### 3.1991 $$\int \frac{(a d e+(c d^2+a e^2) x+c d e x^2)^2}{(d+e x)^{13/2}} \, dx$$

Optimal. Leaf size=83 $\frac{4 c d \left (c d^2-a e^2\right )}{5 e^3 (d+e x)^{5/2}}-\frac{2 \left (c d^2-a e^2\right )^2}{7 e^3 (d+e x)^{7/2}}-\frac{2 c^2 d^2}{3 e^3 (d+e x)^{3/2}}$

[Out]

(-2*(c*d^2 - a*e^2)^2)/(7*e^3*(d + e*x)^(7/2)) + (4*c*d*(c*d^2 - a*e^2))/(5*e^3*(d + e*x)^(5/2)) - (2*c^2*d^2)
/(3*e^3*(d + e*x)^(3/2))

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Rubi [A]  time = 0.0388828, antiderivative size = 83, normalized size of antiderivative = 1., 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} $\frac{4 c d \left (c d^2-a e^2\right )}{5 e^3 (d+e x)^{5/2}}-\frac{2 \left (c d^2-a e^2\right )^2}{7 e^3 (d+e x)^{7/2}}-\frac{2 c^2 d^2}{3 e^3 (d+e x)^{3/2}}$

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)^(13/2),x]

[Out]

(-2*(c*d^2 - a*e^2)^2)/(7*e^3*(d + e*x)^(7/2)) + (4*c*d*(c*d^2 - a*e^2))/(5*e^3*(d + e*x)^(5/2)) - (2*c^2*d^2)
/(3*e^3*(d + e*x)^(3/2))

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]

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])

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)^{13/2}} \, dx &=\int \frac{(a e+c d x)^2}{(d+e x)^{9/2}} \, dx\\ &=\int \left (\frac{\left (-c d^2+a e^2\right )^2}{e^2 (d+e x)^{9/2}}-\frac{2 c d \left (c d^2-a e^2\right )}{e^2 (d+e x)^{7/2}}+\frac{c^2 d^2}{e^2 (d+e x)^{5/2}}\right ) \, dx\\ &=-\frac{2 \left (c d^2-a e^2\right )^2}{7 e^3 (d+e x)^{7/2}}+\frac{4 c d \left (c d^2-a e^2\right )}{5 e^3 (d+e x)^{5/2}}-\frac{2 c^2 d^2}{3 e^3 (d+e x)^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0382827, size = 67, normalized size = 0.81 $-\frac{2 \left (15 a^2 e^4+6 a c d e^2 (2 d+7 e x)+c^2 d^2 \left (8 d^2+28 d e x+35 e^2 x^2\right )\right )}{105 e^3 (d+e x)^{7/2}}$

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)^(13/2),x]

[Out]

(-2*(15*a^2*e^4 + 6*a*c*d*e^2*(2*d + 7*e*x) + c^2*d^2*(8*d^2 + 28*d*e*x + 35*e^2*x^2)))/(105*e^3*(d + e*x)^(7/
2))

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Maple [A]  time = 0.045, size = 73, normalized size = 0.9 \begin{align*} -{\frac{70\,{c}^{2}{d}^{2}{x}^{2}{e}^{2}+84\,acd{e}^{3}x+56\,{c}^{2}{d}^{3}ex+30\,{a}^{2}{e}^{4}+24\,ac{d}^{2}{e}^{2}+16\,{c}^{2}{d}^{4}}{105\,{e}^{3}} \left ( ex+d \right ) ^{-{\frac{7}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2/(e*x+d)^(13/2),x)

[Out]

-2/105/(e*x+d)^(7/2)*(35*c^2*d^2*e^2*x^2+42*a*c*d*e^3*x+28*c^2*d^3*e*x+15*a^2*e^4+12*a*c*d^2*e^2+8*c^2*d^4)/e^
3

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Maxima [A]  time = 1.02218, size = 104, normalized size = 1.25 \begin{align*} -\frac{2 \,{\left (35 \,{\left (e x + d\right )}^{2} c^{2} d^{2} + 15 \, c^{2} d^{4} - 30 \, a c d^{2} e^{2} + 15 \, a^{2} e^{4} - 42 \,{\left (c^{2} d^{3} - a c d e^{2}\right )}{\left (e x + d\right )}\right )}}{105 \,{\left (e x + d\right )}^{\frac{7}{2}} e^{3}} \end{align*}

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)^(13/2),x, algorithm="maxima")

[Out]

-2/105*(35*(e*x + d)^2*c^2*d^2 + 15*c^2*d^4 - 30*a*c*d^2*e^2 + 15*a^2*e^4 - 42*(c^2*d^3 - a*c*d*e^2)*(e*x + d)
)/((e*x + d)^(7/2)*e^3)

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Fricas [A]  time = 1.89575, size = 248, normalized size = 2.99 \begin{align*} -\frac{2 \,{\left (35 \, c^{2} d^{2} e^{2} x^{2} + 8 \, c^{2} d^{4} + 12 \, a c d^{2} e^{2} + 15 \, a^{2} e^{4} + 14 \,{\left (2 \, c^{2} d^{3} e + 3 \, a c d e^{3}\right )} x\right )} \sqrt{e x + d}}{105 \,{\left (e^{7} x^{4} + 4 \, d e^{6} x^{3} + 6 \, d^{2} e^{5} x^{2} + 4 \, d^{3} e^{4} x + d^{4} e^{3}\right )}} \end{align*}

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)^(13/2),x, algorithm="fricas")

[Out]

