3.123 \(\int \frac{(d x)^m}{\left (b x+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=71 \[ -\frac{2 c x^2 (b+c x) (d x)^m \left (-\frac{c x}{b}\right )^{\frac{1}{2}-m} \, _2F_1\left (-\frac{3}{2},\frac{5}{2}-m;-\frac{1}{2};\frac{c x}{b}+1\right )}{3 b^2 \left (b x+c x^2\right )^{5/2}} \]

[Out]

(-2*c*x^2*(-((c*x)/b))^(1/2 - m)*(d*x)^m*(b + c*x)*Hypergeometric2F1[-3/2, 5/2 -
m, -1/2, 1 + (c*x)/b])/(3*b^2*(b*x + c*x^2)^(5/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.0960954, antiderivative size = 71, 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 \[ -\frac{2 c x^2 (b+c x) (d x)^m \left (-\frac{c x}{b}\right )^{\frac{1}{2}-m} \, _2F_1\left (-\frac{3}{2},\frac{5}{2}-m;-\frac{1}{2};\frac{c x}{b}+1\right )}{3 b^2 \left (b x+c x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[(d*x)^m/(b*x + c*x^2)^(5/2),x]

[Out]

(-2*c*x^2*(-((c*x)/b))^(1/2 - m)*(d*x)^m*(b + c*x)*Hypergeometric2F1[-3/2, 5/2 -
m, -1/2, 1 + (c*x)/b])/(3*b^2*(b*x + c*x^2)^(5/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 15.8033, size = 76, normalized size = 1.07 \[ \frac{2 c^{2} x^{- m + \frac{5}{2}} x^{m + \frac{1}{2}} \left (d x\right )^{m} \left (- \frac{c x}{b}\right )^{- m - \frac{1}{2}} \left (b + c x\right ){{}_{2}F_{1}\left (\begin{matrix} - m + \frac{5}{2}, - \frac{3}{2} \\ - \frac{1}{2} \end{matrix}\middle |{1 + \frac{c x}{b}} \right )}}{3 b^{3} \left (b x + c x^{2}\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x)**m/(c*x**2+b*x)**(5/2),x)

[Out]

2*c**2*x**(-m + 5/2)*x**(m + 1/2)*(d*x)**m*(-c*x/b)**(-m - 1/2)*(b + c*x)*hyper(
(-m + 5/2, -3/2), (-1/2,), 1 + c*x/b)/(3*b**3*(b*x + c*x**2)**(5/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0581716, size = 64, normalized size = 0.9 \[ \frac{2 \sqrt{\frac{c x}{b}+1} (d x)^m \, _2F_1\left (\frac{5}{2},m-\frac{3}{2};m-\frac{1}{2};-\frac{c x}{b}\right )}{b^2 (2 m-3) x \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d*x)^m/(b*x + c*x^2)^(5/2),x]

[Out]

(2*(d*x)^m*Sqrt[1 + (c*x)/b]*Hypergeometric2F1[5/2, -3/2 + m, -1/2 + m, -((c*x)/
b)])/(b^2*(-3 + 2*m)*x*Sqrt[x*(b + c*x)])

_______________________________________________________________________________________

Maple [F]  time = 0.033, size = 0, normalized size = 0. \[ \int{ \left ( dx \right ) ^{m} \left ( c{x}^{2}+bx \right ) ^{-{\frac{5}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x)^m/(c*x^2+b*x)^(5/2),x)

[Out]

int((d*x)^m/(c*x^2+b*x)^(5/2),x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d x\right )^{m}}{{\left (c x^{2} + b x\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)^m/(c*x^2 + b*x)^(5/2),x, algorithm="maxima")

[Out]

integrate((d*x)^m/(c*x^2 + b*x)^(5/2), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\left (d x\right )^{m}}{{\left (c^{2} x^{4} + 2 \, b c x^{3} + b^{2} x^{2}\right )} \sqrt{c x^{2} + b x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)^m/(c*x^2 + b*x)^(5/2),x, algorithm="fricas")

[Out]

integral((d*x)^m/((c^2*x^4 + 2*b*c*x^3 + b^2*x^2)*sqrt(c*x^2 + b*x)), x)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d x\right )^{m}}{\left (x \left (b + c x\right )\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)**m/(c*x**2+b*x)**(5/2),x)

[Out]

Integral((d*x)**m/(x*(b + c*x))**(5/2), x)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d x\right )^{m}}{{\left (c x^{2} + b x\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)^m/(c*x^2 + b*x)^(5/2),x, algorithm="giac")

[Out]

integrate((d*x)^m/(c*x^2 + b*x)^(5/2), x)