### 3.1986 $$\int \frac{(a d e+(c d^2+a e^2) x+c d e x^2)^2}{(d+e x)^{3/2}} \, dx$$

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

[Out]

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

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Rubi [A]  time = 0.0391043, 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} $-\frac{4 c d (d+e x)^{5/2} \left (c d^2-a e^2\right )}{5 e^3}+\frac{2 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}{3 e^3}+\frac{2 c^2 d^2 (d+e x)^{7/2}}{7 e^3}$

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)^(3/2),x]

[Out]

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

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

Mathematica [A]  time = 0.0396866, size = 67, normalized size = 0.81 $\frac{2 (d+e x)^{3/2} \left (35 a^2 e^4+14 a c d e^2 (3 e x-2 d)+c^2 d^2 \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )}{105 e^3}$

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)^(3/2),x]

[Out]

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

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Maple [A]  time = 0.043, size = 73, normalized size = 0.9 \begin{align*}{\frac{30\,{c}^{2}{d}^{2}{x}^{2}{e}^{2}+84\,acd{e}^{3}x-24\,{c}^{2}{d}^{3}ex+70\,{a}^{2}{e}^{4}-56\,ac{d}^{2}{e}^{2}+16\,{c}^{2}{d}^{4}}{105\,{e}^{3}} \left ( ex+d \right ) ^{{\frac{3}{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)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.03428, size = 108, normalized size = 1.3 \begin{align*} \frac{2 \,{\left (15 \,{\left (e x + d\right )}^{\frac{7}{2}} c^{2} d^{2} - 42 \,{\left (c^{2} d^{3} - a c d e^{2}\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 35 \,{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}{\left (e x + d\right )}^{\frac{3}{2}}\right )}}{105 \, 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)^(3/2),x, algorithm="maxima")

[Out]

2/105*(15*(e*x + d)^(7/2)*c^2*d^2 - 42*(c^2*d^3 - a*c*d*e^2)*(e*x + d)^(5/2) + 35*(c^2*d^4 - 2*a*c*d^2*e^2 + a
^2*e^4)*(e*x + d)^(3/2))/e^3

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Fricas [A]  time = 1.92341, size = 236, normalized size = 2.84 \begin{align*} \frac{2 \,{\left (15 \, c^{2} d^{2} e^{3} x^{3} + 8 \, c^{2} d^{5} - 28 \, a c d^{3} e^{2} + 35 \, a^{2} d e^{4} + 3 \,{\left (c^{2} d^{3} e^{2} + 14 \, a c d e^{4}\right )} x^{2} -{\left (4 \, c^{2} d^{4} e - 14 \, a c d^{2} e^{3} - 35 \, a^{2} e^{5}\right )} x\right )} \sqrt{e x + d}}{105 \, 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)^(3/2),x, algorithm="fricas")

[Out]

2/105*(15*c^2*d^2*e^3*x^3 + 8*c^2*d^5 - 28*a*c*d^3*e^2 + 35*a^2*d*e^4 + 3*(c^2*d^3*e^2 + 14*a*c*d*e^4)*x^2 - (
4*c^2*d^4*e - 14*a*c*d^2*e^3 - 35*a^2*e^5)*x)*sqrt(e*x + d)/e^3

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Sympy [A]  time = 50.8447, size = 411, normalized size = 4.95 \begin{align*} \begin{cases} - \frac{\frac{2 a^{2} d^{2} e^{2}}{\sqrt{d + e x}} + 4 a^{2} d e^{2} \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right ) + 2 a^{2} e^{2} \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right ) + 4 a c d^{3} \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right ) + 8 a c d^{2} \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right ) + 4 a c d \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right ) + \frac{2 c^{2} d^{4} \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e^{2}} + \frac{4 c^{2} d^{3} \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{2}} + \frac{2 c^{2} d^{2} \left (\frac{d^{4}}{\sqrt{d + e x}} + 4 d^{3} \sqrt{d + e x} - 2 d^{2} \left (d + e x\right )^{\frac{3}{2}} + \frac{4 d \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{2}}}{e} & \text{for}\: e \neq 0 \\\frac{c^{2} d^{\frac{5}{2}} x^{3}}{3} & \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)**(3/2),x)

[Out]

Piecewise((-(2*a**2*d**2*e**2/sqrt(d + e*x) + 4*a**2*d*e**2*(-d/sqrt(d + e*x) - sqrt(d + e*x)) + 2*a**2*e**2*(
d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3) + 4*a*c*d**3*(-d/sqrt(d + e*x) - sqrt(d + e*x)) +
8*a*c*d**2*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3) + 4*a*c*d*(-d**3/sqrt(d + e*x) - 3*d
**2*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5) + 2*c**2*d**4*(d**2/sqrt(d + e*x) + 2*d*sqrt(d +
e*x) - (d + e*x)**(3/2)/3)/e**2 + 4*c**2*d**3*(-d**3/sqrt(d + e*x) - 3*d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2)
- (d + e*x)**(5/2)/5)/e**2 + 2*c**2*d**2*(d**4/sqrt(d + e*x) + 4*d**3*sqrt(d + e*x) - 2*d**2*(d + e*x)**(3/2)
+ 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e**2)/e, Ne(e, 0)), (c**2*d**(5/2)*x**3/3, True))

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Giac [A]  time = 1.20345, size = 143, normalized size = 1.72 \begin{align*} \frac{2}{105} \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} c^{2} d^{2} e^{18} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} c^{2} d^{3} e^{18} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{2} d^{4} e^{18} + 42 \,{\left (x e + d\right )}^{\frac{5}{2}} a c d e^{20} - 70 \,{\left (x e + d\right )}^{\frac{3}{2}} a c d^{2} e^{20} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} e^{22}\right )} e^{\left (-21\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)^(3/2),x, algorithm="giac")

[Out]

2/105*(15*(x*e + d)^(7/2)*c^2*d^2*e^18 - 42*(x*e + d)^(5/2)*c^2*d^3*e^18 + 35*(x*e + d)^(3/2)*c^2*d^4*e^18 + 4
2*(x*e + d)^(5/2)*a*c*d*e^20 - 70*(x*e + d)^(3/2)*a*c*d^2*e^20 + 35*(x*e + d)^(3/2)*a^2*e^22)*e^(-21)