3.322 \(\int \frac{(d+e x)^3}{\left (b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=139 \[ \frac{e \sqrt{b x+c x^2} \left (3 b^2 e^2+2 c e x (2 c d-b e)-6 b c d e+8 c^2 d^2\right )}{b^2 c^2}-\frac{2 (d+e x)^2 (x (2 c d-b e)+b d)}{b^2 \sqrt{b x+c x^2}}+\frac{3 e^2 (2 c d-b e) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{c^{5/2}} \]

[Out]

(-2*(d + e*x)^2*(b*d + (2*c*d - b*e)*x))/(b^2*Sqrt[b*x + c*x^2]) + (e*(8*c^2*d^2
- 6*b*c*d*e + 3*b^2*e^2 + 2*c*e*(2*c*d - b*e)*x)*Sqrt[b*x + c*x^2])/(b^2*c^2) +
(3*e^2*(2*c*d - b*e)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/c^(5/2)

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Rubi [A]  time = 0.287087, antiderivative size = 139, 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 \[ \frac{e \sqrt{b x+c x^2} \left (3 b^2 e^2+2 c e x (2 c d-b e)-6 b c d e+8 c^2 d^2\right )}{b^2 c^2}-\frac{2 (d+e x)^2 (x (2 c d-b e)+b d)}{b^2 \sqrt{b x+c x^2}}+\frac{3 e^2 (2 c d-b e) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{c^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(d + e*x)^2*(b*d + (2*c*d - b*e)*x))/(b^2*Sqrt[b*x + c*x^2]) + (e*(8*c^2*d^2
- 6*b*c*d*e + 3*b^2*e^2 + 2*c*e*(2*c*d - b*e)*x)*Sqrt[b*x + c*x^2])/(b^2*c^2) +
(3*e^2*(2*c*d - b*e)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/c^(5/2)

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Rubi in Sympy [A]  time = 33.5513, size = 133, normalized size = 0.96 \[ - \frac{3 e^{2} \left (b e - 2 c d\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{c^{\frac{5}{2}}} - \frac{2 \left (d + e x\right )^{2} \left (b d - x \left (b e - 2 c d\right )\right )}{b^{2} \sqrt{b x + c x^{2}}} + \frac{e \sqrt{b x + c x^{2}} \left (3 b e \left (b e - 2 c d\right ) + 8 c^{2} d^{2} - 2 c e x \left (b e - 2 c d\right )\right )}{b^{2} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*e**2*(b*e - 2*c*d)*atanh(sqrt(c)*x/sqrt(b*x + c*x**2))/c**(5/2) - 2*(d + e*x)
**2*(b*d - x*(b*e - 2*c*d))/(b**2*sqrt(b*x + c*x**2)) + e*sqrt(b*x + c*x**2)*(3*
b*e*(b*e - 2*c*d) + 8*c**2*d**2 - 2*c*e*x*(b*e - 2*c*d))/(b**2*c**2)

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Mathematica [A]  time = 0.175713, size = 131, normalized size = 0.94 \[ \frac{\sqrt{c} \left (3 b^3 e^3 x+b^2 c e^2 x (e x-6 d)-2 b c^2 d^2 (d-3 e x)-4 c^3 d^3 x\right )-3 b^2 e^2 \sqrt{x} \sqrt{b+c x} (b e-2 c d) \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{b^2 c^{5/2} \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[c]*(-4*c^3*d^3*x + 3*b^3*e^3*x - 2*b*c^2*d^2*(d - 3*e*x) + b^2*c*e^2*x*(-6
*d + e*x)) - 3*b^2*e^2*(-2*c*d + b*e)*Sqrt[x]*Sqrt[b + c*x]*Log[c*Sqrt[x] + Sqrt
[c]*Sqrt[b + c*x]])/(b^2*c^(5/2)*Sqrt[x*(b + c*x)])

