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

Optimal. Leaf size=110 \[ \frac{d \left (2 c d^2-3 a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 c^{3/2}}+\frac{e \sqrt{a+c x^2} \left (4 \left (4 c d^2-a e^2\right )+5 c d e x\right )}{6 c^2}+\frac{e \sqrt{a+c x^2} (d+e x)^2}{3 c} \]

[Out]

(e*(d + e*x)^2*Sqrt[a + c*x^2])/(3*c) + (e*(4*(4*c*d^2 - a*e^2) + 5*c*d*e*x)*Sqr
t[a + c*x^2])/(6*c^2) + (d*(2*c*d^2 - 3*a*e^2)*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^
2]])/(2*c^(3/2))

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Rubi [A]  time = 0.221495, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{d \left (2 c d^2-3 a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 c^{3/2}}+\frac{e \sqrt{a+c x^2} \left (4 \left (4 c d^2-a e^2\right )+5 c d e x\right )}{6 c^2}+\frac{e \sqrt{a+c x^2} (d+e x)^2}{3 c} \]

Antiderivative was successfully verified.

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

[Out]

(e*(d + e*x)^2*Sqrt[a + c*x^2])/(3*c) + (e*(4*(4*c*d^2 - a*e^2) + 5*c*d*e*x)*Sqr
t[a + c*x^2])/(6*c^2) + (d*(2*c*d^2 - 3*a*e^2)*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^
2]])/(2*c^(3/2))

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Rubi in Sympy [A]  time = 25.2743, size = 100, normalized size = 0.91 \[ \frac{e \sqrt{a + c x^{2}} \left (d + e x\right )^{2}}{3 c} - \frac{e \sqrt{a + c x^{2}} \left (4 a e^{2} - 16 c d^{2} - 5 c d e x\right )}{6 c^{2}} - \frac{d \left (3 a e^{2} - 2 c d^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{2 c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e*sqrt(a + c*x**2)*(d + e*x)**2/(3*c) - e*sqrt(a + c*x**2)*(4*a*e**2 - 16*c*d**2
- 5*c*d*e*x)/(6*c**2) - d*(3*a*e**2 - 2*c*d**2)*atanh(sqrt(c)*x/sqrt(a + c*x**2
))/(2*c**(3/2))

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Mathematica [A]  time = 0.0995298, size = 92, normalized size = 0.84 \[ \frac{e \sqrt{a+c x^2} \left (c \left (18 d^2+9 d e x+2 e^2 x^2\right )-4 a e^2\right )+3 \sqrt{c} d \left (2 c d^2-3 a e^2\right ) \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{6 c^2} \]

Antiderivative was successfully verified.

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

[Out]

(e*Sqrt[a + c*x^2]*(-4*a*e^2 + c*(18*d^2 + 9*d*e*x + 2*e^2*x^2)) + 3*Sqrt[c]*d*(
2*c*d^2 - 3*a*e^2)*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/(6*c^2)

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Maple [A]  time = 0.011, size = 126, normalized size = 1.2 \[{{d}^{3}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}}+{\frac{{e}^{3}{x}^{2}}{3\,c}\sqrt{c{x}^{2}+a}}-{\frac{2\,a{e}^{3}}{3\,{c}^{2}}\sqrt{c{x}^{2}+a}}+{\frac{3\,d{e}^{2}x}{2\,c}\sqrt{c{x}^{2}+a}}-{\frac{3\,ad{e}^{2}}{2}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}+3\,{\frac{{d}^{2}e\sqrt{c{x}^{2}+a}}{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d^3*ln(c^(1/2)*x+(c*x^2+a)^(1/2))/c^(1/2)+1/3*e^3*x^2/c*(c*x^2+a)^(1/2)-2/3*e^3*
a/c^2*(c*x^2+a)^(1/2)+3/2*d*e^2*x/c*(c*x^2+a)^(1/2)-3/2*d*e^2*a/c^(3/2)*ln(c^(1/
2)*x+(c*x^2+a)^(1/2))+3*d^2*e/c*(c*x^2+a)^(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 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.234248, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (2 \, c e^{3} x^{2} + 9 \, c d e^{2} x + 18 \, c d^{2} e - 4 \, a e^{3}\right )} \sqrt{c x^{2} + a} \sqrt{c} - 3 \,{\left (2 \, c^{2} d^{3} - 3 \, a c d e^{2}\right )} \log \left (2 \, \sqrt{c x^{2} + a} c x -{\left (2 \, c x^{2} + a\right )} \sqrt{c}\right )}{12 \, c^{\frac{5}{2}}}, \frac{{\left (2 \, c e^{3} x^{2} + 9 \, c d e^{2} x + 18 \, c d^{2} e - 4 \, a e^{3}\right )} \sqrt{c x^{2} + a} \sqrt{-c} + 3 \,{\left (2 \, c^{2} d^{3} - 3 \, a c d e^{2}\right )} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right )}{6 \, \sqrt{-c} c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/12*(2*(2*c*e^3*x^2 + 9*c*d*e^2*x + 18*c*d^2*e - 4*a*e^3)*sqrt(c*x^2 + a)*sqrt
(c) - 3*(2*c^2*d^3 - 3*a*c*d*e^2)*log(2*sqrt(c*x^2 + a)*c*x - (2*c*x^2 + a)*sqrt
(c)))/c^(5/2), 1/6*((2*c*e^3*x^2 + 9*c*d*e^2*x + 18*c*d^2*e - 4*a*e^3)*sqrt(c*x^
2 + a)*sqrt(-c) + 3*(2*c^2*d^3 - 3*a*c*d*e^2)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a))
)/(sqrt(-c)*c^2)]

