3.11.39 \(\int (d+e x)^3 (c d^2+2 c d e x+c e^2 x^2)^{3/2} \, dx\) [1039]

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

[Out]

1/7*(c*e^2*x^2+2*c*d*e*x+c*d^2)^(7/2)/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 = {657, 643} \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 643

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 657

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 = 0.01, 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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Maple [A]
time = 0.57, size = 35, normalized size = 1.03

method	result	size
risch	\(\frac {c \left (e x +d \right )^{6} \sqrt {\left (e x +d \right )^{2} c}}{7 e}\)	\(25\)
default	\(\frac {\left (e x +d \right )^{4} \left (x^{2} c \,e^{2}+2 c d e x +c \,d^{2}\right )^{\frac {3}{2}}}{7 e}\)	\(35\)
gosper	\(\frac {x \left (e^{6} x^{6}+7 d \,e^{5} x^{5}+21 d^{2} e^{4} x^{4}+35 d^{3} e^{3} x^{3}+35 d^{4} e^{2} x^{2}+21 d^{5} e x +7 d^{6}\right ) \left (x^{2} c \,e^{2}+2 c d e x +c \,d^{2}\right )^{\frac {3}{2}}}{7 \left (e x +d \right )^{3}}\)	\(95\)
trager	\(\frac {c x \left (e^{6} x^{6}+7 d \,e^{5} x^{5}+21 d^{2} e^{4} x^{4}+35 d^{3} e^{3} x^{3}+35 d^{4} e^{2} x^{2}+21 d^{5} e x +7 d^{6}\right ) \sqrt {x^{2} c \,e^{2}+2 c d e x +c \,d^{2}}}{7 e x +7 d}\)	\(96\)

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,method=_RETURNVERBOSE)

[Out]

1/7*(e*x+d)^4*(c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2)/e

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (29) = 58\).
time = 0.28, size = 94, normalized size = 2.76 \begin {gather*} \frac {{\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{\frac {5}{2}} x^{2} e}{7 \, c} + \frac {{\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{\frac {5}{2}} d^{2} e^{\left (-1\right )}}{7 \, c} + \frac {2 \, {\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{\frac {5}{2}} d x}{7 \, c} \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*x^2*e^2 + 2*c*d*x*e + c*d^2)^(5/2)*x^2*e/c + 1/7*(c*x^2*e^2 + 2*c*d*x*e + c*d^2)^(5/2)*d^2*e^(-1)/c + 2
/7*(c*x^2*e^2 + 2*c*d*x*e + c*d^2)^(5/2)*d*x/c

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (29) = 58\).
time = 3.35, size = 100, normalized size = 2.94 \begin {gather*} \frac {{\left (c x^{7} e^{6} + 7 \, c d x^{6} e^{5} + 21 \, c d^{2} x^{5} e^{4} + 35 \, c d^{3} x^{4} e^{3} + 35 \, c d^{4} x^{3} e^{2} + 21 \, c d^{5} x^{2} e + 7 \, c d^{6} x\right )} \sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}}}{7 \, {\left (x e + 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*x^7*e^6 + 7*c*d*x^6*e^5 + 21*c*d^2*x^5*e^4 + 35*c*d^3*x^4*e^3 + 35*c*d^4*x^3*e^2 + 21*c*d^5*x^2*e + 7*c
*d^6*x)*sqrt(c*x^2*e^2 + 2*c*d*x*e + c*d^2)/(x*e + d)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 277 vs. \(2 (31) = 62\).
time = 0.28, 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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 122 vs. \(2 (29) = 58\).
time = 1.36, size = 122, normalized size = 3.59 \begin {gather*} \frac {1}{7} \, {\left (c x^{7} e^{6} \mathrm {sgn}\left (x e + d\right ) + 7 \, c d x^{6} e^{5} \mathrm {sgn}\left (x e + d\right ) + 21 \, c d^{2} x^{5} e^{4} \mathrm {sgn}\left (x e + d\right ) + 35 \, c d^{3} x^{4} e^{3} \mathrm {sgn}\left (x e + d\right ) + 35 \, c d^{4} x^{3} e^{2} \mathrm {sgn}\left (x e + d\right ) + 21 \, c d^{5} x^{2} e \mathrm {sgn}\left (x e + d\right ) + 7 \, c d^{6} x \mathrm {sgn}\left (x e + d\right )\right )} \sqrt {c} \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*x^7*e^6*sgn(x*e + d) + 7*c*d*x^6*e^5*sgn(x*e + d) + 21*c*d^2*x^5*e^4*sgn(x*e + d) + 35*c*d^3*x^4*e^3*sg
n(x*e + d) + 35*c*d^4*x^3*e^2*sgn(x*e + d) + 21*c*d^5*x^2*e*sgn(x*e + d) + 7*c*d^6*x*sgn(x*e + d))*sqrt(c)

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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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