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