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

Optimal. Leaf size=34 \[ \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{7/2}}{7 c^2 e} \]

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Rubi [A]  time = 0.02, antiderivative size = 34, 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 = {643, 629} \begin {gather*} \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{7/2}}{7 c^2 e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^3*(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)^(7/2)/(7*c^2*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]

Rule 643

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[e^(m - 1)/c^((m - 1)/2
), Int[(d + e*x)*(a + b*x + c*x^2)^(p + (m - 1)/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 - 1)/2]

Rubi steps

\begin {align*} \int (d+e x)^3 \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2} \, dx &=\frac {\int (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2} \, dx}{c}\\ &=\frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{7/2}}{7 c^2 e}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 27, normalized size = 0.79 \begin {gather*} \frac {(d+e x)^4 \left (c (d+e x)^2\right )^{3/2}}{7 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((d + e*x)^4*(c*(d + e*x)^2)^(3/2))/(7*e)

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IntegrateAlgebraic [A]  time = 0.04, size = 23, normalized size = 0.68 \begin {gather*} \frac {\left (c (d+e x)^2\right )^{7/2}}{7 c^2 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c*(d + e*x)^2)^(7/2)/(7*c^2*e)

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fricas [B]  time = 0.40, size = 103, normalized size = 3.03 \begin {gather*} \frac {{\left (c e^{6} x^{7} + 7 \, c d e^{5} x^{6} + 21 \, c d^{2} e^{4} x^{5} + 35 \, c d^{3} e^{3} x^{4} + 35 \, c d^{4} e^{2} x^{3} + 21 \, c d^{5} e x^{2} + 7 \, c d^{6} x\right )} \sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{7 \, {\left (e x + d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*(c*e^6*x^7 + 7*c*d*e^5*x^6 + 21*c*d^2*e^4*x^5 + 35*c*d^3*e^3*x^4 + 35*c*d^4*e^2*x^3 + 21*c*d^5*e*x^2 + 7*c
*d^6*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 = 0.27, size = 88, normalized size = 2.59 \begin {gather*} \frac {1}{7} \, {\left (c d^{6} e^{\left (-1\right )} + {\left (6 \, c d^{5} + {\left (15 \, c d^{4} e + {\left (20 \, c d^{3} e^{2} + {\left (15 \, c d^{2} e^{3} + {\left (c x e^{5} + 6 \, c d e^{4}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*(c*d^6*e^(-1) + (6*c*d^5 + (15*c*d^4*e + (20*c*d^3*e^2 + (15*c*d^2*e^3 + (c*x*e^5 + 6*c*d*e^4)*x)*x)*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 = 95, normalized size = 2.79 \begin {gather*} \frac {\left (e^{6} x^{6}+7 e^{5} x^{5} d +21 e^{4} x^{4} d^{2}+35 d^{3} e^{3} x^{3}+35 d^{4} e^{2} x^{2}+21 d^{5} e x +7 d^{6}\right ) \left (c \,e^{2} x^{2}+2 c d e x +c \,d^{2}\right )^{\frac {3}{2}} x}{7 \left (e x +d \right )^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*x*(e^6*x^6+7*d*e^5*x^5+21*d^2*e^4*x^4+35*d^3*e^3*x^3+35*d^4*e^2*x^2+21*d^5*e*x+7*d^6)*(c*e^2*x^2+2*c*d*e*x
+c*d^2)^(3/2)/(e*x+d)^3

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maxima [B]  time = 1.50, size = 94, normalized size = 2.76 \begin {gather*} \frac {{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac {5}{2}} e x^{2}}{7 \, c} + \frac {2 \, {\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac {5}{2}} d x}{7 \, c} + \frac {{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac {5}{2}} d^{2}}{7 \, c e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(5/2)*e*x^2/c + 2/7*(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(5/2)*d*x/c + 1/7*(c*e
^2*x^2 + 2*c*d*e*x + c*d^2)^(5/2)*d^2/(c*e)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 3.30, size = 277, normalized size = 8.15 \begin {gather*} \begin {cases} \frac {c d^{6} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{7 e} + \frac {6 c d^{5} x \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{7} + \frac {15 c d^{4} e x^{2} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{7} + \frac {20 c d^{3} e^{2} x^{3} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{7} + \frac {15 c d^{2} e^{3} x^{4} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{7} + \frac {6 c d e^{4} x^{5} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{7} + \frac {c e^{5} x^{6} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{7} & \text {for}\: e \neq 0 \\d^{3} x \left (c d^{2}\right )^{\frac {3}{2}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((c*d**6*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/(7*e) + 6*c*d**5*x*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x
**2)/7 + 15*c*d**4*e*x**2*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/7 + 20*c*d**3*e**2*x**3*sqrt(c*d**2 + 2*c*d*e
*x + c*e**2*x**2)/7 + 15*c*d**2*e**3*x**4*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/7 + 6*c*d*e**4*x**5*sqrt(c*d*
*2 + 2*c*d*e*x + c*e**2*x**2)/7 + c*e**5*x**6*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/7, Ne(e, 0)), (d**3*x*(c*
d**2)**(3/2), True))

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