3.1040 \(\int (d+e x)^2 (c d^2+2 c d e x+c e^2 x^2)^{3/2} \, dx\)

Optimal. Leaf size=39 \[ \frac {(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{6 c e} \]

[Out]

1/6*(e*x+d)*(c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2)/c/e

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {642, 609} \[ \frac {(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{6 c e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((d + e*x)*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2))/(6*c*e)

Rule 609

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p + 1
)), x] /; FreeQ[{a, b, c, p}, x] && EqQ[b^2 - 4*a*c, 0] && NeQ[p, -2^(-1)]

Rule 642

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[e^m/c^(m/2), Int[(a +
b*x + c*x^2)^(p + m/2), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p] && EqQ[
2*c*d - b*e, 0] && IntegerQ[m/2]

Rubi steps

\begin {align*} \int (d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2} \, dx &=\frac {\int \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2} \, dx}{c}\\ &=\frac {(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{6 c e}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 28, normalized size = 0.72 \[ \frac {(d+e x) \left (c (d+e x)^2\right )^{5/2}}{6 c e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((d + e*x)*(c*(d + e*x)^2)^(5/2))/(6*c*e)

________________________________________________________________________________________

fricas [B]  time = 0.84, size = 91, normalized size = 2.33 \[ \frac {{\left (c e^{5} x^{6} + 6 \, c d e^{4} x^{5} + 15 \, c d^{2} e^{3} x^{4} + 20 \, c d^{3} e^{2} x^{3} + 15 \, c d^{4} e x^{2} + 6 \, c d^{5} x\right )} \sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{6 \, {\left (e x + d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(c*e^5*x^6 + 6*c*d*e^4*x^5 + 15*c*d^2*e^3*x^4 + 20*c*d^3*e^2*x^3 + 15*c*d^4*e*x^2 + 6*c*d^5*x)*sqrt(c*e^2*
x^2 + 2*c*d*e*x + c*d^2)/(e*x + d)

________________________________________________________________________________________

giac [B]  time = 0.26, size = 77, normalized size = 1.97 \[ \frac {1}{6} \, {\left (c d^{5} e^{\left (-1\right )} + {\left (5 \, c d^{4} + {\left (10 \, c d^{3} e + {\left (10 \, c d^{2} e^{2} + {\left (c x e^{4} + 5 \, c d e^{3}\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(c*d^5*e^(-1) + (5*c*d^4 + (10*c*d^3*e + (10*c*d^2*e^2 + (c*x*e^4 + 5*c*d*e^3)*x)*x)*x)*x)*sqrt(c*x^2*e^2
+ 2*c*d*x*e + c*d^2)

________________________________________________________________________________________

maple [B]  time = 0.05, size = 84, normalized size = 2.15 \[ \frac {\left (e^{5} x^{5}+6 d \,e^{4} x^{4}+15 e^{3} x^{3} d^{2}+20 d^{3} e^{2} x^{2}+15 d^{4} e x +6 d^{5}\right ) \left (c \,e^{2} x^{2}+2 c d e x +c \,d^{2}\right )^{\frac {3}{2}} x}{6 \left (e x +d \right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*x*(e^5*x^5+6*d*e^4*x^4+15*d^2*e^3*x^3+20*d^3*e^2*x^2+15*d^4*e*x+6*d^5)*(c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2)/(
e*x+d)^3

________________________________________________________________________________________

maxima [A]  time = 1.52, size = 60, normalized size = 1.54 \[ \frac {{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac {5}{2}} x}{6 \, c} + \frac {{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac {5}{2}} d}{6 \, c e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(5/2)*x/c + 1/6*(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(5/2)*d/(c*e)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int {\left (d+e\,x\right )}^2\,{\left (c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2\right )}^{3/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c \left (d + e x\right )^{2}\right )^{\frac {3}{2}} \left (d + e x\right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**2*(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)**2, x)

________________________________________________________________________________________