Optimal. Leaf size=48 \[ \frac{2 x^m (a+b x)^{5/2} \left (-\frac{b x}{a}\right )^{-m} \, _2F_1\left (\frac{5}{2},-m;\frac{7}{2};\frac{b x}{a}+1\right )}{5 b} \]
[Out]
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Rubi [A] time = 0.0388495, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{2 x^m (a+b x)^{5/2} \left (-\frac{b x}{a}\right )^{-m} \, _2F_1\left (\frac{5}{2},-m;\frac{7}{2};\frac{b x}{a}+1\right )}{5 b} \]
Antiderivative was successfully verified.
[In] Int[x^m*(a + b*x)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 6.11254, size = 37, normalized size = 0.77 \[ \frac{2 x^{m} \left (- \frac{b x}{a}\right )^{- m} \left (a + b x\right )^{\frac{5}{2}}{{}_{2}F_{1}\left (\begin{matrix} - m, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{1 + \frac{b x}{a}} \right )}}{5 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**m*(b*x+a)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0717578, size = 83, normalized size = 1.73 \[ \frac{x^{m+1} \sqrt{a+b x} \left (a (m+2) \, _2F_1\left (-\frac{1}{2},m+1;m+2;-\frac{b x}{a}\right )+b (m+1) x \, _2F_1\left (-\frac{1}{2},m+2;m+3;-\frac{b x}{a}\right )\right )}{(m+1) (m+2) \sqrt{\frac{b x}{a}+1}} \]
Antiderivative was successfully verified.
[In] Integrate[x^m*(a + b*x)^(3/2),x]
[Out]
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Maple [F] time = 0.026, size = 0, normalized size = 0. \[ \int{x}^{m} \left ( bx+a \right ) ^{{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^m*(b*x+a)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{\frac{3}{2}} x^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(3/2)*x^m,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x + a\right )}^{\frac{3}{2}} x^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(3/2)*x^m,x, algorithm="fricas")
[Out]
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Sympy [A] time = 23.4338, size = 37, normalized size = 0.77 \[ \frac{a^{\frac{3}{2}} x x^{m} \Gamma \left (m + 1\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{b x e^{i \pi }}{a}} \right )}}{\Gamma \left (m + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**m*(b*x+a)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{\frac{3}{2}} x^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(3/2)*x^m,x, algorithm="giac")
[Out]