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

Optimal. Leaf size=119 \[ \frac{a \left (4 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{8 c^{3/2}}+\frac{x \sqrt{a+c x^2} \left (4 c d^2-a e^2\right )}{8 c}+\frac{5 d e \left (a+c x^2\right )^{3/2}}{12 c}+\frac{e \left (a+c x^2\right )^{3/2} (d+e x)}{4 c} \]

[Out]

((4*c*d^2 - a*e^2)*x*Sqrt[a + c*x^2])/(8*c) + (5*d*e*(a + c*x^2)^(3/2))/(12*c) +
(e*(d + e*x)*(a + c*x^2)^(3/2))/(4*c) + (a*(4*c*d^2 - a*e^2)*ArcTanh[(Sqrt[c]*x
)/Sqrt[a + c*x^2]])/(8*c^(3/2))

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Rubi [A]  time = 0.156887, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ \frac{a \left (4 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{8 c^{3/2}}+\frac{x \sqrt{a+c x^2} \left (4 c d^2-a e^2\right )}{8 c}+\frac{5 d e \left (a+c x^2\right )^{3/2}}{12 c}+\frac{e \left (a+c x^2\right )^{3/2} (d+e x)}{4 c} \]

Antiderivative was successfully verified.

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

[Out]

((4*c*d^2 - a*e^2)*x*Sqrt[a + c*x^2])/(8*c) + (5*d*e*(a + c*x^2)^(3/2))/(12*c) +
(e*(d + e*x)*(a + c*x^2)^(3/2))/(4*c) + (a*(4*c*d^2 - a*e^2)*ArcTanh[(Sqrt[c]*x
)/Sqrt[a + c*x^2]])/(8*c^(3/2))

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Rubi in Sympy [A]  time = 19.8301, size = 104, normalized size = 0.87 \[ - \frac{a \left (a e^{2} - 4 c d^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{8 c^{\frac{3}{2}}} + \frac{5 d e \left (a + c x^{2}\right )^{\frac{3}{2}}}{12 c} + \frac{e \left (a + c x^{2}\right )^{\frac{3}{2}} \left (d + e x\right )}{4 c} - \frac{x \sqrt{a + c x^{2}} \left (a e^{2} - 4 c d^{2}\right )}{8 c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*(a*e**2 - 4*c*d**2)*atanh(sqrt(c)*x/sqrt(a + c*x**2))/(8*c**(3/2)) + 5*d*e*(a
+ c*x**2)**(3/2)/(12*c) + e*(a + c*x**2)**(3/2)*(d + e*x)/(4*c) - x*sqrt(a + c*
x**2)*(a*e**2 - 4*c*d**2)/(8*c)

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Mathematica [A]  time = 0.105428, size = 99, normalized size = 0.83 \[ \frac{\sqrt{c} \sqrt{a+c x^2} \left (a e (16 d+3 e x)+2 c x \left (6 d^2+8 d e x+3 e^2 x^2\right )\right )-3 a \left (a e^2-4 c d^2\right ) \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{24 c^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[c]*Sqrt[a + c*x^2]*(a*e*(16*d + 3*e*x) + 2*c*x*(6*d^2 + 8*d*e*x + 3*e^2*x^
2)) - 3*a*(-4*c*d^2 + a*e^2)*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/(24*c^(3/2))

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Maple [A]  time = 0.01, size = 122, normalized size = 1. \[{\frac{{d}^{2}x}{2}\sqrt{c{x}^{2}+a}}+{\frac{a{d}^{2}}{2}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}}+{\frac{{e}^{2}x}{4\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{a{e}^{2}x}{8\,c}\sqrt{c{x}^{2}+a}}-{\frac{{a}^{2}{e}^{2}}{8}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{2\,de}{3\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*d^2*x*(c*x^2+a)^(1/2)+1/2*d^2*a/c^(1/2)*ln(c^(1/2)*x+(c*x^2+a)^(1/2))+1/4*e^
2*x*(c*x^2+a)^(3/2)/c-1/8*e^2*a/c*x*(c*x^2+a)^(1/2)-1/8*e^2*a^2/c^(3/2)*ln(c^(1/
2)*x+(c*x^2+a)^(1/2))+2/3*d*e*(c*x^2+a)^(3/2)/c

