3.526 \(\int (d+e x) \left (a+c x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=87 \[ \frac{3 a^2 d \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{8 \sqrt{c}}+\frac{1}{4} d x \left (a+c x^2\right )^{3/2}+\frac{3}{8} a d x \sqrt{a+c x^2}+\frac{e \left (a+c x^2\right )^{5/2}}{5 c} \]

[Out]

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

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Rubi [A]  time = 0.0796319, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ \frac{3 a^2 d \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{8 \sqrt{c}}+\frac{1}{4} d x \left (a+c x^2\right )^{3/2}+\frac{3}{8} a d x \sqrt{a+c x^2}+\frac{e \left (a+c x^2\right )^{5/2}}{5 c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.30316, size = 80, normalized size = 0.92 \[ \frac{3 a^{2} d \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{8 \sqrt{c}} + \frac{3 a d x \sqrt{a + c x^{2}}}{8} + \frac{d x \left (a + c x^{2}\right )^{\frac{3}{2}}}{4} + \frac{e \left (a + c x^{2}\right )^{\frac{5}{2}}}{5 c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.11558, size = 88, normalized size = 1.01 \[ \frac{\sqrt{a+c x^2} \left (8 a^2 e+a c x (25 d+16 e x)+2 c^2 x^3 (5 d+4 e x)\right )+15 a^2 \sqrt{c} d \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{40 c} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + c*x^2]*(8*a^2*e + 2*c^2*x^3*(5*d + 4*e*x) + a*c*x*(25*d + 16*e*x)) + 1
5*a^2*Sqrt[c]*d*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/(40*c)

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Maple [A]  time = 0.006, size = 69, normalized size = 0.8 \[{\frac{dx}{4} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,adx}{8}\sqrt{c{x}^{2}+a}}+{\frac{3\,{a}^{2}d}{8}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}}+{\frac{e}{5\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)*(c*x^2+a)^(3/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.249253, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{2} c d \log \left (-2 \, \sqrt{c x^{2} + a} c x -{\left (2 \, c x^{2} + a\right )} \sqrt{c}\right ) + 2 \,{\left (8 \, c^{2} e x^{4} + 10 \, c^{2} d x^{3} + 16 \, a c e x^{2} + 25 \, a c d x + 8 \, a^{2} e\right )} \sqrt{c x^{2} + a} \sqrt{c}}{80 \, c^{\frac{3}{2}}}, \frac{15 \, a^{2} c d \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) +{\left (8 \, c^{2} e x^{4} + 10 \, c^{2} d x^{3} + 16 \, a c e x^{2} + 25 \, a c d x + 8 \, a^{2} e\right )} \sqrt{c x^{2} + a} \sqrt{-c}}{40 \, \sqrt{-c} c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/80*(15*a^2*c*d*log(-2*sqrt(c*x^2 + a)*c*x - (2*c*x^2 + a)*sqrt(c)) + 2*(8*c^2
*e*x^4 + 10*c^2*d*x^3 + 16*a*c*e*x^2 + 25*a*c*d*x + 8*a^2*e)*sqrt(c*x^2 + a)*sqr
t(c))/c^(3/2), 1/40*(15*a^2*c*d*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) + (8*c^2*e*x^
4 + 10*c^2*d*x^3 + 16*a*c*e*x^2 + 25*a*c*d*x + 8*a^2*e)*sqrt(c*x^2 + a)*sqrt(-c)
)/(sqrt(-c)*c)]

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Sympy [A]  time = 18.9216, size = 219, normalized size = 2.52 \[ \frac{a^{\frac{3}{2}} d x \sqrt{1 + \frac{c x^{2}}{a}}}{2} + \frac{a^{\frac{3}{2}} d x}{8 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{3 \sqrt{a} c d x^{3}}{8 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{3 a^{2} d \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{8 \sqrt{c}} + a e \left (\begin{cases} \frac{\sqrt{a} x^{2}}{2} & \text{for}\: c = 0 \\\frac{\left (a + c x^{2}\right )^{\frac{3}{2}}}{3 c} & \text{otherwise} \end{cases}\right ) + c e \left (\begin{cases} - \frac{2 a^{2} \sqrt{a + c x^{2}}}{15 c^{2}} + \frac{a x^{2} \sqrt{a + c x^{2}}}{15 c} + \frac{x^{4} \sqrt{a + c x^{2}}}{5} & \text{for}\: c \neq 0 \\\frac{\sqrt{a} x^{4}}{4} & \text{otherwise} \end{cases}\right ) + \frac{c^{2} d x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{c x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.216918, size = 107, normalized size = 1.23 \[ -\frac{3 \, a^{2} d{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{8 \, \sqrt{c}} + \frac{1}{40} \, \sqrt{c x^{2} + a}{\left ({\left (25 \, a d + 2 \,{\left ({\left (4 \, c x e + 5 \, c d\right )} x + 8 \, a e\right )} x\right )} x + \frac{8 \, a^{2} e}{c}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/8*a^2*d*ln(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/sqrt(c) + 1/40*sqrt(c*x^2 + a)*
((25*a*d + 2*((4*c*x*e + 5*c*d)*x + 8*a*e)*x)*x + 8*a^2*e/c)