-2/105*(35*c^2*d^2*e^2*x^2 + 8*c^2*d^4 + 12*a*c*d^2*e^2 + 15*a^2*e^4 + 14*(2*c^2*d^3*e + 3*a*c*d*e^3)*x)*sqrt(
e*x + d)/(e^7*x^4 + 4*d*e^6*x^3 + 6*d^2*e^5*x^2 + 4*d^3*e^4*x + d^4*e^3)

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Sympy [A]  time = 65.3969, size = 510, normalized size = 6.14 \begin{align*} \begin{cases} - \frac{30 a^{2} e^{4}}{105 d^{3} e^{3} \sqrt{d + e x} + 315 d^{2} e^{4} x \sqrt{d + e x} + 315 d e^{5} x^{2} \sqrt{d + e x} + 105 e^{6} x^{3} \sqrt{d + e x}} - \frac{24 a c d^{2} e^{2}}{105 d^{3} e^{3} \sqrt{d + e x} + 315 d^{2} e^{4} x \sqrt{d + e x} + 315 d e^{5} x^{2} \sqrt{d + e x} + 105 e^{6} x^{3} \sqrt{d + e x}} - \frac{84 a c d e^{3} x}{105 d^{3} e^{3} \sqrt{d + e x} + 315 d^{2} e^{4} x \sqrt{d + e x} + 315 d e^{5} x^{2} \sqrt{d + e x} + 105 e^{6} x^{3} \sqrt{d + e x}} - \frac{16 c^{2} d^{4}}{105 d^{3} e^{3} \sqrt{d + e x} + 315 d^{2} e^{4} x \sqrt{d + e x} + 315 d e^{5} x^{2} \sqrt{d + e x} + 105 e^{6} x^{3} \sqrt{d + e x}} - \frac{56 c^{2} d^{3} e x}{105 d^{3} e^{3} \sqrt{d + e x} + 315 d^{2} e^{4} x \sqrt{d + e x} + 315 d e^{5} x^{2} \sqrt{d + e x} + 105 e^{6} x^{3} \sqrt{d + e x}} - \frac{70 c^{2} d^{2} e^{2} x^{2}}{105 d^{3} e^{3} \sqrt{d + e x} + 315 d^{2} e^{4} x \sqrt{d + e x} + 315 d e^{5} x^{2} \sqrt{d + e x} + 105 e^{6} x^{3} \sqrt{d + e x}} & \text{for}\: e \neq 0 \\\frac{c^{2} x^{3}}{3 d^{\frac{5}{2}}} & \text{otherwise} \end{cases} \end{align*}

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)**(13/2),x)

[Out]

Piecewise((-30*a**2*e**4/(105*d**3*e**3*sqrt(d + e*x) + 315*d**2*e**4*x*sqrt(d + e*x) + 315*d*e**5*x**2*sqrt(d
+ e*x) + 105*e**6*x**3*sqrt(d + e*x)) - 24*a*c*d**2*e**2/(105*d**3*e**3*sqrt(d + e*x) + 315*d**2*e**4*x*sqrt(
d + e*x) + 315*d*e**5*x**2*sqrt(d + e*x) + 105*e**6*x**3*sqrt(d + e*x)) - 84*a*c*d*e**3*x/(105*d**3*e**3*sqrt(
d + e*x) + 315*d**2*e**4*x*sqrt(d + e*x) + 315*d*e**5*x**2*sqrt(d + e*x) + 105*e**6*x**3*sqrt(d + e*x)) - 16*c
**2*d**4/(105*d**3*e**3*sqrt(d + e*x) + 315*d**2*e**4*x*sqrt(d + e*x) + 315*d*e**5*x**2*sqrt(d + e*x) + 105*e*
*6*x**3*sqrt(d + e*x)) - 56*c**2*d**3*e*x/(105*d**3*e**3*sqrt(d + e*x) + 315*d**2*e**4*x*sqrt(d + e*x) + 315*d
*e**5*x**2*sqrt(d + e*x) + 105*e**6*x**3*sqrt(d + e*x)) - 70*c**2*d**2*e**2*x**2/(105*d**3*e**3*sqrt(d + e*x)
+ 315*d**2*e**4*x*sqrt(d + e*x) + 315*d*e**5*x**2*sqrt(d + e*x) + 105*e**6*x**3*sqrt(d + e*x)), Ne(e, 0)), (c*
*2*x**3/(3*d**(5/2)), True))

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Giac [A]  time = 1.24375, size = 146, normalized size = 1.76 \begin{align*} -\frac{2 \,{\left (35 \,{\left (x e + d\right )}^{4} c^{2} d^{2} - 42 \,{\left (x e + d\right )}^{3} c^{2} d^{3} + 15 \,{\left (x e + d\right )}^{2} c^{2} d^{4} + 42 \,{\left (x e + d\right )}^{3} a c d e^{2} - 30 \,{\left (x e + d\right )}^{2} a c d^{2} e^{2} + 15 \,{\left (x e + d\right )}^{2} a^{2} e^{4}\right )} e^{\left (-3\right )}}{105 \,{\left (x e + d\right )}^{\frac{11}{2}}} \end{align*}

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)^(13/2),x, algorithm="giac")

[Out]

-2/105*(35*(x*e + d)^4*c^2*d^2 - 42*(x*e + d)^3*c^2*d^3 + 15*(x*e + d)^2*c^2*d^4 + 42*(x*e + d)^3*a*c*d*e^2 -
30*(x*e + d)^2*a*c*d^2*e^2 + 15*(x*e + d)^2*a^2*e^4)*e^(-3)/(x*e + d)^(11/2)