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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 + a)*(e*x + d)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.248456, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (6 \, c e^{2} x^{3} + 16 \, c d e x^{2} + 16 \, a d e + 3 \,{\left (4 \, c d^{2} + a e^{2}\right )} x\right )} \sqrt{c x^{2} + a} \sqrt{c} - 3 \,{\left (4 \, a c d^{2} - a^{2} e^{2}\right )} \log \left (2 \, \sqrt{c x^{2} + a} c x -{\left (2 \, c x^{2} + a\right )} \sqrt{c}\right )}{48 \, c^{\frac{3}{2}}}, \frac{{\left (6 \, c e^{2} x^{3} + 16 \, c d e x^{2} + 16 \, a d e + 3 \,{\left (4 \, c d^{2} + a e^{2}\right )} x\right )} \sqrt{c x^{2} + a} \sqrt{-c} + 3 \,{\left (4 \, a c d^{2} - a^{2} e^{2}\right )} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right )}{24 \, \sqrt{-c} c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/48*(2*(6*c*e^2*x^3 + 16*c*d*e*x^2 + 16*a*d*e + 3*(4*c*d^2 + a*e^2)*x)*sqrt(c*
x^2 + a)*sqrt(c) - 3*(4*a*c*d^2 - a^2*e^2)*log(2*sqrt(c*x^2 + a)*c*x - (2*c*x^2
+ a)*sqrt(c)))/c^(3/2), 1/24*((6*c*e^2*x^3 + 16*c*d*e*x^2 + 16*a*d*e + 3*(4*c*d^
2 + a*e^2)*x)*sqrt(c*x^2 + a)*sqrt(-c) + 3*(4*a*c*d^2 - a^2*e^2)*arctan(sqrt(-c)
*x/sqrt(c*x^2 + a)))/(sqrt(-c)*c)]

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Sympy [A]  time = 17.5045, size = 184, normalized size = 1.55 \[ \frac{a^{\frac{3}{2}} e^{2} x}{8 c \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{\sqrt{a} d^{2} x \sqrt{1 + \frac{c x^{2}}{a}}}{2} + \frac{3 \sqrt{a} e^{2} x^{3}}{8 \sqrt{1 + \frac{c x^{2}}{a}}} - \frac{a^{2} e^{2} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{8 c^{\frac{3}{2}}} + \frac{a d^{2} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{2 \sqrt{c}} + 2 d e \left (\begin{cases} \frac{\sqrt{a} x^{2}}{2} & \text{for}\: c = 0 \\\frac{\left (a + c x^{2}\right )^{\frac{3}{2}}}{3 c} & \text{otherwise} \end{cases}\right ) + \frac{c e^{2} x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{c x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(3/2)*e**2*x/(8*c*sqrt(1 + c*x**2/a)) + sqrt(a)*d**2*x*sqrt(1 + c*x**2/a)/2 +
3*sqrt(a)*e**2*x**3/(8*sqrt(1 + c*x**2/a)) - a**2*e**2*asinh(sqrt(c)*x/sqrt(a))
/(8*c**(3/2)) + a*d**2*asinh(sqrt(c)*x/sqrt(a))/(2*sqrt(c)) + 2*d*e*Piecewise((s
qrt(a)*x**2/2, Eq(c, 0)), ((a + c*x**2)**(3/2)/(3*c), True)) + c*e**2*x**5/(4*sq
rt(a)*sqrt(1 + c*x**2/a))

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GIAC/XCAS [A]  time = 0.216715, size = 130, normalized size = 1.09 \[ \frac{1}{24} \, \sqrt{c x^{2} + a}{\left ({\left (2 \,{\left (3 \, x e^{2} + 8 \, d e\right )} x + \frac{3 \,{\left (4 \, c^{2} d^{2} + a c e^{2}\right )}}{c^{2}}\right )} x + \frac{16 \, a d e}{c}\right )} - \frac{{\left (4 \, a c d^{2} - a^{2} e^{2}\right )}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{8 \, c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*sqrt(c*x^2 + a)*((2*(3*x*e^2 + 8*d*e)*x + 3*(4*c^2*d^2 + a*c*e^2)/c^2)*x +
16*a*d*e/c) - 1/8*(4*a*c*d^2 - a^2*e^2)*ln(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/c^
(3/2)