3.1083 \(\int (d+e x)^m \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=42 \[ \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2} (d+e x)^{m+1}}{e (m+4)} \]

[Out]

((d + e*x)^(1 + m)*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2))/(e*(4 + m))

_______________________________________________________________________________________

Rubi [A]  time = 0.065936, antiderivative size = 42, 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 \[ \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2} (d+e x)^{m+1}}{e (m+4)} \]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 18.7919, size = 37, normalized size = 0.88 \[ \frac{\left (d + e x\right )^{m + 1} \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{\frac{3}{2}}}{e \left (m + 4\right )} \]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0529508, size = 31, normalized size = 0.74 \[ \frac{\left (c (d+e x)^2\right )^{3/2} (d+e x)^{m+1}}{e (m+4)} \]

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)*(c*(d + e*x)^2)^(3/2))/(e*(4 + m))

_______________________________________________________________________________________

Maple [A]  time = 0.003, size = 41, normalized size = 1. \[{\frac{ \left ( ex+d \right ) ^{1+m}}{e \left ( 4+m \right ) } \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{{\frac{3}{2}}}} \]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.693991, size = 95, normalized size = 2.26 \[ \frac{{\left (c^{\frac{3}{2}} e^{4} x^{4} + 4 \, c^{\frac{3}{2}} d e^{3} x^{3} + 6 \, c^{\frac{3}{2}} d^{2} e^{2} x^{2} + 4 \, c^{\frac{3}{2}} d^{3} e x + c^{\frac{3}{2}} d^{4}\right )}{\left (e x + d\right )}^{m}}{e{\left (m + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(c^(3/2)*e^4*x^4 + 4*c^(3/2)*d*e^3*x^3 + 6*c^(3/2)*d^2*e^2*x^2 + 4*c^(3/2)*d^3*e
*x + c^(3/2)*d^4)*(e*x + d)^m/(e*(m + 4))

_______________________________________________________________________________________

Fricas [A]  time = 0.241326, size = 96, normalized size = 2.29 \[ \frac{{\left (c e^{3} x^{3} + 3 \, c d e^{2} x^{2} + 3 \, c d^{2} e x + c d^{3}\right )} \sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}{\left (e x + d\right )}^{m}}{e m + 4 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.240421, size = 154, normalized size = 3.67 \[ \frac{c^{\frac{3}{2}} x^{4} e^{\left (m{\rm ln}\left (x e + d\right ) + 4\right )} + 4 \, c^{\frac{3}{2}} d x^{3} e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + 6 \, c^{\frac{3}{2}} d^{2} x^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )} + 4 \, c^{\frac{3}{2}} d^{3} x e^{\left (m{\rm ln}\left (x e + d\right ) + 1\right )} + c^{\frac{3}{2}} d^{4} e^{\left (m{\rm ln}\left (x e + d\right )\right )}}{m e + 4 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(c^(3/2)*x^4*e^(m*ln(x*e + d) + 4) + 4*c^(3/2)*d*x^3*e^(m*ln(x*e + d) + 3) + 6*c
^(3/2)*d^2*x^2*e^(m*ln(x*e + d) + 2) + 4*c^(3/2)*d^3*x*e^(m*ln(x*e + d) + 1) + c
^(3/2)*d^4*e^(m*ln(x*e + d)))/(m*e + 4*e)