3.17.74 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^2}{(d+e x)^{11/2}} \, dx\)

Optimal. Leaf size=81 \[ \frac {4 c d \left (c d^2-a e^2\right )}{3 e^3 (d+e x)^{3/2}}-\frac {2 \left (c d^2-a e^2\right )^2}{5 e^3 (d+e x)^{5/2}}-\frac {2 c^2 d^2}{e^3 \sqrt {d+e x}} \]

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Rubi [A]  time = 0.04, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {626, 43} \begin {gather*} \frac {4 c d \left (c d^2-a e^2\right )}{3 e^3 (d+e x)^{3/2}}-\frac {2 \left (c d^2-a e^2\right )^2}{5 e^3 (d+e x)^{5/2}}-\frac {2 c^2 d^2}{e^3 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2/(d + e*x)^(11/2),x]

[Out]

(-2*(c*d^2 - a*e^2)^2)/(5*e^3*(d + e*x)^(5/2)) + (4*c*d*(c*d^2 - a*e^2))/(3*e^3*(d + e*x)^(3/2)) - (2*c^2*d^2)
/(e^3*Sqrt[d + e*x])

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 626

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

Rubi steps

\begin {align*} \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^{11/2}} \, dx &=\int \frac {(a e+c d x)^2}{(d+e x)^{7/2}} \, dx\\ &=\int \left (\frac {\left (-c d^2+a e^2\right )^2}{e^2 (d+e x)^{7/2}}-\frac {2 c d \left (c d^2-a e^2\right )}{e^2 (d+e x)^{5/2}}+\frac {c^2 d^2}{e^2 (d+e x)^{3/2}}\right ) \, dx\\ &=-\frac {2 \left (c d^2-a e^2\right )^2}{5 e^3 (d+e x)^{5/2}}+\frac {4 c d \left (c d^2-a e^2\right )}{3 e^3 (d+e x)^{3/2}}-\frac {2 c^2 d^2}{e^3 \sqrt {d+e x}}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 67, normalized size = 0.83 \begin {gather*} -\frac {2 \left (3 a^2 e^4+2 a c d e^2 (2 d+5 e x)+c^2 d^2 \left (8 d^2+20 d e x+15 e^2 x^2\right )\right )}{15 e^3 (d+e x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2/(d + e*x)^(11/2),x]

[Out]

(-2*(3*a^2*e^4 + 2*a*c*d*e^2*(2*d + 5*e*x) + c^2*d^2*(8*d^2 + 20*d*e*x + 15*e^2*x^2)))/(15*e^3*(d + e*x)^(5/2)
)

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IntegrateAlgebraic [A]  time = 0.09, size = 84, normalized size = 1.04 \begin {gather*} -\frac {2 \left (3 a^2 e^4-6 a c d^2 e^2+10 a c d e^2 (d+e x)+3 c^2 d^4-10 c^2 d^3 (d+e x)+15 c^2 d^2 (d+e x)^2\right )}{15 e^3 (d+e x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2/(d + e*x)^(11/2),x]

[Out]

(-2*(3*c^2*d^4 - 6*a*c*d^2*e^2 + 3*a^2*e^4 - 10*c^2*d^3*(d + e*x) + 10*a*c*d*e^2*(d + e*x) + 15*c^2*d^2*(d + e
*x)^2))/(15*e^3*(d + e*x)^(5/2))

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fricas [A]  time = 0.40, size = 105, normalized size = 1.30 \begin {gather*} -\frac {2 \, {\left (15 \, c^{2} d^{2} e^{2} x^{2} + 8 \, c^{2} d^{4} + 4 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4} + 10 \, {\left (2 \, c^{2} d^{3} e + a c d e^{3}\right )} x\right )} \sqrt {e x + d}}{15 \, {\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2/(e*x+d)^(11/2),x, algorithm="fricas")

[Out]

-2/15*(15*c^2*d^2*e^2*x^2 + 8*c^2*d^4 + 4*a*c*d^2*e^2 + 3*a^2*e^4 + 10*(2*c^2*d^3*e + a*c*d*e^3)*x)*sqrt(e*x +
d)/(e^6*x^3 + 3*d*e^5*x^2 + 3*d^2*e^4*x + d^3*e^3)

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giac [A]  time = 0.22, size = 108, normalized size = 1.33 \begin {gather*} -\frac {2 \, {\left (15 \, {\left (x e + d\right )}^{4} c^{2} d^{2} - 10 \, {\left (x e + d\right )}^{3} c^{2} d^{3} + 3 \, {\left (x e + d\right )}^{2} c^{2} d^{4} + 10 \, {\left (x e + d\right )}^{3} a c d e^{2} - 6 \, {\left (x e + d\right )}^{2} a c d^{2} e^{2} + 3 \, {\left (x e + d\right )}^{2} a^{2} e^{4}\right )} e^{\left (-3\right )}}{15 \, {\left (x e + d\right )}^{\frac {9}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2/(e*x+d)^(11/2),x, algorithm="giac")

[Out]

-2/15*(15*(x*e + d)^4*c^2*d^2 - 10*(x*e + d)^3*c^2*d^3 + 3*(x*e + d)^2*c^2*d^4 + 10*(x*e + d)^3*a*c*d*e^2 - 6*
(x*e + d)^2*a*c*d^2*e^2 + 3*(x*e + d)^2*a^2*e^4)*e^(-3)/(x*e + d)^(9/2)

