### 3.1985 $$\int \frac{(a d e+(c d^2+a e^2) x+c d e x^2)^2}{\sqrt{d+e x}} \, dx$$

Optimal. Leaf size=83 $-\frac{4 c d (d+e x)^{7/2} \left (c d^2-a e^2\right )}{7 e^3}+\frac{2 (d+e x)^{5/2} \left (c d^2-a e^2\right )^2}{5 e^3}+\frac{2 c^2 d^2 (d+e x)^{9/2}}{9 e^3}$

[Out]

(2*(c*d^2 - a*e^2)^2*(d + e*x)^(5/2))/(5*e^3) - (4*c*d*(c*d^2 - a*e^2)*(d + e*x)^(7/2))/(7*e^3) + (2*c^2*d^2*(
d + e*x)^(9/2))/(9*e^3)

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Rubi [A]  time = 0.0389183, 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} $-\frac{4 c d (d+e x)^{7/2} \left (c d^2-a e^2\right )}{7 e^3}+\frac{2 (d+e x)^{5/2} \left (c d^2-a e^2\right )^2}{5 e^3}+\frac{2 c^2 d^2 (d+e x)^{9/2}}{9 e^3}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Mathematica [A]  time = 0.043057, size = 67, normalized size = 0.81 $\frac{2 (d+e x)^{5/2} \left (63 a^2 e^4+18 a c d e^2 (5 e x-2 d)+c^2 d^2 \left (8 d^2-20 d e x+35 e^2 x^2\right )\right )}{315 e^3}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*(d + e*x)^(5/2)*(63*a^2*e^4 + 18*a*c*d*e^2*(-2*d + 5*e*x) + c^2*d^2*(8*d^2 - 20*d*e*x + 35*e^2*x^2)))/(315*
e^3)

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Maple [A]  time = 0.046, size = 73, normalized size = 0.9 \begin{align*}{\frac{70\,{c}^{2}{d}^{2}{x}^{2}{e}^{2}+180\,acd{e}^{3}x-40\,{c}^{2}{d}^{3}ex+126\,{a}^{2}{e}^{4}-72\,ac{d}^{2}{e}^{2}+16\,{c}^{2}{d}^{4}}{315\,{e}^{3}} \left ( ex+d \right ) ^{{\frac{5}{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)^(1/2),x)

[Out]

2/315*(e*x+d)^(5/2)*(35*c^2*d^2*e^2*x^2+90*a*c*d*e^3*x-20*c^2*d^3*e*x+63*a^2*e^4-36*a*c*d^2*e^2+8*c^2*d^4)/e^3

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Maxima [B]  time = 1.00856, size = 378, normalized size = 4.55 \begin{align*} \frac{2 \,{\left (315 \, \sqrt{e x + d} a^{2} d^{2} e^{2} + 42 \,{\left (\frac{{\left (3 \,{\left (e x + d\right )}^{\frac{5}{2}} - 10 \,{\left (e x + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{e x + d} d^{2}\right )} c d}{e} + \frac{5 \,{\left (c d^{2} + a e^{2}\right )}{\left ({\left (e x + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{e x + d} d\right )}}{e}\right )} a d e + \frac{{\left (35 \,{\left (e x + d\right )}^{\frac{9}{2}} - 180 \,{\left (e x + d\right )}^{\frac{7}{2}} d + 378 \,{\left (e x + d\right )}^{\frac{5}{2}} d^{2} - 420 \,{\left (e x + d\right )}^{\frac{3}{2}} d^{3} + 315 \, \sqrt{e x + d} d^{4}\right )} c^{2} d^{2}}{e^{2}} + \frac{18 \,{\left (5 \,{\left (e x + d\right )}^{\frac{7}{2}} - 21 \,{\left (e x + d\right )}^{\frac{5}{2}} d + 35 \,{\left (e x + d\right )}^{\frac{3}{2}} d^{2} - 35 \, \sqrt{e x + d} d^{3}\right )}{\left (c d^{2} + a e^{2}\right )} c d}{e^{2}} + \frac{21 \,{\left (c d^{2} + a e^{2}\right )}^{2}{\left (3 \,{\left (e x + d\right )}^{\frac{5}{2}} - 10 \,{\left (e x + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{e x + d} d^{2}\right )}}{e^{2}}\right )}}{315 \, e} \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)^(1/2),x, algorithm="maxima")

[Out]

2/315*(315*sqrt(e*x + d)*a^2*d^2*e^2 + 42*((3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*c
*d/e + 5*(c*d^2 + a*e^2)*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)/e)*a*d*e + (35*(e*x + d)^(9/2) - 180*(e*x + d)^
(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*c^2*d^2/e^2 + 18*(5*(e*x
+ d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*(c*d^2 + a*e^2)*c*d/e^2 + 2
1*(c*d^2 + a*e^2)^2*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)/e^2)/e

