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3/sqrt(c*x^2 + b*x),x, algorithm="giac")

[Out]

1/24*sqrt(c*x^2 + b*x)*(2*x*(4*x*e^3/c + (18*c^2*d*e^2 - 5*b*c*e^3)/c^3) + 3*(24
*c^2*d^2*e - 18*b*c*d*e^2 + 5*b^2*e^3)/c^3) - 1/16*(16*c^3*d^3 - 24*b*c^2*d^2*e
+ 18*b^2*c*d*e^2 - 5*b^3*e^3)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c)
- b))/c^(7/2)