3.718 \(\int \frac{x^{-3+m}}{\sqrt{a+b x}} \, dx\)

Optimal. Leaf size=51 \[ -\frac{2 b^2 x^m \sqrt{a+b x} \left (-\frac{b x}{a}\right )^{-m} \, _2F_1\left (\frac{1}{2},3-m;\frac{3}{2};\frac{b x}{a}+1\right )}{a^3} \]

[Out]

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

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Rubi [A]  time = 0.0458692, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{2 b^2 x^m \sqrt{a+b x} \left (-\frac{b x}{a}\right )^{-m} \, _2F_1\left (\frac{1}{2},3-m;\frac{3}{2};\frac{b x}{a}+1\right )}{a^3} \]

Antiderivative was successfully verified.

[In]  Int[x^(-3 + m)/Sqrt[a + b*x],x]

[Out]

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

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Rubi in Sympy [A]  time = 8.37523, size = 42, normalized size = 0.82 \[ - \frac{2 b^{2} x^{m} \left (- \frac{b x}{a}\right )^{- m} \sqrt{a + b x}{{}_{2}F_{1}\left (\begin{matrix} - m + 3, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{1 + \frac{b x}{a}} \right )}}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 0.205362, size = 156, normalized size = 3.06 \[ \frac{x^{m-2} \sqrt{\frac{b x}{a}+1} \left (a^3 m \left (m^2-1\right ) \, _2F_1\left (-\frac{1}{2},m-2;m-1;-\frac{b x}{a}\right )-b (m-2) x \left (a^2 m (m+1) \, _2F_1\left (-\frac{1}{2},m-1;m;-\frac{b x}{a}\right )+b (m-1) x \left (b m x \, _2F_1\left (\frac{1}{2},m+1;m+2;-\frac{b x}{a}\right )-a (m+1) \, _2F_1\left (-\frac{1}{2},m;m+1;-\frac{b x}{a}\right )\right )\right )\right )}{a^3 (m-2) (m-1) m (m+1) \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.03, size = 0, normalized size = 0. \[ \int{{x}^{-3+m}{\frac{1}{\sqrt{bx+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-3+m)/(b*x+a)^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m - 3}}{\sqrt{b x + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^(m - 3)/sqrt(b*x + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x^(m - 3)/sqrt(b*x + a), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^(m - 3)/sqrt(b*x + a), x)