3.369 \(\int \left (-\frac{b x^m}{2 (a+b x)^{3/2}}+\frac{m x^{-1+m}}{\sqrt{a+b x}}\right ) \, dx\)

Optimal. Leaf size=13 \[ \frac{x^m}{\sqrt{a+b x}} \]

[Out]

x^m/Sqrt[a + b*x]

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Rubi [C]  time = 0.111998, antiderivative size = 92, normalized size of antiderivative = 7.08, number of steps used = 5, number of rules used = 2, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{x^m \left (-\frac{b x}{a}\right )^{-m} \, _2F_1\left (-\frac{1}{2},-m;\frac{1}{2};\frac{b x}{a}+1\right )}{\sqrt{a+b x}}-\frac{2 m x^m \sqrt{a+b x} \left (-\frac{b x}{a}\right )^{-m} \, _2F_1\left (\frac{1}{2},1-m;\frac{3}{2};\frac{b x}{a}+1\right )}{a} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 14.4416, size = 73, normalized size = 5.62 \[ \frac{x^{m} \left (- \frac{b x}{a}\right )^{- m}{{}_{2}F_{1}\left (\begin{matrix} - m, - \frac{1}{2} \\ \frac{1}{2} \end{matrix}\middle |{1 + \frac{b x}{a}} \right )}}{\sqrt{a + b x}} - \frac{2 m x^{m} \left (- \frac{b x}{a}\right )^{- m} \sqrt{a + b x}{{}_{2}F_{1}\left (\begin{matrix} - m + 1, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{1 + \frac{b x}{a}} \right )}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(-1/2*b*x**m/(b*x+a)**(3/2)+m*x**(-1+m)/(b*x+a)**(1/2),x)

[Out]

x**m*(-b*x/a)**(-m)*hyper((-m, -1/2), (1/2,), 1 + b*x/a)/sqrt(a + b*x) - 2*m*x**
m*(-b*x/a)**(-m)*sqrt(a + b*x)*hyper((-m + 1, 1/2), (3/2,), 1 + b*x/a)/a

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Mathematica [C]  time = 0.0474768, size = 100, normalized size = 7.69 \[ \frac{x^m \sqrt{a+b x} \left (2 a (m+1) \, _2F_1\left (-\frac{1}{2},m;m+1;-\frac{b x}{a}\right )-b x \left (2 m \, _2F_1\left (\frac{1}{2},m+1;m+2;-\frac{b x}{a}\right )+\, _2F_1\left (\frac{3}{2},m+1;m+2;-\frac{b x}{a}\right )\right )\right )}{2 a^2 (m+1) \sqrt{\frac{b x}{a}+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.15, size = 0, normalized size = 0. \[ \int -{\frac{b{x}^{m}}{2} \left ( bx+a \right ) ^{-{\frac{3}{2}}}}+{m{x}^{-1+m}{\frac{1}{\sqrt{bx+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int(-1/2*b*x^m/(b*x+a)^(3/2)+m*x^(-1+m)/(b*x+a)^(1/2),x)

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Maxima [A]  time = 1.53198, size = 15, normalized size = 1.15 \[ \frac{x^{m}}{\sqrt{b x + a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(m*x^(m - 1)/sqrt(b*x + a) - 1/2*b*x^m/(b*x + a)^(3/2),x, algorithm="maxima")

[Out]

x^m/sqrt(b*x + a)

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Fricas [A]  time = 0.227745, size = 15, normalized size = 1.15 \[ \frac{x^{m}}{\sqrt{b x + a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(m*x^(m - 1)/sqrt(b*x + a) - 1/2*b*x^m/(b*x + a)^(3/2),x, algorithm="fricas")

[Out]

x^m/sqrt(b*x + a)

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Sympy [A]  time = 28.0152, size = 73, normalized size = 5.62 \[ \frac{m x^{m} \Gamma \left (m\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, m \\ m + 1 \end{matrix}\middle |{\frac{b x e^{i \pi }}{a}} \right )}}{\sqrt{a} \Gamma \left (m + 1\right )} - \frac{b x x^{m} \Gamma \left (m + 1\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{b x e^{i \pi }}{a}} \right )}}{2 a^{\frac{3}{2}} \Gamma \left (m + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/2*b*x**m/(b*x+a)**(3/2)+m*x**(-1+m)/(b*x+a)**(1/2),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{m x^{m - 1}}{\sqrt{b x + a}} - \frac{b x^{m}}{2 \,{\left (b x + a\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(m*x^(m - 1)/sqrt(b*x + a) - 1/2*b*x^m/(b*x + a)^(3/2),x, algorithm="giac")

[Out]

integrate(m*x^(m - 1)/sqrt(b*x + a) - 1/2*b*x^m/(b*x + a)^(3/2), x)