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

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

[Out]

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

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Rubi [A]  time = 0.0763275, 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 \[ \frac{3 a^2 A \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{8 \sqrt{b}}+\frac{1}{4} A x \left (a+b x^2\right )^{3/2}+\frac{3}{8} a A x \sqrt{a+b x^2}+\frac{B \left (a+b x^2\right )^{5/2}}{5 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.6607, size = 80, normalized size = 0.92 \[ \frac{3 A a^{2} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{8 \sqrt{b}} + \frac{3 A a x \sqrt{a + b x^{2}}}{8} + \frac{A x \left (a + b x^{2}\right )^{\frac{3}{2}}}{4} + \frac{B \left (a + b x^{2}\right )^{\frac{5}{2}}}{5 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0820824, size = 88, normalized size = 1.01 \[ \frac{\sqrt{a+b x^2} \left (8 a^2 B+a b x (25 A+16 B x)+2 b^2 x^3 (5 A+4 B x)\right )+15 a^2 A \sqrt{b} \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{40 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 69, normalized size = 0.8 \[{\frac{Ax}{4} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,aAx}{8}\sqrt{b{x}^{2}+a}}+{\frac{3\,A{a}^{2}}{8}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){\frac{1}{\sqrt{b}}}}+{\frac{B}{5\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(b*x^2+a)^(3/2),x)

[Out]

1/4*A*x*(b*x^2+a)^(3/2)+3/8*a*A*x*(b*x^2+a)^(1/2)+3/8*A*a^2/b^(1/2)*ln(x*b^(1/2)
+(b*x^2+a)^(1/2))+1/5*B*(b*x^2+a)^(5/2)/b

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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((b*x^2 + a)^(3/2)*(B*x + A),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.263626, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, A a^{2} b \log \left (-2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right ) + 2 \,{\left (8 \, B b^{2} x^{4} + 10 \, A b^{2} x^{3} + 16 \, B a b x^{2} + 25 \, A a b x + 8 \, B a^{2}\right )} \sqrt{b x^{2} + a} \sqrt{b}}{80 \, b^{\frac{3}{2}}}, \frac{15 \, A a^{2} b \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) +{\left (8 \, B b^{2} x^{4} + 10 \, A b^{2} x^{3} + 16 \, B a b x^{2} + 25 \, A a b x + 8 \, B a^{2}\right )} \sqrt{b x^{2} + a} \sqrt{-b}}{40 \, \sqrt{-b} b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 9.32045, size = 219, normalized size = 2.52 \[ \frac{A a^{\frac{3}{2}} x \sqrt{1 + \frac{b x^{2}}{a}}}{2} + \frac{A a^{\frac{3}{2}} x}{8 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 A \sqrt{a} b x^{3}}{8 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 A a^{2} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 \sqrt{b}} + \frac{A b^{2} x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} + B a \left (\begin{cases} \frac{\sqrt{a} x^{2}}{2} & \text{for}\: b = 0 \\\frac{\left (a + b x^{2}\right )^{\frac{3}{2}}}{3 b} & \text{otherwise} \end{cases}\right ) + B b \left (\begin{cases} - \frac{2 a^{2} \sqrt{a + b x^{2}}}{15 b^{2}} + \frac{a x^{2} \sqrt{a + b x^{2}}}{15 b} + \frac{x^{4} \sqrt{a + b x^{2}}}{5} & \text{for}\: b \neq 0 \\\frac{\sqrt{a} x^{4}}{4} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a**(3/2)*x*sqrt(1 + b*x**2/a)/2 + A*a**(3/2)*x/(8*sqrt(1 + b*x**2/a)) + 3*A*sq
rt(a)*b*x**3/(8*sqrt(1 + b*x**2/a)) + 3*A*a**2*asinh(sqrt(b)*x/sqrt(a))/(8*sqrt(
b)) + A*b**2*x**5/(4*sqrt(a)*sqrt(1 + b*x**2/a)) + B*a*Piecewise((sqrt(a)*x**2/2
, Eq(b, 0)), ((a + b*x**2)**(3/2)/(3*b), True)) + B*b*Piecewise((-2*a**2*sqrt(a
+ b*x**2)/(15*b**2) + a*x**2*sqrt(a + b*x**2)/(15*b) + x**4*sqrt(a + b*x**2)/5,
Ne(b, 0)), (sqrt(a)*x**4/4, True))

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GIAC/XCAS [A]  time = 0.22258, size = 103, normalized size = 1.18 \[ -\frac{3 \, A a^{2}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{8 \, \sqrt{b}} + \frac{1}{40} \, \sqrt{b x^{2} + a}{\left (\frac{8 \, B a^{2}}{b} +{\left (25 \, A a + 2 \,{\left (8 \, B a +{\left (4 \, B b x + 5 \, A b\right )} x\right )} x\right )} x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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