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Maple [A]  time = 0.013, size = 177, normalized size = 1.3 \[ -2\,{\frac{{d}^{3} \left ( 2\,cx+b \right ) }{{b}^{2}\sqrt{c{x}^{2}+bx}}}+{\frac{{e}^{3}{x}^{2}}{c}{\frac{1}{\sqrt{c{x}^{2}+bx}}}}+3\,{\frac{b{e}^{3}x}{{c}^{2}\sqrt{c{x}^{2}+bx}}}-{\frac{3\,b{e}^{3}}{2}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{5}{2}}}}-6\,{\frac{d{e}^{2}x}{c\sqrt{c{x}^{2}+bx}}}+3\,{\frac{d{e}^{2}}{{c}^{3/2}}\ln \left ({\frac{b/2+cx}{\sqrt{c}}}+\sqrt{c{x}^{2}+bx} \right ) }+6\,{\frac{{d}^{2}ex}{b\sqrt{c{x}^{2}+bx}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*d^3*(2*c*x+b)/b^2/(c*x^2+b*x)^(1/2)+e^3*x^2/c/(c*x^2+b*x)^(1/2)+3*e^3*b/c^2/(
c*x^2+b*x)^(1/2)*x-3/2*e^3*b/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))-6
*d*e^2/c/(c*x^2+b*x)^(1/2)*x+3*d*e^2/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^
(1/2))+6*d^2*e/b/(c*x^2+b*x)^(1/2)*x

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.235058, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (2 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} \sqrt{c x^{2} + b x} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} - 2 \, \sqrt{c x^{2} + b x} c\right ) - 2 \,{\left (b^{2} c e^{3} x^{2} - 2 \, b c^{2} d^{3} -{\left (4 \, c^{3} d^{3} - 6 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} - 3 \, b^{3} e^{3}\right )} x\right )} \sqrt{c}}{2 \, \sqrt{c x^{2} + b x} b^{2} c^{\frac{5}{2}}}, \frac{3 \,{\left (2 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} \sqrt{c x^{2} + b x} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) +{\left (b^{2} c e^{3} x^{2} - 2 \, b c^{2} d^{3} -{\left (4 \, c^{3} d^{3} - 6 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} - 3 \, b^{3} e^{3}\right )} x\right )} \sqrt{-c}}{\sqrt{c x^{2} + b x} b^{2} \sqrt{-c} c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/2*(3*(2*b^2*c*d*e^2 - b^3*e^3)*sqrt(c*x^2 + b*x)*log((2*c*x + b)*sqrt(c) - 2
*sqrt(c*x^2 + b*x)*c) - 2*(b^2*c*e^3*x^2 - 2*b*c^2*d^3 - (4*c^3*d^3 - 6*b*c^2*d^
2*e + 6*b^2*c*d*e^2 - 3*b^3*e^3)*x)*sqrt(c))/(sqrt(c*x^2 + b*x)*b^2*c^(5/2)), (3
*(2*b^2*c*d*e^2 - b^3*e^3)*sqrt(c*x^2 + b*x)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(
c*x)) + (b^2*c*e^3*x^2 - 2*b*c^2*d^3 - (4*c^3*d^3 - 6*b*c^2*d^2*e + 6*b^2*c*d*e^
2 - 3*b^3*e^3)*x)*sqrt(-c))/(sqrt(c*x^2 + b*x)*b^2*sqrt(-c)*c^2)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{3}}{\left (x \left (b + c x\right )\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((d + e*x)**3/(x*(b + c*x))**(3/2), x)

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GIAC/XCAS [A]  time = 0.236309, size = 169, normalized size = 1.22 \[ -\frac{\frac{2 \, d^{3}}{b} - x{\left (\frac{x e^{3}}{c} - \frac{4 \, c^{3} d^{3} - 6 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} - 3 \, b^{3} e^{3}}{b^{2} c^{2}}\right )}}{\sqrt{c x^{2} + b x}} - \frac{3 \,{\left (2 \, c d e^{2} - b e^{3}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{2 \, c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(2*d^3/b - x*(x*e^3/c - (4*c^3*d^3 - 6*b*c^2*d^2*e + 6*b^2*c*d*e^2 - 3*b^3*e^3)
/(b^2*c^2)))/sqrt(c*x^2 + b*x) - 3/2*(2*c*d*e^2 - b*e^3)*ln(abs(-2*(sqrt(c)*x -
sqrt(c*x^2 + b*x))*sqrt(c) - b))/c^(5/2)