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

Optimal. Leaf size=144 \[ \frac{a d \left (4 c d^2-3 a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{8 c^{3/2}}+\frac{e \left (a+c x^2\right )^{3/2} \left (8 \left (6 c d^2-a e^2\right )+21 c d e x\right )}{60 c^2}+\frac{d x \sqrt{a+c x^2} \left (4 c d^2-3 a e^2\right )}{8 c}+\frac{e \left (a+c x^2\right )^{3/2} (d+e x)^2}{5 c} \]

[Out]

(d*(4*c*d^2 - 3*a*e^2)*x*Sqrt[a + c*x^2])/(8*c) + (e*(d + e*x)^2*(a + c*x^2)^(3/
2))/(5*c) + (e*(8*(6*c*d^2 - a*e^2) + 21*c*d*e*x)*(a + c*x^2)^(3/2))/(60*c^2) +
(a*d*(4*c*d^2 - 3*a*e^2)*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(8*c^(3/2))

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Rubi [A]  time = 0.285031, antiderivative size = 144, 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 \[ \frac{a d \left (4 c d^2-3 a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{8 c^{3/2}}+\frac{e \left (a+c x^2\right )^{3/2} \left (8 \left (6 c d^2-a e^2\right )+21 c d e x\right )}{60 c^2}+\frac{d x \sqrt{a+c x^2} \left (4 c d^2-3 a e^2\right )}{8 c}+\frac{e \left (a+c x^2\right )^{3/2} (d+e x)^2}{5 c} \]

Antiderivative was successfully verified.

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

[Out]

(d*(4*c*d^2 - 3*a*e^2)*x*Sqrt[a + c*x^2])/(8*c) + (e*(d + e*x)^2*(a + c*x^2)^(3/
2))/(5*c) + (e*(8*(6*c*d^2 - a*e^2) + 21*c*d*e*x)*(a + c*x^2)^(3/2))/(60*c^2) +
(a*d*(4*c*d^2 - 3*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 = 27.4492, size = 133, normalized size = 0.92 \[ - \frac{a d \left (3 a e^{2} - 4 c d^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{8 c^{\frac{3}{2}}} - \frac{d x \sqrt{a + c x^{2}} \left (3 a e^{2} - 4 c d^{2}\right )}{8 c} + \frac{e \left (a + c x^{2}\right )^{\frac{3}{2}} \left (d + e x\right )^{2}}{5 c} - \frac{e \left (a + c x^{2}\right )^{\frac{3}{2}} \left (8 a e^{2} - 48 c d^{2} - 21 c d e x\right )}{60 c^{2}} \]

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]

-a*d*(3*a*e**2 - 4*c*d**2)*atanh(sqrt(c)*x/sqrt(a + c*x**2))/(8*c**(3/2)) - d*x*
sqrt(a + c*x**2)*(3*a*e**2 - 4*c*d**2)/(8*c) + e*(a + c*x**2)**(3/2)*(d + e*x)**
2/(5*c) - e*(a + c*x**2)**(3/2)*(8*a*e**2 - 48*c*d**2 - 21*c*d*e*x)/(60*c**2)

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Mathematica [A]  time = 0.168252, size = 132, normalized size = 0.92 \[ \frac{\sqrt{a+c x^2} \left (-16 a^2 e^3+a c e \left (120 d^2+45 d e x+8 e^2 x^2\right )+6 c^2 x \left (10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right )\right )+15 a \sqrt{c} d \left (4 c d^2-3 a e^2\right ) \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{120 c^2} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + c*x^2]*(-16*a^2*e^3 + a*c*e*(120*d^2 + 45*d*e*x + 8*e^2*x^2) + 6*c^2*x
*(10*d^3 + 20*d^2*e*x + 15*d*e^2*x^2 + 4*e^3*x^3)) + 15*a*Sqrt[c]*d*(4*c*d^2 - 3
*a*e^2)*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/(120*c^2)