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Sympy [A]  time = 10.8619, size = 216, normalized size = 1.96 \[ \frac{3 \sqrt{a} d e^{2} x \sqrt{1 + \frac{c x^{2}}{a}}}{2 c} - \frac{3 a d e^{2} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{2 c^{\frac{3}{2}}} + d^{3} \left (\begin{cases} \frac{\sqrt{- \frac{a}{c}} \operatorname{asin}{\left (x \sqrt{- \frac{c}{a}} \right )}}{\sqrt{a}} & \text{for}\: a > 0 \wedge c < 0 \\\frac{\sqrt{\frac{a}{c}} \operatorname{asinh}{\left (x \sqrt{\frac{c}{a}} \right )}}{\sqrt{a}} & \text{for}\: a > 0 \wedge c > 0 \\\frac{\sqrt{- \frac{a}{c}} \operatorname{acosh}{\left (x \sqrt{- \frac{c}{a}} \right )}}{\sqrt{- a}} & \text{for}\: c > 0 \wedge a < 0 \end{cases}\right ) + 3 d^{2} e \left (\begin{cases} \frac{x^{2}}{2 \sqrt{a}} & \text{for}\: c = 0 \\\frac{\sqrt{a + c x^{2}}}{c} & \text{otherwise} \end{cases}\right ) + e^{3} \left (\begin{cases} - \frac{2 a \sqrt{a + c x^{2}}}{3 c^{2}} + \frac{x^{2} \sqrt{a + c x^{2}}}{3 c} & \text{for}\: c \neq 0 \\\frac{x^{4}}{4 \sqrt{a}} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*sqrt(a)*d*e**2*x*sqrt(1 + c*x**2/a)/(2*c) - 3*a*d*e**2*asinh(sqrt(c)*x/sqrt(a)
)/(2*c**(3/2)) + d**3*Piecewise((sqrt(-a/c)*asin(x*sqrt(-c/a))/sqrt(a), (a > 0)
& (c < 0)), (sqrt(a/c)*asinh(x*sqrt(c/a))/sqrt(a), (a > 0) & (c > 0)), (sqrt(-a/
c)*acosh(x*sqrt(-c/a))/sqrt(-a), (c > 0) & (a < 0))) + 3*d**2*e*Piecewise((x**2/
(2*sqrt(a)), Eq(c, 0)), (sqrt(a + c*x**2)/c, True)) + e**3*Piecewise((-2*a*sqrt(
a + c*x**2)/(3*c**2) + x**2*sqrt(a + c*x**2)/(3*c), Ne(c, 0)), (x**4/(4*sqrt(a))
, True))

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GIAC/XCAS [A]  time = 0.219587, size = 122, normalized size = 1.11 \[ \frac{1}{6} \, \sqrt{c x^{2} + a}{\left (x{\left (\frac{2 \, x e^{3}}{c} + \frac{9 \, d e^{2}}{c}\right )} + \frac{2 \,{\left (9 \, c^{2} d^{2} e - 2 \, a c e^{3}\right )}}{c^{3}}\right )} - \frac{{\left (2 \, c d^{3} - 3 \, a d e^{2}\right )}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{2 \, c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*sqrt(c*x^2 + a)*(x*(2*x*e^3/c + 9*d*e^2/c) + 2*(9*c^2*d^2*e - 2*a*c*e^3)/c^3
) - 1/2*(2*c*d^3 - 3*a*d*e^2)*ln(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/c^(3/2)