3.330 \(\int (A+B x) \left (a+c x^2\right )^{3/2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0626748, 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 \[ \frac{3 a^2 A \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{8 \sqrt{c}}+\frac{1}{4} A x \left (a+c x^2\right )^{3/2}+\frac{3}{8} a A x \sqrt{a+c x^2}+\frac{B \left (a+c x^2\right )^{5/2}}{5 c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.35005, size = 80, normalized size = 0.92 \[ \frac{3 A a^{2} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{8 \sqrt{c}} + \frac{3 A a x \sqrt{a + c x^{2}}}{8} + \frac{A x \left (a + c x^{2}\right )^{\frac{3}{2}}}{4} + \frac{B \left (a + c x^{2}\right )^{\frac{5}{2}}}{5 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+a)**(3/2),x)

[Out]

3*A*a**2*atanh(sqrt(c)*x/sqrt(a + c*x**2))/(8*sqrt(c)) + 3*A*a*x*sqrt(a + c*x**2
)/8 + A*x*(a + c*x**2)**(3/2)/4 + B*(a + c*x**2)**(5/2)/(5*c)

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Mathematica [A]  time = 0.084575, size = 88, normalized size = 1.01 \[ \frac{\sqrt{a+c x^2} \left (8 a^2 B+a c x (25 A+16 B x)+2 c^2 x^3 (5 A+4 B x)\right )+15 a^2 A \sqrt{c} \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{40 c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 69, normalized size = 0.8 \[{\frac{Ax}{4} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,aAx}{8}\sqrt{c{x}^{2}+a}}+{\frac{3\,A{a}^{2}}{8}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}}+{\frac{B}{5\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+a)^(3/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.355813, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, A a^{2} c \log \left (-2 \, \sqrt{c x^{2} + a} c x -{\left (2 \, c x^{2} + a\right )} \sqrt{c}\right ) + 2 \,{\left (8 \, B c^{2} x^{4} + 10 \, A c^{2} x^{3} + 16 \, B a c x^{2} + 25 \, A a c x + 8 \, B a^{2}\right )} \sqrt{c x^{2} + a} \sqrt{c}}{80 \, c^{\frac{3}{2}}}, \frac{15 \, A a^{2} c \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) +{\left (8 \, B c^{2} x^{4} + 10 \, A c^{2} x^{3} + 16 \, B a c x^{2} + 25 \, A a c x + 8 \, B a^{2}\right )} \sqrt{c x^{2} + a} \sqrt{-c}}{40 \, \sqrt{-c} c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 18.4172, size = 219, normalized size = 2.52 \[ \frac{A a^{\frac{3}{2}} x \sqrt{1 + \frac{c x^{2}}{a}}}{2} + \frac{A a^{\frac{3}{2}} x}{8 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{3 A \sqrt{a} c x^{3}}{8 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{3 A a^{2} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{8 \sqrt{c}} + \frac{A c^{2} x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{c x^{2}}{a}}} + B a \left (\begin{cases} \frac{\sqrt{a} x^{2}}{2} & \text{for}\: c = 0 \\\frac{\left (a + c x^{2}\right )^{\frac{3}{2}}}{3 c} & \text{otherwise} \end{cases}\right ) + B c \left (\begin{cases} - \frac{2 a^{2} \sqrt{a + c x^{2}}}{15 c^{2}} + \frac{a x^{2} \sqrt{a + c x^{2}}}{15 c} + \frac{x^{4} \sqrt{a + c x^{2}}}{5} & \text{for}\: c \neq 0 \\\frac{\sqrt{a} x^{4}}{4} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+a)**(3/2),x)

[Out]

A*a**(3/2)*x*sqrt(1 + c*x**2/a)/2 + A*a**(3/2)*x/(8*sqrt(1 + c*x**2/a)) + 3*A*sq
rt(a)*c*x**3/(8*sqrt(1 + c*x**2/a)) + 3*A*a**2*asinh(sqrt(c)*x/sqrt(a))/(8*sqrt(
c)) + A*c**2*x**5/(4*sqrt(a)*sqrt(1 + c*x**2/a)) + B*a*Piecewise((sqrt(a)*x**2/2
, Eq(c, 0)), ((a + c*x**2)**(3/2)/(3*c), True)) + B*c*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))

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GIAC/XCAS [A]  time = 0.280137, size = 103, normalized size = 1.18 \[ -\frac{3 \, A a^{2}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{8 \, \sqrt{c}} + \frac{1}{40} \, \sqrt{c x^{2} + a}{\left (\frac{8 \, B a^{2}}{c} +{\left (25 \, A a + 2 \,{\left (8 \, B a +{\left (4 \, B c x + 5 \, A c\right )} x\right )} x\right )} x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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