3.121 \(\int \frac{(d x)^m}{\sqrt{b x+c x^2}} \, dx\)

Optimal. Leaf size=65 \[ \frac{2 (b+c x) (d x)^m \left (-\frac{c x}{b}\right )^{\frac{1}{2}-m} \, _2F_1\left (\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{c x}{b}+1\right )}{c \sqrt{b x+c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0893921, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{2 (b+c x) (d x)^m \left (-\frac{c x}{b}\right )^{\frac{1}{2}-m} \, _2F_1\left (\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{c x}{b}+1\right )}{c \sqrt{b x+c x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(d*x)^m/Sqrt[b*x + c*x^2],x]

[Out]

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

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Rubi in Sympy [A]  time = 14.8736, size = 68, normalized size = 1.05 \[ - \frac{2 x^{- m + \frac{1}{2}} x^{m + \frac{1}{2}} \left (d x\right )^{m} \left (- \frac{c x}{b}\right )^{- m - \frac{1}{2}} \left (b + c x\right ){{}_{2}F_{1}\left (\begin{matrix} - m + \frac{1}{2}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{1 + \frac{c x}{b}} \right )}}{b \sqrt{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x)**m/(c*x**2+b*x)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0490515, size = 59, normalized size = 0.91 \[ \frac{2 x \sqrt{\frac{c x}{b}+1} (d x)^m \, _2F_1\left (\frac{1}{2},m+\frac{1}{2};m+\frac{3}{2};-\frac{c x}{b}\right )}{(2 m+1) \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d*x)^m/Sqrt[b*x + c*x^2],x]

[Out]

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

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Maple [F]  time = 0.034, size = 0, normalized size = 0. \[ \int{ \left ( dx \right ) ^{m}{\frac{1}{\sqrt{c{x}^{2}+bx}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x)^m/(c*x^2+b*x)^(1/2),x)

[Out]

int((d*x)^m/(c*x^2+b*x)^(1/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)^m/sqrt(c*x^2 + b*x),x, algorithm="maxima")

[Out]

integrate((d*x)^m/sqrt(c*x^2 + b*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)^m/sqrt(c*x^2 + b*x),x, algorithm="fricas")

[Out]

integral((d*x)^m/sqrt(c*x^2 + b*x), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d x\right )^{m}}{\sqrt{x \left (b + c x\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)**m/(c*x**2+b*x)**(1/2),x)

[Out]

Integral((d*x)**m/sqrt(x*(b + c*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)^m/sqrt(c*x^2 + b*x),x, algorithm="giac")

[Out]

integrate((d*x)^m/sqrt(c*x^2 + b*x), x)