3.1086 \(\int \frac{(d+e x)^m}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=45 \[ -\frac{(d+e x)^{m+1}}{e (2-m) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \]

[Out]

-((d + e*x)^(1 + m)/(e*(2 - m)*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2)))

_______________________________________________________________________________________

Rubi [A]  time = 0.0668953, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ -\frac{(d+e x)^{m+1}}{e (2-m) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-((d + e*x)^(1 + m)/(e*(2 - m)*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2)))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 19.0261, size = 39, normalized size = 0.87 \[ - \frac{\left (d + e x\right )^{m + 1}}{e \left (- m + 2\right ) \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(d + e*x)**(m + 1)/(e*(-m + 2)*(c*d**2 + 2*c*d*e*x + c*e**2*x**2)**(3/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0344759, size = 31, normalized size = 0.69 \[ \frac{(d+e x)^{m+1}}{e (m-2) \left (c (d+e x)^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(d + e*x)^(1 + m)/(e*(-2 + m)*(c*(d + e*x)^2)^(3/2))

_______________________________________________________________________________________

Maple [A]  time = 0.002, size = 41, normalized size = 0.9 \[{\frac{ \left ( ex+d \right ) ^{1+m}}{e \left ( -2+m \right ) } \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(e*x+d)^(1+m)/e/(-2+m)/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2)

_______________________________________________________________________________________

Maxima [A]  time = 0.696291, size = 69, normalized size = 1.53 \[ \frac{{\left (e x + d\right )}^{m} \sqrt{c}}{c^{2} e^{3}{\left (m - 2\right )} x^{2} + 2 \, c^{2} d e^{2}{\left (m - 2\right )} x + c^{2} d^{2} e{\left (m - 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(e*x + d)^m*sqrt(c)/(c^2*e^3*(m - 2)*x^2 + 2*c^2*d*e^2*(m - 2)*x + c^2*d^2*e*(m
- 2))

_______________________________________________________________________________________

Fricas [A]  time = 0.235631, size = 165, normalized size = 3.67 \[ \frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}{\left (e x + d\right )}^{m}}{c^{2} d^{3} e m - 2 \, c^{2} d^{3} e +{\left (c^{2} e^{4} m - 2 \, c^{2} e^{4}\right )} x^{3} + 3 \,{\left (c^{2} d e^{3} m - 2 \, c^{2} d e^{3}\right )} x^{2} + 3 \,{\left (c^{2} d^{2} e^{2} m - 2 \, c^{2} d^{2} e^{2}\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)*(e*x + d)^m/(c^2*d^3*e*m - 2*c^2*d^3*e + (c^
2*e^4*m - 2*c^2*e^4)*x^3 + 3*(c^2*d*e^3*m - 2*c^2*d*e^3)*x^2 + 3*(c^2*d^2*e^2*m
- 2*c^2*d^2*e^2)*x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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