3.1050 \(\int (d+e x)^3 (c d^2+2 c d e x+c e^2 x^2)^{5/2} \, dx\)

Optimal. Leaf size=34 \[ \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{9/2}}{9 c^2 e} \]

[Out]

1/9*(c*e^2*x^2+2*c*d*e*x+c*d^2)^(9/2)/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 = {643, 629} \[ \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{9/2}}{9 c^2 e} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^3*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2),x]

[Out]

(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(9/2)/(9*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 )^{5/2} \, dx &=\frac {\int (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{7/2} \, dx}{c}\\ &=\frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{9/2}}{9 c^2 e}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 27, normalized size = 0.79 \[ \frac {(d+e x)^4 \left (c (d+e x)^2\right )^{5/2}}{9 e} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^3*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2),x]

[Out]

((d + e*x)^4*(c*(d + e*x)^2)^(5/2))/(9*e)

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fricas [B]  time = 0.97, size = 145, normalized size = 4.26 \[ \frac {{\left (c^{2} e^{8} x^{9} + 9 \, c^{2} d e^{7} x^{8} + 36 \, c^{2} d^{2} e^{6} x^{7} + 84 \, c^{2} d^{3} e^{5} x^{6} + 126 \, c^{2} d^{4} e^{4} x^{5} + 126 \, c^{2} d^{5} e^{3} x^{4} + 84 \, c^{2} d^{6} e^{2} x^{3} + 36 \, c^{2} d^{7} e x^{2} + 9 \, c^{2} d^{8} x\right )} \sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{9 \, {\left (e x + d\right )}} \]

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)^(5/2),x, algorithm="fricas")

[Out]

1/9*(c^2*e^8*x^9 + 9*c^2*d*e^7*x^8 + 36*c^2*d^2*e^6*x^7 + 84*c^2*d^3*e^5*x^6 + 126*c^2*d^4*e^4*x^5 + 126*c^2*d
^5*e^3*x^4 + 84*c^2*d^6*e^2*x^3 + 36*c^2*d^7*e*x^2 + 9*c^2*d^8*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 = 0.30, size = 128, normalized size = 3.76 \[ \frac {1}{9} \, {\left (c^{2} d^{8} e^{\left (-1\right )} + {\left (8 \, c^{2} d^{7} + {\left (28 \, c^{2} d^{6} e + {\left (56 \, c^{2} d^{5} e^{2} + {\left (70 \, c^{2} d^{4} e^{3} + {\left (56 \, c^{2} d^{3} e^{4} + {\left (28 \, c^{2} d^{2} e^{5} + {\left (c^{2} x e^{7} + 8 \, c^{2} d e^{6}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}} \]

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)^(5/2),x, algorithm="giac")

[Out]

1/9*(c^2*d^8*e^(-1) + (8*c^2*d^7 + (28*c^2*d^6*e + (56*c^2*d^5*e^2 + (70*c^2*d^4*e^3 + (56*c^2*d^3*e^4 + (28*c
^2*d^2*e^5 + (c^2*x*e^7 + 8*c^2*d*e^6)*x)*x)*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 = 0.04, size = 117, normalized size = 3.44 \[ \frac {\left (e^{8} x^{8}+9 d \,e^{7} x^{7}+36 d^{2} e^{6} x^{6}+84 d^{3} e^{5} x^{5}+126 d^{4} e^{4} x^{4}+126 d^{5} e^{3} x^{3}+84 d^{6} e^{2} x^{2}+36 x \,d^{7} e +9 d^{8}\right ) \left (c \,e^{2} x^{2}+2 c d e x +c \,d^{2}\right )^{\frac {5}{2}} x}{9 \left (e x +d \right )^{5}} \]

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)^(5/2),x)

[Out]

1/9*x*(e^8*x^8+9*d*e^7*x^7+36*d^2*e^6*x^6+84*d^3*e^5*x^5+126*d^4*e^4*x^4+126*d^5*e^3*x^3+84*d^6*e^2*x^2+36*d^7
*e*x+9*d^8)*(c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2)/(e*x+d)^5

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maxima [B]  time = 1.49, size = 94, normalized size = 2.76 \[ \frac {{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac {7}{2}} e x^{2}}{9 \, c} + \frac {2 \, {\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac {7}{2}} d x}{9 \, c} + \frac {{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac {7}{2}} d^{2}}{9 \, c e} \]

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)^(5/2),x, algorithm="maxima")

[Out]

1/9*(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(7/2)*e*x^2/c + 2/9*(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(7/2)*d*x/c + 1/9*(c*e
^2*x^2 + 2*c*d*e*x + c*d^2)^(7/2)*d^2/(c*e)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int {\left (d+e\,x\right )}^3\,{\left (c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2\right )}^{5/2} \,d x \]

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)^(5/2),x)

[Out]

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

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sympy [A]  time = 13.50, size = 374, normalized size = 11.00 \[ \begin {cases} \frac {c^{2} d^{8} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9 e} + \frac {8 c^{2} d^{7} x \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9} + \frac {28 c^{2} d^{6} e x^{2} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9} + \frac {56 c^{2} d^{5} e^{2} x^{3} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9} + \frac {70 c^{2} d^{4} e^{3} x^{4} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9} + \frac {56 c^{2} d^{3} e^{4} x^{5} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9} + \frac {28 c^{2} d^{2} e^{5} x^{6} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9} + \frac {8 c^{2} d e^{6} x^{7} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9} + \frac {c^{2} e^{7} x^{8} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9} & \text {for}\: e \neq 0 \\d^{3} x \left (c d^{2}\right )^{\frac {5}{2}} & \text {otherwise} \end {cases} \]

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)**(5/2),x)

[Out]

Piecewise((c**2*d**8*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/(9*e) + 8*c**2*d**7*x*sqrt(c*d**2 + 2*c*d*e*x + c*
e**2*x**2)/9 + 28*c**2*d**6*e*x**2*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/9 + 56*c**2*d**5*e**2*x**3*sqrt(c*d*
*2 + 2*c*d*e*x + c*e**2*x**2)/9 + 70*c**2*d**4*e**3*x**4*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/9 + 56*c**2*d*
*3*e**4*x**5*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/9 + 28*c**2*d**2*e**5*x**6*sqrt(c*d**2 + 2*c*d*e*x + c*e**
2*x**2)/9 + 8*c**2*d*e**6*x**7*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/9 + c**2*e**7*x**8*sqrt(c*d**2 + 2*c*d*e
*x + c*e**2*x**2)/9, Ne(e, 0)), (d**3*x*(c*d**2)**(5/2), True))

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