Optimal. Leaf size=107 \[ \frac{5 a^3 A \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{16 \sqrt{c}}+\frac{5}{16} a^2 A x \sqrt{a+c x^2}+\frac{1}{6} A x \left (a+c x^2\right )^{5/2}+\frac{5}{24} a A x \left (a+c x^2\right )^{3/2}+\frac{B \left (a+c x^2\right )^{7/2}}{7 c} \]
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Rubi [A] time = 0.0800207, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ \frac{5 a^3 A \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{16 \sqrt{c}}+\frac{5}{16} a^2 A x \sqrt{a+c x^2}+\frac{1}{6} A x \left (a+c x^2\right )^{5/2}+\frac{5}{24} a A x \left (a+c x^2\right )^{3/2}+\frac{B \left (a+c x^2\right )^{7/2}}{7 c} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)*(a + c*x^2)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 10.017, size = 100, normalized size = 0.93 \[ \frac{5 A a^{3} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{16 \sqrt{c}} + \frac{5 A a^{2} x \sqrt{a + c x^{2}}}{16} + \frac{5 A a x \left (a + c x^{2}\right )^{\frac{3}{2}}}{24} + \frac{A x \left (a + c x^{2}\right )^{\frac{5}{2}}}{6} + \frac{B \left (a + c x^{2}\right )^{\frac{7}{2}}}{7 c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(c*x**2+a)**(5/2),x)
[Out]
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Mathematica [A] time = 0.118473, size = 108, normalized size = 1.01 \[ \frac{105 a^3 A \sqrt{c} \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )+\sqrt{a+c x^2} \left (48 a^3 B+3 a^2 c x (77 A+48 B x)+2 a c^2 x^3 (91 A+72 B x)+8 c^3 x^5 (7 A+6 B x)\right )}{336 c} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)*(a + c*x^2)^(5/2),x]
[Out]
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Maple [A] time = 0.007, size = 85, normalized size = 0.8 \[{\frac{Ax}{6} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,aAx}{24} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{a}^{2}Ax}{16}\sqrt{c{x}^{2}+a}}+{\frac{5\,A{a}^{3}}{16}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}}+{\frac{B}{7\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+a)^(5/2),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((c*x^2 + a)^(5/2)*(B*x + A),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.292965, size = 1, normalized size = 0.01 \[ \left [\frac{105 \, A a^{3} c \log \left (-2 \, \sqrt{c x^{2} + a} c x -{\left (2 \, c x^{2} + a\right )} \sqrt{c}\right ) + 2 \,{\left (48 \, B c^{3} x^{6} + 56 \, A c^{3} x^{5} + 144 \, B a c^{2} x^{4} + 182 \, A a c^{2} x^{3} + 144 \, B a^{2} c x^{2} + 231 \, A a^{2} c x + 48 \, B a^{3}\right )} \sqrt{c x^{2} + a} \sqrt{c}}{672 \, c^{\frac{3}{2}}}, \frac{105 \, A a^{3} c \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) +{\left (48 \, B c^{3} x^{6} + 56 \, A c^{3} x^{5} + 144 \, B a c^{2} x^{4} + 182 \, A a c^{2} x^{3} + 144 \, B a^{2} c x^{2} + 231 \, A a^{2} c x + 48 \, B a^{3}\right )} \sqrt{c x^{2} + a} \sqrt{-c}}{336 \, \sqrt{-c} c}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^(5/2)*(B*x + A),x, algorithm="fricas")
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Sympy [A] time = 37.4885, size = 348, normalized size = 3.25 \[ \frac{A a^{\frac{5}{2}} x \sqrt{1 + \frac{c x^{2}}{a}}}{2} + \frac{3 A a^{\frac{5}{2}} x}{16 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{35 A a^{\frac{3}{2}} c x^{3}}{48 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{17 A \sqrt{a} c^{2} x^{5}}{24 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{5 A a^{3} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{16 \sqrt{c}} + \frac{A c^{3} x^{7}}{6 \sqrt{a} \sqrt{1 + \frac{c x^{2}}{a}}} + B a^{2} \left (\begin{cases} \frac{\sqrt{a} x^{2}}{2} & \text{for}\: c = 0 \\\frac{\left (a + c x^{2}\right )^{\frac{3}{2}}}{3 c} & \text{otherwise} \end{cases}\right ) + 2 B a c \left (\begin{cases} - \frac{2 a^{2} \sqrt{a + c x^{2}}}{15 c^{2}} + \frac{a x^{2} \sqrt{a + c x^{2}}}{15 c} + \frac{x^{4} \sqrt{a + c x^{2}}}{5} & \text{for}\: c \neq 0 \\\frac{\sqrt{a} x^{4}}{4} & \text{otherwise} \end{cases}\right ) + B c^{2} \left (\begin{cases} \frac{8 a^{3} \sqrt{a + c x^{2}}}{105 c^{3}} - \frac{4 a^{2} x^{2} \sqrt{a + c x^{2}}}{105 c^{2}} + \frac{a x^{4} \sqrt{a + c x^{2}}}{35 c} + \frac{x^{6} \sqrt{a + c x^{2}}}{7} & \text{for}\: c \neq 0 \\\frac{\sqrt{a} x^{6}}{6} & \text{otherwise} \end{cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(c*x**2+a)**(5/2),x)
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GIAC/XCAS [A] time = 0.277354, size = 136, normalized size = 1.27 \[ -\frac{5 \, A a^{3}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{16 \, \sqrt{c}} + \frac{1}{336} \,{\left (\frac{48 \, B a^{3}}{c} +{\left (231 \, A a^{2} + 2 \,{\left (72 \, B a^{2} +{\left (91 \, A a c + 4 \,{\left (18 \, B a c +{\left (6 \, B c^{2} x + 7 \, A c^{2}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt{c x^{2} + a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^(5/2)*(B*x + A),x, algorithm="giac")
[Out]