3.711 \(\int \frac{x^m}{(a+b x)^{3/2}} \, dx\)

Optimal. Leaf size=46 \[ -\frac{2 x^m \left (-\frac{b x}{a}\right )^{-m} \, _2F_1\left (-\frac{1}{2},-m;\frac{1}{2};\frac{b x}{a}+1\right )}{b \sqrt{a+b x}} \]

[Out]

(-2*x^m*Hypergeometric2F1[-1/2, -m, 1/2, 1 + (b*x)/a])/(b*(-((b*x)/a))^m*Sqrt[a
+ b*x])

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Rubi [A]  time = 0.0392248, antiderivative size = 46, 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 \[ -\frac{2 x^m \left (-\frac{b x}{a}\right )^{-m} \, _2F_1\left (-\frac{1}{2},-m;\frac{1}{2};\frac{b x}{a}+1\right )}{b \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]  Int[x^m/(a + b*x)^(3/2),x]

[Out]

(-2*x^m*Hypergeometric2F1[-1/2, -m, 1/2, 1 + (b*x)/a])/(b*(-((b*x)/a))^m*Sqrt[a
+ b*x])

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Rubi in Sympy [A]  time = 5.97762, size = 39, normalized size = 0.85 \[ - \frac{2 x^{m} \left (- \frac{b x}{a}\right )^{- m}{{}_{2}F_{1}\left (\begin{matrix} - m, - \frac{1}{2} \\ \frac{1}{2} \end{matrix}\middle |{1 + \frac{b x}{a}} \right )}}{b \sqrt{a + b x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*x**m*(-b*x/a)**(-m)*hyper((-m, -1/2), (1/2,), 1 + b*x/a)/(b*sqrt(a + b*x))

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Mathematica [A]  time = 0.034299, size = 53, normalized size = 1.15 \[ \frac{x^{m+1} \sqrt{\frac{a+b x}{a}} \, _2F_1\left (\frac{3}{2},m+1;m+2;-\frac{b x}{a}\right )}{a (m+1) \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

(x^(1 + m)*Sqrt[(a + b*x)/a]*Hypergeometric2F1[3/2, 1 + m, 2 + m, -((b*x)/a)])/(
a*(1 + m)*Sqrt[a + b*x])

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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]

int(x^m/(b*x+a)^(3/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m}}{{\left (b x + a\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m/(b*x + a)^(3/2),x, algorithm="maxima")

[Out]

integrate(x^m/(b*x + a)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{m}}{{\left (b x + a\right )}^{\frac{3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x^m/(b*x + a)^(3/2), x)

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Sympy [A]  time = 6.79872, size = 36, normalized size = 0.78 \[ \frac{x x^{m} \Gamma \left (m + 1\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{b x e^{i \pi }}{a}} \right )}}{a^{\frac{3}{2}} \Gamma \left (m + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*x**m*gamma(m + 1)*hyper((3/2, m + 1), (m + 2,), b*x*exp_polar(I*pi)/a)/(a**(3/
2)*gamma(m + 2))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m}}{{\left (b x + a\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^m/(b*x + a)^(3/2), x)