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Fricas [B]  time = 1.86642, size = 316, normalized size = 3.81 \begin{align*} \frac{2 \,{\left (35 \, c^{2} d^{2} e^{4} x^{4} + 8 \, c^{2} d^{6} - 36 \, a c d^{4} e^{2} + 63 \, a^{2} d^{2} e^{4} + 10 \,{\left (5 \, c^{2} d^{3} e^{3} + 9 \, a c d e^{5}\right )} x^{3} + 3 \,{\left (c^{2} d^{4} e^{2} + 48 \, a c d^{2} e^{4} + 21 \, a^{2} e^{6}\right )} x^{2} - 2 \,{\left (2 \, c^{2} d^{5} e - 9 \, a c d^{3} e^{3} - 63 \, a^{2} d e^{5}\right )} x\right )} \sqrt{e x + d}}{315 \, 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)^(1/2),x, algorithm="fricas")

[Out]

2/315*(35*c^2*d^2*e^4*x^4 + 8*c^2*d^6 - 36*a*c*d^4*e^2 + 63*a^2*d^2*e^4 + 10*(5*c^2*d^3*e^3 + 9*a*c*d*e^5)*x^3
+ 3*(c^2*d^4*e^2 + 48*a*c*d^2*e^4 + 21*a^2*e^6)*x^2 - 2*(2*c^2*d^5*e - 9*a*c*d^3*e^3 - 63*a^2*d*e^5)*x)*sqrt(
e*x + d)/e^3

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Sympy [A]  time = 85.4955, size = 631, normalized size = 7.6 \begin{align*} \begin{cases} - \frac{\frac{2 a^{2} d^{3} e^{2}}{\sqrt{d + e x}} + 6 a^{2} d^{2} e^{2} \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right ) + 6 a^{2} d 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 ) + 2 a^{2} e^{2} \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 ) + 4 a c d^{4} \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right ) + 12 a c d^{3} \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 ) + 12 a c d^{2} \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 ) + 4 a c d \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 ) + \frac{2 c^{2} d^{5} \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{6 c^{2} d^{4} \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{6 c^{2} d^{3} \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}} + \frac{2 c^{2} d^{2} \left (- \frac{d^{5}}{\sqrt{d + e x}} - 5 d^{4} \sqrt{d + e x} + \frac{10 d^{3} \left (d + e x\right )^{\frac{3}{2}}}{3} - 2 d^{2} \left (d + e x\right )^{\frac{5}{2}} + \frac{5 d \left (d + e x\right )^{\frac{7}{2}}}{7} - \frac{\left (d + e x\right )^{\frac{9}{2}}}{9}\right )}{e^{2}}}{e} & \text{for}\: e \neq 0 \\\frac{c^{2} d^{\frac{7}{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)**(1/2),x)

[Out]

Piecewise((-(2*a**2*d**3*e**2/sqrt(d + e*x) + 6*a**2*d**2*e**2*(-d/sqrt(d + e*x) - sqrt(d + e*x)) + 6*a**2*d*e
**2*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3) + 2*a**2*e**2*(-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) + 4*a*c*d**4*(-d/sqrt(d + e*x) - sqrt(d + e*x)) + 12*
a*c*d**3*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3) + 12*a*c*d**2*(-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) + 4*a*c*d*(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) + 2*c**2*d**5*(d**2/sqrt(d + e*x
) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e**2 + 6*c**2*d**4*(-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 + 6*c**2*d**3*(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 + 2*c**2*d**2*(-d**5/sqrt(d + e*x) - 5*d
**4*sqrt(d + e*x) + 10*d**3*(d + e*x)**(3/2)/3 - 2*d**2*(d + e*x)**(5/2) + 5*d*(d + e*x)**(7/2)/7 - (d + e*x)*
*(9/2)/9)/e**2)/e, Ne(e, 0)), (c**2*d**(7/2)*x**3/3, True))

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Giac [B]  time = 1.15654, size = 524, normalized size = 6.31 \begin{align*} \frac{2}{315} \,{\left (21 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{x e + d} d^{2}\right )} c^{2} d^{4} e^{\left (-2\right )} + 18 \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} - 35 \, \sqrt{x e + d} d^{3}\right )} c^{2} d^{3} e^{\left (-2\right )} + 210 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} a c d^{3} +{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} - 180 \,{\left (x e + d\right )}^{\frac{7}{2}} d + 378 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} - 420 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} + 315 \, \sqrt{x e + d} d^{4}\right )} c^{2} d^{2} e^{\left (-2\right )} + 315 \, \sqrt{x e + d} a^{2} d^{2} e^{2} + 84 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{x e + d} d^{2}\right )} a c d^{2} + 210 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} a^{2} d e^{2} + 18 \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} - 35 \, \sqrt{x e + d} d^{3}\right )} a c d + 21 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{x e + d} d^{2}\right )} a^{2} e^{2}\right )} e^{\left (-1\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)^(1/2),x, algorithm="giac")

[Out]

2/315*(21*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*c^2*d^4*e^(-2) + 18*(5*(x*e + d)^(
7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*c^2*d^3*e^(-2) + 210*((x*e + d)^(
3/2) - 3*sqrt(x*e + d)*d)*a*c*d^3 + (35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 42
0*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*c^2*d^2*e^(-2) + 315*sqrt(x*e + d)*a^2*d^2*e^2 + 84*(3*(x*e + d
)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a*c*d^2 + 210*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^2
*d*e^2 + 18*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*a*c*d +
21*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^2*e^2)*e^(-1)