3.122 \(\int \frac{(d x)^m}{\left (b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=66 \[ \frac{2 x (b+c x) (d x)^m \left (-\frac{c x}{b}\right )^{\frac{1}{2}-m} \, _2F_1\left (-\frac{1}{2},\frac{3}{2}-m;\frac{1}{2};\frac{c x}{b}+1\right )}{b \left (b x+c x^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0912, antiderivative size = 66, 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 \[ \frac{2 x (b+c x) (d x)^m \left (-\frac{c x}{b}\right )^{\frac{1}{2}-m} \, _2F_1\left (-\frac{1}{2},\frac{3}{2}-m;\frac{1}{2};\frac{c x}{b}+1\right )}{b \left (b x+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(d*x)^m/(b*x + c*x^2)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 14.7304, size = 73, normalized size = 1.11 \[ - \frac{2 c x^{- m + \frac{3}{2}} x^{m + \frac{1}{2}} \left (d x\right )^{m} \left (- \frac{c x}{b}\right )^{- m - \frac{1}{2}} \left (b + c x\right ){{}_{2}F_{1}\left (\begin{matrix} - m + \frac{3}{2}, - \frac{1}{2} \\ \frac{1}{2} \end{matrix}\middle |{1 + \frac{c x}{b}} \right )}}{b^{2} \left (b x + c x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0548166, size = 61, normalized size = 0.92 \[ \frac{2 \sqrt{\frac{c x}{b}+1} (d x)^m \, _2F_1\left (\frac{3}{2},m-\frac{1}{2};m+\frac{1}{2};-\frac{c x}{b}\right )}{b (2 m-1) \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.032, size = 0, normalized size = 0. \[ \int{ \left ( dx \right ) ^{m} \left ( c{x}^{2}+bx \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x)^m/(c*x^2+b*x)^(3/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((d*x)^m/(c*x^2 + b*x)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((d*x)^m/(c*x^2 + b*x)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((d*x)**m/(x*(b + c*x))**(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((d*x)^m/(c*x^2 + b*x)^(3/2), x)