3.1041 \(\int (d+e x) (c d^2+2 c d e x+c e^2 x^2)^{3/2} \, dx\)

Optimal. Leaf size=34 \[ \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{5 c e} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {629} \[ \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{5 c e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 629

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

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 23, normalized size = 0.68 \[ \frac {\left (c (d+e x)^2\right )^{5/2}}{5 c e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*(d + e*x)^2)^(5/2)/(5*c*e)

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fricas [B]  time = 1.03, size = 79, normalized size = 2.32 \[ \frac {{\left (c e^{4} x^{5} + 5 \, c d e^{3} x^{4} + 10 \, c d^{2} e^{2} x^{3} + 10 \, c d^{3} e x^{2} + 5 \, c d^{4} x\right )} \sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{5 \, {\left (e x + d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.25, size = 66, normalized size = 1.94 \[ \frac {1}{5} \, {\left (c d^{4} e^{\left (-1\right )} + {\left (4 \, c d^{3} + {\left (6 \, c d^{2} e + {\left (c x e^{3} + 4 \, c d e^{2}\right )} x\right )} x\right )} x\right )} \sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.05, size = 73, normalized size = 2.15 \[ \frac {\left (e^{4} x^{4}+5 d \,e^{3} x^{3}+10 d^{2} e^{2} x^{2}+10 d^{3} e x +5 d^{4}\right ) \left (c \,e^{2} x^{2}+2 c d e x +c \,d^{2}\right )^{\frac {3}{2}} x}{5 \left (e x +d \right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.41, size = 30, normalized size = 0.88 \[ \frac {{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac {5}{2}}}{5 \, c e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.56, size = 34, normalized size = 1.00 \[ \frac {{\left (d+e\,x\right )}^2\,{\left (c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2\right )}^{3/2}}{5\,e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((d + e*x)^2*(c*d^2 + c*e^2*x^2 + 2*c*d*e*x)^(3/2))/(5*e)

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sympy [A]  time = 1.21, size = 194, normalized size = 5.71 \[ \begin {cases} \frac {c d^{4} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{5 e} + \frac {4 c d^{3} x \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{5} + \frac {6 c d^{2} e x^{2} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{5} + \frac {4 c d e^{2} x^{3} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{5} + \frac {c e^{3} x^{4} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{5} & \text {for}\: e \neq 0 \\d x \left (c d^{2}\right )^{\frac {3}{2}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((c*d**4*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/(5*e) + 4*c*d**3*x*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x
**2)/5 + 6*c*d**2*e*x**2*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/5 + 4*c*d*e**2*x**3*sqrt(c*d**2 + 2*c*d*e*x +
c*e**2*x**2)/5 + c*e**3*x**4*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/5, Ne(e, 0)), (d*x*(c*d**2)**(3/2), True))

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