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Maple [A]  time = 0.01, size = 164, normalized size = 1.1 \[{\frac{{d}^{3}x}{2}\sqrt{c{x}^{2}+a}}+{\frac{{d}^{3}a}{2}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}}+{\frac{{e}^{3}{x}^{2}}{5\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{2\,{e}^{3}a}{15\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,d{e}^{2}x}{4\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{3\,ad{e}^{2}x}{8\,c}\sqrt{c{x}^{2}+a}}-{\frac{3\,d{e}^{2}{a}^{2}}{8}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{{d}^{2}e}{c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*d^3*x*(c*x^2+a)^(1/2)+1/2*d^3*a/c^(1/2)*ln(c^(1/2)*x+(c*x^2+a)^(1/2))+1/5*e^
3*x^2*(c*x^2+a)^(3/2)/c-2/15*e^3*a/c^2*(c*x^2+a)^(3/2)+3/4*d*e^2*x*(c*x^2+a)^(3/
2)/c-3/8*d*e^2*a/c*x*(c*x^2+a)^(1/2)-3/8*d*e^2*a^2/c^(3/2)*ln(c^(1/2)*x+(c*x^2+a
)^(1/2))+d^2*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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.247654, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (24 \, c^{2} e^{3} x^{4} + 90 \, c^{2} d e^{2} x^{3} + 120 \, a c d^{2} e - 16 \, a^{2} e^{3} + 8 \,{\left (15 \, c^{2} d^{2} e + a c e^{3}\right )} x^{2} + 15 \,{\left (4 \, c^{2} d^{3} + 3 \, a c d e^{2}\right )} x\right )} \sqrt{c x^{2} + a} \sqrt{c} - 15 \,{\left (4 \, a c^{2} d^{3} - 3 \, a^{2} c d e^{2}\right )} \log \left (2 \, \sqrt{c x^{2} + a} c x -{\left (2 \, c x^{2} + a\right )} \sqrt{c}\right )}{240 \, c^{\frac{5}{2}}}, \frac{{\left (24 \, c^{2} e^{3} x^{4} + 90 \, c^{2} d e^{2} x^{3} + 120 \, a c d^{2} e - 16 \, a^{2} e^{3} + 8 \,{\left (15 \, c^{2} d^{2} e + a c e^{3}\right )} x^{2} + 15 \,{\left (4 \, c^{2} d^{3} + 3 \, a c d e^{2}\right )} x\right )} \sqrt{c x^{2} + a} \sqrt{-c} + 15 \,{\left (4 \, a c^{2} d^{3} - 3 \, a^{2} c d e^{2}\right )} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right )}{120 \, \sqrt{-c} c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/240*(2*(24*c^2*e^3*x^4 + 90*c^2*d*e^2*x^3 + 120*a*c*d^2*e - 16*a^2*e^3 + 8*(1
5*c^2*d^2*e + a*c*e^3)*x^2 + 15*(4*c^2*d^3 + 3*a*c*d*e^2)*x)*sqrt(c*x^2 + a)*sqr
t(c) - 15*(4*a*c^2*d^3 - 3*a^2*c*d*e^2)*log(2*sqrt(c*x^2 + a)*c*x - (2*c*x^2 + a
)*sqrt(c)))/c^(5/2), 1/120*((24*c^2*e^3*x^4 + 90*c^2*d*e^2*x^3 + 120*a*c*d^2*e -
16*a^2*e^3 + 8*(15*c^2*d^2*e + a*c*e^3)*x^2 + 15*(4*c^2*d^3 + 3*a*c*d*e^2)*x)*s
qrt(c*x^2 + a)*sqrt(-c) + 15*(4*a*c^2*d^3 - 3*a^2*c*d*e^2)*arctan(sqrt(-c)*x/sqr
t(c*x^2 + a)))/(sqrt(-c)*c^2)]

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Sympy [A]  time = 18.4194, size = 265, normalized size = 1.84 \[ \frac{3 a^{\frac{3}{2}} d e^{2} x}{8 c \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{\sqrt{a} d^{3} x \sqrt{1 + \frac{c x^{2}}{a}}}{2} + \frac{9 \sqrt{a} d e^{2} x^{3}}{8 \sqrt{1 + \frac{c x^{2}}{a}}} - \frac{3 a^{2} d e^{2} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{8 c^{\frac{3}{2}}} + \frac{a d^{3} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{2 \sqrt{c}} + 3 d^{2} e \left (\begin{cases} \frac{\sqrt{a} x^{2}}{2} & \text{for}\: c = 0 \\\frac{\left (a + c x^{2}\right )^{\frac{3}{2}}}{3 c} & \text{otherwise} \end{cases}\right ) + e^{3} \left (\begin{cases} - \frac{2 a^{2} \sqrt{a + c x^{2}}}{15 c^{2}} + \frac{a x^{2} \sqrt{a + c x^{2}}}{15 c} + \frac{x^{4} \sqrt{a + c x^{2}}}{5} & \text{for}\: c \neq 0 \\\frac{\sqrt{a} x^{4}}{4} & \text{otherwise} \end{cases}\right ) + \frac{3 c d e^{2} x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{c x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*a**(3/2)*d*e**2*x/(8*c*sqrt(1 + c*x**2/a)) + sqrt(a)*d**3*x*sqrt(1 + c*x**2/a)
/2 + 9*sqrt(a)*d*e**2*x**3/(8*sqrt(1 + c*x**2/a)) - 3*a**2*d*e**2*asinh(sqrt(c)*
x/sqrt(a))/(8*c**(3/2)) + a*d**3*asinh(sqrt(c)*x/sqrt(a))/(2*sqrt(c)) + 3*d**2*e
*Piecewise((sqrt(a)*x**2/2, Eq(c, 0)), ((a + c*x**2)**(3/2)/(3*c), True)) + e**3
*Piecewise((-2*a**2*sqrt(a + c*x**2)/(15*c**2) + a*x**2*sqrt(a + c*x**2)/(15*c)
+ x**4*sqrt(a + c*x**2)/5, Ne(c, 0)), (sqrt(a)*x**4/4, True)) + 3*c*d*e**2*x**5/
(4*sqrt(a)*sqrt(1 + c*x**2/a))

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GIAC/XCAS [A]  time = 0.218175, size = 194, normalized size = 1.35 \[ \frac{1}{120} \, \sqrt{c x^{2} + a}{\left ({\left (2 \,{\left (3 \,{\left (4 \, x e^{3} + 15 \, d e^{2}\right )} x + \frac{4 \,{\left (15 \, c^{3} d^{2} e + a c^{2} e^{3}\right )}}{c^{3}}\right )} x + \frac{15 \,{\left (4 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )}}{c^{3}}\right )} x + \frac{8 \,{\left (15 \, a c^{2} d^{2} e - 2 \, a^{2} c e^{3}\right )}}{c^{3}}\right )} - \frac{{\left (4 \, a c d^{3} - 3 \, a^{2} d e^{2}\right )}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{8 \, c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/120*sqrt(c*x^2 + a)*((2*(3*(4*x*e^3 + 15*d*e^2)*x + 4*(15*c^3*d^2*e + a*c^2*e^
3)/c^3)*x + 15*(4*c^3*d^3 + 3*a*c^2*d*e^2)/c^3)*x + 8*(15*a*c^2*d^2*e - 2*a^2*c*
e^3)/c^3) - 1/8*(4*a*c*d^3 - 3*a^2*d*e^2)*ln(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/
c^(3/2)