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maple [A]  time = 0.05, size = 73, normalized size = 0.90 \begin {gather*} -\frac {2 \left (15 c^{2} d^{2} e^{2} x^{2}+10 a c d \,e^{3} x +20 c^{2} d^{3} e x +3 a^{2} e^{4}+4 a c \,d^{2} e^{2}+8 c^{2} d^{4}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^2/(e*x+d)^(11/2),x)

[Out]

-2/15/(e*x+d)^(5/2)*(15*c^2*d^2*e^2*x^2+10*a*c*d*e^3*x+20*c^2*d^3*e*x+3*a^2*e^4+4*a*c*d^2*e^2+8*c^2*d^4)/e^3

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maxima [A]  time = 1.04, size = 77, normalized size = 0.95 \begin {gather*} -\frac {2 \, {\left (15 \, {\left (e x + d\right )}^{2} c^{2} d^{2} + 3 \, c^{2} d^{4} - 6 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4} - 10 \, {\left (c^{2} d^{3} - a c d e^{2}\right )} {\left (e x + d\right )}\right )}}{15 \, {\left (e x + d\right )}^{\frac {5}{2}} e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2/(e*x+d)^(11/2),x, algorithm="maxima")

[Out]

-2/15*(15*(e*x + d)^2*c^2*d^2 + 3*c^2*d^4 - 6*a*c*d^2*e^2 + 3*a^2*e^4 - 10*(c^2*d^3 - a*c*d*e^2)*(e*x + d))/((
e*x + d)^(5/2)*e^3)

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mupad [B]  time = 0.08, size = 78, normalized size = 0.96 \begin {gather*} -\frac {\frac {2\,a^2\,e^4}{5}+\frac {2\,c^2\,d^4}{5}-\left (\frac {4\,c^2\,d^3}{3}-\frac {4\,a\,c\,d\,e^2}{3}\right )\,\left (d+e\,x\right )+2\,c^2\,d^2\,{\left (d+e\,x\right )}^2-\frac {4\,a\,c\,d^2\,e^2}{5}}{e^3\,{\left (d+e\,x\right )}^{5/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^2/(d + e*x)^(11/2),x)

[Out]

-((2*a^2*e^4)/5 + (2*c^2*d^4)/5 - ((4*c^2*d^3)/3 - (4*a*c*d*e^2)/3)*(d + e*x) + 2*c^2*d^2*(d + e*x)^2 - (4*a*c
*d^2*e^2)/5)/(e^3*(d + e*x)^(5/2))

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sympy [A]  time = 17.07, size = 388, normalized size = 4.79 \begin {gather*} \begin {cases} - \frac {6 a^{2} e^{4}}{15 d^{2} e^{3} \sqrt {d + e x} + 30 d e^{4} x \sqrt {d + e x} + 15 e^{5} x^{2} \sqrt {d + e x}} - \frac {8 a c d^{2} e^{2}}{15 d^{2} e^{3} \sqrt {d + e x} + 30 d e^{4} x \sqrt {d + e x} + 15 e^{5} x^{2} \sqrt {d + e x}} - \frac {20 a c d e^{3} x}{15 d^{2} e^{3} \sqrt {d + e x} + 30 d e^{4} x \sqrt {d + e x} + 15 e^{5} x^{2} \sqrt {d + e x}} - \frac {16 c^{2} d^{4}}{15 d^{2} e^{3} \sqrt {d + e x} + 30 d e^{4} x \sqrt {d + e x} + 15 e^{5} x^{2} \sqrt {d + e x}} - \frac {40 c^{2} d^{3} e x}{15 d^{2} e^{3} \sqrt {d + e x} + 30 d e^{4} x \sqrt {d + e x} + 15 e^{5} x^{2} \sqrt {d + e x}} - \frac {30 c^{2} d^{2} e^{2} x^{2}}{15 d^{2} e^{3} \sqrt {d + e x} + 30 d e^{4} x \sqrt {d + e x} + 15 e^{5} x^{2} \sqrt {d + e x}} & \text {for}\: e \neq 0 \\\frac {c^{2} x^{3}}{3 d^{\frac {3}{2}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2/(e*x+d)**(11/2),x)

[Out]

Piecewise((-6*a**2*e**4/(15*d**2*e**3*sqrt(d + e*x) + 30*d*e**4*x*sqrt(d + e*x) + 15*e**5*x**2*sqrt(d + e*x))
- 8*a*c*d**2*e**2/(15*d**2*e**3*sqrt(d + e*x) + 30*d*e**4*x*sqrt(d + e*x) + 15*e**5*x**2*sqrt(d + e*x)) - 20*a
*c*d*e**3*x/(15*d**2*e**3*sqrt(d + e*x) + 30*d*e**4*x*sqrt(d + e*x) + 15*e**5*x**2*sqrt(d + e*x)) - 16*c**2*d*
*4/(15*d**2*e**3*sqrt(d + e*x) + 30*d*e**4*x*sqrt(d + e*x) + 15*e**5*x**2*sqrt(d + e*x)) - 40*c**2*d**3*e*x/(1
5*d**2*e**3*sqrt(d + e*x) + 30*d*e**4*x*sqrt(d + e*x) + 15*e**5*x**2*sqrt(d + e*x)) - 30*c**2*d**2*e**2*x**2/(
15*d**2*e**3*sqrt(d + e*x) + 30*d*e**4*x*sqrt(d + e*x) + 15*e**5*x**2*sqrt(d + e*x)), Ne(e, 0)), (c**2*x**3/(3
*d**(3/2)), True))

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